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e-ISSN 2588-9214p-ISSN 1319-5166

Volume 26 Issue 1/2 2020

Arab Journal of Mathematical

Sciences

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Editorialboards

1

Arab Journal of MathematicalSciences

Vol. 26 No. 1/2, 2020p. 1

Emerald Publishing Limitede-ISSN: 2588-9214p-ISSN: 1319-5166

Quarto trim size: 174mm x 240mm

Existence of mild solutions forfractional non-instantaneousimpulsive integro-differential

equations with nonlocal conditionsArshi Meraj and Dwijendra N. Pandey

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

AbstractThis paper is concerned with the existence of mild solutions for a class of fractional semilinear integro-differential equations having non-instantaneous impulses. The result is obtained by using noncompactsemigroup theory and fixed point theorem. The obtained result is illustrated by an example at the end.

Keywords Fractional differential equations, Nonlocal conditions, Fixed point theorem, Noncompact

semigroup, Measure of noncompactness

Paper type Original Article

1. IntroductionThe objective of this paper is to study the existence of mild solutions to the following abstractintegro-differential equations of fractional order with non-instantaneous impulses andnonlocal conditions in a Banach space X:

cDquðtÞ þ AuðtÞ ¼ f

0@t; uðtÞ;

Z t

0

Kðt; sÞuðsÞds1A; t ∈∪m

k¼0ðsk; tkþ1�;

uðtÞ ¼ γkðt; uðtÞÞ; t ∈∪mk¼1ðtk; sk�;

uð0Þ þ gðuÞ ¼ u0; (1.1)

Mild solutionsfor integro-differentialequations

3

JEL Classification — 34A08, 34A12, 34A37, 34K30, 45J05© Arshi Meraj and Dwijendra N. Pandey. Published in the Arab Journal of Mathematical Sciences.

Published by Emerald Publishing Limited. This article is published under the Creative CommonsAttribution (CCBY4.0) license. Anyonemay reproduce, distribute, translate and create derivativeworksof this article (for both commercial and non-commercial purposes), subject to full attribution to theoriginal publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

The authors would like to express thanks to the editor and referees for their careful reading of themanuscript and valuable comments. The work of first author is supported by the “Ministry of HumanResource Development, India under Grant Number: MHR-01-23-200-428”.

The publisher wishes to inform readers that the article “Existence of mild solutions for fractionalnon-instantaneous impulsive integro-differential equations with nonlocal conditions” was originallypublished by the previous publisher of theArab Journal of Mathematical Sciences and the pagination ofthis article has been subsequently changed. There has been no change to the content of the article. Thischange was necessary for the journal to transition from the previous publisher to the new one. Thepublisher sincerely apologises for any inconvenience caused. To access and cite this article, pleaseuse Meraj, A., Pandey, D.N. (2018), “Existence of mild solutions for fractional non-instantaneousimpulsive integro-differential equations with nonlocal conditions”, Arab Journal of MathematicalSciences, Vol. 26 No. 1/2, pp. 3-13. The original publication date for this paper was 27/11/2018.

The current issue and full text archive of this journal is available on Emerald Insight at:

https://www.emerald.com/insight/1319-5166.htm

Received 6 August 2018Revised 4 October 2018

Accepted 11 November 2018


Vol. 26 No. 1/2, 2020pp. 3-13


DOI 10.1016/j.ajmsc.2018.11.002

where cDq is the Caputo fractional derivative of order qð0 < q < 1Þ; A : DðAÞ⊂X →X isclosed linear operator, −A is the infinitesimal generator of an equicontinuous and uniformlybounded C0 semigroup TðtÞðt ≥ 0Þ on X ; J ¼ ½0; a�; a > 0 is a constant, 0 < t1 < t2 < � � � <tm < tmþ1 :¼ a; s0 :¼ 0 and sk ∈ ðtk; tkþ1Þ for each k ¼ 1; 2; . . . ;m; f : J 3X 3X →X ;g : PCðJ ;XÞ→X are given functions satisfying certain assumptions, γk : ðtk; sk�3X →Xare non-instantaneous impulsive functions for all k ¼ 1; 2; . . . ;m and K ∈C ðD;ℝþÞwhereD :¼ fðt; sÞ : 0 ≤ s < t ≤ ag and u0 ∈X :

In the past decades, many researchers paid attention to study the differential equationswith instantaneous impulses, which have been used to describe abrupt changes such asshocks, harvesting and natural disasters. Particularly, the theory of instantaneous impulsiveequations have wide applications in control, mechanics, electrical engineering, biological andmedical fields. For more details on the differential equations with instantaneous impulses onemay see [2,4,7,14,15].

It seems that models with instantaneous impulses could not explain the certain dynamicsof evolution process in pharmacotherapy. For example, one considers the hemodynamicequilibrium of a person, the introduction of the drugs in bloodstream and the consequentabsorption for the body are gradual and continuous process. Hern�andez andO’Regan [12] andPierri et al. [18], initially studied Cauchy problems for first order evolution equations withnon-instantaneous impulses. The recent results for evolution equations with non-instantaneous impulses can be found in [1,8,13,19–21] and the references therein.

The nonlocal problemwasmotivated by physical problems. Indeed it is demonstrated thatthe nonlocal problems have better effects in applications than the classical Cauchy problems.For example it is used to represent mathematical models for evolution of various phenomenasuch as nonlocal neutral networks, nonlocal pharmacokinetics, nonlocal pollution andnonlocal combustion (see [16]). The existence results to evolution equations with nonlocalconditions in Banach space were first studied by Byszewski [6]. Deng [9] used the nonlocalcondition to describe the diffusion phenomenon of a small amount of gas in atransparent tube.

To the best of our knowledge, there is no work yet reported on fractional non-instantaneous impulsive integro-differential equations with nonlocal conditions (1.1) whenthe corresponding semigroup TðtÞðt ≥ 0Þ is noncompact. Therefore inspired by the previousworks, wewill study the existence of PC-mild solutions for (1.1) under the assumption that thecorresponding C0 semigroup is noncompact, by using the properties of Kuratowski measureof noncompactness, and ρ-set contractionmapping fixed point theorem (see Lemma 2.10). Weconclude this section by summarizing the contents of this paper. In the next section, we willintroduce some basic definitions, notations and preliminary lemmas. In Section 3, we willprove existence of mild solutions for the problem (1.1) also we will give an example toillustrate the feasibility of our abstract result.

2. PreliminariesLet X be a Banach space with norm k$k, we use θ to denote the zero function in PCðJ ;XÞ andJ ¼ ½0; a� for any constant a > 0. Let CðJ ;XÞ be a Banach space of all continuous functionsfrom J into X endowed with supremum norm kukC ¼ supt∈JkuðtÞk. Consider the spacePC ðJ ;XÞ ¼ fu : J →X : u is continuous at t ≠ tk; u ðtk−Þ ¼ u ðtkÞ and u ðtkþÞ exists for all k ¼1; 2; . . . :mg, which is a Banach space endowed with supremum norm kukPC ¼ supt∈JkuðtÞk.For each finite constant r > 0, letΩr ¼ fu∈PCðJ ;XÞ : kuðtÞk≤ r; t ∈ Jg. Let LpðJ ;XÞð1≤ p < ∞Þbe the Banach space of all X-valued Bochner integrable functions defined on J with norm

kukLpðJ ;XÞ ¼ ð R a

0 kuðtÞkpdtÞ1p. Denote Gu ðtÞ :¼ R t

0 K ðt; sÞ u ðsÞ ds, and let G* ¼ supt∈JR t

0 K ðt; sÞds < ∞: Let M ¼ supt∈JkTðtÞkLðXÞ, where LðXÞ stands for the Banach space of all linear

AJMS26,1/2

4

and bounded operators on X, note that M ≥ 1. A C0-semigroup TðtÞðt ≥ 0Þ is calledequicontinuous if the operator TðtÞ is continuous by the operator norm for every t > 0.

Lemma2.1 ([10]). If hsatisfies a uniformH€older continuity with exponent β∈ ð0; 1�, then theunique solution of the following linear Cauchy problem:

cDquðtÞ þ AuðtÞ ¼ hðtÞ; t ∈ J ; (2.1)

uð0Þ ¼ x0 ∈X ;

is given by

uðtÞ ¼ UðtÞx0 þZ t

0

ðt � sÞq�1V ðt � sÞ h ðsÞ ds; (2.2)

where

UðtÞ ¼Z ∞

0

ζqðθÞT ðtqθÞ dθ; V ðtÞ ¼ q

Z ∞

0

θζqðθÞT ðtqθÞ dθ; (2.3)

ζqðθÞ ¼1

qθ−1−

1qρqðθ

−1q Þ; ρqðθÞ ¼

1

π

X∞n¼0

ð−1Þn−1θ−qn−1Γðnqþ 1Þn!

sinðnπqÞ; θ∈ ð0;∞Þ; (2.4)

ζqðθÞ is a probability density function defined on ð0;∞Þ.Remark 2.2. ζqðθÞ≥ 0; θ∈ ð0;∞Þ; R∞

0 ζqðθÞdθ ¼ 1;R∞

0 θζqðθÞdθ ¼ 1Γð1þqÞ:

Lemma 2.3 ([22]). The operators UðtÞðt ≥ 0Þ and V ðtÞðt ≥ 0Þ have the following properties:(i) For any fixed t ≥ 0, UðtÞ and VðtÞ are strongly continuous.(ii) For any fixed t ≥ 0, UðtÞ and V ðtÞ are linear bounded operators, moreover for any

u∈X,

kUðtÞuk≤Mkuk; kV ðtÞuk≤ M

ΓðqÞ kuk:

(iii) If TðtÞðt ≥ 0Þ is an equicontinuous semigroup, then UðtÞ and VðtÞ are continuous fort > 0 by the operator norm, which means that for 0 < t0 < t00 ≤ a, we have

kUðt00Þ � Uðt0Þk→ 0 and kVðt00Þ � V ðt0Þk→ 0 as t00 → t0:

Definition 2.4 ([13]). A function u∈PCðJ ;XÞ is said to be a mild solution of the problem(1.1) if uð0Þ ¼ u0 − gðuÞ, uðtÞ ¼ γkðt; uðtÞÞ for all t ∈∪m

k¼1ðtk; sk�, and

uðtÞ ¼

8>>>>><>>>>>:

UðtÞðu0 � gðuÞÞ þZ t

0

ðt � sÞq�1V ðt � sÞ f ðs; uðsÞ; GuðsÞÞds; t ∈ ð0; t1�;

Uðt � skÞγkðsk; uðskÞÞ þZ t

sk

ðt � sÞq�1Vðt � sÞ f ðs; uðsÞ; GuðsÞÞds;

t ∈ ðsk; tkþ1�; k ¼ 1; 2; : : : ;m:

Now, we recall some properties of measure of noncompactness which are useful to prove ourmain result. For the details about measure of noncompactness, one may see [3,11]. Let αð$Þdenotes the Kuratowski measure of noncompactness of the bounded set.


5

Lemma 2.5 ([3]). Let X be a Banach space, and U ⊂CðJ ;XÞ, UðtÞ ¼ fuðtÞ : u∈Ugðt ∈ JÞ.If U is bounded and equicontinuous in CðJ ;XÞ, then αðUðtÞÞ is continuous on J,and αðUÞ ¼ maxt∈JαðUðtÞÞ.Lemma 2.6 ([11]). If X be a Banach space and D ¼ fung∞n¼1 ⊂PCðJ ;XÞ be a bounded andcountable set, then αðDðtÞÞ is Lebesgue integrable on J, and

α

0@8<:

Z t

0

unðsÞds9=;

∞

n¼1

1A≤ 2

Z t

0

αðfunðsÞg∞n¼1Þds:

Lemma 2.7 ([5]). Let X be a Banach space and U is bounded subset of X, then there exists acountable set D ¼ fung∞n¼1 ⊂U such that αðUÞ≤ 2αðDÞ.Lemma 2.8 ([3]). Let X and E be Banach spaces and Q : DðQÞ⊂E→X is Lipschitzcontinuous with constant L, then αðQðVÞÞ≤LαðV Þ for any bounded subset V ⊂DðQÞ.Definition 2.9 ([8]). Let X be a Banach space, and S be a nonempty subset of X. Acontinuous map Q : S→X is called ρ-set contractive if there exists a constant ρ∈ ½0; 1Þ suchthat for every bounded set Ω⊂ S,

αðQðΩÞÞ≤ ραðΩÞ:

Lemma 2.10 ([8]). Let X be a Banach space, Ω⊂X be a closed bounded and convex subset,and the operator Q : Ω→Ω is ρ-set contractive, then Q has at least one fixed point in Ω:

3. Main result and exampleIn this section, we will discuss the existence of mild solutions for the system (1.1), then we willpresent an example to illustrate our proved result. Let us introduce the required assumptionswhich are needed to prove our main result:

(H1) For each t ∈ J, the function f ðt; $; $Þ : X 3X →X is continuous and for allðx; yÞ∈X 3X, the function f ð$; x; yÞ : J →X is Lebesgue measurable.

(H2) There exist a continuous nondecreasing function ψ : ½0;∞Þ→ ð0;∞Þ, a constantq1 ∈ ð0; qÞ, and a function f∈L

1q1ðJ ;ℝþÞ such that

kf ðt; x; yÞk≤fðtÞψðkxkÞ; ∀x; y∈X ; t ∈ J :

(H3) g : PCðJ ;XÞ→X is continuous and there exists a constant α* > 0 such that

kgðxÞ � gðyÞk≤ α*kx� yk; ∀x; y∈PCðJ ;XÞ:

(H4) γk : ½tk; sk�3X →X are continuous and there exist constants Kγk > 0; k ¼ 1; 2; . . . ;msuch that

kγkðt; xÞ � γkðt; yÞk≤Kγkkx� yk; ∀x; y∈X ; t ∈ ½tk; sk�:

(H5) There exist positive constants Lk and Nk; k ¼ 0; 1; 2; . . . ;m such that for anycountable sets D1;D2 ⊂X,

αðf ðt;D1;D2ÞÞ≤LkαðD1Þ þ NkαðD2Þ; ∀t ∈ ðsk; tkþ1�; k ¼ 0; 1; 2; . . . ;m:

AJMS26,1/2

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Let us denote:

K ¼ maxk¼1;2;...;m

Kγk; K* ¼ maxfK; α*g;

L ¼ maxk¼0;1;2;...;m

ðLk þ NkG*Þðtkþ1 � skÞq: (3.1)

Theorem 3.1. Assume that the semigroup TðtÞðt ≥ 0Þ generated by −A is equicontinuous,the functions gðθÞ and γkð$; θÞ are bounded for k ¼ 1; 2; . . . ;m, and the assumptions (H1)–(H5) are satisfied, then the system (1.1) has at least one PC- mild solution provided that

maxfΛ1;Λ2g< 1; (3.2)

where Λ1 ¼ Mðα* þ KÞ and Λ2 ¼ M ðK* þ 4LΓðqþ1ÞÞ.

Proof. Define the operator F : PCðJ ;XÞ→PCðJ ;XÞ asðFuÞðtÞ ¼ ðF1uÞðtÞ þ ðF2uÞðtÞ; (3.3)

where

ðF1uÞðtÞ ¼8<:

UðtÞðu0 � gðuÞÞ; t ∈ ½0; t1�;γkðt; uðtÞÞ; t ∈ ðtk; sk�; k ¼ 1; 2; . . . ;m;Uðt � skÞγkðsk; uðskÞÞ; t ∈ ðsk; tkþ1�; k ¼ 1; 2; . . . ;m:

(3.4)

ðF2uÞðtÞ ¼

8>><>>:

Z t

sk

tðt � sÞq−1V ðt � sÞf ðs; uðsÞ;GuðsÞÞds;t ∈ ðsk; tkþ1�; k ¼ 0; 1; 2; . . . ;m;

0; otherwise

(3.5)

It is easy to see that F is well defined. From Definition 2.4, one can easily see that the PC-mildsolution of the system (1.1) is equivalent to a fixed point of the operator F defined by (3.3).Now, we will prove that the operator F has a fixed point.

Let u∈ΩR for some R > 0, q2 ¼ q− 11− q1

∈ ð−1; 0Þ and M1 ¼ ψðRÞkfkL

1q1 ðJ ;RþÞ

, by using

H€older inequality and (H2), we obtain

Z t

0

kðt � sÞq−1f ðs; uðsÞ;GuðsÞÞkds#�Z t

0

ðt � sÞq2ds�1−q1

ψðRÞkfkL

1q1 ðJ ;RþÞ

#M1

ð1þ q2Þ1−q1að1þq2Þð1−q1Þ:

(3.6)

Now, we divide the proof into the following steps:

Step I: We prove that there exists a constant R > 0 such that FðΩRÞ⊂ΩR.

If this is not true, then for each r > 0, there will exist ur ∈Ωr and tr ∈ J such thatkðFurðtrÞÞk > r. If tr ∈ ½0; t1�, then by (3.3), (3.6), and (H3) we have


7

kðFurÞðtrÞk≤Mðku0k þ α*kur � θk þ kgðθÞk þ MM1

ΓðqÞð1þ q2Þ1−q1að1þq2Þð1−q1Þ

≤Mðα*r þ ku0k þ kgðθÞk þ MM1

ΓðqÞð1þ q2Þ1−q1að1þq2Þð1−q1Þ:

(3.7)

If tr ∈ ðtk; sk�; k ¼ 1; 2; . . . ;m, then by (3.4) and (H4), we obtain

kðFurÞðtrÞk ¼ ðγkðtr; urðtrÞÞÞ≤KγkkurðtrÞk þ kγkðtr; θÞk≤Kγkr þ β;

(3.8)

where β ¼ maxk¼1;2;...;mfsupt∈Jkγkðt; θÞkg. If tr ∈ ðsk; tkþ1�; k ¼ 1; 2; . . . ;m; then by (3.3), (3.6),and (H4) we have

kðFurÞðtrÞk≤ MðKγkr þ βÞ þM

Z tr

sk

ðtr � sÞq−1kf ðs; urðsÞ;GurðsÞÞkds

≤ MðKγkr þ βÞ þ MM1

ΓðqÞð1þ q2Þ1−q1að1þq2Þð1−q1Þ

(3.9)

Combining (3.7)–(3.9) with the fact r < kðFurÞðtrÞk, we obtainr < kðFurÞðtrÞk≤Mðα*r þ ku0k þ kgðθÞk þMðKr þ βÞ þ MM1

ΓðqÞð1þ q2Þ1−q1að1þq2Þð1−q1Þ:

(3.10)

Dividing both sides of (3.10) by r and taking limit as r→∞, we have

1≤Mðα* þ KÞ; (3.11)

which contradicts (3.2).

Step II: We prove that the operator F1 : ΩR →ΩR is Lipschitz continuous.

For t ∈ ½0; t1� and u; v∈ΩR, using (3.4) and (H3) we have

kðF1uÞðtÞ � ðF1vÞðtÞk≤MkgðuÞ � gðvÞk≤Mα*ku� vk: (3.12)

For t ∈ ðtk; sk�; k ¼ 1; 2; . . . ;m and u; v∈ΩR, by (3.4) and the assumption (H4), we obtain

kðF1uÞðtÞ � ðF1vÞðtÞk ≤ KγkkuðtÞ � vðtÞk≤MKku� vk: (3.13)

For t ∈ ðsk; tkþ1�; k ¼ 1; 2; . . . ;m and u; v∈ΩR, using (H4), we have

kðF1uÞðtÞ � ðF1vÞðtÞk≤Mkγkðsk; uðskÞÞ � γkðsk; vðskÞÞk≤MKku� vk: (3.14)

From (3.12)–(3.14), we obtain

kF1u� F1vk≤MK*ku� vk; (3.15)

where K* :¼ mfK; α*g.

AJMS26,1/2

8

Step III: In this step, we prove that F2 is continuous on ΩR.

Let fung be a sequence in ΩR such that limn→∞un ¼ u in ΩR. By the continuity of nonlinearterm f with respect to second and third variables, for each s∈ J, we have

limn→∞

f ðs; unðsÞ;GunðsÞÞ ¼ f ðs; uðsÞ;GuðsÞÞ: (3.16)

So, we can conclude that

supt∈J

kf ðs; unðsÞ;GunðsÞÞ � f ðs; uðsÞ;GuðsÞÞk→ 0 as n→∞: (3.17)

For s∈ ½sk; t� and t ∈ ðsk; tkþ1�; k ¼ 0; 1; 2; . . . ;m; by (3.16) and (3.17), we obtain

kðF2unÞðtÞ � ðF2uÞðtÞk

≤M

ΓðqÞZ t

sk

ðt � sÞq−1kf ðs; unðsÞ;GunðsÞÞ � f ðs; uðsÞ;GuðsÞÞkds

≤Maq

Γðqþ 1Þ supt∈J

kf ðs; unðsÞ;GunðsÞÞ � f ðs; uðsÞ;GuðsÞÞk

(3.18)

→ 0 as n→∞: (3.19)

Hence,kF2un � F2uk→ 0 as n→∞; (3.20)

which means that F2 is continuous on ΩR.

Step IV: Now, we show F2 : ΩR →ΩR is equicontinuous.

For any u∈ΩR and sk ≤ t0 < t00 ≤ tkþ1 for k ¼ 0; 1; 2; . . . ;m, we have

kðF2uÞðt00Þ � ðF2uÞðt0Þk ¼ kZ t00

sk

ðt00 � sÞq−1Vðt00 � sÞf ðs; uðsÞ;GuðsÞÞds

�Z t0

sk

ðt0 � sÞq−1Vðt00 � sÞf ðs; uðsÞ;GuðsÞÞdsk

≤ kZ t00

t0

ðt00 � sÞq−1Vðt00 � sÞf ðs; uðsÞ;GuðsÞÞdsk

þ kZ t0

sk

½ðt00 � sÞq−1 � ðt0 � sÞq−1�Vðt00�sÞf ðs; uðsÞ;GuðsÞÞdsk

þ kZ t0

sk

ðt0 � sÞq−1½V ðt00�sÞ � V ðt0 � sÞ�f ðs; uðsÞ;GuðsÞÞdsk

¼ I1 þ I2 þ I3;

where,

I1 ¼ kZ t00

t0ðt00 � sÞq−1Vðt00 � sÞf ðs; uðsÞ;GuðsÞÞdsk;

I2 ¼ kZ t0

sk

½ðt00 � sÞq−1 � ðt0 � sÞq−1�V ðt00 � sÞf ðs; uðsÞ;GuðsÞÞdsk;


9

I3 ¼ kZ t0

sk

ðt0 � sÞq−1½Vðt00 � sÞ � V ðt0 � sÞ�f ðs; uðsÞ;GuðsÞÞdsk:

Now, we only need to check that I1; I2 and I3 tend to 0 independently of u∈ΩR when t00 → t0.By (3.6), we have

I1 ≤M1M

ΓðqÞð1þ q2Þ1−q1ðt00 � t0Þð1þq2Þð1−q1Þ

→ 0 as t00 → t0:

For I2; by (H2), Lemma 2.3, H€older inequality, and [22], we get that

I2 ≤M

ΓðqÞ

0B@

Z t0

sk

½ðt00 � sÞq�1 � ðt0 � sÞq�1� 11�q1ds

1CA

1−q1

ψðRÞkfkL

1q1 ðJ ;ℝÞ

≤M1M

ΓðqÞ�Z t0

sk

½ðt0 � sÞq2 � ðt00 � sÞq2 �ds�1−q1

≤M1M

ΓðqÞð1þ q2Þ1−q1½ðt0Þ1þq2 � ðt00Þ1þq2 þ ðt00 � t0Þ1þq2 �1−q1

≤M1M

ΓðqÞð1þ q2Þ1−q1ðt00 � t0Þð1þq2Þð1−q1Þ

→ 0 as t00 → t0:

For t0 ¼ sk; it is easy to see that I3 ¼ 0: For t

0> sk and e > 0 small enough, by (H2), Lemma

2.3, and the equicontinuity of TðtÞ , we estimate

I3 ≤ kZ t0−e

sk

ðt0 � sÞq−1½V ðt00 � sÞ � Vðt0 � sÞ�f ðs; uðsÞ;GuðsÞÞdsk

þkZ t0

t0−eðt0 � sÞq−1½V ðt00 � sÞ � V ðt0 � sÞ�f ðs; uðsÞ;GuðsÞÞdsk

≤

Z t0−e

sk

kðt0 � sÞq−1f ðs; uðsÞ;GuðsÞÞkds sups∈½sk;t0−e�

kV ðt00 � sÞ � Vðt0 � sÞk

þ 2M

ΓðqÞZ t0

t0−ekðt0 � sÞq−1f ðs; uðsÞ;GuðsÞÞkds

≤M1

ð1þ q2Þ1−q1ððt0Þ1þq2 � e1þq2Þ1−q1 sup

s∈½sk;t0−e�kVðt00 � sÞ � V ðt0 � sÞk

þ 2M1M

ΓðqÞð1þ q2Þ1−q1eð1þq2Þð1−q1Þ

→ 0 as t00 → t0:

As a result, kðF2uÞðt00Þ− ðF2uÞðt0Þk→ 0 independently of u∈ΩR as t00→ t0, whichmeans that

F2 : ΩR →ΩR is equicontinuous.

Step V: We show that F : ΩR →ΩR is a ρ-set contractive map.

AJMS26,1/2

10

For any bounded set D⊂ΩR, by Lemma 2.7, we know that there exists a countable setD0 ¼ fung⊂D such that

αðF2ðDÞÞ≤ 2αðF2ðD0ÞÞ: (3.21)

Since F2ðD0Þ⊂F2ðΩRÞ is bounded and equicontinuous, by Lemma 2.5, we get

αðF2ðD0ÞÞ ¼ maxt∈½sk;tkþ1 �;k¼0;1;2;...;m

αðF2ðD0ÞðtÞÞ: (3.22)

For every t ∈ ½sk; tkþ1�; k ¼ 0; 1; 2; . . . ;m, by Lemma 2.6, the assumption (H5) and (3.1), wehave

αðF2ðD0ÞðtÞÞ ¼ α��Z t

sk

ðt � sÞq−1V ðt � sÞf ðs; unðsÞ;GunðsÞÞds��

≤2M

ΓðqÞZ t

sk

ðt � sÞq−1αðff ðs; unðsÞ;GunðsÞÞgÞds

≤2M

ΓðqÞZ t

sk

ðt � sÞq−1½LkαðD0ðsÞÞ þ NkαðGD0ðsÞÞ�ds:

(3.23)

Meanwhile, we have

αðGD0ðsÞÞ≤ αðGD0Þ≤ kGkαðD0Þ≤G*αðD0Þ≤G*αðDÞ: (3.24)

Therefore,

αðF2ðD0ÞðtÞÞ≤ 2M

Γðqþ 1Þ ðLk þ NkG*Þðtkþ1 � skÞqαðDÞ≤ 2ML

Γðqþ 1Þ αðDÞ: (3.25)

From (3.21) and (3.25), we obtain

αðF2ðDÞÞ≤ 4ML

Γðqþ 1Þ αðDÞ: (3.26)

From (3.15) and Lemma 2.8, we know that for any bounded set D⊂ΩR,

αðF1ðDÞÞ≤MK*αðDÞ: (3.27)

Therefore, by (3.26) and (3.27), we obtain

αðFðDÞÞ≤ αðF1ðDÞÞ þ αðF2ðDÞÞ≤M

�K* þ 4L

Γðqþ 1Þ�αðDÞ ¼ Λ2αðDÞ: (3.28)

Now combining (3.28) with (3.2) and Definition 2.9, we get that F : ΩR →ΩR is a ρ-set-contractive map with ρ ¼ Λ2. Hence Lemma 2.10 implies that F has at least one fixed pointu∈ΩR, which is a PC-mild solution of (1.1). ,

Next, we present an example to illustrate our main result.

Example. Consider the following fractional partial differential system with non-instantaneous impulses and nonlocal conditions:


11

cD12uðt; xÞ þ v2

vx2uðt; xÞ

¼ 1

25

e−t

1þ etuðt; xÞ þ

Z t

0

1

50e−suðs; xÞds; x∈ ð0; 1Þ; t ∈ ð0; 1

3�∪ð2

3; 1�;

uðt; 0Þ ¼ uðt; 1Þ ¼ 0; t ∈ ½0; 1�;

uðt; xÞ ¼ e−ðt−13Þ

4

juðt; xÞj1þ juðt; xÞj; x∈ ð0; 1Þ; t ∈ ð1

3;2

3�;

uð0; xÞ þX2

i¼1

1

3iuð1

i; xÞ ¼ u0ðxÞ; x∈ ½0; 1�:

8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:

(3.29)

Let X ¼ L2½0; 1� and Au ¼ u00with DðAÞ ¼ fu∈X : u; u

0are absolutely continuous and

u00∈X ; uð0Þ ¼ uð1Þ ¼ 0g. It is well known by [17], that −A generates an equicontinuous

C0-semigroup TðtÞðt ≥ 0Þ on X, and kTðtÞk≤ 1, for any t ≥ 0. Let a ¼ t2 ¼ 1; t0 ¼ s0 ¼ 0;t1 ¼ 1

3; s1 ¼ 23. By putting

uðtÞ ¼ uðt; $Þ;

f ðt; uðtÞ;GuðtÞÞ ¼ 1

25

e−t

1þ etuðt; $Þ þ

Z t

0

1

50e−suðs; $Þds;

GuðtÞ ¼Z t

0

1

50e−suðs; $Þds;

γ1ðt; uðtÞÞ ¼e−ðt−

13Þ

4

juðt; $Þj1þ juðt; $Þj;

gðuÞ ¼X2

i¼1

1

3iu

�1

i; $

�;

the parabolic partial differential equation (3.29) can be rewritten into the abstract form of (1.1)for m ¼ 1: It is easy to verify that the assumptions (H1)–(H5) and condition (3.2) hold with

q ¼ 1

2;M ¼ 1; fðtÞ ¼ 1

25

e−t

1þ etþ 1

50; ψ ; ðrÞ ¼ r;

α� ¼ 4

9; K ¼ Kγ1 ¼

1

4; L ¼ 0:02; Λ1 ¼ 0:69 < 1; Λ2 ¼ 0:53 < 1:

Therefore, Theorem 3.1 is applicable, so the system (3.29) has at least one PC-mild solution.

References

[1] L. Bai, J.J. Nieto, Variational approach to differential equations with not instantaneous impulses,Appl. Math. Lett. 73 (2017) 44–48.

[2] D.D. Bainov, V. Lakshmikantham, P.S. Simeonov, Theory of Impulsive Differential Equations, in:Series in Modern Applied Mathematics, World Scientific, Singapore, 1989.

[3] J. Banas, K. Goebel, Measure of Noncompactness in Banach Space, Marcal Dekker Inc., NewYork, 1980.

AJMS26,1/2

12

[4] M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclusions, in:Contemp. Math. Appl., Hindawi Publ. Corp., New York, 2006.

[5] D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108(1998) 109–138.

[6] L. Byszewski, Theorem about the existence and uniqueness of solutions of a semilinear evolutionnonlocal Cauchy problem, J. Math. Appl. Anal. 162 (1991) 494–505.

[7] P. Chen, Y. Li, Mixed monotone iterative technique for a class of semilinear impulsive evolutionequations in Banach spaces, Nonlinear Anal. 74 (2011) 3578–3588.

[8] P. Chen, X. Zhang, Y. Li, Existence of mild solutions to partial differential equations with non-instantaneous impulses, Electron. J. Differential Equations 241 (2016) 1–11.

[9] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initialconditions, J. Math. Anal. Appl. 179 (1993) 630–637.

[10] M.M. El-Borai, Some probability densities and fundamental solutions of fractional evolutionequations, Chaos Solitons Fractals 14 (2002) 433–440.

[11] H. Heinz, On the behaviour of measures of noncompactness with respect to differentiation andintegration of vector valued functions, Nonlinear Anal. 7 (1983) 1351–1371.

[12] E. Hern�andez, D. O’Regan, On a new class of abstract impulsive differential equations, Proc.Amer. Math. Soc. 141 (2013) 1641–1649.

[13] P. Kumar, D.N. Pandey, D. Bahuguna, On a new class of abstract impulsive functional differentialequations of fractional order, J. Nonlinear Sci. Appl. 7 (2014) 102–114.

[14] J. Liang, J.H. Liu, T.J. Xiao, Nonlocal impulsive problems for integrodifferential equations, Math.Comput. Modelling 49 (2009) 789–804.

[15] S. Liang, R. Mei, Existence of mild solutions for fractional impulsive neutral evolution equationswith nonlocal conditions, Adv. Differential Equations (2014) http://dx.doi.org/10.1186/1687-1847-2014-101.

[16] M. McKibben, Discovering Evolution Equations with Applications, Chapman and Hall/CRC, BocaRaton, 2011.

[17] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, in:Applied Mathematical Sciences, Springer-Verlag, Berlin, 1983.

[18] M. Pierri, D. O’Regan, V. Rolnik, Existence of solutions for semi-linear abstract differentialequations with non instantaneous impulses, Appl. Math. Comput. 219 (2013) 6743–6749.

[19] J. Wang, X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput. 46 (2014) 321–334.

[20] D. Yang, J. Wang, Integral boundary value problems for nonlinear non-instataneous impulsivedifferential equations, J. Appl. Math. Comput. (2016) http://dx.doi.org/10.1007/s12190-016-1025-8.

[21] X. Yu, J. Wang, Periodic boundary value problems for nonlinear impulsive evolution equations onBanach spaces, Commun. Nonlinear Sci. Numer. Simul. 22 (2015) 980–989.

[22] X. Zhang, P. Chen, Fractional evolution equation nonlocal problems with noncompactsemigroups, Opuscula Math. 36 (2016) 123–137.

Corresponding authorArshi Meraj can be contacted at: [email protected]

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13


Estimation of different entropiesvia Abel–Gontscharoff

Green functions andFink’s identityusing Jensen type functionals

Khuram Ali KhanDepartment of Mathematics, University of Sargodha, Sargodha, Pakistan

Tasadduq NiazDepartment of Mathematics, University of Sargodha, Sargodha, Pakistan andDepartment of Mathematics, The University of Lahore, Sargodha-Campus,

Sargodha, Pakistan

Ðilda Pe�cari�cCatholic University of Croatia, Zagreb, Croatia, and

Josip Pe�cari�cRUDN University, Moscow, Russia

AbstractIn this work, we estimated the different entropies like Shannon entropy, R�enyi divergences, Csisz�ardivergence by using Jensen’s type functionals. The Zipf’s–Mandelbrot law and hybrid Zipf’s–Mandelbrotlaw are used to estimate the Shannon entropy. The Abel–Gontscharoff Green functions and Fink’s Identityare used to construct new inequalities and generalized them for m-convex function.

Keywordsm-convex function, Jensen’s inequality, Shannon entropy, f- and R�enyi divergence, Fink’s identity,

Abel–Gontscharoff Green function, Entropy


Estimation ofdifferententropies

15

© Khuram Ali Khan, Tasadduq Niaz, Ðilda Pe�cari�c and Josip Pe�cari�c. Published in Arab Journal ofMathematical Sciences. Published by Emerald Publishing Limited. This article is published under theCreative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate andcreate derivative works of this article (for both commercial and non-commercial purposes), subject to fullattribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

The research of 4th author was supported by the Ministry of Education and Science of the RussianFederation (the Agreement number No. 02.a03.21.0008).

The authors wish to thank the anonymous referees for their very careful reading of the manuscriptand fruitful comments and suggestions.

Authors contribution: All authors jointly worked on the results and they read and approved the finalmanuscript.

Competing interests: The authors declare that there is no conflict of interest regarding the publication ofthis paper.

The publisher wishes to inform readers that the article “Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals”was originally published bythe previous publisher of theArab Journal ofMathematical Sciences and the pagination of this article has beensubsequently changed. There has been no change to the content of the article. This changewas necessary forthe journal to transition from thepreviouspublisher to thenewone.Thepublisher sincerely apologises for anyinconvenience caused. To access and cite this article, please use Khan, K.A., Niaz, T., Pe�cari�c, Ð., Pe�cari�c, J.(2018), “Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity usingJensen type functionals” Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 15-39. The originalpublication date for this paper was 31/12/2018.



Received 7 November 2018Revised 15 December 2018

Accepted 18 December 2018


Vol. 26 No. 1/2, 2020pp. 15-39


DOI 10.1016/j.ajmsc.2018.12.002

1. Introduction and preliminary resultsIn recent years many researchers generalized different inequalities using different identitiesinvolving green functions, for example in [24] Nasir et al. generalized the Popoviciu inequalityusing Mongomery identity along with the new green function. Also in [25] Niaz et al. usedFink’s identity along with new Abel–Gontscharoff type Green functions for ‘two point rightfocal’ to generalize the refinement of Jensen inequality.

Themost commonly usedwords, the largest cities of countries, income of billionaire can bedescribed in terms of Zipf’s law. The f -divergence means the distance between twoprobability distributions by making an average value, which is weighted by a specifiedfunction. As f -divergence, there are other probability distributions like Csisz�ar f -divergence[11,12], some special case of which is Kullback–Leibler-divergence used to find theappropriate distance between the probability distributions (see [20,21]). The notion ofdistance is stronger than divergence because it gives the properties of symmetry and triangleinequalities. Probability theory has application in many fields and the divergence betweenprobability distribution has many applications in these fields.

Many natural phenomena like distribution of wealth and income in a society, distribution offace book likes, distribution of football goals follow power law distribution (Zipf’s Law). Likeabove phenomena, distribution of city sizes also follows Power Law distribution. Auerbach [3]first time gave the idea that the distribution of city size can be well approximated with the helpofPareto distribution (PowerLawdistribution). This ideawaswell refinedbymany researchersbut Zipf [32] worked significantly in this field. The distribution of city sizes is investigated bymany scholars of the urban economics, like Rosen and Resnick [29], Black and Henderson [4],Ioannides andOverman [19], Soo [30], Anderson andGe [2] andBosker et al. [5]. Zipf’s law statesthat: “The rank of cities with a certain number of inhabitants varies proportional to the citysizes with some negative exponent, say that is close to unit”. In other words, Zipf’s Law statesthat the product of city sizes and their ranks appear roughly constant. This indicates that thepopulation of the second largest city is one half of the population of the largest city and the thirdlargest city equal to the one third of the population of the largest city and the population of nthcity is 1

nof the largest city population. This rule is called rank, size rule and also named as Zipf’s

Law. Hence Zip’s Law not only shows that the city size distribution follows the Paretodistribution, but also shows that the estimated value of the shape parameter is equal to unity.

In [18] L. Horv�ath et al. introduced some new functionals based on the f -divergencefunctionals and obtained some estimates for the new functionals. They obtained f -divergenceand R�enyi divergence by applying a cyclic refinement of Jensen’s inequality. They alsoconstruct some new inequalities for R�enyi and Shannon entropies and used Zipf–Mandelbrotlaw to illustrate the results.

The inequalities involving higher order convexity are used by many physicists in higherdimension problems since the founding of higher order convexity by T. Popoviciu (see [27,p. 15]). It is quite interesting fact that there are some results that are true for convex functionsbut when we discuss them in higher order convexity they do not remain valid.

In [27, p. 16], the following criteria are given to check the m-convexity of the function.If f ðmÞ exists, then f is m-convex if and only if f ðmÞ ≥ 0.In recent years many researchers have generalized the inequalities for m-convex

functions; like S. I. Butt et al. generalized the Popoviciu inequality for m-convex functionusing Taylor’s formula, Lidstone polynomial, Montgomery identity, Fink’s identity,Abel–Gontscharoff interpolation and Hermite interpolating polynomial (see [6–10]).

Since many years Jensen’s inequality has of great interest. The researchers have given therefinement of Jensen’s inequality by defining some new functions (see [16,17]). Like manyresearchers L. Horv�ath and J. Pe�cari�c in [14,17], see also [15, p. 26], gave a refinement ofJensen’s inequality for convex function. They defined some essential notions to prove therefinement given as follows:

AJMS26,1/2

16

Let X be a set, and:PðXÞ :¼ Power set of X,jX j :¼ Number of elements of X,ℕ :¼ Set of natural numbers with 0.Consider q≥ 1 and r≥ 2 be fixed integers. Define the functions

Fr;S : f1; . . . ; qgr → f1; . . . ; qgr−1 1≤ S≤ r;

Fr : f1; . . . ; qgr →P�f1; . . . ; qgr−1�;

and

Tr : Pðf1; . . . ; qgrÞ→P�f1; . . . ; qgr−1�;

by

Fr;Sði1; . . . ; irÞ :¼ ði1; i2; . . . ; iS−1; iSþ1; . . . ; irÞ 1≤ S≤ r;

Frði1; . . . ; irÞ ¼[rS¼1

�Fr;Sði1; . . . ; irÞ

�;

and

TrðIÞ ¼8<:

f; I ¼ f;[ði1 ;...;irÞ∈I

Frði1; . . . ; irÞ; I ≠f:

9=;

Next let the function

αr;i : f1; . . . ; qgr →ℕ 1≤ i≤ q

defined by

αr;i ði1; . . . ; irÞ is the number of occurrences of i in the sequence ði1; . . . ; irÞ:For each I ∈P ðf1; . . . ; qgrÞ let

αI ;i :¼X

ði1 ;...irÞ∈ I

αr;iði1; . . . ; irÞ 1≤ i≤ q:

ðH1Þ Let n;m be fixed positive integers such that n≥ 1, m≥ 2 and let Im be a subset off1; . . . ; ngm such that

αIm ;i≥ 1 1≤ i≤ n:

Introduce the sets Il ⊂ f1; . . . ; ngl ðm− 1≥ l ≥ 1Þ inductively by

Il−1 :¼ TlðIlÞ m≥ l ≥ 2:

Obviously the sets I1 ¼ f1; . . . ; ng, by ðH1Þ and this insures that αI1;i ¼ 1ð1≤ i≤ nÞ. FromðH1Þwe have αIl ;i≥ 1ðm− 1≥ l ≥ 1; 1≤ i≤ nÞ.

For m≥ l ≥ 2, and for any ðj1; . . . ; jl−1Þ∈ Il−1, let

H Ilðj1; . . . ; jl−1Þ :¼�ðði1; . . . ; ilÞ; kÞ3 f1; . . . ; lgjFl;kði1; . . . ; ilÞ ¼ ðj1; . . . ; jl−1Þg:


17

With the help of these sets they define the functions ηIm;l : Il →ℕðm≥ l ≥ 1Þ inductively by

ηIm ;mði1; . . . ; imÞ :¼ 1 ði1; . . . ; imÞ∈ Im;

ηIm;l−1ðj1; . . . ; jl−1Þ :¼X

ðði1 ;...il Þ;kÞ∈H Ilðj1 ;...;jl�1Þ

ηIm ;lði1; . . . ; ilÞ:

They define some special expressions for 1≤ l ≤m, as follows

A m;l ¼ A m;lðIm; x1; . . . ; xn; p1; . . . ; pn; f Þ :¼ ðm� 1Þ!ðl � 1Þ!

Xði1 ;...il Þ∈Il

ηIm ;lði1; . . . ; ilÞ

3

Xl

j¼1

pijαIm ;ij

!f

0BBB@Pl

j¼1

pijαIm;ij

xijPl

j¼1

pijαIm ;ij

1CCCA

and prove the following theorem.

Theorem 1.1. Assume ðH1Þ, and let f : I →R be a convex function where I ⊂ℝ is aninterval. If x1; . . . ; xn ∈ I and p1; . . . ; pn are positive real numbers such that

PnS¼1pS ¼ 1, then

f

XnS¼1

pSxS

!≤A m;m ≤A m;m−1 ≤ � � � ≤A m;2 ≤A m;1 ¼

XnS¼1

pS f ðxSÞ: (1)

We define the following functionals by taking the differences of refinement of Jensen’sinequality given in (1).

Θ1ðf Þ ¼ A m;r � f

XnS¼1

pSxS

!; r ¼ 1; . . . ;m; (2)

Θ2ðf Þ ¼ A m;r � A m;k; 1≤ r < k≤m: (3)

Under the assumptions of Theorem 1.1, we have

Θiðf Þ≥ 0; i ¼ 1; 2: (4)

Inequalities (4) are reversed if f is concave on I.In [26], the green function G : ½α1; α2�3 ½α1; α2�→R is defined as

Gðu; vÞ ¼

8>><>>:

ðu� α2Þðv� α1Þα2 � α1

; α1 ≤ v≤ u;

ðv� α2Þðu� α1Þα2 � α1

; u≤ v≤ α2:

(5)

The function G is convex with respect to v and due to symmetry also convex with respectto u. One can also note that G is continuous function.

In [31] it is given that any function f : ½α1; α2�→R, such that f ∈C2ð½α1; α2�Þ can bewritten as

f ðuÞ ¼ α2 � u

α2 � α1

f ðα1Þ þ u� α1

α2 � α1

f ðα2Þ þZ α1

α2

Gðu; vÞf 00 ðvÞdv: (6)

AJMS26,1/2

18

2. Inequalities for Csisz�ar divergenceIn [11,12] Csisz�ar introduced the following notion.

Definition 1. Let f : ℝþ→ℝþ be a convex function, let r ¼ ðr1; . . . ; rnÞ and

q ¼ ðq1; . . . ; qnÞ be positive probability distributions. Then f -divergence functional isdefined by

If ðr; qÞ :¼Xni¼1

qi f

�ri

qi

�: (7)

And he stated that by defining

f ð0Þ :¼ limx→0þ

f ðxÞ; 0 f

�0

0

�:¼ 0; 0 f

�a0

:¼ lim

x→0þx f�a0

; a > 0; (8)

we can also use the nonnegative probability distributions as well.In [18], L. Horv�ath, et al. gave the following functional based on the previous definition.

Definition 2. Let I ⊂ℝ be an interval and let f : I →ℝ be a function, letr ¼ ðr1; . . . ; rnÞ∈Rn and q ¼ ðq1; . . . ; qnÞ∈ ð0;∞Þn such that

rS

qS

∈ I ; S ¼ 1; . . . ; n:

Then they define the sum bI f ðr; qÞ as

bI f ðr; qÞ :¼XnS¼1

qS f

�rS

qS

�: (9)

We apply Theorem 1.1 to bI f ðr; qÞTheorem 2.1. Assume ðH1Þ, let I ⊂ℝ be an interval and let r ¼ ðr1; . . . ; rnÞ andq ¼ ðq1; . . . ; qnÞ are in ð0;∞Þn such that

rS

qS

∈ I ; S ¼ 1; . . . ; n:

ðiÞ If f : I →ℝ is a convex function, then

bI f ðr; qÞ ¼XnS¼1

qS f�rS

qS

�¼ A

½1�m;1 ≥ A

½1�m;2 ≥ � � � ≥ A

½1�m;m−1 ≥ A

½1�m;m

≥ f�Pn

S¼1rSPn

S¼1qS

�XnS¼1

qS:(10)

where

A½1�m;l ¼

ðm� 1Þ!ðl � 1Þ!

Xði1 ;...;il Þ∈Il

ηIm ;lði1; . . . ; ilÞ Xl

j¼1

qijαIm;ij

!f

0BB@Pl

j¼1

rijαIm ;ijPl

j¼1

qijαIm ;ij

1CCA (11)

If f is a concave function, then inequality signs in (10) are reversed.


19

ðiiÞ If f : I →R is a function such that x→ xf ðxÞðx∈ IÞ is convex, then Xn

S¼1

rS

!f

XnS¼1

rSPn

S¼1qS

!≤ A½2�

m;m ≤A½2�m;m−1 ≤ � � � ≤A

½2�m;2 ≤A

½2�m;1

¼XnS¼1

rS f

�rS

qS

�¼ bI idf ðr; qÞ

(12)

where

A½2�m;l ¼


Xði1;...;il Þ∈Il

ηIm;lði1; . . . ; ilÞ Xl

j¼1

qijαIm ;ij

!0BB@Pl

j¼1

rijαIm ;ijPl

j¼1

qijαIm ;ij

1CCA3 f

0BB@Pl

j¼1

rijαIm ;ijPl

j¼1

qijαIm ;ij

1CCA

Proof. ðiÞ Consider pS ¼ qSPn

S¼1qS

and xS ¼ rS

qS

in Theorem 1.1, we have

f

XnS¼1

qSPn

S¼1qS

rS

qS

!≤ � � � ≤ ðm� 1Þ!

ðl � 1Þ!X

ði1 ;...;il Þ∈IlηIm ;lði1; . . . ; ilÞ

3

0BB@Xl

j¼1

qijXn

S¼1qS

αIm ;ij

1CCAf

0BBBBBBB@

Pl

j¼1

qijXn

i¼1qi

αIm ;ij

rijqij

Pl

j¼1

qijXn

i¼1qi

αIm ;ij

1CCCCCCCA

≤ . . . ≤XnS¼1

qSPn

i¼1qS

f

�rS

qS

� (13)

And taking the sumPn

S¼1qi we have (10).

ðiiÞ Using f :¼ idf (where “id” is the identity function) in Theorem 1.1, we have

XnS¼1

pSxS f

XnS¼1

pSxS

!≤ � � � ≤ ðm� 1Þ!

ðl � 1Þ!X


3

Xl

j¼1

pijαIm ;ij

!0BBB@Pl

j¼1

pijαIm ;ij

xijPl

j¼1

pijαIm ;ij

1CCCA f

0BBB@Pl

j¼1

pijαIm ;ij

xijPl

j¼1

pijαIm ;ij

1CCCA

≤ . . . ≤XnS¼1

pSxS f ðxSÞ

(14)

Now on using pS ¼ qSPn

S¼1qS

and xS ¼ rSqS

; S ¼ 1; . . . ; n, we get

AJMS26,1/2

20

XnS¼1

qSPn

S¼1qS

rS

qS

f

XnS¼1

qSPn

S¼1qS

rS

qS

!≤ � � � ≤ ðm� 1Þ!

ðl � 1Þ!X


3

0BB@Xl

j¼1

qijXn

S¼1qS

αIm ;ij

1CCA

0BBBBBBB@

Pl

j¼1

qijXn

S¼1qS

αIm ;ij

rijqij

Pl

j¼1

qijXn

S¼1qS

αIm ;ij

1CCCCCCCA

f

0BBBBBBB@

Pl

j¼1

qijXn

S¼1qS

αIm ;ij

rijqij

Pl

j¼1

qijXn

S¼1qS

αIm ;ij

1CCCCCCCA

≤XnS¼1

qSPn

S¼1qS

rS

qS

f

�rS

qS

�

(15)

On taking sumPn

S¼1qS on both sides, we get (12). ,

3. Inequalities for Shannon Entropy

Definition 3 (See [18]). The Shannon entropy of positive probability distributionr ¼ ðr1; . . . ; rnÞ is defined by

S :¼ −XnS¼1

rSlogðrSÞ: (16)

Corollary 3.1. Assume ðH1Þ.ðiÞ If q ¼ ðq1; . . . ; qnÞ∈ ð0;∞Þn, and the base of log is greater than 1, then

S ≤A½3�m;m ≤A

½3�m;m−1 ≤ � � � ≤A

½3�m;2 ≤A

½3�m;1 ¼ log

�nPn

S¼1qS

�XnS¼1

qS; (17)

where

A½3�m;l ¼ −

ðm� 1Þ!!

ðl � 1Þ!X


Xl

j¼1

qijαIm ;ij

!log

Xl

j¼1

qijαIm ;ij

!: (18)

If the base of log is between 0 and 1, then inequality signs in (17) are reversed.ðiiÞ If q ¼ ðq1; . . . ; qnÞ is a positive probability distribution and the base of log is greater than 1,then we have the estimates for the Shannon entropy of q

S ≤A½4�m;m ≤A

½4�m;m−1 ≤ � � � ≤A

½4�m;2 ≤A

½4�m;1 ¼ log ðnÞ; (19)

where

A½4�m;l ¼ −




j¼1

qijαIm ;ij

!log

Xl

j¼1

qijαIm ;ij

!:


21

Proof. ðiÞ Using f :¼ log and r ¼ ð1; . . . ; 1Þ in Theorem 2.1 ðiÞ, we get (17).ðiiÞ It is the special case of ðiÞ. ,Definition 4 (See [18])The Kullback–Leibler divergence between the positive probability distributionr ¼ ðr1; . . . ; rnÞ and q ¼ ðq1; . . . ; qnÞ is defined by

Dðr; qÞ :¼XnS¼1

rilog

�ri

qi

�: (20)

Corollary 3.2. Assume ðH1Þ.ðiÞ Let r ¼ ðr1; . . . ; rnÞ∈ ð0;∞Þn and q :¼ ðq1; . . . ; qnÞ∈ ð0;∞Þn. If the base of log isgreater than 1, then

XnS¼1

rSlog

XnS¼1

rSPn

S¼1qS

!≤A½5�

m;m ≤A½5�m;m−1 ≤ � � � ≤A

½5�m;2 ≤A

½5�m;1

¼XnS¼1

rSlog

�rS

qS

�¼ Dðr; qÞ;

(21)

where

A½5�m;l ¼




j¼1

qijαIm ;ij

!0BB@Pl

j¼1

rijαIm ;ijPl

j¼1

qijαIm ;ij

1CCA3 log

0BB@Pl

j¼1

rijαIm ;ijPl

j¼1

qijαIm ;ij

1CCA:

If the base of log is between 0 and 1, then inequality in (21) is reversed.ðiiÞ If r and q are positive probability distributions, and the base of l is greater than 1, then wehave

Dðr; qÞ ¼ A½6�m;1 ≥A

½6�m;2 ≥ � � � ≥A

½6�m;m−1 ≥A½6�

m;m ≥ 0; (22)

where

A½6�m;l ¼




j¼1

qijαIm ;ij

!0BB@Pl

j¼1

rijαIm ;ijPl

j¼1

qijαIm ;ij

1CCA3 log

0BB@Pl

j¼1

rijαIm ;ijPl

j¼1

qijαIm ;ij

1CCA

If the base of log is between 0 and 1, then inequality signs in (22) are reversed.

Proof. ðiÞ On taking f :¼ log in Theorem 2.1 ðiiÞ, we get (21).ðiiÞ Since r and q are positive probability distributions therefore

PnS¼1rS ¼

PnS¼1qS ¼ 1, so

the smallest term in (21) is given as

XnS¼1

rSlog

XnS¼1

rSPn

S¼1qS

!¼ 0: (23)

Hence for positive probability distribution r and q the (21) will become (22). ,

AJMS26,1/2

22

4. Inequalities for R�enyi Divergence and EntropyThe R�enyi divergence and entropy come from [28].

Definition 5. Let r ¼ ðr1; . . . ; rnÞ and q :¼ ðq1; . . . ; qnÞ be positive probabilitydistributions, and let λ≥ 0, λ≠ 1.

ðaÞ The R�enyi divergence of order λ is defined by

Dλðr; qÞ :¼ 1

λ� 1log

Xni¼1

qi

�ri

qi

�λ!: (24)

ðbÞ The R�enyi entropy of order λ of r is defined by

HλðrÞ :¼ 1

1� λlog

Xni¼1

rλi

!: (25)

The R�enyi divergence and the R�enyi entropy can also be extended to non-negativeprobability distributions. If λ→ 1 in (24), we have the Kullback–Leibler divergence, and ifλ→ 1 in (25), then we have the Shannon entropy. In the next two results, inequalities can befound for the R�enyi divergence.

Theorem 4.1. Assume ðH1Þ, let r ¼ ðr1; . . . ; rnÞ and q ¼ ðq1; . . . ; qnÞ are probabilitydistributions.ðiÞ If 0≤ λ≤ μ such that λ; μ≠ 1, and the base of log is greater than 1, then

Dλðr; qÞ≤A½7�m;m ≤A

½7�m;m−1 ≤ � � � ≤A

½7�m;2 ≤A

½7�m;1 ¼ Dμðr; qÞ; (26)

where

A½7�m;l ¼

1

μ� 1log

0BBBBBBB@ðm� 1Þ!ðl � 1Þ!



j¼1

rijαIm;ij

!3

0BBB@

Pl

j¼1

rijαIm ;ij

rijqij

!λ�1

Pl

j¼1

rijαIm ;ij

1CCCA

μ−1λ−1

1CCCCCCCA

The reverse inequalities hold in (26) if the base of log is between 0 and 1.ðiiÞ If 1 < μ and the base of log is greater than 1, then

D1ðr; qÞ ¼ Dðr; qÞ ¼XnS¼1

rSlog

�rS

qS

�≤A½8�

m;m ≤A½8�m;m−1 ≤ � � � ≤A

½8�m;2 ≤A

½8�m;1 ¼ Dμðr; qÞ;

(27)

where

A½8�m;l ¼ ≤

1

μ� 1log

0BB@ðm� 1Þ!

ðl � 1Þ!X

ði1 ;...;il Þ∈IlηIm;lði1; . . . ; ilÞ

Xl

j¼1

rijαIm ;ij

!3 exp

ðμ� 1ÞPl

j¼1

rijαIm ;ij

log

rijqij

!

Pl

j¼1

rijαIm ;ij

1CCA

0BBBB@

1CCCCA

here the base of exp is the same as the base of log, and the reverse inequalities hold if the base oflog is between 0 and 1.ðiiiÞ If 0≤ λ < 1, and the base of log is greater than 1, then


23

Dλðr; qÞ≤A½9�m;m ≤A

½9�m;m−1 ≤ � � � ≤A

½9�m;2 ≤A

½9�m;1 ¼ D1ðr; qÞ; (28)

where

A½9�m;l ¼

1

λ� 1




j¼1

rijαIm ;ij

!3 log

Pl

j¼1

rijαIm ;ij

rijqij

!λ−1

Pl

j¼1

rijαIm ;ij

0BBBBB@

1CCCCCA(29)

Proof. By applying Theorem 1.1 with I ¼ ð0;∞Þ, f : ð0;∞Þ→R, f ðtÞ ¼ tμ−1λ−1

pS :¼ rS; xS :¼�rS

qS

�λ−1

; S ¼ 1; . . . ; n;

we have

XnS¼1

qS

�rS

qS

�λ!μ−1

λ−1

¼ Xn

S¼1

rS

�rS

qS

�λ!μ−1

λ−1

≤ . . . ≤ðm� 1Þ!ðl � 1Þ!



j¼1

rijαIm ;ij

!

3

0BBB@

Pl

j¼1

rijαIm ;ij

rijqij

!λ�1

Pl

j¼1

rijαIm ;ij

1CCCA

μ−1λ−1

≤ . . . ≤XnS¼1

rS

��rS

qS

�λ�1�μ−1λ−1

(30)

if either 0≤ λ < 1 < β or 1 < λ≤ μ, and the reverse inequality in (30) holds if 0≤ λ≤ β < 1.By raising to power 1

μ− 1, we have from all

XnS¼1

qS

�rS

qS

�λ! 1

λ−1

≤ . . . ≤

0BBBBBBB@ðm� 1Þ!ðl � 1Þ!

Xði1 ;...;il Þ∈ Il


j¼1

rijαIm;ij

!3

0BBB@

Pl

j¼1

rijαIm ;ij

rijqij

!λ�1

Pl

j¼1

rijαIm ;ij

1CCCA

μ�1λ�1

1CCCCCCCA

1μ−1

≤ . . . ≤

0B@X

n

S¼1

rS

��rS

qS

�λ�1�μ�1λ�1

1CA

1μ−1

¼ Xn

S¼1

qS

�rS

qS

�μ! 1

μ−1

(31)

Since log is increasing if the base of log is greater than 1, it now follows (26). If the base of logis between 0 and 1, then log is decreasing and therefore inequality in (26) is reversed. If λ ¼ 1and β ¼ 1, we have ðiiÞ and ðiiiÞ respectively by taking limit, when λ goes to 1. ,

Theorem 4.2. Assume ðH1Þ, let r ¼ ðr1; . . . ; rnÞ and q ¼ ðq1; . . . ; qnÞ are probabilitydistributions. If either 0≤ λ < 1 and the base of log is greater than 1, or 1 < λ and the base oflog is between 0 and 1, then

AJMS26,1/2

24

1

Pn

S¼1qS

�rSqS

�λ

XnS¼1

qS

�rS

qS

�λ

log

�rS

qS

�

¼ A½10�m;1 ≤A

½10�m;2 ≤ � � � ≤A

½10�m;m−1 ≤A½10�

m;m ≤Dλðr;qÞ≤A½11�m;m

≤A½11�m;m ≤ � � � ≤A

½11�m;2 ≤A

½11�m;1 ¼ D1ðr;qÞ

(32)

where

A½10�m;m ¼ 1

ðλ� 1ÞPn

S¼1qS

�rSqS

�λðm� 1Þ!ðl � 1Þ!


ηIm ;lði1; . . . ilÞ

3

Xl

j¼1

rijαIm ;ij

rijqij

!λ−1!log

0BBB@

Pl

j¼1

rijαIm ;ij

rijqij

!λ−1

Pl

j¼1

rijαIm ;ij

1CCCA

and

A½11�m;m ¼ 1

λ� 1



ηIm ;lði1; . . . ilÞ Xl

j¼1

rijαIm ;ij

!3 log

0BBB@

Pl

j¼1

rijαIm ;ij

rijqij

!λ−1

Pl

j¼1

rijαIm ;ij

1CCCA:

The inequalities in (32) are reversed if either 0≤ λ < 1 and the base of log is between 0 and 1,or 1 < λ and the base of l is greater than 1.

Proof. We prove only the case when 0≤ λ < 1 and the base of log is greater than 1 and theother cases can be proved similarly. Since 1

λ− 1 < 0and the function log is concave then choose

I ¼ ð0;∞Þ, f :¼ log, pS ¼ rS, xS :¼ ðrS

qS

Þλ−1 in Theorem 1.1, we have

Dλðr; qÞ ¼ 1

λ� 1log

XnS¼1

qS

�rS

qS

�λ!

¼ 1

λ� 1log

XnS¼1

rS

�rS

qS

�λ−1!

≤ � � � ≤ 1

λ� 1




j¼1

rijαIm;ij

!log

Pl

j¼1

rijαIm ;ij

rijqij

!λ−1

Pl

j¼1

rijαIm ;ij

0BBBBB@

1CCCCCA

≤ � � � ≤ 1

λ� 1

XnS¼1

rSlog

��rS

qS

�λ−1�¼XnS¼1

rSlog

�rS

qS

�¼ D1ðr; qÞ

(33)

and this gives the upper bound for Dλðr; qÞ.


25

Since the base of log is greater than 1, the function x↦ xf ðxÞ ðx > 0Þ is convex therefore1

1− λ < 0 and Theorem 1.1 gives

Dλðr; qÞ ¼ 1

λ� 1log

XnS¼1

qS

�rS

qS

�λ!

¼ 1

λ� 1

�Pn

S¼1qS

�rSqS

�λ� Xn

S¼1

qS

�rS

qS

�λ!log

XnS¼1

qS

�rS

qS

�λ!

≥ � � � ≥ 1

λ� 1

�Pn

S¼1qS

�rSqS

�λ� ðm� 1Þ!ðl � 1Þ!



j¼1

rijαIm;ij

!

0BBB@

Pl

j¼1

rijαIm ;ij

rijqij

!λ−1

Pl

j¼1

rijαIm ;ij

1CCCAlog

0BBB@

Pl

j¼1

rijαIm ;ij

rijqij

!λ−1

Pl

j¼1

rijαIm ;ij

1CCCA

¼ 1

λ� 1

�Pn

S¼1 qS

�rSqS

�λ� ðm� 1Þ!ðl � 1Þ!


ηIm ;lði1; . . . ; ilÞ

Xl

j¼1

rijαIm;ij

rijqij

!λ−1!log

0BBB@

Pl

j¼1

rijαIm ;ij

rijqij

!λ−1

Pl

j¼1

rijαIm ;ij

1CCCA

≥ � � � ≥ 1

λ� 1

XnS¼1

rS

�rS

qS

�λ−1

log

�rS

qS

�λ−11

Pn

S¼1 rS

�rSqS

�λ−1

¼ 1

Pn

S¼1qS

�rSqS

�λ

XnS¼1

qS

�rS

qS

�λ

log

�rS

qS

�

(34)

which give the lower bound of Dλðr; qÞ. ,By using Theorems 4.1, 4.2 and Definition 5, some inequalities of R�enyi entropy are

obtained. Let 1n¼�1n; . . . ; 1

n

be a discrete probability distribution.

Corollary 4.3. Assume ðH1Þ, let r ¼ ðr1; . . . ; rnÞ and q ¼ ðq1; . . . ; qnÞ are positiveprobability distributions.

ðiÞ If 0≤ λ≤ μ, λ; μ≠ 1, and the base of log is greater than 1, then

HλðrÞ ¼ logðnÞ � Dλ

�r;1

n

�≥A½12�

m;m ≥A½12�m;m ≥ � � �A½12�

m;2 ≥A½12�m;1 ¼ HμðrÞ; (35)

AJMS26,1/2

26

where

A½12�m;l ¼

1

1� μlog

0BBBBBB@ðm� 1Þ!ðl � 1Þ!


ηIm ;lði1; . . . ; ilÞ3 Xl

j¼1

rijαIm;ij

!3

0BB@Pl

j¼1

rλij

αIm ;ijPl

j¼1

rijαIm ;ij

1CCA

μ−1λ−1

1CCCCCCA:

The reverse inequalities hold in (35) if the base of log is between 0 and 1.

ðiiÞ If 1 < μ and base of log is greater than 1, then

S ¼ −XnS¼1

pilogðpiÞ≥A½13�m;m ≥A

½13�m;m−1 ≥ � � � ≥A

½13�m;2 ≥A

½13�m;1 ¼ HμðrÞ (36)

where

A½13�m;l ¼ logðnÞ þ 1

1� μlog

0BB@ðm� 1Þ!

ðl � 1Þ!X

ði1;...;il Þ∈IlηIm;lði1; . . . ; ilÞ

Xl

j¼1

rijαIm ;ij

!

3 exp

0BB@ðμ� 1ÞPl

j¼1

rijαIm ;ij

logðnrijÞPl

j¼1

rijαIm ;ij

1CCA1CCA;

the base of exp is the same as the base of log. The inequalities in (36) are reversed if the base oflog is between 0 and 1.

ðiiiÞ If 0≤ λ < 1, and the base of log is greater than 1, then

HλðrÞ≥A½14�m;m ≥A

½14�m;m−1 ≥ � � � ≥A

½14�m;2 ≤A

½14�m;1 ¼ S; (37)

where

A½14�m;m ¼ 1

1� λðm� 1Þ!ðl � 1Þ!



j¼1

rijαIm ;ij

!3 log

0BB@Pl

j¼1

rλij

αIm ;ijPl

j¼1

rijαIm ;ij

1CCA: (38)

The inequalities in (37) are reversed if the base of log is between 0 and 1.

Proof. ðiÞ Suppose q ¼ 1nthen from (24), we have

Dλðr; qÞ ¼ 1

λ� 1log

XnS¼1

nλ−1rλS

!¼ logðnÞ þ 1

λ� 1log

XnS¼1

rλS

!; (39)

therefore we have


�r;1

n

�: (40)


27

Now using Theorem 4.1 ðiÞ and (40), we get


�r;1

n

�≥ � � � ≥ logðnÞ � 1

μ� 1

3 log

0BBBBBB@nμ−1



ηIm ;lði1; . . . ; ilÞ3 Xl

j¼1

rijαIm ;ij

!0BB@Pl

j¼1

rλij

αIm ;ijPl

j¼1

rijαIm ;ij

1CCA

μ−1λ−1

1CCCCCCA

≥ � � � ≥ logðnÞ � Dμðr; qÞ ¼ HμðrÞ;

(41)

ðiiÞ and ðiiiÞ can be proved similarly. ,

Corollary 4.4. Assume ðH1Þ and let r ¼ ðr1; . . . ; rnÞ and q ¼ ðq1; . . . ; qnÞ are positiveprobability distributions.

If either 0≤ λ < 1 and the base of log is greater than 1, or 1 < λ and the base of log isbetween 0 and 1, then

−1Pn

S¼1rλS

XnS¼1

rλSlogðrSÞ ¼ A

½15�m;1 ≥A

½15�m;2 ≥ � � � ≥A

½15�m;m−1 ≥A½15�

m;m

≥HλðrÞ≥A½16�m;m ≥A

½16�m;m−1 ≥ � � �A½16�

m;2 ≥A½16�m;1 ¼ HðrÞ;

(42)

where

A½15�m;l ¼ 1

ðλ� 1ÞPn

S¼1rλS


Xði1 ;...;il ÞIIl


j¼1

rλijαIm ;ij

!log

0BB@nλ�1

Pl

j¼1

rλijαIm;ijPl

j¼1

rijαIm;ij

1CCA

and

A½16�m;1 ¼ 1

1� λðm� 1Þ!ðl � 1Þ!



j¼1

rijαIm;ij

!log

0BB@Pl

j¼1

rλijαIm ;ijPl

j¼1

rijαIm ;ij

1CCA:

The inequalities in (42) are reversed if either 0≤ λ < 1and the base of log is between 0 and 1, or1 < λ and the base of log is greater than 1.

Proof. The proof is similar to Corollary 4.3 by using Theorem 4.2. ,

5. Inequalities by using Zipf–Mandelbrot lawIn probability theory and statistics, the Zipf–Mandelbrot law is a distribution. It is a powerlaw distribution on ranked data, named after the linguist G. K. Zipf who suggests a simplerdistribution called Zipf’s law. The Zipf’s law is defined as follows (see [32]).

AJMS26,1/2

28

Definition 6. LetN be a number of elements, S be their rank and t be the value of exponentcharacterizing the distribution. Zipf’s law then predicts that out of a population of Nelements, the normalized frequency of element of rank S, f ðS;N ; tÞ is

f ðS;N ; tÞ ¼1StPN

j¼11jt

: (43)

The Zipf–Mandelbrot law is defined as follows (see [22]).

Definition 7. Zipf–Mandelbrot law is a discrete probability distribution depending onthree parameters N ∈ f1; 2; . . . ;g; q∈ ½0;∞Þ and t > 0, and is defined by

f ðS;N ; q; tÞ :¼ 1

ðS þ qÞtHN ;q;t

; S ¼ 1; . . . ;N ; (44)

where

HN ;q;t ¼XNj¼1

1

ðjþ qÞt: (45)

If the total mass of the law is taken over allN, then for q≥ 0, t > 1, S∈N, density function ofZipf–Mandelbrot law becomes

f ðS; q; tÞ ¼ 1

ðS þ qÞtHq;t

; (46)

where

Hq;t ¼X∞j¼1

1

ðjþ qÞt: (47)

For q ¼ 0, the Zipf–Mandelbrot law (44) becomes Zipf’s law (43).

Conclusion 5.1. Assume ðH1Þ, let rbe a Zipf–Mandelbrot law, by Corollary 4.3 ðiiiÞ, we get: If0≤ λ < 1, and the base of log is greater than 1, then

HλðrÞ ¼ 1

1� λlog

1

H λN ;q;t

XnS¼1

1

ðS þ qÞλS!≥ � � � ≥

1

1� λðm� 1Þ!ðl � 1Þ!



j¼1

1

αIm ;ijðij þ qÞHN :q;t

!

3 log

0BBB@

1

H λ−1N ;q;t

Pl

j¼11

αIm ;ijðij�qÞλSPl

j¼11

αIm ;ijðij�qÞS

1CCCA≥ � � � ≥

t

HN ;q;t

XNS¼1

logðS þ qÞðS þ qÞt þ logðHN ;q;tÞ ¼ S:

(48)

The inequalities in (48) are reversed if the base of log is between 0 and 1.


29

Conclusion 5.2. Assume ðH1Þ, let r1 and r2 be the Zipf–Mandelbort law with parametersN ∈ f1; 2; . . .g, q1; q2 ∈ ½0;∞Þ and S1; S2 > 0, respectively, then from Corollary 3.2 ðiiÞ, wehave if the base of l is greater than 1, then

Dðr1; r2Þ ¼XnS¼1

1

ðS þ q1Þt1HN ;q1 ;t1

log

ðS þ q2Þt2HN ;q2;t2

ðS þ q1Þt1HN ;q2;t1

!≥ � � �

≥ðm� 1Þ!ðl � 1Þ!


ηIm;lði1; . . . ; ilÞ

3

0BB@Xl

j¼1

1ðijþq2Þt2HN ;q2 ;t2

αIm;ij

1CCA

0BBBBBBB@

Pl

j¼1

1

ðijþq1Þt1HN ;q1 ;t1

αIm ;ij

Pl

j¼1

1


αIm ;ij

1CCCCCCCA

3 log

0BBBBBBB@

Pl

j¼1

1


αIm ;ij

Pl

j¼1

1


αIm ;ij

1CCCCCCCA

≥ � � � ≥ 0:

(49)

The inequalities in (49) are reversed if the base of l is between 0 and 1.

6. Shannon entropy, Zipf–Mandelbrot law and hybrid Zipf–Mandelbrot lawHere we maximize the Shannon entropy using method of Lagrange multiplier under someequations constraints and get the Zipf–Mandelbrot law.

Theorem 6.1. If J ¼ f1; 2; . . . ;Ng, for a given q≥ 0 a probability distribution thatmaximizes the Shannon entropy under the constraintsX

S ∈ J

rS ¼ 1;XS∈ J

rSðInðS þ qÞÞ :¼ ψ ;

is Zipf–Mandelbrot law.

Proof. If J ¼ f1; 2; . . . ;Ng, we set the Lagrange multipliers λ and t and consider theexpression

~S ¼ −XNS¼1

rS ln rS � λ

XNS¼1

rS � 1

!� t

XNS¼1

rSlnðS þ qÞ � ψ

!

Just for the sake of convenience, replace λ by ln λ− 1, thus the last expression gives

~S ¼ −XNS¼1

rSln rS � ðln λ� 1Þ XN

S¼1

rS � 1

!� t

XNS¼1

rSlnðS þ qÞ � ψ

!

From ~SrS ¼ 0, for S ¼ 1; 2; . . . ;N, we get

rS ¼ 1

λðS þ qÞt;

and on using the constraintPN

S¼1rS ¼ 1, we have

AJMS26,1/2

30

λ ¼XNS¼1

�1

ðS þ 1Þt�

where t > 0, concluding that

rS ¼ 1

ðS þ qÞtHN ;q;t

; S ¼ 1; 2; . . . ;N :,

Remark6.2. Observe that the Zipf–Mandelbrot law and ShannonEntropy can be boundedfrom above (see [23]).

S ¼ −XNS¼1

f ðS;N ; q; tÞln f ðS;N ; q; tÞ≤ �XNS¼1

f ðS;N ; q; tÞln qS

where ðq1; . . . ; qN Þ is a positive N-tuple such thatPN

S¼1qS ¼ 1.

Theorem 6.3. If J ¼ f1; . . . ;Ng, then probability distribution that maximizes Shannonentropy under constraintsX

S ∈ J

rS :¼ 1;XS ∈ J

rS lnðS þ qÞ :¼ Ψ ;XS ∈ J

SrS :¼ η

is hybrid Zipf–Mandelbrot law given as

rS ¼ wS

ðS þ qÞkΦ*ðk; q;wÞ; S∈ J ;

where

ΦJ ðk; q;wÞ ¼XS∈ J

wS

ðS þ qÞk:

Proof. First consider J ¼ f1; . . . ;Ng, we set the Lagrange multiplier and consider theexpression

~S ¼ −XNS¼1

rS ln rS þ lnw

XNS¼1

SrS � η

!� ðln λ� 1Þ

XNS¼1

rS � 1

!� k

XNS¼1

rS lnðS þ qÞ � Ψ

!:

On setting ~SrS ¼ 0, for S ¼ 1; . . . ;N, we get

−ln rS þ S lnw� ln λ� k lnðS þ qÞ ¼ 0;

after solving for rS, we get λ ¼PN

S¼1wS

ðSþqÞk; and we recognize this as the partial sum of Lerch’s

transcendent that we will denote by

Φ�N ðk; q;wÞ ¼

XNS¼1

wS

ðS þ qÞk withw≥ 0; k > 0:

,

Remark 6.4. Observe that for Zipf–Mandelbrot law, Shannon entropy can be boundedfrom above (see [23]).


31

S ¼ −XNS¼1

fhðS;N ; q; kÞln fhðS;N ; q; kÞ≤ �XNS¼1

fhðS;N ; q; kÞln qS

where ðq1; . . . ; qN Þ is any positive N-tuple such thatPN

S¼1qS ¼ 1.Under the assumption of Theorem 2.1 ðiÞ, define the non-negative functionals as follows:

Θ3ðf Þ ¼ A ½1�m;r � f

�Pn

S¼1rSPn

S¼1qS

�XnS¼1

qS; r ¼ 1; . . . ;m; (50)

Θ4ðf Þ ¼ A ½1�m;r � A ½1�

m;k; 1≤ r < k≤m: (51)

Under the assumption of Theorem 2.1 ðiiÞ, define the non-negative functionals as follows:

Θ5ðf Þ ¼ A ½2�m;r �

XnS¼1

rS

!f

�Pn

S¼1rSPn

S¼1qS

�; r ¼ 1; . . . ;m; (52)

Θ6ðf Þ ¼ A ½2�m;r � A ½2�

m;k; 1≤ r < k≤m: (53)

Under the assumption of Corollary 3.1 ðiÞ, define the following non-negative functionals

Θ7ðf Þ ¼ A½3�m;r þ

Xni¼1

qilogðqiÞ; r ¼ 1; . . . ; n (54)

Θ8ðf Þ ¼ A½3�m;r � A

½3�m;k; 1≤ r < k≤m: (55)

Under the assumption of Corollary 3.1 ðiiÞ, define the following non-negative functionals as

Θ9ðf Þ ¼ A½4�m;r � S; r ¼ 1; . . . ;m (56)

Θ10ðf Þ ¼ A½4�m;r � A

½4�m;k; 1≤ r < k≤m: (57)

Under the assumption of Corollary 3.2 ðiÞ, let us define the non-negative functionals asfollows:

Θ11ðf Þ ¼ A½5�m;r �

XnS¼1

rS log

XnS¼1

logrnPn

S¼1qS

!; r ¼ 1; . . . ;m

(58)

Θ12ðf Þ ¼ A½5�m;r � A

½5�m;k; 1≤ r < k≤m: (59)

Under the assumption of Corollary 3.2 ðiiÞ, define the non-negative functionals as followsΘ13ðf Þ ¼ A½6�

m;r � A½6�m;k; 1≤ r < k≤m: (60)

Under the assumption of Theorem 4.1 ðiÞ, consider the following functionals

Θ14ðf Þ ¼ A½7�m;r � Dλðr;qÞ; r ¼ 1; . . . ;m (61)

Θ15ðf Þ ¼ A½7�m;r � A

½7�m;k; 1≤ r < k≤m: (62)

AJMS26,1/2

32

Under the assumption of Theorem 4.1 ðiiÞ, consider the following functionals:

Θ16ðf Þ ¼ A½8�m;r � D1ðr;qÞ; r ¼ 1; . . . ;m (63)

Θ17ðf Þ ¼ A½8�m;r � A

½8�m;k; 1≤ r < k≤m: (64)

Under the assumption of Theorem 4.1 ðiiiÞ, consider the following functionals:


Θ19ðf Þ ¼ A½9�m;r � A

½9�m;k; 1≤ r < k≤m: (66)

Under the assumption of Theorem 4.2 consider the following non-negative functionals

Θ20ðf Þ ¼ Dλðr;qÞ � A½10�m;r; r ¼ 1; . . . ;m (67)

Θ21ðf Þ ¼ A½10�m;k � A½10�

m;r; 1≤ r < k≤m: (68)


Θ23ðf Þ ¼ A½11�m;r � A½11�

m;r; 1≤ r < k≤m: (70)

Θ24ðf Þ ¼ A½11�m;r � A

½10�m;k; r ¼ 1; . . . ;m; k ¼ 1; . . . ;m: (71)

Under the assumption of Corollary 4.3 (i), consider the following non-negative functionals

Θ25ðf Þ ¼ HλðrÞ � A½12�m;r; r ¼ 1; . . . ;m (72)

Θ26ðf Þ ¼ A½12�m;k � A½12�

m;r; 1≤ r < k≤m: (73)

Under the assumption of Corollary 4.3 (ii), consider the following functionals

Θ27ðf Þ ¼ S � A½13�m;r; r ¼ 1; . . . ;m (74)

Θ28ðf Þ ¼ A½13�m;k � A½13�

m;r; 1≤ r < k≤m: (75)

Under the assumption of Corollary 4.3 (iii), consider the following functionals

Θ29ðf Þ ¼ HλðrÞ � A½14�m;r; r ¼ 1; . . . ;m (76)

Θ30ðf Þ ¼ A½14�m;k � A½14�

m;r; 1≤ r < k≤m: (77)

Under the assumption of Corollary 4.4, define the following functionals

Θ31 ¼ A½15�m;r � HλðrÞ; r ¼ 1; . . . ;m (78)

Θ32 ¼ A½15�m;r � A

½15�m;k; 1≤ r < k≤m: (79)

Θ33 ¼ HλðrÞ � A½16�m;r; r ¼ 1; . . . ;m (80)


33

Θ34 ¼ A½16�m;k � A½16�

m;r; 1≤ r < k≤m: (81)

Θ35 ¼ A½15�m;r � A

½16�m;k; r ¼ 1; . . . ;m; k ¼ 1; . . . ;m: (82)

7.Generalization of refinement of Jensen’s, R�enyi andShannon type inequalitiesFink’s Identity and Abel–Gontscharoff Green functionIn [13], A. M. Fink gave the following result.

Let f : ½α1; α2�→ℝ, where ½α1; α2� be an interval, is a function such that f ðn−1Þ isabsolutely continuous then the following identity holds

f ðzÞ ¼ n

α2 � α1

Z α2

α1

f ðζÞdζ þXn−1λ¼1

n� λλ!

f ðλ−1Þðα2Þðz� α2Þλ � f ðλ−1Þðα1Þðz� α1Þλ

α2 � α1

!

þ 1

ðn� 1Þ!ðα2 � α1ÞZ α2

α1

ðz� ζÞn−1Fα2α1ðζ; zÞf ðnÞðζÞdζ; (83)

where

Fα2α1ðζ; zÞ ¼

ζ � α1; α1 ≤ ζ≤ z≤ α2;ζ � α2; α1 ≤ z < ζ≤ α2:

(84)

The complete reference about Abel–Gontscharoff polynomial and theorem for ‘two-pointright focal’ problem is given in [1].

TheAbel–Gontscharoff polynomial for ‘two-point right focal’ interpolating polynomial forn ¼ 2 can be given as

f ðzÞ ¼ f ðα1Þ þ ðz� α1Þf 0 ðα2Þ þZ α2

α1

G1ðz;wÞf 00 ðwÞdw; (85)

where

G1ðz;wÞ ¼α1 � w; α1 ≤w≤ z;α1 � z; z≤w≤α2:

(86)

In [8], S. I. Butt et al. gave some new types of Green functions defined as

G2ðz;wÞ ¼α2 � z; α1 ≤w≤ z;α2 � w; z≤w≤α2;

(87)

G3ðz;wÞ ¼z� α1; α1 ≤w≤ z;w� α1; z≤w≤α2;

(88)

G4ðz;wÞ ¼α2 � w; α1 ≤w≤ z;α2 � z; z≤w≤α2;

(89)

Figure 1 shows the graph of Green functions Giðz;wÞ; i ¼ 1; 2; 3; 4 defined in (86)–(89)respectively for fixed value of w. They also introduced some new Abel–Gontscharoff typeidentities by using these new Green functions in the following lemma.

Lemma A. Let f : ½α1; α2� be a twice differentiable function and Gk ðk ¼ 2; 3; 4Þ be the ‘two-point right focal problem’-type Green functions defined by (87)–(89). Then the followingidentities hold:

f ðzÞ ¼ f ðα2Þ � ðα2 � zÞf 0 ðα1Þ �Z α2

α1


AJMS26,1/2

34

f ðzÞ ¼ f ðα2Þ � ðα2 � α1Þf 0 ðα2Þ þ ðz� α1Þ f 0 ðα1Þ þZ α2

α1


f ðzÞ ¼ f ðα1Þ þ ðα2 � α1Þ f 0 ðα1Þ � ðα2 � zÞ f 0 ðα2Þ þZ α2

α1

G4ðz;wÞ f 00 ðwÞdw: (92)

Theorem 7.1. Assume (H1), and let f : I ¼ ½α1; α2�→R be a function such that for m≥ 3

(an integer) f ðm−1Þ is absolutely continuous. Also, let x1; . . . ; xn ∈ I, p1; . . . ; pn, be positive realnumbers such that

Pni¼1pi¼ 1. Assume that Fα2

α1 , Gk ðk ¼ 1; 2; 3; 4Þ and Θi ði ¼ 1; . . . ; 35Þ arethe same as defined in (84), (86)–(89), (2), (3), (50)–(82) respectively.Then:

(1) For k ¼ 1; 3; 4 we have the following identities:

Θiðf Þ ¼ ðm� 2Þ�f0 ðα2Þ � f

0 ðα1Þα2 � α1

�Z α2

α1

ΘiðGkð$;wÞÞdwþ 1

α2 � α1

Z α2

α1

ΘiðGkð$;wÞÞ

3Xm−3

λ¼1

�m� 2� λ

λ!

��f ðλþ1Þðα2Þðw� α2Þλ � f ðλþ1Þðα1Þðw� α1Þλ

�dw

þ 1

ðm� 3Þ!ðα2 � α1ÞZ α2

α1

f ðmÞðζÞ

3

�Z α2

α1

ΘiðGkð$;wÞÞðw� ζÞm−3Fα2α1α2ðζ;wÞdw

�dζ; i ¼ 1; . . . ; 35:

(93)

Figure 1.Graph of Green

functions for fix w.


35

(2) For k ¼ 2 we have

Θiðf Þ ¼ ð−1Þðm� 2Þ�f0 ðα2Þ � f

0 ðα1Þα2 � α1

�Z α2

α1

ΘiðG2ð$;wÞÞdw

þ ð−1Þα2 � α1

Z α2

α1

ΘiðG2ð$;wÞÞ3Xm−3

λ¼1

�m� 2� λ

λ!

��f ðλþ1Þðα2Þðw� α2Þλ

� f ðλþ1Þðα1Þðw� α1Þλ�dwþ ð−1Þ


α1

f ðmÞðζÞ

3

�Z α2

α1

ΘiðG2ð$;wÞÞðw� ζÞm−3Fα2α1ðζ;wÞdw

�dζ;

i ¼ 1; . . . ; 35:

(94)

Proof. (i) Using Abel–Gontsharoff-typeidentities (85), (91), (92) in Θiðf Þ, i ¼ 1; . . . ; 35, andusing properties of Θiðf Þ, we get

Θiðf Þ ¼Z α2

α1

ΘiðGkð$;wÞÞf 00 ðwÞdw; i ¼ 1; 2: (95)

From identity (83), we get

f0 ðwÞ ¼ ðm� 2Þ

�f0 ðα2Þ � f

0 ðα1Þα2 � α1

�þXm−3

λ¼1

�m� 2� λ

λ!

�

3

�f ðλÞðα2Þðw� α2Þλ−1 � f ðλÞðα2Þðw� α2Þλ−1

α2 � α1

�

þ 1


α1

ðw� ζÞm−3Fα2α1ðζ;wÞf ðmÞðζÞdζ: (96)

Using (95) and (96) and applying Fubini’s theorem we get the result (93) for k ¼ 1; 3; 4.

(ii) Substituting Abel–Gontscharoff-typeinequality (90) in Θiðf Þ, i ¼ 1; . . . ; 35, andfollowing similar steps to (i), we get (94). ,

Theorem 7.2. Assume (H1), and let f : I ¼ ½α1; α2�→R be a function such that for m≥ 3

(an integer) f ðm−1Þ is absolutely continuous. Also, let x1; . . . ; xn ∈ I, p1; . . . ; pn are positive realnumbers such that

Pni¼1pi ¼ 1. Assume that Fα2

α1 , Gk ðk ¼ 1; 2; 3; 4Þ and Θi (i ¼ 1; 2) are the

same as defined in (84), (86)–(89), (2), (3), (50)–(82) respectively. For m≥ 3 assume that

Z α2

α1

ΘiðGkð$; ζÞÞðw� ζÞm−3Fα2α1ðζ;wÞdw≥ 0; ζ∈ ½α1; α2�; i ¼ 1; . . . ; 35; (97)

for k ¼ 1; 3; 4. If f is an m-convex function, then

AJMS26,1/2

36

(i) For k ¼ 1; 3; 4, the following holds:

Θið f Þ≥ ðm� 2Þ�f0 ðα2Þ � f

0 ðα1Þα2 � α1

�Z α2

α1

ΘiðGkð$;wÞÞdw

þ 1

α2 � α1

Z α2

α1

ΘiðGkð$;wÞÞ3Xm−3

λ¼1

�m� 2� λ

λ!


� f ðλþ1Þðα1Þðw� α1ÞλÞdw;i ¼ 1; . . . ; 35:

(98)

(ii) For k ¼ 2, we have

Θiðf Þ≤ ð−1Þðm� 2Þ�f0 ðα2Þ � f

00 ðα1Þα2 � α1

�Z α2

α1

ΘiðG2ð$;wÞÞdw

þ ð−1Þα2 � α1

Z α2

α1

ΘiðG2ð$;wÞÞ3Xm−3

λ¼1

�m� 2� λ

λ!


� f ðλþ1Þðα1Þðw� α1ÞλÞdw;i ¼ 1; . . . ; 35:

(99)

Proof. (i) Since f ðm−1Þ is absolutely continuous on ½α1; α2�, f ðmÞ exists almost everywhere.Also, since f is m-convex therefore we have f ðmÞðζÞ≥ 0 for a.e. on ½α1; α2�. So, applyingTheorem 1.1, we obtain (98).

(ii) Similar to (i). ,

RemarkA. We can investigate the bounds for the identities related to the generalization ofrefinement of Jensen inequality using inequalities for the C�ebys�ev functional and someresults relating to the G€russ and Ostrowski type inequalities can be constructed as given inSection 3 of [6]. Also we can construct the non-negative functionals from inequalities (98)–(99)and give related mean value theorems and we can construct the new families ofm-exponentially convex functions and Cauchy means related to these functionals as givenin Section 4 of [6].

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[26] J. Pe�cari�c, K.A. Khan, I. Peri�c, Generalization of Popoviciu type inequalities for symmetric meansgenerated by convex functions, J. Math. Comput. Sci. 4 (6) (2014) 1091–1113.

[27] J. Pe�cari�c, F. Proschan, Y.L. Tong, Convex Functions, Partial Orderings and StatisticalApplications, Academic Press, New York, 1992.

[28] A. R�enyi, On measure of information and entropy, in: Proceeding of the Fourth BerkelySymposium on Mathematics, Statistics and Probability, 1960, pp. 547–561.

[29] K.T. Rosen, M. Resnick, The size distribution of cities: an examination of the Pareto law andprimacy, J. Urban Econ. 8 (2) (1980) 165–186.

AJMS26,1/2

38

[30] K.T. Soo, Zipf’s Law for cities: a cross-country investigation, Reg. Sci. Urban Econ. 35 (3) (2005)239–263.

[31] D.V. Widder, Completely convex function and Lidstone series, Trans. Amer. Math. Soc. 51 (1942)(1942) 387–398.

[32] G.K. Zipf, Human Behaviour and the Principle of Least-Effort, Addison-Wesley, Cambridge MAedn. Reading, 1949.

Corresponding authorTasadduq Niaz can be contacted at: [email protected]



39


Some new fractional integralinequalities for generalizedrelative semi-m-(r ; h1, h2)-

preinvex mappings via generalizedMittag-Leffler function

Artion Kashuri and Rozana LikoDepartment of Mathematics, Faculty of Technical Science, University Ismail Qemali,

Vlora, Albania

AbstractThe authors discover a new identity concerning differentiable mappings defined onm-invex set via fractionalintegrals. By using the obtained identity as an auxiliary result, some fractional integral inequalities forgeneralized relative semi-m-ðr; h1; h2Þ-preinvexmappings by involving generalizedMittag-Leffler function arepresented. It is pointed out that some new special cases can be deduced from main results of the paper. Alsothese inequalities have some connections with known integral inequalities. At the end, some applications tospecial means for different positive real numbers are provided as well.

KeywordsHermite–Hadamard inequality, H€older’s inequality,Minkowski inequality, Powermean inequality,

Generalized Mittag-Leffler function, Fractional integrals, m-invex


1. IntroductionThe following double inequality is known as Hermite–Hadamard inequality.

Theorem 1.1. Let f : I ⊆ℝ→ℝ be a convex mapping on an interval I of real numbers anda; b∈ I with a < b: Then the subsequent double inequality holds:

New fractionalintegral

inequalities

41

JEL Classification — primary 26A51, secondary 26A33, 26D07, 26D10, 26D15, 33E12© Artion Kashuri and Rozana Liko. Published in the Arab Journal of Mathematical Sciences.


We thank anonymous referee for his/her valuable suggestion regarding the manuscript.The publisher wishes to inform readers that the article “Some new fractional integral inequalities for

generalized relative semi-m-(r; h1, h2)-preinvex mappings via generalized Mittag-Leffler function” wasoriginally published by the previous publisher of the Arab Journal of Mathematical Sciences and thepagination of this article has been subsequently changed. There has been no change to the content of thearticle. This change was necessary for the journal to transition from the previous publisher to the newone. The publisher sincerely apologises for any inconvenience caused. To access and cite this article,please use Kashuri, A., Liko, R. (2019), “Some new fractional integral inequalities for generalized relativesemi-m-(r; h1, h2)-preinvex mappings via generalized Mittag-Leffler function”, Arab Journal ofMathematical Sciences, Vol. 26 No. 1/2, pp. 41-55, The original publication date for this paper was02/01/19.



Received 24 August 2018Revised 19 October 2018



Vol. 26 No. 1/2, 2020pp. 41-55


DOI 10.1016/j.ajmsc.2018.12.003

f

�aþ b

2

�≤

1

b� a

Z b

a

f ðxÞdx≤ f ðaÞ þ f ðbÞ2

: (1.1)

For recent results concerning Hermite–Hadamard type inequalities through various classesof convex functions the readers are referred to [3–5,7,8,12–,20,21,23–25,29,32] and thereferences mentioned in these papers.

Let us recall some special functions and evoke some basic definitions as follows.

Definition 1.2 ([20]). Let f ∈L½a; b�: The Riemann–Liouville integrals Jαaþ f and Jαb− f oforder α > 0 with a ≥ 0 are defined by

Jαaþ f ðxÞ ¼ 1

ΓðαÞZ x

a

ðx� tÞα−1f ðtÞdt; x > a

and

Jαb− f ðxÞ ¼1

ΓðαÞZ b

x

ðt � xÞα−1f ðtÞdt; b > x;

where ΓðαÞ ¼ Rþ∞

0e−uuα−1du: Here J 0aþ f ðxÞ ¼ J 0b− f ðxÞ ¼ f ðxÞ:

Note that α ¼ 1; the fractional integral reduces to the classical integral.

Definition 1.3 ([27]). Let μ; ν; k; l; γ be positive real numbers and ω ∈ ℝ: Then thegeneralized fractional integral operators containing Mittag-Leffler function eγ;δ;kμ;ν;l;ω;aþ andeγ;δ;kμ;ν;l;ω;b−

for a real valued continuous function f are defined by:�eγ;δ;kμ;ν;l;ω;aþ f

�ðxÞ ¼

Z x

a

ðx� tÞv−1Eγ;δ;kμ;ν;l ðωðx� tÞμÞ f ðtÞdt (1.2)

and�eγ;δ;kμ;ν;l;ω;b− f

�ðxÞ ¼

Z b

x

ðt � xÞv−1Eγ;δ;kμ;ν;l ðωðt � xÞμÞf ðtÞdt;

where the function Eγ;δ;kμ;ν;l is the generalized Mittag-Leffler function defined as

Eγ;δ;kμ;ν;l ðtÞ ¼

X∞0

ðγÞkntnΓðμnþ νÞðδÞln

(1.3)

and ðaÞn is the Pochhammer symbol, it defined as

ðaÞn ¼ aðaþ 1Þðaþ 2Þ$ . . . $ðaþ n� 1Þ; ðaÞ0 ¼ 1:

For ω ¼ 0 in (1.2), integral operator eγ;δ;kμ;ν;l;ω;aþ reduces to the Riemann–Liouville fractionalintegral operator.

In [27,30] properties of generalized integral operator and generalized Mittag-Leffler

functions are studied in detail. In [27] it is proved that Eγ;δ;kμ;ν;l ðtÞ is absolutely convergent for

k < l þ μ:Let S be the sum of series of absolute terms ofEγ;δ;kμ;ν;l ðtÞ:Wewill use this property of

Mittag-Leffler function in sequel.

Definition 1.4 ([1]). A set K ⊆ ℝn is said to be invex with respect to the mappingΛ : K3K→ ℝn; if xþ t Λðy; xÞ∈K for every x; y∈K and t ∈ ½0; 1�:

AJMS26,1/2

42

Definition 1.5 ([7]). A non-negative function f : I ⊆ ℝ → ½0;þ∞Þ is said to beP-function, if

f ðtxþ ð1� tÞyÞ≤ f ðxÞ þ f ðyÞ; ∀x; y∈ I ; t ∈ ½0; 1�:

Definition 1.6 ([22]). Let h : ½0; 1�→ ℝ be a non-negative function and h ≠ 0. Thefunction f on the invex set K is said to be h-preinvex with respect to Λ, if

f ðxþ tΛðy; xÞÞ ≤ hð1� tÞ f ðxÞ þ hðtÞ f ðyÞ (1.4)

for each x; y∈ K and t ∈ ½0; 1�where f ð$Þ > 0.

Definition 1.7 ([31]). Let f : K ⊆ ℝ→ ℝ be a non-negative function. A functionf : K→ ℝ is said to be a tgs-convex on K if the inequality

f ðð1� tÞxþ tyÞ≤ tð1� tÞ½ f ðxÞ þ f ðyÞ� (1.5)

holds for all x; y ∈ K and t ∈ ð0; 1Þ.Definition 1.8 ([19]). A function f : I ⊆ ℝ→ ℝ is said to be MT-convex, if it is non-negative and ∀x; y ∈ I and t ∈ ð0; 1Þ satisfies the subsequent inequality:

f ðtxþ ð1� tÞyÞ≤ffiffit

p

2ffiffiffiffiffiffiffiffiffiffi1� t

p f ðxÞ þffiffiffiffiffiffiffiffiffiffi1� t

p

2ffiffit

p f ðyÞ: (1.6)

Definition 1.9 ([25]).A function: f : I ⊆ ℝ → ℝ is said to bem-MT-convex, if f is positiveand for ∀x; y∈ I, and t ∈ ð0; 1Þ, among m∈ ð0; 1�, satisfies the following inequality

f ðtxþmð1� tÞyÞ≤ffiffit

p

2ffiffiffiffiffiffiffiffiffiffi1� t

p f ðxÞ þmffiffiffiffiffiffiffiffiffiffi1� t

p

2ffiffit

p f ðyÞ: (1.7)

Definition 1.10 ([8]). A set K ⊆ℝn is named as m-invex with respect to the mappingΛ : K3 K→ℝn for some fixed m ∈ ð0; 1�; if mxþ tΛðy;mxÞ ∈ K holds for each x; y ∈ Kand any t ∈ ½0; 1�:Remark 1.11. In Definition 1.10, under certain conditions, the mapping Λðy;mxÞ could bereduced to Λðy; xÞ: For example when m ¼ 1; then the m-invex set degenerates an invex seton K:

Definition 1.12 ([26]). Let K ⊆ ℝ be an open m-invex set with respect to the mappingΛ : K 3 K→ℝ and h1; h2 : ½0; 1�→ ½0;þ∞Þ. A function f : K→ ℝ is said to begeneralized ðm; h1; h2Þ-preinvex, if

f ðmxþ tΛðy;mxÞÞ≤mh1ðtÞf ðxÞ þ h2ðtÞf ðyÞ (1.8)

is valid for all x; y ∈ K and t ∈ ½0; 1�, for some fixed m ∈ ð0; 1�.Motivated by the above literatures, the main objective of this paper is to establish in

Section 2, some new fractional integral inequalities for generalized relative semi-m-ðr; h1; h2Þ-preinvex mappings by involving generalized Mittag-Leffler function. It ispointed out that some new special cases will be deduced from main results of the paper.Also we will see that these inequalities have some connections with known integralinequalities. In Section 3, some applications to special means for different positive realnumbers will be given.


inequalities

43

2. Main resultsThe following definitions will be used in this section.

Definition 2.1. Let m: ½0; 1�→ ð0; 1� be a function. A set K ⊆ ℝn is named as m-invexwith respect to the mapping Λ : K 3 K → ℝn; ifmðtÞxþ ξΛðy;mðtÞxÞ∈ K holds for eachx; y ∈ K and any t; ξ ∈ ½0; 1�:Remark 2.2. In Definition 2.1, under certain conditions, the mapping Λðy;mðtÞxÞ for anyt; ξ ∈ ½0; 1� could be reduced to Λðy;mxÞ: For example whenmðtÞ ¼ m for all t ∈ ½0; 1�; thenthe m-invex set degenerates to an m-invex set on K:

We next introduce the notion of generalized relative semi-m-ðr; h1; h2Þ-preinvex mappings.

Definition 2.3. Let K ⊆ ℝ be an open m-invex set with respect to the mappingΛ : K3K → ℝ:Suppose h1; h2: ½0; 1�→ ½0;þ∞Þ;ψ : I → K are continuous functions andm: ½0; 1�→ ð0; 1�: A mapping f : K→ ð0;þ∞Þ is said to be generalized relative semi-m-ðr; h1; h2Þ-preinvex, if

f�mðtÞψðxÞ þ ξΛðψðyÞ;mðtÞψðxÞÞ� ≤ hmðξÞh1ðξÞf rðxÞ þ h2ðξÞf rðyÞ

i1r

(2.1)

holds for all x; y ∈ I and t; ξ ∈ ½0; 1�;where r ≠ 0:

Remark 2.4. In Definition 2.3, if we choosem ¼ m ¼ r ¼ 1; this definition reduces to thedefinition considered by Noor in [23] and Preda et al. in [11].

Remark 2.5. In Definition 2.3, if we choose m ¼ m ¼ r ¼ 1 and ψðxÞ ¼ x; then we getDefinition 1.12.

Remark 2.6. Let us discuss some special cases in Definition 2.3 as follows.

(I) Taking h1ðtÞ ¼ h2ðtÞ ¼ 1; then we get the generalized relative semi-ðm;PÞ-preinvexmappings.

(II) Taking h1ðtÞ ¼ ð1− tÞs and h2ðtÞ ¼ ts for s∈ ð0; 1�; then we get the generalizedrelative semi-ðm; sÞ-Breckner-preinvex mappings.

(III) Taking h1ðtÞ ¼ ð1− tÞ−s and h2ðtÞ ¼ t−s for s∈ ð0; 1�; then we get the generalizedrelative semi-ðm; sÞ-Godunova–Levin–Dragomir-preinvex mappings.

(IV) Taking h1ðtÞ ¼ hð1− tÞ and h2ðtÞ ¼ hðtÞ, then we get the generalized relative semi-ðm; hÞ-preinvex mappings.

(V) Taking h1ðtÞ ¼ h2ðtÞ ¼ tð1− tÞ, then we get the generalized relative semi-ðm; tgsÞ-preinvex mappings.

(VI) Taking h1ðtÞ ¼ffiffiffiffiffiffiffi1− t

p2ffiffit

p and h2ðtÞ ¼ffiffit

p2ffiffiffiffiffiffiffi1− t

p , then we get the generalized relative semi-m-MT-preinvex mappings.

It is worthmentioning here that to the best of our knowledge all the special cases discussedabove are new in the literature.

For establishing our main results we need to prove the following lemma.

Lemma 2.7. Let ψ : I → K and g : K → ℝ are continuous functions andm : ½0; 1�→ ð0; 1�: Suppose K ¼ ½mðtÞψðaÞ;mðtÞψðaÞ þ ΛðψðbÞ;mðtÞψðaÞÞ� ⊆ ℝ be anopen m-invex subset with respect to Λ : K3K→ ℝ for ΛðψðbÞ;mðtÞψðaÞÞ > 0 and

∀t ∈ ½0; 1�:Assume that f : K→ ℝ be a differentiable mapping on K+: If f0; g ∈ LðKÞ; then

the following equality for ν > 0 holds:

AJMS26,1/2

44

�Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞgðsÞEγ;δ;k

μ;ν;l ðωsμÞds�ν

3hf ðmðtÞψðaÞÞ þ f ðmðtÞψðaÞ þ ΛðψðbÞ;mðtÞψðaÞÞÞ

i

�νZ mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞ

�Z ξ


μ;ν;l ðωsμÞds�ν−1

3 gðξÞEγ;δ;kμ;ν;l ðωξμÞf ðξÞdξ


mðtÞψðaÞ


ξ

gðsÞEγ;δ;kμ;ν;l ðωsμÞds

�ν−1


¼Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞ

�Z ξ



f ’ðξÞdξ


mðtÞψðaÞ


ξ


�ν

3 f ’ðξÞdξ:

(2.2)

We denote

If ;g;E;Λ;ψ ;mðν; a; bÞ :¼Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞ

�Z ξ



3 f ’ðξÞdξ


mðtÞψðaÞ


ξ


�ν

3 f0 ðξÞdξ:

(2.3)

Proof. Integrating by parts, we get

If ;g;E;Λ;ψ ;mðν; a; bÞ ¼�Z ξ



f ðξÞmðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞ


mðtÞψðaÞ

�Z ξ


μ;ν; l ðωsμÞds�v−1


��Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

ξ


�ν

f ðξÞmðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞ


mðtÞψðaÞ


ξ


�v−1


¼�Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

ξ


�ν

3hf ðmðtÞψðaÞÞ þ f ðmðtÞψðaÞ þ ΛðψðbÞ;mðtÞψðaÞÞÞ

i


mðtÞψðaÞ

�Z ξ


μ;ν;l ðωsμÞds�v−1



mðtÞψðaÞ


ξ


�v−1

3 gðξÞEγ;δ;kμ;ν;l ðωξμÞf ðξÞdξ:

This completes the proof of the lemma.


inequalities

45

Using Lemma 2.7, we now state the following theorems for the corresponding version forpower of first derivative.

Theorem 2.8. Let h1; h2 : ½0; 1�→ ½0;þ∞Þ;ψ : I → K and g: K→ ℝ are continuousfunctions and m : ½0; 1�→ ð0; 1�: Suppose K ¼ ½mðtÞψðaÞ;mðtÞψðaÞþ ΛðψðbÞ; mðtÞψðaÞÞ�⊆ℝ be an open m-invex subset with respect to Λ : K3K→ ℝ for ΛðψðbÞ;mðtÞψðaÞÞ > 0 and ∀t ∈ ½0; 1�: Assume that f : K→ ð0;þ∞Þ be a differentiable mapping

on K+ such that f0; g ∈ LðKÞ: If ðf 0 ðxÞÞq is generalized relative semi-m-ðr; h1; h2Þ-preinvex

mapping, 0 < r≤ 1; k < l þ μ; q > 1; p−1 þ q−1 ¼ 1 and jjgjj∞ ¼ sups∈K jgðsÞj; then thefollowing inequality for ν > 0 holds:

jIf ;g;E;Λ;ψ ;mðν; a; bÞj≤ 2jjgjjν∞SνΛνþ1ðψðbÞ;mðtÞψðaÞÞffiffiffiffiffiffiffiffiffiffiffiffiffi

pνþ 1pp

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrqI r1 ðh1ðξÞ;mðξÞ; rÞ þ ðf 0 ðbÞÞrqI r2 ðh2ðξÞ; rÞrq

q;

(2.4)

where

I1ðh1ðξÞ;mðξÞ; rÞ :¼Z 1

0

m1rðξÞh1

r

1ðξÞdξ; I2ðh2ðξÞ; rÞ :¼Z 1

0

h1r

2ðξÞdξ:

Proof. From Lemma 2.7, the generalized relative semi-m-ðr; h1; h2Þ-preinvexity of ðf 0 ðxÞÞq;H€older inequality, Minkowski inequality, absolute convergence of Mittag-Leffler function,properties of the modulus, the fact gðsÞ ≤ kgk∞; ∀s ∈ K and changing the variableu ¼ mðtÞψðaÞ þ ξΛðψðbÞ;mðtÞψðaÞÞ; ∀t ∈ ½0; 1�;we have

jIf ;g;E;Λ;ψ ;mðν; a; bÞj≤Z ðmðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞ

Z ξ


μ;ν;l ðωsμÞdsv

3 jf 0 ðξÞjdξ

þZ ðmðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞ

Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

ξ


v

3 jf 0 ðξÞjdξ

≤


mðtÞψðaÞ

Z ξ


μ;ν;l ðωsμÞdspv

dξ

�1p

3

�Z ðmðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞðf 0 ðξÞÞqdξ

�1q

þ�Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞ


ξ


pv

dξ

�1p

3



�1q

AJMS26,1/2

46

≤ kgkv∞Sv 3



�1q

3

8>><>>:�Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞðξ� ðmðtÞψðaÞÞpvdξ

�1p

þ�Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞðmðtÞψðaÞ þ ΛðψðbÞ;mðtÞψðaÞÞ � ξÞpvdξ

�1p

9>>=>>;

¼ 2kgkv∞SvΛvþ1ðψðbÞ;mðtÞψðaÞÞffiffiffiffiffiffiffiffiffiffiffiffiffi

pvþ 1pp

3

�Z 1

0

ðf 0 ðmðtÞψðaÞ þ ξΛðψðbÞ;mðtÞψðaÞÞÞÞqdξ�1

q

≤2kgkv

∞SvΛvþ1ðψðbÞ;mðtÞψðaÞÞffiffiffiffiffiffiffiffiffiffiffiffiffi

pvþ 1pp

3

0@Z 1

0

½mðξÞh1ðξÞðf 0 ðaÞÞrq þ h2ðξÞðf 0 ðbÞÞrq�1rdξ1A

1q

≤2kgkv

∞SvΛvþ1ðψðbÞ;mðtÞψðaÞÞffiffiffiffiffiffiffiffiffiffiffiffiffi

pvþ 1pp

3

"�Z 1

0

m1rðξÞðf 0 ðaÞÞqh1

r

1ðξÞdξ�r

þ�Z 1

0

ðf 0 ðbÞÞqh1r

2ðξÞdξ�r# 1

rq

¼ 2kgkv∞SvΛvþ1ðψðbÞ;mðtÞψðaÞÞffiffiffiffiffiffiffiffiffiffiffiffiffi

pvþ 1pp

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrqI r1 ðh1ðξÞ;mðξÞ; rÞ þ ðf 0 ðbÞÞrqI r2 ðh2ðξÞ; rÞ:rq

q

So, the proof of this theorem is completed.

Remark 2.9. In Theorem 2.8, for h1ðtÞ ¼ t; h2ðtÞ ¼ 1− t; r ¼ 1; if we choose ΛðψðbÞ;mðtÞψðaÞÞ ¼ ψðbÞ−mðtÞψðaÞ;where mðtÞ≡ 1; ∀t ∈ ½0; 1� and ψðxÞ ¼ x; ∀x∈ I ; then

(1) If we put ω ¼ 0, we get [[28], Theorem 7].

(2) If we put ω ¼ 0 along with ν ¼ αk, we get [[10], Theorem 2.5].

(3) If we put gðsÞ ¼ 1 and ω ¼ 0, we get [[6], Theorem 2.3].

(4) If we put ω ¼ 0 and ν ¼ 1, we get [[6], Corollary 3].


inequalities

47

Remark 2.10. In Theorem 2.8, for h1ðtÞ ¼ t; h2ðtÞ ¼ 1− t; r ¼ 1; if we chooseΛðψðbÞ;mðtÞψðaÞÞ ¼ ψðbÞ−mðtÞψðaÞ; where mðtÞ≡ 1; ∀t ∈ ½0; 1� and ψðxÞ ¼ x; ∀x∈ I ;we get [[9], Corollary 3.8].

We point out some special cases of Theorem 2.8.

Corollary 2.11. In Theorem 2.8 for p ¼ q ¼ 2; we get the following inequality:

jIf ;g;E;Λ;ψ ;mðν; a; bÞj≤ 2kgkν∞SνΛνþ1ðψðbÞ;mðtÞψðaÞÞffiffiffiffiffiffiffiffiffiffiffiffiffi

2νþ 1p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞ2rI r1 ðh1ðξÞ;mðξÞ; rÞ þ ðf 0 ðbÞÞ2rI r2 ðh2ðξÞ; rÞ2r

q:

(2.5)

Corollary 2.12. In Theorem 2.8 for gðsÞ ¼ 1; we get the following inequality:

jIf ;E;Λ;ψ ;mðν; a; bÞj

¼� Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞEγ;δ;kμ;v;l ðωsμÞds

�νhf ðmðtÞψðaÞÞ þ f ðmðtÞψðaÞ

þΛðψðbÞ;mðtÞψðaÞÞÞi


mðtÞψðaÞ

�Z ξ

mðtÞψðaÞEγ;δ;kμ;ν;l ðωsμÞds

�v−1

Eγ;δ;kμ;ν;l ðωξμÞf ðξÞdξ


mðtÞψðaÞ


ξ

Eγ;δ;kμ;ν;l ðωsμÞds

�v−1

3Eγ;δ;kμ;ν;l ðωsμÞf ðξÞdξ

≤2SvΛvþ1ðψðbÞ;mðtÞψðaÞÞffiffiffiffiffiffiffiffiffiffiffiffiffi

pvþ 1pp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrqI r1 ðh1ðξÞ;mðξÞ; rÞ þ ðf 0 ðbÞÞrqI r2 ðh2ðξÞ; rÞ:rq

q

(2.6)

Corollary 2.13. In Theorem 2.8 for h1ðtÞ ¼ h2ðtÞ ¼ 1 and mðtÞ ¼ m∈ ð0; 1� for allt ∈ ½0; 1�; we get the following inequality for the generalized relative semi-ðm;PÞ-preinvexmappings: (2.7)

If ;g;E;Λ;ψ ;mðν; a; bÞ≤ 2kgkν∞SνΛνþ1ðψðbÞ;mψðaÞÞffiffiffiffiffiffiffiffiffiffiffiffiffi

pνþ 1pp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrq þ ðf 0 ðbÞÞrqrq

q: (2.7)

Corollary 2.14. In Theorem 2.8 for h1ðtÞ ¼ hð1− tÞ; h2ðtÞ ¼ hðtÞ and mðtÞ ¼ m∈ ð0; 1�for all t ∈ ½0; 1�;we get the following inequality for the generalized relative semi-ðm; hÞ-preinvexmappings:

If ;g;E;Λ;ψ ;mðν; a; bÞ ≤ 2kgkν∞SνΛνþ1ðψðbÞ;mψðaÞÞffiffiffiffiffiffiffiffiffiffiffiffiffi

pνþ 1pp 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI2ðhðξÞ; rÞq

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrq þ ðf 0 ðbÞÞrqrq

q:

(2.8)

AJMS26,1/2

48

Corollary 2.15. In Corollary 2.14 for h1ðtÞ ¼ ð1− tÞs and h2ðtÞ ¼ ts; we get the followinginequality for the generalized relative semi-ðm; sÞ-Breckner-preinvex mappings:


pνþ 1pp 3

ffiffiffiffiffiffiffiffiffiffir

r þ s

q

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrq þ ðf 0 ðbÞÞrqrq

q:

(2.9)

Corollary 2.16. In Corollary 2.14 for h1ðtÞ ¼ ð1− tÞ−s; h2ðtÞ ¼ t−s and 0 < s < r; we getthe following inequality for the generalized relative semi-ðm; sÞ-Godunova–Levin–Dragomir-preinvex mappings:


pνþ 1pp 3

ffiffiffiffiffiffiffiffiffiffir

r � s

q

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrq þ ðf 0 ðbÞÞrqrq

q:

(2.10)

Corollary 2.17. In Theorem 2.8 for h1ðtÞ ¼ h2ðtÞ ¼ tð1− tÞ andmðtÞ ¼ m∈ ð0; 1� for allt ∈ ½0; 1�; we get the following inequality for the generalized relative semi-ðm; tgsÞ-preinvexmappings:


pνþ 1pp 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ

�1þ 1

r; 1þ 1

r

�q

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrq þ ðf 0 ðbÞÞrqrq

q:

(2.11)

Corollary 2.18. In Corollary 2.14 for h1ðtÞ ¼ffiffiffiffiffiffiffi1− t

p2ffiffit

p ; h2ðtÞ ¼ffiffit


p and r∈ ; ð12; 1�we get thefollowing inequality for the generalized relative semi-m-MT-preinvex mappings:


pνþ 1pp

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ

�1� 1

2r; 1þ 1

2r

�q

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrq þ ðf 0 ðbÞÞrqrq

q:

(2.12)

Theorem 2.19. Let h1; h2 : ½0; 1�→ ½0;þ∞Þ;ψ : I → K and g : K→ ℝ are continuousfunctions and m : ½0; 1�→ ð0; 1�: Suppose K ¼ ½mðtÞψðaÞ;mðtÞψðaÞ þ ΛðψðbÞ; mðtÞψðaÞÞ�⊆ℝ be an open m-invex subset with respect to Λ : K3K→ ℝ for ΛðψðbÞ;mðtÞψðaÞÞi0and∀t ∈ ½0; 1�:Assume that f : K→ ð0;þ∞Þbe a differentiablemapping onK+ suchthat f

0; g∈LðKÞ: If ðf 0 ðxÞÞq is the generalized relative semi-m-ðr; h1; h2Þ-preinvex mapping,

0 < r≤ 1; k < l þ μ; q≥ 1 and kgk∞ ¼ sups∈K jgðsÞj; then the following inequality for ν > 0holds:

If ;g;E;Λ;ψ ;mðν; a; bÞ≤ kgkν∞SνΛνþ1ðψðbÞ;mðtÞψðaÞÞ

ðνþ 1Þ1−1q

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrqI r1 ðh1ðξÞ;mðξÞ; ν; rÞ þ ðf 0 ðbÞÞrqI r2 ðh2ðξÞ; ν; rÞrq

q

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrqI1rðh1ðξÞ;mðξÞ; ν; rÞ þ ðf 0 ðbÞÞrqI2rðh2ðξÞ; ν; rÞrq

q �;

(2.13)


inequalities

49

whereI1ðh1ðξÞ;mðξÞ; ν; rÞ :¼

Z 1

0

m1rðξÞξνh1

r

1ðξÞdξ; I2ðh2ðξÞ; ν; rÞ :¼Z 1

0

ξνh1r

2ðξÞdξ

and

I1ðh1ðξÞ;mðξÞ; ν; rÞ :¼Z 1

0

m1rðξÞð1� ξÞνh1

r

1ðξÞdξ; I2ðh2ðξÞ; ν; rÞ :¼Z 1

0

ð1� ξÞνh1r

2ðξÞdξ:

Proof. From Lemma 2.7, the generalized relative semi-m-ðr; h1; h2Þ-preinvexity of ðf 0 ðxÞÞq;the well-known power mean inequality, Minkowski inequality, absolute convergence ofMittag-Leffler function, properties of the modulus, the fact gðsÞ ≤ kgk∞; ∀s ∈ K andchanging the variable u ¼ mðtÞψðaÞ þ ξΛðψðbÞ;mðtÞψðaÞÞ; ∀t ∈ ½0; 1�;we haveIf ;g;E;Λ;ψ ;mðν; a; bÞ≤


mðtÞψðaÞ

Z ξ


μ;v;l ðωsμÞdsv

3 jf 0 ðξÞjdξ

þZ mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞ


ξ

gðsÞEγ;δ;kμ;v;l ðωsμÞds

v

3 jf 0 ðξÞjdξ

≤


mðtÞψðaÞ

Z ξ


μ;v;l ðωsμÞdsv�1−1

q

3


mðtÞψðaÞ

Z ξ


μ;v;l ðωsμÞdsv

ðf 0 ðξÞÞqdξ�1

q

þ Z mðtÞψðaÞþΛðψðbÞ;mðtÞψðaÞÞ

mðtÞψðaÞ


ξ


v

dξ

!1−1q

3


mðtÞψðaÞ


ξ


v

3 ðf 0 ðξÞÞqdξ!1

q

≤kgkv

∞SvΛvþ1ðψðbÞ;mðtÞψðaÞÞ

ðvþ 1Þ1−1q

3

�Z 1

0

ξvðf 0 ðmðtÞψðaÞ þ ξΛðψðbÞ;mðtÞψðaÞÞÞÞqdξ 1

q

þ� Z 1

0

ð1� ξÞvðf 0 ðmðtÞψðaÞ þ ξΛðψðbÞ;mðtÞψðaÞÞÞÞqdξ 1

q�

≤kgkv


ðvþ 1Þ1−1q

AJMS26,1/2

50

3

�Z 1

0

ξv½mðξÞh1ðf 0 ðaÞÞrq þ h2ðξÞðf 0 ðaÞÞrq�1rdξ 1

q

þ� Z 1

0

ð1� ξÞv½mðξÞh1ðf 0 ðaÞÞrq þ h2ðξÞðf 0 ðaÞÞrq�1rdξ 1

q�

≤kgkv


ðvþ 1Þ1−1q

3

(" Z 1

0

m1rðξÞðf 0 ðaÞÞqξvh1

r

1ðξÞdξ!r

þ Z 1

0

ðf 0 ðbÞÞqξvh1r

2ðξÞdξ!r# 1

rq

þ" Z 1

0

m1rðξÞðf 0 ðaÞÞqð1� ξÞvh1

r

1ðξÞdξ!r

þ Z 1

0

ðf 0 ðbÞÞqð1� ξÞvh1r

2ðξÞdξ!r# 1

rq)

¼ kgkv∞SvΛvþ1ðψðbÞ;mðtÞψðaÞÞ

ðvþ 1Þ1−1q

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrqI r1 ðh1ðξÞ;mðξÞ; v; rÞ þ ðf 0 ðaÞÞrqI r2 ðh2ðξÞ; v; rÞrq

q

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrqI1rðh1ðξÞ;mðξÞ; v; rÞ þ ðf 0 ðaÞÞrqI2rðh2ðξÞ; v; rÞrq

q �:

So, the proof of this theorem is completed.We point out some special cases of Theorem 2.19.

Corollary 2.20. In Theorem 2.19 for q ¼ 1; we get the following inequality:If ;g;E;Λ;ψ ;mðν; a; bÞ≤ kgkν∞SνΛνþ1ðψðbÞ;mðtÞψðaÞÞ

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrI r1 ðh1ðξÞ;mðξÞ; ν; rÞ þ ðf 0 ðbÞÞrI r2 ðh2ðξÞ; ν; rÞr

q

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrI1rðh1ðξÞ;mðξÞ; ν; rÞ þ ðf 0 ðbÞÞrI2rðh2ðξÞ; ν; rÞr

q �:

(2.14)

Corollary 2.21. In Theorem 2.19 for gðsÞ ¼ 1; we get the following inequality:

If ;E;Λ;ψ ;mðν; a; bÞ≤ SνΛνþ1ðψðbÞ;mðtÞψðaÞÞðνþ 1Þ1−1

q

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrqI r1 ðh1ðξÞ;mðξÞ; ν; rÞ þ ðf 0 ðbÞÞrqI r2 ðh2ðξÞ; ν; rÞrq

q

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrqI1rðh1ðξÞ;mðξÞ; ν; rÞ þ ðf 0 ðbÞÞrqI2rðh2ðξÞ; ν; rÞrq

q �:

(2.15)


inequalities

51

Corollary 2.22. In Theorem 2.19 for h1ðtÞ ¼ h2ðtÞ ¼ 1 and mðtÞ ¼ m∈ ð0; 1� for allt ∈ ½0; 1�; we get the following inequality for the generalized relative semi-ðm;PÞ-preinvexmappings:

If ; g;E;Λ;ψ ;mðν; a; bÞ ≤ 2kgkν∞SνΛνþ1ðψðbÞ;mψðaÞÞ

νþ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrq þ ðf 0 ðbÞÞrqrq

q: (2.16)

Corollary 2.23. In Theorem 2.19 for h1ðtÞ ¼ hð1− tÞ; h2ðtÞ ¼ hðtÞ andmðtÞ ¼ m∈ ð0; 1�for all t ∈ ½0; 1�;we get the following inequality for the generalized relative semi- ðm; hÞ-preinvexmappings:

If ; g;E;Λ;ψ ;mðν; a; bÞ ≤ kgkν∞SνΛνþ1ðψðbÞ;mψðaÞÞ

ðνþ 1Þ1−1q

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrqI r2 ðhð1� ξÞ; ν; rÞ þ ððf 0 ðbÞÞrqI r2 ðhðξÞ; ν; rÞrq

q

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrqI1rðhð1� ξÞ; ν; rÞ þ ððf 0 ðbÞÞrqI2rðhðξÞ; ν; rÞrq

q �:

(2.17)

Corollary 2.24. In Corollary 2.23 for h1ðtÞ ¼ ð1− tÞs and h2ðtÞ ¼ ts; we get the followinginequality for the generalized relative semi-ðm; sÞ-Breckner-preinvex mappings:

If ;g;E;Λ;ψ ;mðν; a; bÞ≤ kgkν∞SνΛνþ1ðψðbÞ;mψðaÞÞ

ðνþ 1Þ1−1q

3

8><>:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrqβr

�srþ 1; vþ 1

�þ ðf 0 ðbÞÞrq

1

srþ νþ 1

!r

rq

vuuuut

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrq

1

srþ vþ 1

!r

þ ðf 0 ðbÞÞrqβr�srþ 1; vþ 1

�rq

vuuuut9>=>;:

(2.18)

Corollary 2.25. In Corollary 2.23 for h1ðtÞ ¼ ð1− tÞ−s; h2ðtÞ ¼ t−s and 0 < s < r; we getthe following inequality for the generalized relative semi-ðm; sÞ-Godunova–Levin–Dragomir-preinvex mappings:

If ;g;E;Λ;ψ ;mðν; a; bÞ ≤ kgkν∞SνΛνþ1ðψðbÞ;mψðaÞÞ

ðvþ 1Þ1−1q

3

8><>:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrqβr

�1� s

r; νþ 1

�þ ðf 0 ðbÞÞrq

1

ν� srþ 1

!r

rq

vuuuut

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðf 0 ðaÞÞrq

1

v� stþ 1

!r

þ ðf 0 ðbÞÞrqβr�1� s

r; vþ 1

�rq

vuuuut9>=>;:

(2.19)

AJMS26,1/2

52

Corollary 2.26. In Theorem 2.19 for h1ðtÞ ¼ h2ðtÞ ¼ tð1− tÞ andmðtÞ ¼ m∈ ð0; 1� for allt ∈ ½0; 1�; we get the following inequality for the generalized relative semi-ðm; tgsÞ-preinvexmappings:

If ;g;E;Λ;ψ ;mðν; a; bÞ ≤ 2kgkν∞SνΛνþ1ðψðbÞ;mψðaÞÞðνþ 1Þ1−1

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ

�1þ 1

r; νþ 1

rþ 1

�q

s

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimð f 0 ðaÞÞrq þ ð f 0 ðbÞÞrqrq

q:

(2.20)

Corollary 2.27. In Corollary 2.23 for h1ðtÞ ¼ffiffiffiffiffiffiffi1− t

p2ffiffit

p ; h2ðtÞ ¼ffiffit


p and r∈ ð12; 1�; we get thefollowing inequality for the generalized relative semi-m- MT-preinvex mappings:

If ;g;E;Λ;ψ ;mðν; a; bÞ≤ kgkν∞SνΛνþ1ðψðbÞ;mψðaÞÞ

ðvþ 1Þ1−1q

3

( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimð f 0 ðaÞÞrqβr

�ν� 1

2rþ 1; 1þ 1

2r

�þ ð f 0 ðbÞÞrqβr

�νþ 1

2rþ 1; 1� 1

2r

�rq

s

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimð f 0 ðaÞÞrqβr

�νþ 1

2rþ 1; 1� 1

2r

�þ ð f 0 ðbÞÞrqβr

�ν� 1

2rþ 1; 1þ 1

2r

�rq

s ):

(2.21)

Remark2.28. By taking particular values of parameters used inMittag-Leffler function inTheorems 2.8 and 2.19, several fractional integral inequalities can be obtained.

Remark 2.29. Also, applying our Theorems 2.8 and 2.19, for f0 ðxÞ ≤ K; for all x∈ I ;we

can get some new fractional integral inequalities.

3. Applications to special means

Definition 3.1. ([2]). A function M : ℝ2þ → ℝþ, is called a Mean function if it has the

following properties:

(1) Homogeneity: Mðax; ayÞ ¼ aMðx; yÞ; for all a > 0;

(2) Symmetry: Mðx; yÞ ¼ Mðy; xÞ;(3) Reflexivity: Mðx; xÞ ¼ x;

(4) Monotonicity: If x≤ x0and y≤ y

0; then Mðx; yÞ≤Mðx0

; y0 Þ;

(5) Internality: mfx; yg≤Mðx; yÞ≤mfx; yg.Let us consider some special means for arbitrary positive real numbers α≠ β as follows: Thearithmetic mean A :¼ Aðα; βÞ; The geometric mean G :¼ Gðα; βÞ; The harmonic meanH :¼ Hðα; βÞ; The power mean Pr :¼ Prðα; βÞ; The identric mean I :¼ Iðα; βÞ; Thelogarithmic mean L :¼ Lðα; βÞ; The generalized log-mean Lp :¼ Lpðα; βÞ; The weightedp-power meanM ¼ Mp. Now, let a and b be positive real numbers such that a < b: Considerthe function M :¼ MðψðaÞ;ψðbÞÞ : ½ψðaÞ;ψðaÞ þ ΛðψðbÞ;ψðaÞÞ�3 ½ψðaÞ;ψðaÞ þ ΛðψðbÞ;ψðaÞÞ�→ ℝþ;which is one of the above mentioned means, therefore one can obtain variousinequalities using the results of Section 2 for these means as follows: Replace


inequalities

53

ΛðψðyÞ;mðtÞψðxÞÞ with ΛðψðyÞ;ψðxÞÞ where mðtÞ ≡ 1; for all t ∈ ½0; 1� and settingΛðψðyÞ;ψðxÞÞ ¼ MðψðxÞ;ψðyÞÞ for all x; y∈ I ; in (2.4) and (2.13), one can obtain the followinginteresting inequalities involving means:

If ;g;E;M ;ψðν;a;bÞ

≤ 2kgkν∞SνM

νþ1

ffiffiffiffiffiffiffiffiffiffiffiffipvþ1p

p 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf 0 ðaÞÞrqI r2 ðh1ðξÞ;rÞþðf 0 ðbÞÞrqI r2 ðh2ðξÞ;rÞrq

q; (3.1)

If ;g;E;M ;ψðν;a;bÞ

≤kgkν∞SvM

νþ1

ðvþ1Þ1−1q

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið f 0 ðaÞÞrqI r2 ðh1ðξÞ;ν;rÞþð f 0 ðbÞÞrqI r2 ðh2ðξÞ;ν;rÞrq

q

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið f 0 ðaÞÞrqI1rðh1ðξÞ;ν;rÞþð f 0 ðbÞÞrqI2rðh2ðξÞ;ν;rÞrq

q �:

(3.2)

Letting M :¼ A;G;H ;Pr; I ;L;Lp;Mp in (3.1) and (3.2), we get the inequalities involving

means for particular choices of ð f 0 ðxÞÞq that are the generalized relative semi-1-ðr; h1; h2Þ-preinvex mappings.

Remark3.2. Also, applying ourTheorems 2.8 and 2.19 for appropriate choices of functionsh1 and h2 (see Remark 2.6) such that ð f 0 ðxÞÞq to be the generalized relative semi-1-ðr; h1; h2Þ-preinvex mappings (see examples: f ðxÞ ¼ xα, where α > 1; ∀x > 0; f ðxÞ ¼ 1

x;

∀x > 0; f ðxÞ ¼ ex; ∀x∈ℝ; f ðxÞ ¼ −lnx; ∀x > 0; etc.), we can deduce some new inequalitiesusing above special means. The details are left to the interested reader.

References

[1] T. Antczak, Mean value in invexity analysis, Nonlinear Anal. 60 (2005) 1473–1484.

[2] P.S. Bullen, Handbook of Means and their Inequalities, Kluwer Academic Publishers, Dordrecht, 2003.

[3] F.X. Chen, S.H. Wu, Several complementary inequalities to inequalities of Hermite-Hadamardtype for s-convex functions, J. Nonlinear Sci. Appl. 9 (2) (2016) 705–716.

[4] Y.-M. Chu, M.A. Khan, T.U. Khan, T. Ali, Generalizations of Hermite-Hadamard type inequalitiesfor MT-convex functions, J. Nonlinear Sci. Appl. 9 (5) (2016) 4305–4316.

[5] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractionalintegration, Ann. Funct. Anal. 1 (1) (2010) 51–58.

[6] S.S. Dragomir, R.P. Agarwal, Two inequalities for differentiable mappings and applications tospecial means of real numbers and trapezoidal formula, Appl. Math. Lett. 11 (5) (1998) 91–95.

[7] S.S. Dragomir, J. Pe�cari�c, L.E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21(1995) 335–341.

[8] T.S. Du, J.G. Liao, Y.J. Li, Properties and integral inequalities of Hadamard-Simpson type for thegeneralizedðs;mÞ-preinvex functions, J. Nonlinear Sci. Appl. 9 (2016) 3112–3126.

[9] G. Farid, G. Abbas, Generalizations of some fractional integral inequalities for m-convexfunctions via generalized Mittag-Leffler function, Stud. Univ. BabeS-Bolyai Math. 63 (1) (2018)23–35.

[10] G. Farid, A.U. Rehman, Generalizations of some integral inequalities for fractional integrals, Ann.Math. Sil. 31 (2017) 14.

[11] C. Fulga, V. Preda, Nonlinear programming with f-preinvex and local f-preinvex functions,European J. Oper. Res. 192 (2009) 737–743.

[12] A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for generalizedðr; s;m;fÞ-preinvex functions, Eur. J. Pure Appl. Math. 10 (3) (2017) 495–505.

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[13] A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for MTðm;wÞ-preinvexfunctions, Stud. Univ. BabeS-Bolyai, Math. 62 (4) (2017) 439–450.

[14] A. Kashuri, R. Liko, Hermite-Hadamard type inequalities for generalized ðs;m;wÞ-preinvexfunctions via k-fractional integrals, Tbil. Math. J. 10 (4) (2017) 73–82.

[15] M.A. Khan, Y.-M. Chu, A. Kashuri, R. Liko, Hermite-Hadamard type fractional integralinequalities forMTðr;g;m;fÞ-preinvex functions, J. Comput. Anal. Appl. 26 (8) (2019) 1487–1503.

[16] M.A. Khan, Y.-M. Chu, A. Kashuri, R. Liko, G. Ali, New Hermite-Hadamard inequalities forconformable fractional integrals, J. Funct. Spaces (2018) 6928130 pp. 9.

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[18] W.J. Liu, Some Simpson type inequalities for h-convex and ðα;mÞ-convex functions, J. Comput.Anal. Appl. 16 (5) (2014) 1005–1012.

[19] W. Liu, W. Wen, J. Park, Ostrowski type fractional integral inequalities for MT-convex functions,Miskolc Math. Notes 16 (1) (2015) 249–256.

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[22] M. Matloka, Inequalities for h-preinvex functions, Appl. Math. Comput. 234 (2014) 52–57.

[23] M.A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, J. Math. Anal.Approx. Theory 2 (2007) 126–131.

[24] M.A. Noor, K.I. Noor, M.U. Awan, S. Khan, Hermite-Hadamard type inequalities for differentiablehw-preinvex functions, Arab. J. Math. 4 (2015) 63–76.

[25] O. Omotoyinbo, A. Mogbodemu, Some new Hermite-Hadamard integral inequalities for convexfunctions, Int. J. Sci. Innov. Tech. 1 (1) (2014) 1–12.

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Corresponding authorArtion Kashuri can be contacted at: [email protected]



inequalities

55


Unbalanced multi-drawing urnwith random addition matrix

Aguech RafikDepartment of Statistics and Operation Research, King Saud University,

Riyadh, Saudi Arabia, and

Selmi OlfaUniversity of Monastir, Monastir, Tunisia

AbstractIn this paper, we consider a two color multi-drawing urn model. At each discrete time step, we draw uniformlyat random a sample of m balls ðm≥ 1Þ and note their color, they will be returned to the urn together with arandom number of balls depending on the sample’s composition. The replacement rule is a 2 3 2 matrixdepending on bounded discrete positive random variables. Using a stochastic approximation algorithm andmartingales methods, we investigate the asymptotic behavior of the urn after many draws.

Keywords Central limit theorem, Unbalanced urn, Martingale, Stochastic algorithm


1. IntroductionThe classical P�olya urn was introduced by P�olya and Eggenberger [7] describing contagiousdiseases. The first model is as follows: An urn contains balls of two colors at the start, whiteand blue. At each step, one picks a ball randomly and returns it to the urn with a ball of thesame color. Afterwards, there weremany generalizations and urnmodel become a simple toolto describe several models such finance, clinical trials (see [19,22]), biology (see [11]), computersciences, internet (see [8,18]), etc...

Recently,Mahmoud, Chen,Wei, Kuba and Sulzbach [4,5,12–15], have focused on themulti-drawing urn. Instead of picking a ball,one picks a sample of m balls (m≥ ‘), say ‘white andðm− ‘Þblue balls. The pick is returned back to the urn together with am−‘white and bm−‘ blueballs, where a‘ and b‘; 0≤ ‘≤m are integers. At first, they treated two particular cases when{am−‘ ¼ c3 ‘ and bm−‘ ¼ c3 ðm− ‘Þ} and when {am−‘ ¼ c3 ðm− ‘Þ and bm−‘ ¼ c3 ‘},where c is a positive constant. By different methods as martingales andmoment methods, the

Unbalancedmulti-drawing

urn

57

© Aguech Rafik and Selmi Olfa. Published in the Arab Journal of Mathematical Sciences. Published byEmerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (forboth commercial and non-commercial purposes), subject to full attribution to the original publicationand authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

The first author is grateful to the King Saud University, Deanship of Scientific Research, College ofScience Research Center. The authors also thank two anonymous referees for their valuable commentsand suggestions.

The publisher wishes to inform readers that the article “Unbalanced multi-drawing urn with randomadditionmatrix”was originally published by the previous publisher of theArab Journal ofMathematicalSciences and the pagination of this article has been subsequently changed. There has been no change tothe content of the article. This change was necessary for the journal to transition from the previouspublisher to the new one. The publisher sincerely apologises for any inconvenience caused. To accessand cite this article, please use Rafik, A., Olfa, S. (2019), “Unbalanced multi-drawing urn with randomaddition matrix” Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 57-74. The originalpublication date for this paper was 11/01/2019.



Received 11 October 2018Revised 23 December 2018



Vol. 26 No. 1/2, 2020pp. 57-74


DOI 10.1016/j.ajmsc.2018.12.004

authors described the asymptotic behavior of the urn composition. When considering thegeneral case and in order to ensure the existence of a martingale, they supposed thatWn, thenumber of white balls in the urn after n draws, satisfies the affinity condition i.e, there existtwo deterministic sequences ðαnÞ and ðβnÞ such that, for all n≥ 0, E½Wnþ1jF n� ¼ αnWn þ βn.Under this condition, the authors focused on small and large index urns. Later, the affinitycondition was removed in the work of Lasmer, Mailler and Selmi [16], they generalized thismodel and looked at the case of more than two colors.

This paper contains the first results about multi drawing P�olya urns with randomreplacement rule. Even in the classical P�olya urn, where one ball is picked at every timestep very few results cover the unbalanced case: exceptions are the works of Janson andAguech. In [9] Janson studied a generalized urn model containing q different colors(q≥ 1) with a q3 q replacement matrix A with random entries such that Ai; j ≥ 0 andEðA2

i; jÞ < ∞ for all i; j ¼ 1; . . . ; q. Janson considered the case when the mean of A is anirreducible matrix. Using the method of embedding in continuous time of Athrea andKarlin [3], he gave explicit formulas for the asymptotic variances and covariances aswell as functional limit theorems for the urn. Then, Janson [10] considered a particulartwo color P�olya urn model evolving according to a triangular replacement matrix (thematrix in non irreducible) with deterministic entries. He established theorems describingthe asymptotic behavior of the composition of the urn after n draws. Afterwards,Aguech [1] extended some results and studied two colors urn model with triangularreplacement matrix. The entries of such a matrix, Xn; Yn and Cn, are positive randomvariables with finite means and variances. The embedding in continuous times’ methodwere successful once again and he gave theorems about the asymptotic behavior of theurn’s composition after a long time.

In this paper, we deal with a two color unbalanced urn class with multiple drawing andrandom addition matrix. Consider X and Y two discrete-valued random variables. Weassume that there exists two constantsU > 0 and L > 0 such that L≤X ≤U and L≤Y ≤L.Let ðXnÞn≥0 (resp ðYn≥0Þn≥0) be a sequence of independent random variables distributed likeX(resp Y). The sequences Xn and Yn are not assumed to be independent.

The model we study is defined as follows: An urn contains initiallyW0 white balls and B0

blue balls, we fix an integer m≥ 1, at a discrete step n≥ 1, we draw uniformly at random asample ofmballs, we denote by ξn the number of white balls among thosemballs (we assumethat the initial composition of the urn is more than m to make the first draw possible). Wereturn the drawn sample together with Qnðξn; m− ξnÞt balls, where Qn is a 2 3 2 matrixdepending on the random variables Xn and Yn. Let us denote byWn (resp Bn) the number ofwhite balls (resp blue balls), Tn the total number of balls and by Zn the proportion of whiteballs in the urn at time n. In other words, the process is defined recursively as follows: for alln≥ 1 �

Wn

Bn

�D�Wn−1

Bn−1

�þ Qn

�ξn

m� ξn

�: (1)

Let F n be the σ-field generated by the first n draws. Note that, with these notations, we havefor k∈ f0; . . . ;mg;

ℙ½ξn ¼ kjF n−1� ¼ðWn�1

k

ÞðBn�1m−k

ÞðTn�1

mÞ : (2)

Thus, conditioning on F n−1 the variable ξn has an hypergeometric distribution withparameters m; Zn−1 and Tn−1. Some particular cases were the interest of recent works [4,15]

AJMS26,1/2

58

and [2], where the authors characterized the urn models defined by Eq. (1) for the followingcases

Qn ∈

��a 00 a

�;

�0 a

a 0

�;

�a 00 b

�;

�0 a

b 0

��;

where a; bare strictly positive integers. To generalize the previousworks, we consider the urnmodels evolving according to Eq. (1) with

Qn ∈

��Xn 00 Yn

�;

�0 Xn

Yn 0

�;

�Xn 00 Xn

�;

�0 Xn

Xn 0

��:

The main idea is to use the stochastic algorithms and martingales in order to prove that thenumber of white balls in the urn converges almost surely and to study its fluctuations aroundits limit whenever it is possible.

The paper is organized as follows. In Section 2, we give the main results of the paper.Section 3 is devoted to the details of the stochastic approximation algorithm’s method. Theproofs of the main results are detailed in Section 4.

2. Main resultsWe start with some notations. The notation a:s: stands for almost surely. For a randomvariable R, we denote by

μR ¼ EðRÞ and σ2R ¼ VarðRÞ;

by μX :¼ μX1(respectively μY :¼ μY1

) and σ2X :¼ σ2

X1(respectively σ2Y :¼ σ2

Y1). For xn and yn

two sequences of real numbers such that yn ≠ 0 for all n, we denote xn ¼ oðynÞ (respectivelyxn ¼ oðynÞ; a:s) if limn→þ∞xn=yn ¼ 0 (if limn→þ∞xn=yn ¼ 0; a:swhen xn and yn are random).

In this section we state our main result. As mentioned in the introduction, we study urnmodels evolving according to Eq. (1). Recall that in the whole of paper we consider ðXnÞn≥1(resp ðYnÞn≥1), a sequence of independent random variables distributed like X (resp Y).

The present theorem deals with an urn evolving with an anti-diagonal replacementmatrix. The model is then opposite reinforced, i.e the more color is drawn the more itreinforces the opposite color.

Theorem 1. Let z :¼ffiffiffiffiμX

pffiffiffiffiμX

p þ ffiffiffiffiμY

p and consider the urn model evolving by the matrix

Qn ¼�0 Xn

Yn 0

�. We have the following results:

(1) The total number of balls in the urn after n draws satisfies

Tn ¼ ffiffiffiffiffiffiffiffiffiffiffiμXμY

pm nþ oðnÞ; a:s: (3)

and the number of white and blue balls in the urn after n draws satisfy

Wn ¼ μX ð1� zÞm nþ oðnÞ; a:s:Bn ¼ μY z m nþ oðnÞ; a:s:

(2) Furthermore, with GðxÞ ¼P4i¼0 aix

i, the normalized number of white balls in the urnsatisfies the central limit theorem

Wn � zTnffiffiffin

p →

D N�0;

GðzÞ3

�; as n→ þ∞: (4)


urn

59

(3) Furthermore, when Yn ¼ Xn for all n≥ 0, the total number of balls in the urn after n

draws satisfies, for any δ > 12

Tn ¼ mμXnþ oð ffiffiffin

plnδ nÞ; a:s:

The number of white balls Wn and blue balls Bn in the urn after n draws satisfy for anyδ > 1

2;

Wn ¼ mμX2

nþ oð ffiffiffin

plnδ nÞ; a:s;

Bn ¼ mμX2

nþ oð ffiffiffin

plnδ nÞ; a:s:

We have the convergence in distribution:

limn→þ∞

Wn � 12Tn

Σffiffiffin

p ¼ Nð0; 1Þ and limn→þ∞

Wn � EðWnÞΣ1

ffiffiffin

p ¼ Nð0; 1Þ;where

Σ ¼ m

12

�σ2X þ μX2

�and Σ1 ¼ m

12

�σ2X þ μX2

�þm2σ2X

:

Example 1. Let Xn ¼ a and Yn ¼ b (where a and b are not random), then z ¼ffiffia

pffiffia

p þffiffib

p . Thiscase was studied in [2] and the authors proved the following

ffiffiffin

p �Wn

Tn

� z

�→

D N ð0;ffiffiffiffiffiab

p

3mð ffiffiffia

p þ ffiffiffib

p Þ2Þ; as n→∞:

Under the notation of Theorem 1, we easily compute GðzÞ ¼ mabzð1− zÞ and then theparticular case is proved again.

Example 2. Let Xn ¼ Yn ¼ C (non random), the urn is balanced and the total number ofballs is deterministic and satisfies Tn ¼ T0 þ Cmn. Furthermore, we have μX ¼ C andσ2X ¼ 0, applying Theorem 1ð3Þwe obtain the following limit:

Wn � Cmn2ffiffiffi

np →

D N ð0; mC2

12Þ; as n→∞:

Kuba et al. [15] studied this particular case and established such a result via two differentmethods: The recursion formulas permit to derive the expression of the higher moments ofthe number of white balls and then to conclude functional limit theorem. The same result wasproved via martingales method.

In the following theorem, we consider a diagonal replacement matrixQn. The model is selfreinforced since the rich gets richer. As the particular case when m ¼ 1, we compare μX

μYwith 1, we will distinguish different phases.

Theorem 2. Consider the urn evolving by the matrix Qn ¼�Xn 00 Yn

�:

(1) If μX > μY, then the total number of balls in the urn after n draws satisfies

Tn ¼ mμXnþ oðnÞ; a:s:;

and the asymptotic composition of the urn is

AJMS26,1/2

60

Wn ¼ mμXnþ oðnÞ; Bn ¼ B∞nρ þ oðnρÞ; a:s:

where ρ ¼ μYμXand B∞ is a positive random variable.

(2) If μX ¼ μY, the composition of the urn after n draws satisfies

Tn ¼ mμXnþ oðnÞ; a:s:

In addition, there exists a positive random variable W∞ such that,

Wn ¼ W∞nþ oðnÞ and Bn ¼ ðμxm�W∞Þ nþ oðnÞ; a:s:

(3) Furthermore, if for all n≥ 0, Yn ¼ Xn, the distribution of the random variable W∞ isabsolutely continuous.

Remark. The case when μX < μY is obtained by interchanging the colors. In fact we havethe following almost sure results:

Tn ¼ mμYnþ oðnÞ; Wn ¼ W∞nσ þ oðnÞ and Bn ¼ mμYnþ oðnÞ;

where W∞ is a positive random variable and σ ¼ μXμY:

Example 3. Aguech [1] studied the particular case when m ¼ 1 and considered thefollowing triangular replacement matrix�

Xn 0Cn Yn

�;

where Xn;Yn and Cn are independent positive random variables with finite means andvariances. Via embedding in continuous timemethod andmartingales, the author proved, forCn ¼ 0, the following almost sure results:

(a) If μX > μY ,

Wn ¼ μXnþ oðnÞ; Bn ¼ Dnρ and Tn ¼ μXnþ oðnÞ;where ρ ¼ μY

μXand D is a positive random variable.

(b) If μX ¼ μY ,

Wn ¼ μXW

W þ Bnþ oðnÞ and Bn ¼ μX

B

W þ Bnþ oðnÞ;

where W and B are the almost sure limit of a continuous time martingale.We prove again these results in Theorem 2 using stochastic approximation algorithm.

Example 4. Chen and Kuba [4] studied the case when Xn ¼ Yn ¼ C (C is non random) andm≥ 1. They gave explicit expressions of moment of all order of Wn=n and proved that itsalmost sure limit,W∞ cannot be an ordinary Beta distribution, unlike the original P�olya urnmodel [7] when X ¼ C and m ¼ 1, Eggenberger and P�olya proved in 1923 that the randomvariable W∞=C has a Beta distribution with parameters ðB0=C; W0=CÞ. Unfortunately, inour model we cannot yet derive the expression of higher moments of Wn=n since therecurrence formulas are too intricate.

3. Some results on stochastic approximation algorithmThe stochastic algorithm approximation plays a crucial role in the proofs in order to describethe asymptotic composition of the urn. As many versions of the stochastic algorithm exist inthe literature (see [6] for example), we adapt the version of Renlund in [20,21].


urn

61

Definition 1. A stochastic approximation algorithm ðUnÞn≥0 is a stochastic process takingvalues in ½0; 1� and adapted to a filtration F n that satisfies

Unþ1 � Un ¼ γnþ1ðf ðUnÞ þ ΔMnþ1Þ; (5)

where ðγnÞn≥1 and ðΔMnÞn≥1 are two F n-measurable sequences of random variables, f is afunction from ½0; 1� into ℝ such that f ð0Þ≥ 0, f ð1Þ≤ 0 and the following conditions holdalmost surely: There exists constants c1; c2; KΔ; and Kf positive real numbers such that forany n≥ 1,

(i) c1n≤ γn ≤

c2n;

(ii) EððΔMnþ1Þ2jF nÞ ≤ KΔ;

(iii) j f ðUnÞj ≤ Kf ;

(iv) E½γnþ1ΔMnþ1jF n� ¼ 0.

Definition 2. Let Zf ¼ fx∈ ½0; 1�; f ðxÞ ¼ 0g. A zero p∈Zf will be called stable if thereexists a neighborhood N p of p such that f ðxÞðx− pÞ < 0 whenever x∈N pnfpg: If f isdifferentiable, then f 0ðpÞ is sufficient to determine that p is stable.

Remark. Note that Assumption ðiiÞ in Definition 1 is not stated as in [20] where it isassumed that there exists a positive constant KΔ such that jΔMnj≤KΔ.

We have the following result about the process defined by Eq. (5)

Proposition 1. Let ðUnÞn≥0 be a stochastic algorithm defined by Eq. (5). If f is continuous,then limn→þ∞Un exists almost surely and is a stable zero of f .

The following lemmas will be useful for the proof of Proposition 1.

Lemma 1. Define Vn ¼Pn

i¼1 γiΔMi. Under the assumptions of Proposition 1, Vn convergesalmost surely.

Proof. Under the assumptions mentioned in Definition 1, we have

EðVnþ1jF nÞ ¼ Vn þ Eðγnþ1ΔMnþ1jF nÞ ¼ Vn:

We deduce that ðVn; F nÞn is a martingale. On the other hand,

E�V 2

n

� ¼Xni¼1

E�γ2i ðΔMiÞ2Þ≤

Xni¼1

c22i2E�ðΔMiÞ2Þ≤KΔc

22

Xni¼1

1

i2< ∞:

It follows that ðVnÞn is an L2- bounded martingale, and thus, it converges almost surely. ,

Next lemma ensures that, under the assumptions of Proposition 1, all possible candidatesfor the almost sure limit of Un are necessary among the zeros of f .

Lemma 2 ([20] ). Let Zf ¼ fx ; f ðxÞ ¼ 0g be the set of zeros of f and let CðUnÞ be the setof limit points of fUng defined by

CðUnÞ ¼\n≥1

fUn; Unþ1; . . . g;

where A denotes the closure of a set A. Under the assumptions of Proposition 1, if f iscontinuous, then,

ℙðCðUnÞ⊆Zf Þ ¼ 1:

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62

Lemma 3 ([20] ). Suppose that f ðxÞ < − δ (or f ðxÞ > δ) for some δ > 0, wheneverx∈ ða0; b0Þ. Then,

CðUnÞ\

ða0; b0Þ ¼ 0= a:s:;

and either lim supn Un ≤ a0 or lim infn Un ≥ b0:We are now able to handle the proof of Proposition 1.

Proof of Proposition 1. The proof is close to Theorem 1 in [20], for the convenience ofthe reader, we resume the proof and we mention the main steps. If limn→þ∞Un does not exist,we can find two rational numbers in the open interval

�lim inf n→þ∞Un; lim sup n→þ∞Un½. Let lim infUn < p < q < lim supUn be two arbitrarydifferent rational numbers. If we can show that

ℙðflim infUn ≤ pg∩flim supUn ≥ qgÞ ¼ 0;

then, the existence of the limit will be established and the claim of the proposition followsfrom Lemma 2. For this reason, we need to distinguish two different cases whether or not pand q are in the same connected component of Zf .

Case 1: p and q are not in the same connected component of Zf : Since Zf is closedand f is continuous there must exist ½a; b� ⊆ ½p; q�TZc

f such that f is non-zero and hasa constant sign for all x∈ ða; bÞ. By Lemma 3, it is impossible to have lim infn Un ≤ aand lim supn Un ≥ b.

Case 2: p and q are in the same connected component of Zf : In all the cases of ourframeworkZf is a set of two isolated points, therefore we are not interested to the case when pand q are not in the same connected component.

To establish that the almost sure limit of Un is among the stable point set, we refer thereader to [20] to see a detailed proof. ,

Next result is due to Renlund [21] which will be used in the proofs of Theorems 1 and 2.

Theorem 3 ([21]). Let ðUnÞn≥0 satisfy Eq. (5) and that limn→þ∞ Un ¼ U*. Let

bγn :¼ nγn bf ðUn−1Þ; where bf ðxÞ ¼ f ðxÞU * � x

:

If bγn converges almost surely to some limit bγ > 12 and if E½ðnγnΔMnÞ2jF n−1�→ σ2 > 0; then,

we have the convergence in distribution

ffiffiffin

p ðUn � U *Þ→D N�0;

σ2

2bγ � 1

�:

4. Proof of the main results4.1 Prerequisite for the proofs of the main resultsWe show in the following that the stochastic approximation algorithm is a fruitful methodto study unbalanced urn models. Although there are few versions of such a method thatpermit to γn to be random, the version of Renlund [20] and [21] applies to our model.

Under the assumptions of Theorem 1 and according to Eq. (1), the compositions of the urnsatisfy the following recursions:

Wnþ1 ¼ Wn þ Xnþ1ðm� ξnþ1Þ (6)

and


urn

63

Tnþ1 ¼ Tn þmXnþ1 þ ξnþ1ðYnþ1 � Xnþ1Þ: (7)

We start with first results that will be useful for the proof of Theorem 2.

Lemma 4 (Technical Lemma). For all integers m;A;B such that m ≤ Aþ B we have

Xmm¼0

k

�A

k

��B

m� k

�¼ A

�Aþ B� 1m� 1

�

and

Xmm¼0

k2�A

k

��B

m� k

�¼ AðA� 1Þ

�Aþ B� 2m� 2

�þ A

�Aþ B� 1m� 1

�

Remark. Since conditioning on F n−1 the variable ðξnÞ has an hypergeometric distributionwith parameters m, Zn−1 and Tn−1, it follows from Lemma 4 the following:

EðξnjF n−1Þ ¼ mZn;

and

VarðξnjF n−1Þ ¼ mZn−1ð1� Zn−1ÞTn−1 �m

Tn−1 � 1:

Lemma 5. Under the assumptions of Theorem 1, the proportion of white balls after n draws,Zn, satisfies the stochastic algorithm defined by (5), where γn ¼ 1

Tn,

f ðxÞ ¼ mðμX � μY Þx2 � 2μXmxþ μXm;

and

ΔMnþ1 ¼ Dnþ1 � E½Dnþ1jF n�;with

Dnþ1 ¼ ξnþ1ðZnðXnþ1 � Ynþ1Þ � Xnþ1Þ þmXnþ1ð1� ZnÞ:Proof. In view of the recursions in Equations (6), (7) we have

Znþ1 � Zn ¼ 1

Tnþ1

½Wn þ Xnþ1ðm� ξnþ1Þ � ZnðTn þmXnþ1 þ ξnþ1ðYnþ1 � Xnþ1ÞÞ�¼ 1

Tnþ1

½Xnþ1ðm� ξnþ1Þ � ZnðmXnþ1 þ ξnþ1ðYnþ1 � Xnþ1ÞÞ�¼ Dnþ1

Tnþ1

:

An easy computation shows that EðDnþ1jF nÞ ¼ mðμX − μY ÞZ 2n − 2mμXZn þmμX . ,

Using Proposition 1,we show that the almost sure limit of the proportion of white balls in theurn depends on the means of the variables Xn and Yn:

Proposition 2. The proportion of white balls in the urn after ndraws, under the assumptionsof Theorem 1, satisfies

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64

limn→þ∞

Zn ¼ z :¼ffiffiffiffiffiffiμX

pffiffiffiffiffiffiμX

p þ ffiffiffiffiffiffiμY

p ; a:s: (8)

Proof. In view of Lemma 5, we check the assumptions of Definition 1, indeed,

(i) an easy computation shows that

Tn ¼ T0 þmXni¼1

ðm� ξiÞXi þXni¼1

ξiYi: (9)

Since for all n≥ 1 we have 0 ≤ ξn ≤ m, L ≤ Xn ≤ U and L ≤ Yn ≤ U, then

mnL ≤ Tn ≤ T0 þmnU :

Then the following bound holds, for all n ≥ 1

c1

n≤

1

Tn

≤c2

n; (10)

with c1 ¼ 1T0þmU

and c2 ¼ 1mL

:(ii)

EðΔMnþ1Þ2jF n

≤�μðX�Y Þ2 þ 3μX

��mþm2

�þ 5m2μX2 þ 2m2μXμY

þm2ðjμX � μY j þ 3μX Þ ¼ KΔ;

(iii) jf ðZnÞj≤mðjμY − μX j þ 3μX Þ ¼ Kf ;

(iv) Eh

1Tnþ1

ΔMnþ1jF n

i≤ 1

TnE½ΔMnþ1jF n� ¼ 0:

Since the function f , defined in Lemma 5, is continuous, we conclude by Proposition 1, that theprocess Zn converges a:s: to

z ¼ffiffiffiffiffiffiμX

pffiffiffiffiffiffiμX

p þ ffiffiffiffiffiffiμY

p ;

which is the unique zero of f with negative derivative. ,The following Lemma will intervene in the proof of Theorem.

Lemma 6. Under the assumptions of Theorem 1, the total number of balls after n drawssatisfies

limn→þ∞

Tn

n¼ m

ffiffiffiffiffiffiffiffiffiffiffiμXμY

p; a:s:

Proof. Let Gn ¼Pn

i¼1½ξiðYi −XiÞ � E½ξiðYi −XiÞjF i�1��; by the recursive Eq. (7), we haveTn

n¼ T0

nþm

n

Xni¼1

Xi þmðμY � μX Þn

Xni¼1

Zi−1 þ Gn

n:


urn

65

Since ðXiÞi≥1 are i.i.d. random variables, then by the strong law of large numbers we have

m

n

Xni¼1

Xi !a:s mμX :

Via Proposition 2 and Ces�aro lemma, we conclude that 1n

Pni¼1Zi−1 converges a:s:, as ngoes to

infinity, to z. Finally, we prove that the last term in the right side tends a:s: to zero, as n tendsto infinity. In fact, ðGn; F nÞ is a martingale difference sequence with quadratic variationgiven by

hGin ¼Xni¼1

E½ð∇GiÞ2jF i−1�;

where ∇Gn ¼ Gn −Gn−1 ¼ ξnðYn −XnÞ− E½ξnðYn −XnÞjF n−1�. By a simple computation, wehave the almost sure convergence

limn→þ∞

E½ð∇GnÞ2jF n−1� ¼�mzð1� zÞ þm2z2Þ�σ2

Y þ σ2X�:

Therefore, Ces�aro lemma ensures that a:s:

limn→þ∞

hGinn

¼ �mzð1� zÞ þm2z2Þ�σ2Y þ σ2X

�:

It follows that Gn

n !a:s 0. Thus, for n large enough, we haveTn

n!a:s m ffiffiffiffiffiffiffiffiffiffiffi

μXμYp

: , (11)

Remark. The convergence in Proposition 2 holds also in L2.

Under the hypothesis of Theorem 2, the process of the urn satisfies the following recursions:

Wnþ1 ¼ Wn þ Xnþ1ξnþ1 and Tnþ1 ¼ Tn þmYnþ1 þ ξnþ1ðXnþ1 � Ynþ1Þ: (12)

Next results will be used in the proof of Theorem 2.

Lemma 7. Under the assumptions of Theorem 2, if μX ≠ μY, the proportion of white balls inthe urn after n draws satisfies the stochastic algorithm defined by Eq. (5) where γn ¼ 1=Tn,

f ðxÞ ¼ mðμY � μX Þxðx� 1Þ;and

ΔMnþ1 ¼ Dnþ1 � E½Dnþ1jF n�;with

Dnþ1 ¼ ξnþ1ðZnðYnþ1 � Xnþ1Þ þ Xnþ1Þ �mZnYnþ1:

Proof. We check that, if μX ≠ μY , the assumptions of Definition 1 hold. Indeed,

(i) Eq. (12) shows that

Tn ¼ T0 þmXni¼1

Yi þXni¼1

ξiðXi � YiÞ; (13)

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66

since the expression ofTn is similar to that in Equation (9), we have the same bound ofγn ¼ 1

Tndefined in Eq. (10).

(ii)

EðΔMnþ1Þ2jF n

≤�2mþm2

��4μX2 þ μY 2

�þ 3m2μY 2 þ 2m2μX

þ 2m2μXμY þ 4m2ðμX � μY Þ2 ¼ KΔ:

(iii) jf ðZnÞj ¼ jmðμY − μX ÞZnðZn − 1Þj≤ 2mjμY − μX j ¼ Kf ;

(iv) E½γnþ1ΔMnþ1jF n�≤ 1TnE½ΔMnþ1jF n� ¼ 0: ,

Proposition 3. Under the assumptions of Theorem 2, the proportion of white balls in the urnafter n draws, Zn, satisfies a:s:

limn→þ∞

Zn ¼8<:

0; if μX < μY ;1; if μX > μY ;~Z∞; if μX ¼ μY ;

where ~Z∞ is a positive random variable.

Proof. Recall that, if μX ≠ μY , Zn satisfies the stochastic algorithm of Lemma 7. As thefunction f is continuous, by Theorem 3we conclude that Zn converges a:s: to the stable zero ofthe function hwith a negative derivative, which is 1 if μX > μY and 0 if μX < μY :

In the case when μX ¼ μY , we have Znþ1 ¼ Zn þ Pnþ1

Tnþ1, where

Pnþ1 ¼ Xnþ1ξnþ1 � ZnðmYnþ1 þ ξnþ1ðXnþ1 � Ynþ1ÞÞ:

Since E½Pnþ1jF n� ¼ 0, then Zn is a positive martingale which converges a:s: to a positiverandom variable ~Z∞. ,

As a consequence of Proposition 3, we have

Corollary 1. Suppose that μX ≥ μY, the total number of balls in the urn, Tn, satisfies as ntends to infinity

limn→þ∞

Tn

n¼ mμX ; a:s:

Remark. The convergence in Corollary 1 holds also in L2.

Proof. We have

Tn

n¼ T0

nþm

n

Xni¼1

Yi þ 1

n

Xni¼1

ξiðXi � YiÞ

¼ T0

nþm

n

Xni¼1

Yi þmðμX � μY Þn

Xni¼1

Zi−1 � Gn

n;

where Gn ¼Pn

i¼1½ξiðYi −XiÞ− EðξiðYi −XiÞjF nÞ� is the martingale difference defined in theproof of Lemma 6. Recall that Gn=n converges a:s: to 0 and that Zn converges a:s: to 1 when


urn

67

μX > μY , . Then, using Ces�aro lemma, we obtain the limits requested. If μX ¼ μY , we have1n

Pn

i¼1Yi converges to μX . ,

For the particular case when Xn ¼ Yn for all n, we have the following results

Proposition 4 ([5] ). Let ðΩlÞl≥0 be a sequence of increasing events such that ℙð∪l≥0ΩlÞ ¼ 1.If there exists nonnegative Borel measurable function fflgl≥1 such that for all Borel sets B

ℙ�Ωl ∩ W−1

∞ðBÞ� ¼

ZB

flðxÞdx

then, f ¼ liml→þ∞ fl exists almost everywhere and f is the density of W∞.

Lemma 8. Define the events

Ωl :¼ fWl ≥ mU and Bl ≥ mUg;then, ðΩlÞl≥0 is a sequence of increasing events, moreover we have ℙð∪l≥0ΩlÞ ¼ 1.

Let ðpcÞc∈suppðXÞ the distribution of X.

Lemma 9. For a fixed l > 0, there exists a positive constant κ, such that, for everyc∈ suppðXÞ, n≥ l þ 1, Um ≤ j ≤ Tl−1 and k ≤ Umðnþ 1Þ, we haveXm

i¼0

ℙðWnþ1 ¼ jþ kjWn ¼ jþ k� ciÞ≤ pcð1� 1

nþ κ

n2Þ: (14)

Proof. According to Lemma 4.1 in [5], for Um≤ j≤Tl−1, n≥ l and k≤Umðnþ 1Þ, thefollowing holds:

Xmi¼0

�jþ cðk� iÞ

i

��Tn � j� cðk� iÞ

m� i

�¼ Tm

n

m!þ ð1�m� 2cÞTm−1

n

2ðm� 1Þ! þ � � � ; (15)

which is a polynomial inTn of degreemwith coefficients depending onW0; B0; mand conly.Let un; kðcÞ ¼

Pmi¼0 ℙðWnþ1 ¼ jþ kjWn ¼ jþ k− icÞ. Applying Eq. (15) to our model we

have almost surely

un; kðcÞ ¼ pcXmi¼0

jþ k

i

! Tn � j� k

m� i

! Tn

m

!−1

¼ pc

Tn

m

!−1�Tm

n

m!þ ð1�m� 2cÞ

ðm� 1Þ! Tm−1n þ � � �

�3

�Tm

n

m!þ ð1�mÞ2ðm� 1Þ!T

m−1n þ � � �

�−1

¼ pc

�1� 1

nþ O

�1

n2

��: ,

4.2 Proof of Theorem 1Recall that ðXiÞi≥1 (resp ðYiÞi≥1) is a sequence of random variable distributed like X (resp Y).

We consider the urn model evolving by the anti-diagonal matrix Qn ¼�0 Xn

Yn 0

�.

Proof of claim 1 Theorem 1. In order to describe the asymptotic of the urn’s compositionwe use Lemma 6 which gives the estimate of Tn, the total number of balls in the urn after ndraws. For the number of white and blue balls we have, a:s:

Wn

n¼ Wn

Tn

Tn

nand

Bn

n¼ Bn

Tn

Tn

n;

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68

using Eqs. (8), (11) and Slutsky theorem, we have almost surely, as n goes to infinity,

Wn

n→m


pz and

Bn

n→m


p ð1� zÞ:These convergence hold also in L2.

Proof of claim 2 Theorem 1. To establish a central limit theorem, we aim to applyTheorem 3. Recall that in our model, we have γn ¼ 1=Tn, then we need to find the followinglimits:

limn→þ∞

E½�

n

Tn

�2

ΔM 2nþ1jF n� and lim

n→þ∞

n

Tn

f0 ðZnÞ:

In fact, in view of Lemma 6, we have n=Tn converges a:s: to ðm ffiffiffiffiffiffiffiffiffiffiffiμXμY

p Þ−1 andEðΔMnþ1Þ2jF n

¼ EðDnþ1Þ2jF n

þ E½Dnþ1jF n�2:Since E½Dnþ1jF n�2 converges a:s: to ðf ðzÞÞ2 ¼ 0, we have,

E�ðΔMnþ1Þ2jF n

¼ EZ 2n ðXnþ1 � Ynþ1Þ2 � 2ZnXnþ1 þ Xnþ1jF n

Eξ2nþ1jF n

þm2E�X 2�

þ 2m2�Z 2n ðEðX 2

�� μXμY Þ � ZnE�X 2�Þ:

Using the fact that

Eξ2nþ1jF n

¼ mZnð1� ZnÞTn �m

Tn � 1þm2Z 2

n

and that Zn converges a:s: to z, we conclude that E½D2nþ1jF n� converges a:s: to GðzÞ > 0:

Applying Theorem 3, we obtain the following

ffiffiffin

p ðZn � zÞ→L N�0;

GðzÞ3m2μXμY

�:

Since we have

Wn � zTnffiffiffin

p ¼ ffiffiffin

p �Wn

Tn

� z

�Tn

n;

Slutsky theorem is enough to conclude the proof.

Proof of claim 3 Theorem 1. In this particular case, the claims (1) and (2) apply and thealmost sure limit of the urn’s composition follows immediately as well as a central limittheorem. Furthermore, as such a case is easier, we can obtain a finer rate of convergence of thenormalized number of balls in the urn. We also give another version of central limit theoremsatisfied byWn using the weak dependence between the variables ðξiÞi≥0 and the Bernstein’smethod.

Recall that when Yn ¼ Xn for all n≥ 0, the urn is evolving according to Eq. (1) with areplacement matrix given by

Qn ¼�0 Xn

Xn 0

�:

Theorem 1ð1Þ applies for z ¼ 1=2 and the following almost sure results follows:

Tn ¼ mμXnþ oðnÞ; Wn ¼ mμX2

nþ oðnÞ and Bn ¼ mμX2

nþ oðnÞ:On the other hand, the total number of balls in the urn is a sum of i.i.d. random variablesTn ¼ T0 þ

Pni¼1Xi. According to the strong law of large number we get a finer rate of

convergence of Tn, we have for δ > 12


urn

69

Tn ¼ mμXnþ o� ffiffiffi

np

lnδn�: (16)

Using Wn

n¼ Wn

Tn

Tn

nand Eq. (16), we have

Wn

n¼a:s ð1

2þ oð1ÞÞ

�μXmþ o

�lnδnffiffiffin

p��

:

We conclude that the number of white balls in the urn after n draws, Wn, satisfies almostsurely for n large enough

Wn ¼ μXm2

nþ o� ffiffiffi

np

lnδ n�; δ >

1

2:

Remark. In such amodel, the proportion of white balls in the urn, Zn, satisfies the stochasticapproximation algorithm defined by Eq. (5) with γn ¼ 1=Tn,

f ðxÞ ¼ μXmð1� 2xÞand

ΔMnþ1 ¼ Xnþ1ðm� ξnþ1 �mZnÞ � μXmð1� 2ZnÞ:Moreover, we propose the following result about the variance of Wn.

Proposition 5. Under the hypothesis of Theorem 1, with Yn ¼ Xn for all n≥ 0, the varianceof Wn satisfies for every δ > 1

2;

VarðWnÞ ¼m�σ2X þ μ2X

�þm2σ2X12

nþ o� ffiffiffi

np

lnδn�: (17)

Proof. Because the number of white balls in the urn satisfies Eq. (6), we write

VarðWnþ1Þ ¼ VarðWnÞ þVarðXnþ1ðm� ξnþ1ÞÞ þ 2 ℂovðWn;Xnþ1ðm� ξnþ1ÞÞ:We have

VarðXnðm� ξnÞÞ ¼ E�X 2�Varðm� ξnþ1Þ þVarðXÞE�ðm� ξnþ1Þ2Þ

¼ �σ2X þ μ2X�½EðVarðξnþ1jF nÞÞ þVarðEðξnþ1jF nÞÞ� þ σ2

xE�ðm� ξnþ1Þ2Þ

¼ �σ2x þ μX2

��VarðmZnÞ þ E

�mZnð1� ZnÞTn �m

Tn � 1

��þ σ2XEðm� ξnÞ2:

(18)

On the other hand, since the variables ðXiÞi≥0 are independent then Xnþ1 and Wn areindependent, thus it follows

ℂovðWn; Xnþ1ðm� ξnþ1ÞÞ ¼ ℂovðWn; mXnþ1Þ � ℂovðWn; Xnþ1ξnþ1Þ¼ −ℂovðWn; Xnþ1ξnþ1Þ

¼ −mμX ½EðWn

Wn

Tn

Þ þ EðWnÞEðWn

Tn

Þ�

¼ −mμX ð1

mμXð1þ oðln

δnffiffiffin

p ÞÞVarðWnÞÞ

(19)

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70

Using Eqs. (18) and (19) and the fact that Zn →a:s 1

2 as n goes to infinity, we obtain

VarðWnþ1Þ ¼ ð1� 2

nþ oðlnδn

n32

ÞÞVarðWnÞ þm�σ2X þ μX2

�þm2σ2X4

þ o

�lnδnffiffiffin

p�

¼ anVarðWnÞ þ bn;

where an ¼ ð1− 2nþ oðlnδn

n32

ÞÞ and bn ¼ mðσ2Xþμ

X2 Þþm2σ2X

4þ o

�lnδnffiffi

np�:

Thus,

VarðWnÞ ¼ ðYnk¼1

akÞðVarðW0Þ þXn�1

k¼0

bkYkj¼0

ajÞ:

There exists a constant a such thatQn

k¼1 ak ¼ ea

n2

�1þ o

�lnδnffiffi

np��

, which leads to

VarðWnÞ ¼m�σ2X þ μX2

�þm2σ2X12

nþ o� ffiffiffi

np

lnδn�; δ >

1

2: ,

In this particular case, two versions of the central limit theorem for the number of white ballsare proved. The first version is deduced by Theorem 1(2) and the second one is proved usingthe weak dependence between the variables ðξiÞi≥1 together with Bernstein’s Method.

Applying Theorem 1(2), we have Yn ¼ Xn, it follows that μY ¼ μX , by a simplecomputation for the coefficients ai for i∈ f0; . . . ; 4gwe have for z ¼ 1

2 :

Gð12Þ ¼ m

4

�σ2X þ μX2

�:

We conclude that, in distribution we have

Wn � 12Tnffiffiffi

np →N

�0;

m

12ðσ2

X þ μX2

��:

A second central limit theorem is satisfied by Wn. As the proof is close to that of Lemma 3and Theorem 4 in [2], we will mention only the main steps and we refer the reader to [2] forthe details. The idea of the proof is the following: Once we prove that the variablesðXnðm − ξnÞÞn≥0 are α-mixing variables with a strongmixing coefficient αðnÞ ¼ oðlnδn= ffiffiffi

np Þ,

δ > 1=2 (see Lemma 3 in [2] for detailed computations), Bernstein’s method (see [17]) will besuitable. Consider the same notations as in Theorem 4 in [2] with

~ξi ¼ Xiðm� ξiÞ � μX ðm� EðξiÞÞ; Sn ¼ 1ffiffiffin

pXni¼1

~ξi

and N is the centered normal random variable with variance

σ2 ¼ m

12

�σ2X þ μX2

�þm2σ2X

:

Actually, all that remains in this case, is to compute the variance of Wn. For that, we useProposition 5. As a conclusion,

Wn � EðWnÞffiffiffin

p →

D N�0;

m

12ðσ2X þ μX2

�þm2σ2X

�:

4.3 Proof of Theorem 2Theorem 2 deals with unbalanced urn model with diagonal replacement matrix. We appliedProposition 1 to find the almost sure limit of the proportion of white balls in the urn. The


urn

71

stochastic algorithm applies only to the case when μX ≠ μY , because when μX ¼ μY we fall onthe case f ≡ 0. Furthermore, Theorem 3 does not work, in fact, by a simple computation weobtain σ ¼ 0. Such a result is expected since that even for the case Xn ¼ Yn ¼ C(C isconstant) andm > 1, the fluctuations ofWn=n around its limit has not a normal distribution.

Consider the urn model defined by Eq. (1) with Qn ¼�Xn 00 Yn

�.

Proof of claims 1 and 2 Theorem 2. Corollary 1 ensures that, if μX ≥ μY we have

Tn ¼ mμXnþ oðnÞ:Indeed,

� If μX > μY , we have, a.s.,

limn→þ∞

Wn

n¼ lim

n→þ∞

Wn

Tn

Tn

n¼ mμX :

Moreover, let ~Gn ¼�Qn�1

i¼1 ð1þ mμYTi

��−1

Bn; then ð~Gn; F nÞ is a positive martingale. There

exists a positive number A such thatQn�1

i¼1

�1þ mμY

Ti

�’ Anρ where ρ ¼ μY

μX. Then, as n tends

to infinity we have

Bn

nρ!a:s B∞;

where B∞ is a positive random variable.

• If μX ¼ μY , the sequences�Qn�1

i¼1 ð1þ mμXTi

Þ�−1

Wn and�Qn�1

i¼1 ð1þ mμYTi

Þ�−1

Bn areF n

-martingales such that�Qn�1

i¼1 ð1þ mμXTi

Þ�−1

’ Bn; where B > 0, then, as n tends to

infinity, we have

Wn

n→W∞ and

Bn

n→

~B∞; a:s:;

where W∞ and ~B∞ are positive random variables satisfying ~B∞ ¼ mμX −W∞:

Proof of claim 3 Theorem 2.We consider the case when Yn ¼ Xn for all n ≥ 0, The urnmodel is then evolving according to the recursive Eq. (1) with the replacement matrix

Qn ¼�Xn 00 Xn

�:

Since Theorem 2ð2Þ applies to that case, we obtain the following strong law of large number

Wn

n!a:sW∞ and

Bn

n!a:sðμXm�W∞Þ;

where W∞ is a positive random variable. Furthermore, as Tn is a sum of i.i.d. randomvariables then Tn satisfies for every δ > 1

2

Tn ¼a:s μXm2

nþ o� ffiffiffi

np

lnδn�; a:s: (20)

To prove thatW∞ is absolutely continuous, we follow the proof of Theorem 4.2 in [5] and wegive the main steps. The idea is the following: given the sequence of increasing event Ωl

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72

defined in Lemma 8, if we show that the restriction of W∞ on every Ωl; j ¼ fω; WlðωÞ ¼ jghas a density for each j, withUm ≤ j ≤ Tl−1, then Proposition 4 ensures the existence of thedensity of W∞ almost every where. In fact, for a fixed l and n≥ l þ 1, we denote byvn; j ¼ max

0≤k≤UmnℙðWlþn ¼ jþ kjWl ¼ jÞ. We have the following inequality:

vnþ1; j ≤ max0≤k≤Umðnþ1Þ

(Xmi¼0

Xc∈suppðXÞ

ℙðWlþnþ1 ¼ jþ kjWlþn ¼ jþ k� ciÞ)

≤ max0≤k≤Umðnþ1Þ

(Xmi¼0

Xc∈suppðXÞ

ℙðWlþnþ1 ¼ jþ kjWlþn ¼ jþ k� ciÞ 3 ℙðWlþn ¼ jþ k� cijWl ¼ jÞ)

≤ max0≤k≤Umðnþ1Þ

Xmi¼0

Xc∈suppðXÞ

ℙðWlþnþ1 ¼ jþ kjWlþn ¼ jþ k� ciÞ

3 max0≤~k≤Umn

ℙðWlþn ¼ jþ ~kjWl ¼ jÞ≤X

c∈suppðXÞpc

�1� 1

nþ lþ κ

ðnþ lÞ2�vn;j

¼�1� 1

nþ lþ κ

ðnþ lÞ2�vn;j:

This implies that there exists some positive constantCðlÞ, depending on l only, such that, for afixed l and for all n≥ l þ 1, we get

max0≤k≤mðn−lÞ

ℙðWn ¼ jþ kjWl ¼ jÞ≤Yni¼l

�1� 1

iþ κ

i2

�≤CðlÞn

: (21)

Let ε > 0and δ ¼ εCðlÞ, and setting x1 < x

01 ≤ x2 < x

02 ≤ . . . ≤ xr < x

0r such that

Pri¼1jx

0i − xij

≤ δ: By Fatou’s lemma we have

Xri¼1

ℙðfxi ≤W∞ ≤ x0 g ∩ Ωl; jÞ≤

Xri¼1

lim inf ℙ

�xi ≤

Wn

n≤ x

0ijWl ¼ j

�ℙðΩl; jÞ

≤Xri¼1

lim inf

��x0i � xi

�nþ 1ÞCðlÞ

n

�

≤Xri¼1

�x0i � xi

�CðlÞ ¼ ε:

Then the proof follows.Outlook:We suggest that if we replace the boundedness hypothesis of the variablesX andY bythe assumption that X and Y have finite moments of order 2, our results remain true.

References

[1] R. Aguech, Limit theorems for random triangular urns schemes, J. Appl. Probab. 46 (3) (2009)827–843.

[2] R. Aguech, et al., A generalized urn model with multiple drawing and random addition, Ann. Inst.Statist. Math. (2018) 1–20.

[3] K.B. Athrea, S. Karlin, Embedding of urns schemes into continuous time Markov branchingprocesses and related limit theorems, Ann. Math. Stat. 39 (1968) 1801–1817.


urn

73

[4] M.R. Chen, M. Kuba, On generalized polya urn models, J. Appl. Probab. 50 (4) (2013) 909–1216.

[5] M.R. Chen, C.Z. Wei, A new urn model, J. Appl. Probab. 42 (4) (2005) 964–976.

[6] M. Duflo, Random Iterative Models, Springer-Verlag, Berlin, 1997.

[7] F. Eggenberger, G. P�olya, €Uber die statistik verkeletter vorgange, Z. Angew. Math. Mech. 3 (1923)279–289.

[8] R.N. Goldman, Polya’s urn model and computer aided geometric design, SIAM J. Algebr. DiscreteMethods 6 (1) (1985) 1–28.

[9] S. Janson, Functionnal limt theorems for multitype branching processes and generalized p�olyaurns, Stochastic Process. Appl. 110 (2) (2004) 177–245.

[10] S. Janson, Limit theorems for triangular urn schemes, Probab. Theory Related Fields 134 (3)(2006) 417–452.

[11] N.L. Johnson, S. Kotz, Urn Models and Their Application, John Wiley & Sons, 1977.

[12] M. Kuba, H. Mahmoud, Two-colour balanced affine urn models with multiple drawings II: large-index and triangular urns, 2016, arXiv:150909053.

[13] M. Kuba, H. Mahmoud, Two-colour balanced affine urn models with multiple drawings I: Centrallimit theorem, Adv. Appl. Probab. 90 (2017) 1–26.

[14] M. Kuba, H. Sulzbach, On martingale tail sums in affine two-color urn models with multipledrawings, J. Appl. Probab. 54 (2017) 1–21.

[15] M. Kuba, et al., Analysis of a generalized Friedman’s urn with multiple drawings, Discrete Appl.Math. 161 (18) (2013) 2968–2984.

[16] N. Lasmar, et al., Multiple drawing multi-colour urns by stochastic approximation, J. Appl.Probab. 55 (1) (2018) 254–281.

[17] Z. Lin, C. Lu, Limit Theory for Mixing Dependent Random Variables, Kluwer AcademicPublishers, Boston, 1996.

[18] H. Mahmoud, Random spouts as internet model and p�olya processes, Acta inform. 41 (1) (2004)1–18.

[19] G. Pag�es, S. Laruelle, Randomized urns models revisited using stochastic approximation, Ann.Appl. Probab. 23 (4) (2013) 1409–1436.

[20] H. Renlund, Generalized polya urns via stochastic approximation, 2010, arxiv:10023716v1.

[21] H. Renlund, Limit theorem for Stochastic approximation algorithm, 2011, arxiv:11024741v1.

[22] L.J. Wei, An application of an urn model to the design of sequential controlled clinical trials, J.Am. Stat. Assoc. 73 (1978) 559–563.

Corresponding authorAguech Rafik can be contacted at: [email protected]


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Approximation of fixed point ofmultivalued ρ-quasi-contractive

mappings in modularfunction spaces

Godwin Amechi OkekeDepartment of Mathematics, School of Physical Sciences,Federal University of Technology, Owerri, Nigeria, and

Safeer Hussain KhanDepartment of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar

AbstractThe purpose of this paper is to extend the recent results of Okeke et al. (2018) to the class of multivaluedρ-quasi-contractive mappings in modular function spaces. We approximate fixed points of this class ofnonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts ofT -stability,almost T -stability and summably almost T -stability to modular function spaces and give some results.

KeywordsMultivalued ρ-quasi-contractive mappings, Multivalued mappings, Approximation of fixed point,

Modular function spaces, S-iterative process, ρ-T -stable, ρ-almost T -stable, ρ-summably almost T -stable


1. IntroductionIt is known that there is a close relationship between the problem of solving a nonlinearequation and that of approximating fixed points of a corresponding contractive type operator(see, e.g. [4,17]). Hence, there is a practical and theoretical interest in approximating fixedpoints of several contractive type operators. For over a century now, the study of fixed pointtheory of multivalued nonlinear mappings has attracted many well-known mathematiciansand mathematical scientists (see, e.g. Khan et al. [13]). The motivation for such studies stemsmainly from the usefulness of fixed point theory results in real-world applications, as inGame

Approximationin modular

function spaces

75

JEL Classification — 47H09, 47H10, 49M05, 54H25© Godwin Amechi Okeke and Safeer Hussain Khan. Published in the Arab Journal of Mathematical

Sciences. Published by Emerald Publishing Limited. This article is published under the CreativeCommons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and createderivative works of this article (for both commercial and non-commercial purposes), subject to fullattribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

Conflicts of interest: The authors declare that they do not have any conflicts of interest.Authors’ contributions: All authors contributed equally in writing this research paper. Each author

read and approved the final manuscript.The publisher wishes to inform readers that the article “Approximation of fixed point of multivalued

ρ-quasi-contractive mappings in modular function spaces” was originally published by the previouspublisher of the Arab Journal of Mathematical Sciences and the pagination of this article has beensubsequently changed. There has been no change to the content of the article. This change wasnecessary for the journal to transition from the previous publisher to the new one. The publishersincerely apologises for any inconvenience caused. To access and cite this article, please use AmechiOkeke, G., Hussain Khan, S. (2019), “Approximation of fixed point of multivalued ρ-quasi-contractivemappings in modular function spaces”Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 75-93.The original publication date for this paper was 08/02/2019.



Received 11 November 2018Accepted 3 February 2019


Vol. 26 No. 1/2, 2020pp. 75-93


DOI 10.1016/j.ajmsc.2019.02.001

Theory and Market Economy and in other areas of mathematical sciences such as inNonsmooth Differential Equations.

Modular function spaces are natural generalizations of both function and sequencevariants of several important, from application perspective, spaces like Musielak–Orlicz,Orlicz, Lorentz, Orlicz–Lorentz, Kothe, Lebesgue, Calderon–Lozanovskii spaces andseveral others. Interest in quasi-nonexpansive mappings in modular function spaces stemsmainly in the richness of structure of modular function spaces, that – besides being Banachspaces (or F-spaces in a more general settings) – are equipped with modular equivalents ofnorm or metric notions and also equipped with almost everywhere convergence andconvergence in submeasure. It is known that modular type conditions are much morenatural as modular type assumptions can be more easily verified than their metric or normcounterparts, particularly in applications to integral operators, approximation and fixedpoint results. Moreover, there are certain fixed point results that can be proved only usingthe apparatus of modular function spaces. Hence, fixed point theory results in modularfunction spaces, in this perspective, should be considered as complementary to the fixedpoint theory in normed and metric spaces (see, e.g. [10]). Several authors have proved veryinteresting fixed points results in the framework of modular function spaces, (see, e.g.[10,11,15,18]).

It is our purpose in the present paper to extend the recent results of Okeke et al. [17] to theclass of multivalued ρ-quasi-contractive mappings, which is known to be wider thanthe class of Zamfirescu operators (see, e.g. [5]) in modular function spaces. We approximatethe fixed point of these classes of nonlinear multivalued mappings in modular functionspaces. Moreover, we extend the concepts of T -stability, almost T -stability and summablyalmost T -stability to modular function spaces. Consequently, we define the concepts ofρ-T -stable, ρ-almostT -stable and ρ-summably almostT -stable in modular function spaces.We prove that some fixed point iterative processes are ρ-summably almost T -stable withrespect to T, where T is a multivalued ρ-quasi-contractive mapping in modular functionspaces.

2. PreliminariesIn this study, we letΩdenote a nonempty set and Σ a nontrivial σ-algebra of subsets ofΩ. LetP be a δ-ring of subsets of Ω, such that E ∩ A∈P for any E ∈P and A∈Σ. Let us assumethat there exists an increasing sequence of sets Kn ∈P such that Ω ¼ ∪Kn (for instance, Pcan be the class of sets of finite measure in a σ-finite measure space). By 1A, we denote thecharacteristic function of the setA inΩBy εwe denote the linear space of all simple functionswith supports fromP. ByM∞we denote the space of all extended measurable functions, i.e.,all functions f : Ω→ ½−∞;∞� such that there exists a sequence fgng⊂ ε, jgnj≤ jf j andgnðωÞ→ f ðωÞ for each ω∈Ω.

Definition 2.1. Let ρ : M∞ → ½0;∞�be a nontrivial, convex and even function.We say thatρ is a regular convex function pseudomodular if

(1) ρð0Þ ¼ 0;

(2) ρ is monotone, i.e.,j f ðωÞj≤ jgðωÞj for any ω∈Ω implies ρð f Þ≤ ρðgÞ, wheref ; g∈M∞;

(3) ρ is orthogonally subadditive, i.e., ρð f 1A∪BÞ≤ ρð f 1AÞ þ ρð f 1BÞ for any A;B∈Σsuch that A∩B≠ 0=, f ∈M∞;

(4) ρ has Fatou property, i.e.,j fnðωÞj↑j f ðωÞj for all ω∈Ω implies ρð fnÞ↑ρð f Þ, wheref ∈M∞;

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(5) ρ is order continuous in ε, i.e., gn ∈ ε and j gnðωÞj↓0 implies ρðgnÞ↓0.A set A∈Σ is said to be ρ-null if ρðg1AÞ ¼ 0 for every g ∈ ε. A property pðωÞ is said to holdρ-almost everywhere (ρ-a.e.) if the set {ω∈Ω : pðωÞ does not hold} is ρ-null. As usual, weidentify any pair of measurable sets whose symmetric difference is ρ-null as well as any pairof measurable functions differing only on a ρ-null set. With this in mind we define

MðΩ;Σ;P; ρÞ ¼ f f ∈M∞ : j f ðωÞj < ∞ ρ-a:e:g;where f ∈MðΩ;Σ;P; ρÞ is actually an equivalence class of functions equal ρ-a.e. rather thanan individual function. Where no confusion exists, we shall write M insteadof MðΩ;Σ;P; ρÞ.

The following definitions were given in [12].

Definition 2.2. Let ρ be a regular function pseudomodular;

(a) we say that ρ is a regular convex function modular if ρð f Þ ¼ 0 implies f ¼ 0 ρ-a.e.

(b) we say that ρ is a regular convex function semimodular if ρðα f Þ ¼ 0 for every α > 0implies f ¼ 0 ρ-a.e.

It is known (see, e.g. [10]) that ρ satisfies the following properties:

(1) ρð0Þ ¼ 0 iff f ¼ 0 ρ-a.e.

(2) ρðα f Þ ¼ ρð f Þ for every scalar αwith jαj ¼ 1 and f ∈M.

(3) ρðα f þ βgÞ≤ ρð f Þ þ ρðgÞ if αþ β ¼ 1, α; β ≥ 0 and f ; g∈M.

ρ is called a convex modular if, in addition, the following property is satisfied:ð30 Þ ρðα f þ βgÞ≤ αρð f Þ þ βρðgÞ if αþ β ¼ 1, α; β ≥ 0 and f ; g ∈M.The class of all nonzero regular convex function modulars on Ω is denoted by ℜ.

Definition 2.3. The convex function modular ρ defines the modular function space Lρ as

Lρ ¼ f f ∈M; ρðλf Þ→ 0 as λ→ 0g:Generally, the modular ρ is not subadditive and therefore does not behave as a norm or adistance. However, the modular space Lρ can be equipped with an F-norm defined by

k fkρ ¼ inf

�α > 0 : ρ

�f

α

�≤ α�:

In the case ρ is convex modular,

k fkρ ¼ inf

�α > 0 : ρ

�f

α

�≤ 1

�:

defines a norm on the modular space Lρ, and it is called the Luxemburg norm.

Lemma 2.1 ([10]). Let ρ∈ℜ. Defining L0ρ ¼ ff ∈Lρ; ρðf ; :Þ is order continuousg and

Eρ ¼ f f ∈Lρ; λf ∈L0ρ for every λ > 0g, we have

(i) Lρ � L0ρ � Eρ;

(ii) Eρ has the Lebesgue property, i.e., ρðα f ;DkÞ→ 0, for α > 0, f ∈Eρ and Dk ↓∅;

(iii) Eρ is the closure of ε (in the sense of k:kρ ).


function spaces

77

Definition 2.4. A nonzero regular convex function ρ is said to satisfy the Δ2-condition, ifsupn≥ 1ρð2fn;DkÞ→ 0 as k→∞ whenever fDkg decreases to � and supn≥ 1ρðfn;DkÞ→ 0as k→∞.

If ρ is convex and satisfies Δ2-condition, then Lρ ¼ Eρ.The following uniform convexity type properties of ρ can be found in [6].

Definition 2.5. Let ρ be a nonzero regular convex function modular defined on Ω(i) Let r > 0, e > 0. Define

D1ðr; eÞ ¼ fð f ; gÞ : f ; g ∈Lρ; ρð f Þ≤ r; ρðgÞ≤ r; ρð f � gÞ≥ erg:Let

δ1ðr; eÞ ¼ inf

�1� 1

rρ�f þ g

2

�: ðf ; gÞ∈D1ðr; eÞ

�if D1ðr; eÞ≠ 0=;

and δ1ðr; eÞ ¼ 1 if D1ðr; eÞ ¼ 0=. We say that ρ satisfies ðUC1Þ if for every r > 0, e > 0,δ1ðr; eÞ > 0. Observe that for every r > 0, D1ðr; eÞ≠ 0=, for e > 0 small enough.

(ii) We say that ρ satisfies ðUUC1Þ if for every s≥ 0, e > 0, there exists η1ðs; eÞ > 0depending only on s and e such that δ1ðr; eÞ > η1ðs; eÞ > 0 for any r > s.

(iii) Let r > 0, e > 0. Define

D2ðr; eÞ ¼�ðf ; gÞ : f ; g∈Lρ; ρðf Þ≤ r; ρðgÞ≤ r; ρ

�f � g

2

�≥ er

�:

Let

δ2ðr; eÞ ¼ inf

�1� 1

rρ�f þ g

2

�: ðf ; gÞ∈D2ðr; eÞ

�; if D2ðr; eÞ≠ 0=;

and δ2ðr; eÞ ¼ 1 if D2ðr; eÞ ¼ 0=. We say that ρ satisfies ðUC2Þ if for every r > 0, e > 0,δ2ðr; eÞ > 0. Observe that for every r > 0, D2ðr; eÞ≠ 0=, for e > 0 small enough.

(iv) We say that ρ satisfies ðUUC2Þ if for every s≥ 0, e > 0, there exists η2ðs; eÞ > 0depending only on s and e such that δ2ðr; eÞ > η2ðs; eÞ > 0 for any r > s.

(v) We say that ρ is strictly convex ðSCÞ, if for every f ; g ∈Lρ such that ρðf Þ ¼ ρðgÞ andρ�

fþg2

�¼ ρðf ÞþρðgÞ

2 , there holds f ¼ g.

Proposition 2.1. ([10]).The following conditions characterize relationship between the abovedefined notions:

(i) ðUUCiÞ0ðUCiÞ for i ¼ 1; 2.

(ii) δ1ðr; eÞ≤ δ2ðr; eÞ.(iii) ðUC1Þ0ðUC2Þ.(iv) ðUUC1Þ0ðUUC2Þ.(v) If ρ is homogeneous (e.g. it is a norm), then all the conditions ðUC1Þ,ðUC2Þ,ðUUC1Þ,

ðUUC2Þ are equivalent and δ1ðr; 2eÞ ¼ δ1ð1; 2eÞ ¼ δ2ð1; eÞ ¼ δ2ðr; eÞ.Definition 2.6. Let Lρ be a modular space. The sequence ffng⊂Lρ is called:

(1) ρ-convergent to f ∈Lρ if ρðfn − f Þ→ 0 as n→∞;

(2) ρ-Cauchy, if ρðfn − fmÞ→ 0 as n and m→∞.

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Observe that ρ-convergence does not imply ρ-Cauchy since ρ does not satisfy the triangleinequality. In fact, one can easily show that this will happen if and only if ρ satisfies theΔ2-condition.

Kilmer et al. [14] defined ρ-distance from an f ∈Lρ to a set D⊂Lρ as follows:

distρðf ;DÞ ¼ inffρðf � hÞ : h∈Dg:

Definition 2.7. A subset D⊂Lρ is called:

(1) ρ-closed if the ρ-limit of a ρ-convergent sequence of D always belongs to D;

(2) ρ-a.e. closed if the ρ-a.e. limit of a ρ-a.e. convergent sequence ofD always belongs toD;

(3) ρ-compact if every sequence in D has a ρ-convergent subsequence in D;

(4) ρ-a.e. compact if every sequence in D has a ρ-a.e. convergent subsequence in D;

(5) ρ-bounded if

diamρðDÞ ¼ supfρðf � gÞ : f ; g ∈Dg < ∞:

The following famous result was proved by Zamfirescu [19]

Theorem 2.1. ([19]). Let ðX ; dÞ be a complete metric space, and let T : X →X be a mappingfor which there exist real numbers a; b and c satisfying 0 < a < 1, 0 < b; c < 1

2 such that foreach pair x; y∈X at least one of the following is true:

(z1) dðTx;TyÞ≤ adðx; yÞ,(z2) dðTx;TyÞ≤ b½dðx;TxÞ þ dðy;TyÞ�,(z3) dðTx;TyÞ≤ c½dðx;TyÞ þ dðy;TxÞ�.

Then T has a unique fixed point p and the Picard iteration process fxng defined by

xnþ1 ¼ Txn; n ¼ 0; 1; 2; . . .

converges to p for any x0 ∈X.

Remark2.1.Any operatorTwhich satisfies the contractive conditions (z1)–(z3) of Theorem2.1 is called a Zamfirescu operator (see e.g. [5]) and is denoted by Z .

The following class of quasi-contractive operators was introduced on a normed spaceE byBerinde [5]:

kTx� Tyk≤ δkx� yk þ LkTx� xk;for any x; y∈E, 0≤ δ < 1 and L≥ 0. He proved that this class is wider than the class ofZamfirescu operators.

A set D⊂Lρ is called ρ-proximinal if for each f ∈Lρ there exists an element g∈D such thatρðf − gÞ ¼ distρðf ;DÞ. We shall denote the family of nonempty ρ-bounded ρ-proximinal subsetsofD by PρðDÞ, the family of nonempty ρ-closed ρ-bounded subsets ofD by CρðDÞ and the familyof ρ-compact subsets of D by KρðDÞ. Let Hρð:; :Þbe the ρ-Hausdorff distance on CρðLρÞ, that is,

HρðA;BÞ ¼ max

�supf∈A

distρðf ;BÞ; supg∈B

distρðg;AÞ�;A;B∈CρðLρÞ:

A multivalued map T : D→CρðLρÞ is said to be:

(a) ρ-contraction mapping if there exists a constant k∈ ½0; 1Þ such that

HρðTf ;TgÞ≤ kρðf � gÞ; for all f ; g∈D: (2.1)


function spaces

79

(b) ρ-nonexpansive (see, e.g. Khan and Abbas [12]) if

HρðTf ;TgÞ≤ ρðf � gÞ; for all f ; g ∈D: (2.2)

(c) ρ-quasi-nonexpansive mapping if

HρðTf ; pÞ≤ ρðf � pÞ for all f ∈D and p∈FρðTÞ: (2.3)

(d) ρ-quasi-contractive mapping if

HρðTf ;TgÞ≤ δρðf � gÞ þ LρðTf � f Þ; for all f ; g ∈D; 0≤ δ < 1 and L≥ 0: (2.4)

A sequence ftng⊂ ð0; 1Þ is called bounded away from 0 if there exists a > 0 such that tn ≥ afor every n∈ℕ. Similarly, ftng⊂ ð0; 1Þ is called bounded away from 1 if there exists b < 1such that tn ≤ b for every n∈ℕ.

Recently, Okeke et al. [17] approximated the fixed point of multivalued ρ-quasi-nonexpansivemappings using the Picard–Krasnoselskii hybrid iterative process. It is knownthat this iteration process converges faster than all of Picard, Mann, Krasnoselskii andIshikawa iterative processes when applied to contraction mappings (see, Okeke and Abbas[16]). The following is the analogue of the Picard–Krasnoselskii hybrid iterative process inmodular function spaces: Let T : D→PρðDÞ be a multivalued mapping and ffng⊂D bedefined by the following iteration process:(

fnþ1 ∈PTρ ðgnÞ

gn ¼ ð1� λÞfn þ λPTρ ðvnÞ; n∈ℕ;

(2.5)

where vn ∈PTρ ðfnÞ and 0 < λ < 1. It is our purpose in the present paper to prove some new

fixed point theorems using this iteration process in the framework of modular functionspaces.

The following is the analogue of the S-iteration, introduced by Agarwal et al. [1] inmodular function spaces. 8<

:f0 ∈D

fnþ1 ¼ ð1� αnÞun þ αnvngn ¼ ð1� βnÞ fn þ βnun;

(2.6)

where un ∈PTρ ðfnÞ, vn ∈PT

ρ ðgnÞ, the sequences fαng; fβng⊂ ð0; 1Þ are bounded away fromboth 0 and 1. It is known (see, e.g. [9]) that the S-iteration converges faster than the Manniteration process and the Ishikawa iteration process for Zamfirescu operators.

Definition 2.8. A sequence ffng⊂D is said to be Fej�er monotone with respect to subsetPρðDÞ of D if ρðfnþ1 − pÞ≤ ρðfn − pÞ, for all p∈PT

ρ ðDÞ of D, n∈ℕ.

Definition 2.9. ([12]). A multivalued mappingT : D→CρðDÞ is said to satisfy condition (I)if there exists a nondecreasing function l : ½0;∞Þ→ ½0;∞Þ with lð0Þ ¼ 0, lðrÞ > 0 for allr∈ ð0;∞Þ such that distρðf ;Tf Þ≥ lðdistρðf ;FρðTÞÞÞ for all f ∈D.

The following Lemma will be needed in this study.

Lemma 2.2. ([2]). Let ρ∈ℜ satisfy the Δ2-condition. Let ffng andfgng be two sequences inLρ. Then

limn→∞

ρðgnÞ ¼ 00lim supn→∞

ρðfn þ gnÞ ¼ lim supn→∞

ρðfnÞ

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and

limn→∞

ρðgnÞ ¼ 00lim infn→∞

ρðfn þ gnÞ ¼ lim infn→∞

ρðfnÞ:

Lemma 2.3. ([6]). Let ρ satisfyðUUC1Þ and letftkg⊂ ð0; 1Þ be bounded away from 0 and 1.If there exists R > 0 such that

lim supn→∞

ρðfnÞ≤R; lim supn→∞

ρðgnÞ≤R

and

limn→∞

ρðtnfn þ ð1� tnÞgnÞ ¼ R;

then limn→∞ ρðfn − gnÞ ¼ 0.A function f ∈Lρ is called a fixed point of T : Lρ →PρðDÞ if f ∈Tf . The set of all fixed

points of T will be denoted by FρðTÞ.Lemma 2.4. ([12]). Let T : D→PρðDÞ be a multivalued mapping and

PTρ ðf Þ ¼ fg ∈Tf : ρðf � gÞ ¼ distρ ðf ;Tf Þg:

Then the following are equivalent:

(1) f ∈FρðTÞ, that is, f ∈Tf.

(2) PTρ ðf Þ ¼ ffg, that is,f ¼ g for each g ∈PT

ρ ðf Þ.(3) f ∈FðPT

ρ ðf ÞÞ, that is, f ∈PTρ ðf Þ. Further FρðTÞ ¼ FðPT

ρ ðf ÞÞ where FðPTρ ðf ÞÞ

denotes the set of fixed points of PTρ ðf Þ.

Lemma 2.5. ([3]). Let fang∞n¼0,fbng∞n¼0 be sequences of nonnegative numbers and 0≤ q < 1,such that

anþ1 ≤ qan þ bn; for all n≥ 0:

(i) If limn→∞bn ¼ 0, then limn→∞an ¼ 0.

(ii) IfP∞

n¼0bn < ∞, thenP∞

n¼0an < ∞.

3. Approximation of fixed points in modular function spacesWe begin this section with the following proposition

Proposition 3.1. Let ρ satisfy ðUUC1Þ and Δ2-condition. Let D be a nonempty ρ-closed,ρ-bounded and convex subset of Lρ. Let T : D→PρðDÞ be a multivalued mapping such that PT

ρis a ρ-quasi-contractive mapping, satisfying contractive condition (2.4) and FρðTÞ≠ 0=. Letffng⊂Dbe defined by the two step S-iterative process (2.6), such that the sequencesfαng⊂ ð0; 1Þandfβng⊂ ð0; 1Þare bounded away from both 0 and 1. Then the S-iterative process (2.6) is Fej�ermonotone with respect toFρðTÞ.Proof. Let p∈FρðTÞ. By Lemma 2.4, PT

ρ ðpÞ ¼ fpg and FρðTÞ ¼ FðPTρ Þ. Using relation (2.4)

and (2.6), we obtain the following estimate:

ρðfnþ1 � pÞ ¼ ρ½ð1� αnÞun þ αnvn � p�¼ ρ½ð1� αnÞðun � pÞ þ αnðvn � pÞ�: (3.1)


function spaces

81

The convexity of ρ implies

ρðfnþ1 � pÞ≤ ð1� αnÞρðun � pÞ þ αnρðvn � pÞ≤ ð1� αnÞHρðPT

ρ ðfnÞ;PTρ ðpÞÞ þ αnHρðPT

ρ ðgnÞ;PTρ ðpÞÞ:

(3.2)

From relation (2.4), with f ¼ p, g ¼ fn and also f ¼ p, g ¼ gn, then we obtain the followingestimates from relation (3.2):

HρðPTρ ðfnÞ;PT

ρ ðpÞÞ≤ δρðfn � pÞ: (3.3)

HρðPTρ ðgnÞ;PT

ρ ðpÞÞ≤ δρðgn � pÞ: (3.4)

Using (3.3), (3.4) and the fact that 0≤ δ < 1 in (3.2), we have

ρðfnþ1 � pÞ≤ ð1� αnÞδρðfn � pÞ þ αnδρðgn � pÞ≤ ð1� αnÞρðfn � pÞ þ αnρðgn � pÞ: (3.5)

Next, we have

ρðgn � pÞ ¼ ρ½ð1� βnÞfn þ βnun � p�¼ ρ½ð1� βnÞðfn � pÞ þ βnðun � pÞ�: (3.6)

By convexity of ρ, we have

ρðgn � pÞ≤ ð1� βnÞρðfn � pÞ þ βnHρðPTρ ðfnÞ;PT

ρ ðpÞÞ: (3.7)

Using (2.4) with f ¼ p and g ¼ fn and the fact that 0≤ δ < 1, relation (3.7) yields:

ρðgn � pÞ≤ ð1� βnÞρðfn � pÞ þ βnδρðfn � pÞ≤ ð1� βnÞρðfn � pÞ þ βnρðfn � pÞ¼ ρðfn � pÞ:

(3.8)

Using (3.8) in (3.5), we obtain: (3.9)

ρðfnþ1 � pÞ≤ ð1� αnÞρðfn � pÞ þ αnρðfn � pÞ¼ ρðfn � pÞ: (3.9)

Hence, the S-iteration (2.6) is Fej�er monotone with respect to FρðTÞ. The proof of Proposition3.1 is completed. ,

Next, we prove the following proposition.

Proposition 3.2. Let ρ satisfy the ðUUC1Þ and Δ2-condition. Suppose that D is a nonemptyρ-closed, ρ-bounded and convex subset of Lρ. LetT : D→PρðDÞ be a multivalued mapping such

that PTρ is a ρ-quasi-contractive mapping, satisfying contractive condition (2.4) and FρðTÞ≠ 0=.

Let ffng⊂D be defined by the two step S-iterative process (2.6), such that thesequencesfαng⊂ ð0; 1Þ andfβng⊂ ð0; 1Þ are bounded away from both 0 and 1. Then

(i) the sequence ffng is bounded.(ii) for each f ∈D,fρðfn − f Þg converges.

Proof. Since ffng is Fej�er monotone as shown in Proposition 3.1. Using the fact that ρ satisfiesthe Δ2-condition, we can easily show (i) and (ii). This completes the proof of Proposition 3.2.,

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Theorem 3.1. Let ρ satisfy ðUUC1Þ and Δ2-condition. Let D be a ρ-closed, ρ-bounded andconvex subset of a ρ-complete modular space Lρ and T : D→PρðDÞ be a multivalued mapping

such that PTρ is a ρ-quasi-contractive mapping, satisfying contractive condition (2.4) and

FρðTÞ≠ 0=. Let ffng⊂Dbe defined by the two step S-iterative process (2.6) and f0 ∈D, where thesequences fαng,fβng⊂ ð0; 1Þ are bounded away from both 0 and 1, satisfying

P∞

n¼0αn ¼ ∞.Then ffng converges strongly to the fixed point of T.

Proof. Let p∈FρðTÞ. By Lemma 2.4, PTρ ðpÞ ¼ fpg and FρðTÞ ¼ FðPT

ρ Þ. Using relation (2.4)and (2.6), we obtain the following estimate:


The convexity of ρ implies (3.11)




(3.11)


HρðPTρ ðfnÞ;PT


HρðPTρ ðgnÞ;PT


Using (3.12) and (3.13) in (3.11), we have

ρðfnþ1 � pÞ≤ ð1� αnÞδρðfn � pÞ þ αnδρðgn � pÞ: (3.14)

Next, we have



ρðgn � pÞ≤ ð1� βnÞρðfn � pÞ þ βnHρðPTρ ðfnÞ;PT

ρ ðpÞÞ: (3.16)

Using (2.4) with f ¼ p and g ¼ fn, then relation (3.16) yields:

ρðgn � pÞ≤ ð1� βnÞρðfn � pÞ þ βnδρðfn � pÞ: (3.17)

Using (3.17) in (3.14), we have

ρðfnþ1 � pÞ≤ ð1� αnÞδρðfn � pÞ þ αnδð1� βnð1� δÞÞρðfn � pÞ≤ ½1� αnð1� δð1� βnð1� δÞÞÞ�ρðfn � pÞ: (3.18)

Using (3.18), we inductively obtain

ρðfnþ1 � pÞ≤Ynk¼0

½1� αkð1� δð1� βkð1� δÞÞÞ�ρðf0 � pÞ;

n ¼ 0; 1; 2; 3; . . .

(3.19)

Using the fact that 0≤ δ < 1, fαng; fβng⊂ ð0; 1Þ are bounded away from both 0 and 1,satisfying

P∞

n¼0αn ¼ ∞, relation (3.19) yields


function spaces

83

limn→∞

Ynk¼0

½1� αkð1� δð1� βkð1� δÞÞÞ� ¼ 0; (3.20)

which implies that (3.19) becomes:

limn→∞

ρðfnþ1 � pÞ ¼ 0: (3.21)

Consequently, fn → p∈FρðTÞ. The proof of Theorem 3.1 is completed. ,

Theorem 3.2. Let ρ satisfy ðUUC1Þ and Δ2-condition. Let D be a nonempty ρ-closed,ρ-bounded and convex subset of Lρ. Let T : D→PρðDÞ be a multivalued mapping such that PT

ρis a ρ-quasi-contractive mapping, satisfying contractive condition (2.4) and FρðTÞ≠ 0=. Letffng⊂D be defined by the two step S-iterative process (2.6) and f0 ∈D, where the sequencesfαng,fβng⊂ ð0; 1Þ are bounded away from both 0 and 1. Then limn→∞ρðfn − pÞ exists for allp∈FρðTÞ and limn→∞distρðfn;PT

ρ ðfnÞÞ ¼ 0.

Proof. Let p∈FρðTÞ. By Lemma 2.4, PTρ ðpÞ ¼ fpg and FρðTÞ ¼ FðPT

ρ Þ. Using relation (2.4)and (2.6), we obtain the following estimate:


The convexity of ρ implies




(3.23)


HρðPTρ ðfnÞ;PT


HρðPTρ ðgnÞ;PT


Using (3.24), (3.25) and the fact that 0≤ δ < 1 in (3.23), we have

ρðfnþ1 � pÞ≤ ð1� αnÞδρðfn � pÞ þ αnδρðgn � pÞ≤ ð1� αnÞρðfn � pÞ þ αnρðgn � pÞ: (3.26)

Next, we have



ρðgn � pÞ≤ ð1� βnÞρðfn � pÞ þ βnHρ

�PTρ ðfnÞ;PT

ρ ðpÞ�: (3.28)

Using (3.25) with f ¼ p and g ¼ fn and the fact that 0≤ δ < 1, relation (3.28) yields:

ρðgn � pÞ≤ ð1� βnÞρðfn � pÞ þ βnδρðfn � pÞ≤ ð1� βnÞρðfn � pÞ þ βnρðfn � pÞ¼ ρðfn � pÞ:

(3.29)

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Using (3.29) in (3.26), we obtain:

ρðfnþ1 � pÞ≤ ð1� αnÞρðfn � pÞ þ αnρðfn � pÞ¼ ρðfn � pÞ: (3.30)

This implies that limn→∞ρðfn − pÞ exists for all p∈FρðTÞ.Let

limn→∞

ρðfn � pÞ ¼ K; where K ≥ 0: (3.31)

Now, we show that

limn→∞

distρ�fn;P

Tρ ðfnÞ

� ¼ 0: (3.32)

Since distρðfn;PTρ ðfnÞÞ≤ ρðfn − unÞ, it suffices to show that

limn→∞

ρðfn � unÞ ¼ 0: (3.33)

Now,

ρðun � pÞ≤Hρ

�PTρ ðfnÞ;PT

ρ ðpÞ�≤ ρðfn � pÞ: (3.34)

This implies that

lim supn→∞

ρðun � pÞ≤ lim supn→∞

ρðfn � pÞ: (3.35)

By (3.31), we have

limn→∞

sup ρðun � pÞ≤K: (3.36)

Also from (3.29), we have

lim supn→∞

ρðgn � pÞ≤ lim supn→∞

ρðfn � pÞ; (3.37)

so that

lim supn→∞

ρðgn � pÞ≤K: (3.38)

Moreover, the inequality

ρðvn � pÞ≤HρðPTρ ðgnÞ;PT

ρ ðpÞÞ≤ ρðgn � pÞ≤ ρðfn � pÞ; (3.39)

this implies that

lim supn→∞

ρðvn � pÞ≤ lim supn→∞

ρðfn � pÞ; (3.40)

hence,

lim supn→∞

ρðvn � pÞ≤K: (3.41)

Now,

limn→∞

ρðfnþ1 þ pÞ ¼ limn→∞

ρ½ð1� αnÞun þ αnvn � p�¼ lim

n→∞ρ½ð1� αnÞðun � pÞ þ αnðvn � pÞ�

¼ K:

(3.42)


function spaces

85

Using (3.35), (3.41), (3.42) and Lemma 2.3, we have

limn→∞

ρðvn � unÞ ¼ 0: (3.43)

Now,

ρðfnþ1 � pÞ ¼ ρ½ð1� αnÞun þ αnvn � p�¼ ρ½ðun � pÞ þ αnðvn � unÞ�:

(3.44)

Using Lemma 2.2 and (3.44), we have

K ¼ lim infn→∞

ρðfnþ1 � pÞ ¼ lim infn→∞

ρ½ðun � pÞ þ αnðvn � unÞ�¼ lim inf

n→∞ρðun � pÞ: (3.45)

This means that

K ¼ lim infn→∞

ρðun � pÞ: (3.46)

Using (3.35) and (3.46), we have

limn→∞

ρðun � pÞ ¼ K: (3.47)

Using (3.43), we have

lim infn→∞

ρðun � pÞ ¼ lim infn→∞

ρ½ðun � vnÞ þ ðvn � pÞ� ¼ lim infn→∞

ρðvn � pÞ: (3.48)

But

ρðvn � pÞ≤HρðPTρ ðgnÞ;PT

ρ ðpÞÞ≤ ρðgn � pÞ: (3.49)

Hence,

lim infn→∞

ρðvn � pÞ≤ lim infn→∞

ρðgn � pÞ: (3.50)

By (3.41), we have

K ≤ lim infn→∞

ρðgn � pÞ: (3.51)

From (3.41) and (3.51), we have

limn→∞

ρðgn � pÞ ¼ K: (3.52)

Since

limn→∞

ρðgn � pÞ ¼ limn→∞

ρ½ð1� βnÞfn þ βnun � p�¼ lim

n→∞ρ½ð1� βnÞðfn � pÞ þ βnðun � pÞ� ¼ K:

(3.53)

Using (3.31), (3.35) and Lemma 2.3, we have

limn→∞

ρðfn � unÞ ¼ 0: (3.54)

Hence,

limn→∞

distρ�fn;P

Tρ ðfnÞ

� ¼ 0: (3.55)

The proof of Theorem 3.2 is completed. ,

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Theorem 3.3. Let ρ satisfy ðUUC1Þ and Δ2-condition. Let D be a nonempty ρ-compact,

ρ-bounded and convex subset of Lρ. Let T : D→PρðDÞ be a multivalued mapping such that PTρ

is a ρ-quasi-contractive mapping, satisfying contractive condition (2.4) and FρðTÞ≠ 0=. Letffng⊂D be defined by the two step S-iterative process (2.6) and f0 ∈D, where the sequencesfαng,fβng⊂ ð0; 1Þare bounded away from both 0 and 1. Then ffngρ -converges to a fixed pointof T.

Proof. Using relation (2.4) with f ¼ q, g ¼ fnk and the fact that 0≤ δ < 1. Since D isρ-compact, there exists a subsequence ffnkg of ffng such that limn→∞ðfnk– qÞ ¼ 0 for some

q∈D. Next, we show that q is a fixed point ofT. Suppose t is an arbitrary point in PTρ ðqÞ and

f ∈PTρ ðfnkÞ. Observe that

ρ�q� t

3

�¼ ρ�q� fnk

3þ fnk � f

3þ f � t

3

�

≤1

3ρ�q� fnk

�þ 1

3ρðfnk � f Þ þ 1

3ρðf � tÞ

≤ ρ�q� fnk

�þ distρ�fnk;P

Tρ

�fnk��þ distρ

�PTρ ðfnk

�; tÞ

≤ ρ�q� fnk


Tρ ðfnk

��þ Hρ

�PTρ ðfnk

�;PT

ρ ðqÞÞ

≤ ρ�q� fnk


Tρ ðfnk

��þ δρ�q� fnk

�

≤ ρ�q� fnk


Tρ ðfnk

��þ ρ�q� fnk

�:

(3.56)

By Theorem 3.2, we obtain limn→∞distρðfn;PTρ ðfnÞÞ ¼ 0. So that ρ

�q− t3

� ¼ 0. Therefore, q is a

fixed point of PTρ . By Lemma 2.4, we see that the set of fixed points of PT

ρ is the same as that of

T, hence, we have that ffng ρ-converges to a fixed point of T. The proof of Theorem 3.3 iscompleted. ,

Theorem 3.4. Let ρ satisfy ðUUC1Þ and Δ2-condition. Let D be a nonempty ρ-closed,ρ-bounded and convex subset of Lρ. Let T : D→PρðDÞ be a multivalued mapping satisfying

condition (I) such that PTρ is a ρ-quasi-contractivemapping, satisfying contractive condition (2.4)

and FρðTÞ≠ 0=. Let ffng⊂Dbe defined by the two step S-iterative process (2.6) and f0 ∈D, wherethe sequences fαng,fβng⊂ ð0; 1Þ are bounded away from both 0 and 1. Then ffng ρ-convergesto a fixed point of T.

Proof. The proof of Theorem 3.4 is similar to the proof of Theorem 3 of Khan and Abbas[12]. ,

4. ρ-Stability of fixed point iterations in modular function spacesIn this section, we define the concepts of ρ-T -stable, ρ-almost T -stable and ρ-summablyalmost T -stable in modular function spaces. We prove that some fixed point iterativeprocesses are ρ-summably almost T -stable with respect to T, where T is a multivaluedρ-quasi-contractive mapping in modular function spaces.

Let ρ satisfy ðUUC1Þ and D a nonempty ρ-closed, ρ-bounded and convex subset of Lρ. LetT : D→PρðDÞ be a mapping with FρðTÞ≠ 0=. Suppose that ffng∞n¼0 is a fixed point iterativeprocess, i.e. a sequence ffng∞n¼0 defined by f0 ∈D and (4.1)


function spaces

87

fnþ1 ¼ FðT; fnÞ; n ¼ 0; 1; 2; 3; . . . ; (4.1)

where F is a given function.Several fixed point iterations exist in literature. For instance, Mann iteration, with

FðT; fnÞ ¼ ð1− αnÞfn þ αnTfn, where fαng⊂ ½0; 1� such that fαng is bounded away fromboth 0 and 1. The Ishikawa iteration, with FðT; fnÞ ¼ ð1− αnÞfn þ αnT½ð1− βnÞfn þ βnTfn�,such that fαng∞n¼0; fβng∞n¼0 ⊂ ½0; 1� are both bounded away from both 0 and 1.

Let ffng∞n¼0 converge strongly to some p∈FρðTÞ. In practice, we compute ffng∞n¼0 asfollows:

(i) Choose the initial guess (approximation) f0 ∈D;

(ii) Compute f1 ¼ FðT; f0Þ. However, as a result of various errors that occur duringcomputations (numerical approximations of functions, rounding errors, derivatives,integration, etc.), we do not obtain the exact value of f1, but a different one, say , which isclose enough to f1, this means that h1 ≈ f1;

(iii) Therefore, during the computation of f2 ¼ FðT; f1Þwe have

f2 ¼ FðT; h1Þ: (4.2)

This means that instead of the theoretical value of f2, we expect another value h2 will beobtained, and h2 being close enough to f2, i.e. h2 ≈ f2, and so on.

Continuing this process, we see that instead of the theoretical sequence ffng∞n¼0 defined bythe fixed point iteration (4.1), we obtain practically an approximate sequence fhng∞n¼0.

The fixed point iteration (4.1) is considered to be numerically stable if and only if for hnclose enough to fn at each stage, we have that the approximate fhng∞n¼0 still converges to thefixed point p of FρðTÞ.

Next, we give the following definition, which is the analogue of the concept of T -stabilityintroduced by Harder and Hicks (see, [7,8]) in modular function spaces.

Definition 4.1. Let ρ satisfy ðUUC1Þ and D a nonempty ρ-closed, ρ-bounded and convexsubset of Lρ. Let T : D→PρðDÞ be a mapping with FρðTÞ≠ 0=. Suppose that the fixed pointiterative process (4.1) converges to a fixed point p of T. Let fhng∞n¼0 be an arbitrary sequencein D and set

εn ¼ ρðhnþ1 � FðT; hnÞÞ; n ¼ 0; 1; 2; 3; . . . (4.3)

The fixed point iterative process (4.1) is said to be ρ-T -stable, or ρ-stable or ρ-stable with respectto T if and only if

limn→∞

εn ¼ 00 limn→∞

hn ¼ p: (4.4)

Definition 4.2. Let ρ satisfy ðUUC1Þ and D a nonempty ρ-closed, ρ-bounded and convexsubset of Lρ. Let T : D→PρðDÞ be a mapping with FρðTÞ≠ 0=. Suppose that the fixed pointiterative process (4.1) converges to a fixed point p of T. Let fhng∞n¼0 be an arbitrary sequencein D and let fεng∞n¼0 be defined by (4.3). The fixed point iterative process (4.1) is said to beρ-almost T-stable or ρ-almost stable with respect to T if and only if

X∞n¼0

εn < ∞0 limx→∞

hn ¼ p: (4.5)

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Remark4.1. It is clear from the definitions that any ρ-stable fixed point iteration ffng is alsoρ-almost stable.

A sharper concept of almost stability was introduced by Berinde [4]. He showed somealmost stable fixed point iterations which are also summably almost stable with respect tosome classes of contractive operators. We next define the analogue of this concept in modularfunction spaces.

Definition 4.3. Let ρ satisfy ðUUC1Þ and D a nonempty ρ-closed, ρ-bounded and convexsubset of Lρ. LetT : D→PρðDÞbe a mapping with FρðTÞ≠ 0=. Suppose that the fixed pointiterative process (4.1) converges to a fixed point p of T. Let fhng∞n¼0 be an arbitrarysequence in D and let fεng∞n¼0 be defined by (4.3). The fixed point iterative process (4.1) issaid to be ρ-summably almost T-stable or ρ-summably almost stable with respect to T if andonly if

X∞n¼0

εn < ∞0X∞n¼0

ρðhn � pÞ < ∞: (4.6)

Remark 4.2. Clearly, any fixed point iteration ffng that is ρ-almost stable is alsoρ-summably almost stable, since

X∞n¼0

ρðhn � pÞ < ∞0 limn→∞

hn ¼ p:

However, we show that the converse is generally not true (see Example 4.1 below).

Example 4.1. Let the real number system ℝ be the space modulared as follows:

ρðf Þ ¼ jf jk; k≥ 1:

LetD ¼ ff ∈Lρ : 0≤ f ðxÞ≤ 1g. LetT : D→PρðDÞbe amultivaluedmapping such thatPTρ is

ρ-nonexpansive satisfying Tf ¼ f . Let ffng be the Picard iteration. Then ffng is notρ-summably almost T -stable.

Clearly, D is a nonempty ρ-compact, ρ-bounded and convex subset of Lρ ¼ ℝ whichsatisfies UC1 condition. Moreover, ρðf Þ ¼ jf jk, k≥ 1 is homogeneous and it is of degree k,hence by Proposition 2.1 ðUUC1Þ hold. Clearly, FρðTÞ ¼ ½0; 1�. Suppose p ¼ 0. Take hn ¼ 1

n,

for each n≥ 1. Hence, limn→∞hn ¼ 0, we see that

εn ¼ ρðhnþ1 � FðT; hnÞÞ ¼ distρ

�1

nþ 1;1

n

�

¼�� 1

nþ 1� 1

n

��k

¼�� 1

nðnþ 1Þ�� ¼ 1

nðnþ 1Þ :

Hence,P∞

n¼0εn < ∞.However, we have

X∞n¼0

ρðhn � pÞ ¼X∞n¼0

distρ

�1

n; 0

�¼X∞n¼0

��1n� 0

��k

¼X∞n¼0

��1n�� ¼

X∞n¼0

1

n¼ ∞:

This means that the Picard iteration ffng is not ρ-summably almost T -stable.It is known that the Picard iteration is not T -stable and hence not almost T -stable (see,

e.g. [4]).


function spaces

89

Next, we prove the following results.

Theorem 4.1. Let ρ satisfyðUUC1Þ and Δ2-condition. Let D be a ρ-closed, ρ-bounded andconvex subset of a ρ-complete modular space Lρ and T : D→PρðDÞ be a multivalued mapping


FρðTÞ≠ 0=. Let ffng⊂D be defined by the two step S-iterative process as follows

8<:

f0 ∈D

fnþ1 ¼ ð1� αnÞun þ αnvngn ¼ ð1� βnÞfn þ βnun;

(4.7)

where un ∈PTρ ðfnÞ,vn ∈PT

ρ ðgnÞ, the sequencesfαng; fβng⊂ ð0; 1Þ are bounded away from both

0 and 1. Then ffng is ρ -summably almost stable with respect to T.

Proof. Suppose p∈FρðTÞ and fhng is an arbitrary sequence. Define

�sn ¼ ð1� βnÞhn þ βnwn;εn ¼ ρðhnþ1 � ð1� αnÞwn � αnznÞ; (4.8)

where wn ∈PTρ ðhnÞ, zn ∈PT

ρ ðsnÞ, the sequences fαng; fβng⊂ ð0; 1Þ are bounded away fromboth 0 and 1.

Using the convexity of ρ, we have the following estimates:

ρðhnþ1 � pÞ ¼ ρðhnþ1 � ð1� αnÞwn � αnzn þ ð1� αnÞðwn � pÞ þ αnðzn � pÞÞ≤ εn þ ð1� αnÞρðwn � pÞ þ αnρðzn � pÞ≤ εn þ ð1� αnÞHρðPT

ρ ðhnÞ;PTρ ðpÞÞ þ αnHρðPT

ρ ðsnÞ;PTρ ðpÞÞ:

(4.9)

Using (4.9), relation (2.4) with f ¼ p, g ¼ hn and also f ¼ p, g ¼ sn, we have

ρðhnþ1 � pÞ≤ εn þ ð1� αnÞδρðhn � pÞ þ αnδρðsn � pÞ: (4.10)

Next, by convexity of ρwe have

ρðsn � pÞ ¼ ρðð1� βnÞhn þ βnwn � pÞ≤ ð1� βnÞρðhn � pÞ þ βnHρðPT

ρ ðhnÞ;PTρ ðpÞ

�≤ ð1� βnÞρðhn � pÞ þ βnδρðhn � pÞ≤ ð1� βnÞρðhn � pÞ þ βnρðhn � pÞ¼ ρðhn � pÞ:

(4.11)

Using (4.11) in (4.10), we obtain

ρðhnþ1 � pÞ≤ εn þ ð1� αnÞδρðhn � pÞ þ αnδρðhn � pÞ¼ εn þ δρðhn � pÞ: (4.12)

By Lemma 2.5, we have that the two step S-iteration (4.7) is ρ-summably almost stable withrespect to T. The proof of Theorem 4.1 is completed. ,

AJMS26,1/2

90

Theorem 4.2. Let ρ satisfyðUUC1Þ and Δ2-condition. Let D be a ρ-closed, ρ-bounded andconvex subset of a ρ-complete modular space Lρ and T : D→PρðDÞ be a multivalued mapping


FρðTÞ≠ 0=. Let ffng⊂D be defined by the following iterative process

(f0 ∈D

fnþ1 ∈PTρ ðunÞ

(4.13)

where un ∈PTρ ðfnÞ. Then ffng is ρ -summably almost stable with respect to T.

Proof. Let p∈FρðTÞ and fhng be an arbitrary sequence. Define

εn ¼ ρðhnþ1 �mnÞ; (4.14)

wheremn ∈PTρ ðhnÞ. Using (4.13), (4.14), relation (2.4) with f ¼ p, g ¼ hn and the convexity of ρ,

we have the following estimate:

ρðhnþ1 � pÞ ¼ ρðhnþ1 �mn þmn � pÞ≤ ρðhnþ1 �mnÞ þ ρðmn � pÞ≤ εn þ HρðPT

ρ ðhnÞ;PTρ ðpÞÞ

≤ εn þ δρðhn � pÞ:

(4.15)

By Lemma 2.5, it follows that the fixed point iteration (4.13) is ρ-summably almost stable withrespect to T. The proof of Theorem 4.2 is completed. ,

Theorem 4.3. Let ρ satisfy ðUUC1Þ and Δ2-condition. Let D be a ρ-closed, ρ-bounded andconvex subset of a ρ-complete modular space Lρ and T : D→PρðDÞ be a multivalued

mapping such that PTρ is a ρ-quasi-contractive mapping, satisfying contractive condition

(2.4) and FρðTÞ≠ 0=. Let ffng⊂D be defined by the two step S-iterative process asfollows

8>><>>:

f0 ∈D

fnþ1 ∈Xki¼0

αiuin; n≥ 0; αi ≥ 0; α1 > 0;

Xki¼0

αi ¼ 1:(4.16)

where uin ∈PTi

ρ ðfnÞ. Then ffng is ρ-summably almost stable with respect to T.

Proof. Let p∈FρðTÞ and fhng be any given sequence in D and define

εn ¼ ρ

hnþ1 �

Xki¼0

αizin

!; (4.17)

where zin ∈PTi

ρ ðhnÞ. Using (4.16), (4.17), relation (2.4) with f ¼ p, g ¼ hn and the convexity of ρ,we have the following estimate:


function spaces

91

ρðhnþ1 � pÞ ¼ ρ

hnþ1 �

Xki¼0

αizin þ

Xki¼0

αizin � p

!

≤ ρ

hnþ1 �

Xki¼0

αizin

!þ ρ

Xki¼0

αizin � p

!

≤ εn þ ρ

Xki¼0

αizin � p

!

≤ εn þ Hρ

Xki¼0

αiPTi


!

≤ εn þXki¼0

αiHρðPTi


≤ εn þ Xk

i¼0

αiδi

!ρðhn � pÞ

¼ εn þ qρðhn � pÞ;

(4.18)

where q ¼Pk

i¼0αiδi < 1. Hence, by Lemma 2.5 it follows that the fixed point iteration (4.16) is

ρ-summably almost stable with respect to T. The proof of Theorem 4.3 is completed. ,

References

[1] R.P. Agarwal, D. O’Regan, D.R. Sahu, Iterative construction of fixed points of nearlyasymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (1) (2007) 61–79.

[2] T.D. Benavides, M.A. Khamsi, S. Samadi, Asymptotically non-expansive mappings in modularfunction spaces, J. Math. Anal. Appl. 265 (2002) 249–263.

[3] V. Berinde, Iterative Approximation of Fixed Points, Editura Efemeride, Baia Mare, 2002.

[4] V. Berinde, Summable almost stability of fixed point iteration procedures, Carpathian J. Math. 19(2) (2003) 81–88.

[5] V. Berinde, A convergence theorem for some mean value fixed point iteration procedures,Demonstr. Math. 38 (1) (2005) 177–184.

[6] B.A.B. Dehaish, W.M. Kozlowski, Fixed point iteration for asymptotic pointwise nonexpansivemappings in modular function spaces, Fixed Point Theory Appl. 2012 (2012) 118.

[7] A.M. Harder, T.L. Hicks, Stability results for fixed point iteration procedures, Math. Japon. 33 (5)(1988) 693–706.

[8] A.M. Harder, T.L. Hicks, A stable iteration procedure for nonexpansive mappings, Math. Japon.33 (5) (1988) 687–692.

[9] N. Hussain, A. Rafiq, B. Damjanovi�c, R. Lazovi�c, On rate of convergence of various iterativeschemes, Fixed Point Theory Appl. 2011 (45) (2011) 6.

[10] M.A. Khamsi, W.M. Kozlowski, Fixed Point Theory in Modular Function Spaces, SpringerInternational Publishing, Switzerland, 2015.

[11] S.H. Khan, Approximating fixed points of ðλ; ρÞ-firmly nonexpansive mappings in modularfunction spaces, Arab. J. Math. (2018) 7, http://dx.doi.org/10.1007/s40065-018-0204-x.

[12] S.H. Khan, M. Abbas, Approximating fixed points of multivalued ρ-nonexpansive mappings inmodular function spaces, Fixed Point Theory Appl. 2014 (2014) 34.

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92

[13] S.H. Khan, M. Abbas, S. Ali, Fixed point approximation of multivalued ρ-quasi-nonexpansivemappings in modular function spaces, J. Nonlinear Sci. Appl. 10 (2017) 3168–3179.

[14] S.J. Kilmer, W.M. Kozlowski, G. Lewicki, Sigma order continuity and best approximation inLρ-spaces, Comment. Math. Univ. Carolin. 3 (1991) 2241–2250.

[15] M.A. Kutbi, A. Latif, Fixed points of multivalued mappings in modular function spaces, FixedPoint Theory Appl. (2009) 786357, 12 pages.

[16] G.A. Okeke, M. Abbas, A solution of delay differential equations via Picard-Krasnoselskii hybriditerative process, Arab. J. Math. 6 (2017) 21–29.

[17] G.A. Okeke, S.A. Bishop, S.H. Khan, Iterative approximation of fixed point of multivalued ρ-quasi-nonexpansive mappings in modular function spaces with applications, J. Funct. Spaces 2018(2018) Article ID 1785702, 9 pages.

[18] M. €Ozt€urk, M. Abbas, E. Girgin, Fixed points of mappings satisfying contractive condition ofintegral type in modular spaces endowed with a graph, Fixed Point Theory Appl. 2014 (2014) 220.

[19] T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math. 23 (1992) 292–298.

Corresponding authorGodwin Amechi Okeke can be contacted at: [email protected]



function spaces

93


The implicit midpoint rule fornonexpansive mappings in

2-uniformly convexhyperbolic spacesH. Fukhar-ud-din and A.R. Khan

Department of Mathematics and Statistics,King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

AbstractThe purpose of this paper is to introduce the implicit midpoint rule (IMR) of nonexpansive mappings in2- uniformly convex hyperbolic spaces and study its convergence. Strong and△-convergence theorems basedon this algorithm are proved in this new setting. The results obtained hold concurrently in uniformly convexBanach spaces, CATð0Þ spaces and Hilbert spaces as special cases.

Keywords Uniformly convex hyperbolic space, Nonexpansive mapping, Midpoint rule, Fixed point,

Condition(A), Convergence


1. IntroductionThe iterativemethods for approximating fixed points of nonexpansivemappings have receiveda great attention due to the fact that in many practical problems, the controlling operators arenonexpansive (cf. [16]). The iterativemethods ofMann [17] andHalpern [9] are verypopular (seealso [20]). An implicit iterative method was proposed [25] and studied in [7,12]. The IMR is apowerful numerical method for solving ordinary differential equations and differentialalgebraic equations. For related works in this context, we refer the reader to [2,5,20,22].

For the ordinary differential equation

y0 ðtÞ ¼ gðtÞ; y0 ¼ yð0Þ; (1.1)

Implicit midpointrule for

nonexpansivemappings

95

JEL Classification — 47H09, 47H10© H. Fukhar-ud-din and A.R. Khan. Published in the Arab Journal of Mathematical Sciences.


The authors are grateful to King Fahd University of Petroleum & Minerals (KFUPM) for supportingthis research.

The publisher wishes to inform readers that the article “The implicit midpoint rule for nonexpansivemappings in 2-uniformly convex hyperbolic spaces”was originally published by the previous publisher ofthe Arab Journal of Mathematical Sciences and the pagination of this article has been subsequentlychanged.Therehas beennochange to the content of the article. This changewasnecessary for the journal totransition from the previous publisher to the new one. The publisher sincerely apologises for anyinconvenience caused. To access and cite this article, please use Fukhar-ud-din, H., Khan, A.R. (2019), “Theimplicit midpoint rule for nonexpansive mappings in 2-uniformly convex hyperbolic spaces”Arab Journalof Mathematical Sciences, Vol. 26 No. 1/2, pp. 95-105. The original publication date for this paper was22/02/2019.



Received 24 December 2018Revised 14 February 2019

Accepted 17 February 2019


Vol. 26 No. 1/2, 2020pp. 95-105


DOI 10.1016/j.ajmsc.2019.02.002

IMR generates a sequence fyng via the relation1

hðynþ1 � ynÞ ¼ g

�ynþ1 þ yn

2

�

where h > 0 is a step size. It is well known that if g: ℝk→ℝk is Lipschitzian continuous and

sufficiently smooth, then the sequence fyng converges to the exact solution of (1.1) as h→ 0uniformly over t ∈ ½0; a� for any fixed a > 0.

Based on the above fact, Alghamdi et al. [1] presented the following IMR for nonexpansivemappings in the setting of a Hilbert space H:

ynþ1 ¼ ð1� tnÞyn þ tnT�ynþ1 þ yn

2

�(1.2)

where tn ∈ ð0; 1Þ and T : H →H is a nonexpansive mapping and established weakconvergence of (1.2) to the fixed point of T under some control conditions on ftng.

The extension of a linear version of a known result (usually in Banach spaces or Hilbertspaces) to metric spaces is very important. As an IMR for nonexpansive mappings involvesgeneral convex combinations, sowe need some convex structure in ametric space to define anIMR on a nonlinear domain.

Let C be a nonempty subset of a metric space ðM ; dÞ and T : C→C a mapping. SetF ðTÞ ¼ fx ∈ M : Tx ¼ xg. The mapping T is: (i) nonexpansive if dðT x;T yÞ≤ dðx; yÞfor all x; y∈C (ii) quasi-nonexpansive if dðTx; yÞ≤ dðx; yÞ for all x∈C and y∈FðTÞ (iii)semi-compact if for any bounded sequence fxng in C satisfying dðxn;T xnÞ→ 0, thereexists a subsequence fxnigof fxngsuch that xni → x∈C (iv) completely continuous if everybounded sequence fxng in C implies that fT xng has a convergent subsequence. Asequence fxng is Fej�er monotone with respect to a subset C ofM if dðxnþ1; xÞ≤ dðxn; xÞ forall x∈C:

For a bounded sequence fxng in a metric space M, set

rðx; fxngÞ ¼ lim supn→∞

dðx; xnÞfor all x∈M.

The asymptotic radius of fxngwith respect to C ⊆ M is defined as

rðfxngÞ ¼ infx∈C

r ðx; fxngÞ:

A point y∈C is called the asymptotic centre of fxngwith respect to C ⊆ M if

rðy; fxngÞ≤ rðx; fxngÞ for all x∈C:

The set of all asymptotic centres of fxng is denoted by AðfxngÞ.A sequence fxng in M, is △-convergent to x∈M ð△− limn xn ¼ xÞ if x is the unique

asymptotic centre of fung for every subsequence fung of fxng. It has been observed that△-convergence in metric spaces constitutes an analogue of weak convergence in Hilbertspaces and both coincide in Hilbert spaces.

Let ðM ; dÞbe a metric space. Suppose that there exists a family F of metric segments suchthat any two points x; y inM are endpoints of a unique metric segment ½x; y�∈ F (½x; y� is anisometric image of the real line interval ½0; dðx; yÞ�). We denote by z the unique pointαx⊕ ð1−αÞy of ½x; y�which satisfies

dðx; zÞ ¼ ð1� αÞdðx; yÞ and dðz; yÞ ¼ αdðx; yÞ for α∈ I ¼ ½0; 1�:

AJMS26,1/2

96

Such metric spaces are usually called convex metric spaces [18]. A convex metric spaceM is hyperbolic if

dðαx⊕ ð1� αÞy; αz⊕ ð1� αÞwÞ≤ αdðx; zÞ þ ð1� αÞ dðy;wÞ (1.3)

for all x; y; z;w∈M and α∈ I.For z ¼ w, the hyperbolic inequality reduces to convex structure of Takahashi [23]

dðαx⊕ ð1� αÞy; zÞ≤αdðx; zÞ þ ð1� αÞdðy; zÞ:

A nonempty subset C of a hyperbolic space M is convex if αx⊕ ð1− αÞy∈C for all x; y∈Cand α∈ I . A few examples of nonlinear hyperbolic spaces are Hadamard manifolds [4], theHilbert open unit ball equipped with the hyperbolic metric [8] and the CATð0Þ spaces [14,15]while normed spaces and their subsets are linear hyperbolic spaces. Throughout this paper,we denote 1

2 x⊕12 y by

x⊕ y2 .

A hyperbolic space M is uniformly convex if

δðr; εÞ ¼ inf

�1� 1

rd�a;x⊕ y

2

�: dða; xÞ≤ r; dða; yÞ≤ r; dðx; yÞ≥ rε

�> 0;

for any a∈M, r > 0 and ε > 0.Xu [24], extensively used the concept of p-uniform convexity; its nonlinear version in

hyperbolic spaces for p ¼ 2 has been introduced by Khamsi and Khan [13] as under:For a fixed a∈M ; r > 0; ε > 0, define

ψðr; εÞ ¼ inf

�1

2dða; xÞ2 þ 1

2dða; yÞ2 � d

�a;x⊕ y

2

�2�

where the infimum is taken over all x; y∈M such that dða; xÞ≤ r; dða; yÞ≤ r and dðx; yÞ≥ rε.We say that M is 2-uniformly convex if

cM ¼ infnψðr; εÞ

r2ε2: r > 0; ε > 0

o> 0:

It has been shown in [13] that any CATð0Þ space is 2-uniformly convex hyperbolic spacewith cM ¼ 1

4 .From now onwards we assume that M is a uniformly convex hyperbolic space with the

property that for every s≥ 0; ε > 0, there exists ηðs; εÞ > 0 depending on s and ε such thatδðr; εÞ > ηðs; εÞ > 0 for any r > s.

Using the concept ofmetric segment ½x; y�, we translate (1.2) for nonexpansivemappings ina hyperbolic space as follows:

x0 ¼ x∈C;

xnþ1 ¼ αnT�xn ⊕ xnþ1

2

�⊕ ð1� αnÞxn;

(1.4)

where fαng is the sequence in ð0; 1Þ satisfying (C1): lim infn→∞αn > 0 and (C2): α2nþ1 ≤ λα2

n

for some λ > 0:The following known results are needed in the sequel.

Lemma 1.1 ([3]). Let C be a nonempty closed subset of a complete metric space ðM ; dÞ andfxng be a Fej�er monotone with respect to C. Then fxng strongly converges to x∈C if and onlyif limn→∞dðxn;CÞ ¼ 0.



97

Lemma 1.2 ([6]). Let C be a nonempty closed and convex subset of a complete uniformlyconvex hyperbolic space M. Then every bounded sequence fyng in M has a unique asymptoticcentre with respect to C that lies in C.Lemma 1.3 ([10]). Suppose that M is a 2-uniformly convex hyperbolic space. Then for anyθ∈ ð0; 1Þ, we have that

dðu; θx⊕ ð1� θÞyÞ2 ≤ θdðu; xÞ2 þ ð1� θÞdðu; yÞ2 � 4cM min�θ2; ð1� θÞ2�dðx; yÞ2;

for all u; x; y∈M and cM is the number as given above.Our purpose in this paper is to approximate fixed point of nonexpansive mappings using

iterative method (1.4) in a 2-uniformly convex hyperbolic spaces. This work provides aunified approach to convergence results in Hilbert spaces, uniformly convex Banach spacesand CATð0Þ spaces.

2. Convergence in 2-uniformly convex hyperbolic spacesLemma 2.1. Let C be a nonempty convex subset of a complete hyperbolic space M andT : C→C a nonexpansive mapping. Then the sequence fxng in (1.4) is well defined.

Proof. Define S : C→C by

Sx ¼ α0T�x0 ⊕ x

2

�⊕ ð1� α0Þx0:

With the help of (1.3), we have

dðSx; SyÞ ¼ d�α0T

�x0 ⊕ x

2

�⊕ ð1� α0Þx0; α0T

�x0 ⊕ y

2

�⊕ ð1� α0Þx0

�

≤ α0d�T�x0 ⊕ x

2

�;T

�x0 ⊕ y

2

��

≤ α0d�x0 ⊕ x

2;x0 ⊕ y

2

�

≤α0

2dðx; yÞ:

This gives that S is a contraction with contraction constant α02 ∈ ð0; 1Þ. Therefore by Banach

contraction principle, there is a unique element x1 ∈C such that x1 ¼ Sx1 ¼ α0T�x0 ⊕ x1

2

�⊕

ð1− α0Þx0. Hence x1 is achieved. Similarly, we can find x2 and so on. So in general,

xnþ1 ¼ αnT�xn ⊕ xnþ1

2

�⊕ ð1� αnÞxn: ,

Lemma 2.2. Let C be a nonempty convex subset of a complete 2-uniformly convex hyperbolicspace M and T : C→C a nonexpansive mapping such that FðTÞ≠f. Then for the sequencefxng in (1.4), we have the following: (i) limn→∞dðxn; pÞ exists for all p∈FðTÞ

(ii)P∞

n¼1αnd ðxn; xnþ1Þ < ∞

(iii)P∞

n¼1α2nð1− αnÞ2d

�xn;Tðxn ⊕ xnþ1

2

��2

< ∞.

Proof. Let p∈FðTÞ. Applying Lemma 1.3 to (1.4), we have that

AJMS26,1/2

98

dðxnþ1; pÞ2 ¼ d�αnT

�xn ⊕ xnþ1

2

�⊕ ð1� αnÞxn; p

�2

≤ αnd�T�xn ⊕ xnþ1

2

�; p�2

þ ð1� αnÞdðxn; pÞ2

� 4cM min�α2n; ð1� αnÞ2

�d�xn;T

�xn ⊕ xnþ1

2

��2

≤ αnd�xn ⊕ xnþ1

2; p�2

þ ð1� αnÞdðxn; pÞ2

� 4cMα2nð1� αnÞ2d

�xn;T

�xn ⊕ xnþ1

2

��2

≤ αnd

�1

2dðxn; pÞ2 þ 1

2dðxnþ1; pÞ2 � CM

4dðxn; xnþ1Þ2

þ ð1� αnÞdðxn; pÞ2 � 4cMα2nð1� αnÞ2d

�xn;T

�xn ⊕ xnþ1

2

��2

:

That is, �1� αn

2

�dðxnþ1; pÞ≤

�1� αn

2

�dðxn; pÞ � αnCM

4dðxn; xnþ1Þ2

� 4cMα2nð1� αnÞ2d

�xn;T

�xn ⊕ xnþ1

2

��2

which further implies that

dðxnþ1; pÞ≤ dðxn; pÞ � αnCM

2ð2� αnÞ dðxn; xnþ1Þ:

� 8cMα2nð1� αnÞ2

2ð2� αnÞ d�xn;T

�xn ⊕ xnþ1

2

��2

:

The above inequality provides the following three inequalities:

dðxnþ1; pÞ≤ dðxn; pÞ; (2.1)

αnCM

2ð2� αnÞ dðxn; xnþ1Þ≤ dðxn; pÞ � dðxnþ1; pÞ (2.2)

and

8cMα2nð1� αnÞ2

2ð2� αnÞ d�xn;T

�xn ⊕ xnþ1

2

��2

≤ dðxn; pÞ � dðxnþ1; pÞ: (2.3)

From (2.1), it follows that limn→∞dðxn; pÞ exists, that is, (i) holds.Since αn ∈ ð0; 1Þ, therefore αn ≤ αn

2ð2− αnÞ. Hence (2.2) becomes

αndðxn; xnþ1Þ≤ 1

CM

½dðxn; pÞ � dðxnþ1; pÞ�: (2.4)

Let m≥ 1 be any positive integer. Then from (2.4), we have thatXmn¼1

αndðxn; xnþ1Þ ≤1

CM

½dðx1; pÞ � dðxmþ1; pÞ�≤ dðx1; pÞCM

:



99

Let m→∞. Then X∞n¼1

αndðxn; xnþ1Þ ≤dðx1; pÞCM

< ∞:

That is,

X∞n¼1

αndðxn; xnþ1Þ < ∞;

proving (ii). Similarly, from (2.3), we have

X∞n¼1

α2nð1� αnÞ2d

�xn;T

�xn ⊕ xnþ1

2

��2

< ∞: ,

Lemma 2.3. Let C be a nonempty convex subset of a complete 2-uniformly convex hyperbolicspace M and T : C→C a nonexpansive mapping such that FðTÞ≠f. Then for the sequencefxng in (1.4), we have that limn→∞dðxn; xnþ1Þ ¼ 0.

Proof. Consider

dðxnþ1; xnþ2Þ ¼ d�αnþ1T

�xnþ1 ⊕ xnþ2

2

�⊕ ð1� αnþ1Þxnþ1; xnþ1

�

≤ αnþ1d�xnþ1;T

�xnþ1 ⊕ xnþ2

2

��


�xn ⊕ xnþ1

2

��

þ αnþ1d�T

�xn ⊕ xnþ1

2

�;T

�xnþ1 ⊕ xnþ2

2

��


�xn ⊕ xnþ1

2

��

þ αnþ1d�xn ⊕ xnþ1

2;xnþ1 ⊕ xnþ2

2

�

≤ αnþ1ð1� αnÞd�xn;T

�xn ⊕ xnþ1

2

��

þ αnþ1d�xn ⊕ xnþ1

2;xnþ1 ⊕ xnþ2

2

�

≤ αnþ1ð1� αnÞ d�xn;T

�xn ⊕ xnþ1

2

��

þ αnþ1

2ðdðxn; xnþ1Þ þ dðxnþ1; xnþ2ÞÞ:

Therefore

�1� αnþ1

2

�dðxnþ2; xnþ1Þ≤ αnþ1ð1� αnÞd

�xn;T

�xn ⊕ xnþ1

2

��þ αnþ1

2dðxn; xnþ1Þ

which further implies that

AJMS26,1/2

100

dðxnþ1; xnþ2Þ≤ 2αnþ1ð1� αnÞ2� αnþ1

d�xn;T

�xn ⊕ xnþ1

2

��

þ αnþ1

2� αnþ1

dðxn; xnþ1Þ

≤ 2αnþ1ð1� αnÞd�xn;T

�xn ⊕ xnþ1

2

��

þ αnþ1d ðxn; xnþ1Þ:For some A > 0;B > 0 and using the assumption α2nþ1 ≤ λα2n, we further derive that

dðxnþ1; xnþ2Þ2 ≤ 4Aα2nþ1ð1� αnÞ2d

�xn;T

�xn ⊕ xnþ1

2

��2

þ Bα2nþ1dðxn; xnþ1Þ2

≤ 4Aλα2nþ1ð1� αnÞ2d

�xn;T

�xn ⊕ xnþ1

2

��2

þ Bα2nþ1dðxn; xnþ1Þ2

≤ 4Aλα2nð1� αnÞ2d

�xn;T

�xn ⊕ xnþ1

2

��2

þ Bλα2ndðxn; xnþ1Þ2

≤ 4Aλα2nð1� αnÞ2d

�xn;T

�xn ⊕ xnþ1

2

��2

þ Bλαndðxn; xnþ1Þ2:Hence by Lemma 2.2(ii)–(iii), we have that

X∞n¼1

dðxnþ1; xnþ2Þ2 < ∞:

This in turn implies that

limn→∞

dðxn; xnþ1Þ ¼ 0: , (2.5)

Lemma 2.4. Let C be a nonempty closed and convex subset of a complete 2-uniformly convexhyperbolic space M and T : C→C a nonexpansive mapping such that FðTÞ≠f. Then for thesequence fxng in (1.4), we have that limn→∞dðxn;TxnÞ ¼ 0.

Proof. The condition lim infn→∞αn > 0 implies that 0 < 1αn

≤ 1α for sufficiently large n.

The inequality

d�xn;T

�xn ⊕ xnþ1

2

��≤ dðxn; xnþ1Þ þ d

�xnþ1;T

�xn ⊕ xnþ1

2

��

≤ dðxn; xnþ1Þ þ ð1� αnÞd�xn;T

�xn ⊕ xnþ1

2

��



101

implies that

d�xn;T

�xn ⊕ xnþ1

2

��≤

1

αn

dðxn; xnþ1Þ≤ 1

αdðxn; xnþ1Þ:

By taking lim supn→∞ on both sides in the above inequality and then appealing to Lemma 2.3,we get that

limn→∞

d�xn;T

�xn ⊕ xnþ1

2

��¼ 0: (2.6)

Finally, the inequality

dðxn;T xnÞ≤ d�xn;T

�xn ⊕ xnþ1

2

��þ d

�T�xn ⊕ xnþ1

2

�;T xn

�

≤ d�xn;T

�xn ⊕ xnþ1

2

��þ d

�xn ⊕ xnþ1

2; xn

�

≤ d�xn;T

�xn ⊕ xnþ1

2

��þ 1

2dðxnþ1; xnÞ

together with (2.5) and (2.6) provides that

limn→∞

dðxn;TxnÞ ¼ 0: , (2.7)

The following concept is needed to establish strong convergence of (1.4).Let f be a nondecreasing function on ½0;∞Þwith f ð0Þ ¼ 0 and f ðtÞ > 0 for all t ∈ ð0;∞Þ.

Then the mapping T : C→C with FðTÞ≠f; satisfies condition (A) [21] if

dðx;T xÞ≥ f ðdðx;FðTÞÞÞ for x∈C;

where dðx;FðTÞÞ ¼ inffdðx; yÞ : y∈FðTÞg.Using condition(A) and Lemma 2.4, we obtain the following strong convergence result.Theorem 2.5. Let C be a nonempty closed and convex subset of a complete 2-uniformly

convex hyperbolic space M and T : C→C a nonexpansive mapping such that FðTÞ≠f. If themapping T : C→C satisfies condition(A), then the sequence fxng in (1.4), strongly converges toa fixed point of T.

Proof. By Lemma 2.4, limn→∞dðxn;T xnÞ ¼ 0. Now condition(A) implies thatlimn→∞dðxn;FðTÞÞ ¼ 0. Finally, by Lemma 1.1, fxng strongly converges to a fixed pointof T: ,

Here are our other strong convergence results.Theorem 2.6. Let C be a nonempty closed and convex subset of a complete 2-uniformly

convex hyperbolic space M and T : C→C a nonexpansive mapping such that FðTÞ≠f. If T issemi-compact, then the sequence fxng in (1.4) strongly converges to a fixed point of T.

Proof. By Lemma 2.4, we have that limn→∞dðxn;T xnÞ ¼ 0. Since limn→∞dðxn; pÞ existsfor each p∈FðTÞ, fxng is bounded. As limn→∞dðxn;T xnÞ ¼ 0 and T is semi-compact, sothere is a subsequence fxnig of fxng such that xni → q∈C and hence Txni →Tq. Therefore,limi→∞dðxni;TxniÞ ¼ 0 implies that dðTq; qÞ ¼ 0. That is, q∈FðTÞ. Since limn→∞dðxn; pÞexists and xni → q, xn → q: ,

Theorem 2.7. Let C be a nonempty closed and convex subset of a complete 2-uniformlyconvex hyperbolic space M and T : C→C a nonexpansive mapping such that FðTÞ≠f. If T iscompletely continuous, then the sequence fxng in (1.4), strongly converges to a fixed point of T.

Proof. Since fxng is bounded and T is completely continuous, fTxng has a convergentsubsequence say fTxnig. Therefore by (2.7), fxnigconverges. Let limi→∞xni ¼ υ. By continuity

AJMS26,1/2

102

of T and (2.7), we have that Tυ ¼ υ. By Lemma 2.2, limn→∞dðxn; υÞ exists and so fxngstrongly converges to υ: ,

We now present our △-convergence result.Theorem 2.8. Let C be a nonempty closed and convex subset of a complete 2-uniformly

convex hyperbolic space M and T : C→C a nonexpansive mapping such that FðTÞ≠f. Thenthe sequence fxng in (1.4), △-converges to a fixed point of T.

Proof. It follows from Lemma 2.1 that fxng is bounded in C. By Lemma 1.2, fxng has aunique asymptotic centre, that is,ACðfxngÞ ¼ fyg. Let fwngbe any subsequence of fxngsuchthat ACðfwngÞ ¼ fwg. We claim that w∈FðTÞ. By Lemma 2.4, we have that

limn→∞

dðwn;TwnÞ ¼ 0:

The nonexpansive mapping T satisfies the following inequality:

dðwn;TwÞ≤ dðwn;TwnÞ þ dðwn;wÞwhich further implies that

lim supn→∞

dðwn;TwÞ≤ lim supn→∞

dðwn;TwnÞ þ lim supn→∞

dðwn;wÞ ¼ lim supn→∞

dðwn;wÞ:

By the uniqueness of asymptotic centre, we have Tw ¼ w. Therefore FðTÞ≠f. If y≠w, thenby the uniqueness of asymptotic centre and the fact that limn→∞ dðxn; xÞ exists for eachx∈FðTÞ, we have that

lim supn→∞

dðwn;wÞ < lim supn→∞

dðwn; yÞ≤ lim sup

n→∞

dðxn; yÞ< lim sup

n→∞

dðxn;wÞ¼ lim sup

n→∞

dðwn;wÞ:

This is a contradiction and therefore y ¼ w. This proves that fxng, △-convergesto x∈FðTÞ: ,

Remark 2.9. (1) All the results of this paper instantly hold in Hilbert spaces, uniformlyconvex Banach spaces satisfying Opial property and CAT(0) spaces; (2) The results ofAlghamdi et al. [1] are corollaries of our corresponding results; (3) The interested reader isreferred to [11] for another notion of p-uniformly convex metric spaces; (4) The two controlconditions: (C1)and (C2) in our algorithm (1.4) are satisfied by the sequence αn ¼ 1− 1

nþ1 .

3. ApplicationWeknow thatL2½0; 1� is a Hilbert space and hence it is a 2-uniformly convex hyperbolic space.Suppose that h: ½0; 1�→ ½0; 1� and F : ½0; 1�3 ½0; 1�3ℝ→ℝ are continuous functions and Fsatisfies the Lipschitz continuity condition, i.e.,

jFðt; λ; xÞ � Fðt; s; yÞj≤ jx� yj for t; s∈ ½0; 1� and x; y∈ℝ:

Consider a Fredholm integral equation of the form

xðtÞ ¼ hðtÞ þZ 1

0

Fðt; s; xðsÞÞds for t ∈ ½0; 1�: (3.1)

It has been shown in [19] that the solution of Eq. (3.1) exists in L2½0; 1�. To find anapproximate solution of this equation, we define S : L2½0; 1�→L2½0; 1� by



103

SxðtÞ ¼ hðtÞ þZ 1

0

Fðt; s; xðsÞÞds for t ∈ ½0; 1�:

For x; y∈L2 ½0; 1�, we calculate

jjSx� Syjj2 ¼Z 1

0

jSxðtÞ � SyðtÞj2dt

¼Z 1

0

Z 1

0

ðFðt; s; xðsÞÞ � Fðt; s; yðsÞÞÞds2

dt

≤

Z 1

0

Z 1

0

jxðsÞ � yðsÞjds2

dt

≤

Z 1

0

jxðsÞ � yðsÞj2ds ¼ jjx� yjj2:

So S is nonexpansive. For any function x0 ∈L2½0; 1�, we define a sequence of functions fxng inL2½0; 1� by

xnþ1 ¼ αnS�xn þ xnþ1

2

�þ ð1� αnÞxn

where αn ∈ ð0; 1Þ such that lim infn→∞αn > 0 and α2nþ1 ≤ λα2

n for some λ > 0. Now byTheorem 2.8, fxngweakly converges to the fixed point of S which is a solution of Eq. (3.1).

References

[1] M.A. Alghamdi, M.A. Alghamdi, N. Shahzad, H.K. Xu, The implicit midpoint rule fornonexpansive mappings, Fixed Point Theory Appl. 2014 (2014) 96.

[2] G. Bader, P. Deuflhard, A semi-implicit mid-point rule for stiff systems of ordinary differentialequations, Numer. Math. 41 (1983) 373–398.

[3] H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone Operator Theory in HilbertSpaces, Springer-Verlag, New York, 2011.

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Corresponding authorH. Fukhar-ud-din can be contacted at: [email protected]




105


On abstract Hilfer fractionalintegrodifferential equations with

boundary conditionsSabri T.M. Thabet

Department of Mathematics, University of Aden, Aden, Yemen

Bashir AhmadNonlinear Analysis and Applied Mathematics (NAAM)-Research Group,

Department of Mathematics, Faculty of Science, King Abdulaziz University,Jeddah, Saudi Arabia, and

Ravi P. AgarwalDepartment of Mathematics, Texas A&M University, Kingsville, Texas, USA

AbstractIn this paper, we study a Cauchy-type problem for Hilfer fractional integrodifferential equationswith boundaryconditions. The existence of solutions for the given problem is proved by applyingmeasure of noncompactnesstechnique in an abstract weighted space. Moreover, we use generalized Gronwall inequality with singularity toestablish continuous dependence and uniqueness of e-approximate solutions.

Keywords Hilfer fractional integrodifferential equations, Boundary conditions, M€onch fixed point theorem,

Measure of noncompactness, Existence, Continuous dependence


1. IntroductionFractional calculus has emerged as a powerful tool to study complex phenomena in numerousscientific and engineering disciplines such as viscoelasticity, fluid mechanics, physics andheat conduction in materials with memory. For examples and applications, see [2,14,17–21]and references cited therein. Many authors focused on Riemann–Liouville and Caputo typederivatives in investigating fractional differential equations. In [7], Hilfer introduced a newconcept of generalized Riemann–Liouville derivative (Hilfer derivative) of order α and type β.This definition facilitated dynamic modeling of non-equilibrium processes based on

Hilferfractional

integrodifferentialequations

107

JEL Classification — 26A33, 34A08, 34B15, 34A12, 47H08© Sabri T.M. Thabet, Bashir Ahmad and Ravi P. Agarwal. Published in the Arab Journal of

Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under theCreative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate andcreate derivative works of this article (for both commercial and non-commercial purposes), subject to fullattribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

The publisher wishes to inform readers that the article “On abstract Hilfer fractionalintegrodifferential equations with boundary conditions” was originally published by the previouspublisher of the Arab Journal of Mathematical Sciences and the pagination of this article has beensubsequently changed. There has been no change to the content of the article. This change wasnecessary for the journal to transition from the previous publisher to the new one. The publishersincerely apologises for any inconvenience caused. To access and cite this article, please useThabet, S.T.M., Ahmad, B. and Agarwal, R.P. (2019), “On abstract Hilfer fractional integrodifferentialequations with boundary conditions”, Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 107-125. The original publication date for this paper was 14/03/2019.



Received 23 December 2018Accepted 4 March 2019


Vol. 26 No. 1/2, 2020pp. 107-125


DOI 10.1016/j.ajmsc.2019.03.001

interpolation with respect to parameter of the Riemann–Liouville and Caputo type operators;for instance, see [1,4,6,8,10,11].

Furati et al. [6] established the existence and uniqueness of solutions for the problem:(Dα; β

aþ yðtÞ ¼ f ðt; yðtÞÞ; t ∈ J ¼ ða; b�; 0 < α < 1; 0≤ β≤ 1;

I 1−γaþ yðaþÞ ¼ w; α ≤ γ ¼ αþ β � αβ;

by applying Banach fixed point theorem in weighted space Cγ1−γ½ J ;ℝ�. Abbas et al. [1]

discussed the above problem by using Kuratowski measure of noncompactness.Motivated by the works [1,6], we will study a more general problem of Hilfer fractional

integrodifferential equations with boundary conditions given by(Dα; β

aþ yðtÞ ¼ f ðt; yðtÞ; ðSyÞðtÞÞ; t ∈ J ¼ ða; b�; 0 < α < 1; 0≤ β≤ 1;

I 1−γaþ ½uyðaþÞ þ vyðb−Þ� ¼ w; α≤ γ ¼ αþ β � αβ;(1.1)

where Dα; βaþ is the left-sided Hilfer fractional derivative of order α and type β,

f : J 3X 3X →X, X is an abstract Banach space, u; v;w∈ℝ; uþ v≠ 0, and S is a linear

integral operator defined by ðSyÞðtÞ ¼ R t

akðt; sÞyðsÞds with ζ ¼ maxf R t

akðt; sÞds : ðt; sÞ∈

J 3 Jg, k∈ ðJ 3 J ;ℝÞ.This article is constructed as follows: In Section 2, we recall some preliminaries. Section 3

contains the existence result obtained by usingmeasure of noncompactness andM€onch fixedpoint theorem. We discuss the e-approximate solution of Hilfer fractional integrodifferentialequations in Section 4.

2. PreliminariesIn this section, we present some necessary definitions, notations and preliminaries, whichwillbe used throughout this work.

For −∞ < a < b < ∞, let C½J ;X � denote the space of all continuous functions on J into Xendowed with supremum norm kxkC :¼ supfkxðtÞk : t ∈ Jg. Define by C1−γ ½J ;X � ¼ ff ðxÞ :ða; b�→X jðx− aÞ1−γf ðxÞ∈C½J ;X �g the weighted space of the abstract continuous functions.Obviously, C1−γ ½J ;X � is a Banach space equipped with the norm kfkC1− γ

¼ ��ðx− aÞ1− γf ðxÞ��

C,

and Cn1−γ ½J ;X � ¼ ff ∈Cn−1½J ;X � : f ðnÞ ∈C1−γ ½J ;X �g is the Banach space endowed with the

norm

kfkCn1−γ

¼Xn−1i¼0

kf ðkÞkC þ kf ðnÞkC1�γ; n∈ℕ;

where, C01−γ :¼ C1−γ

Definition 2.1 (See [13]). The left-sided Riemann–Liouville fractional integral of orderα > 0 of function f : ½a;∞Þ→ℝ is defined by

�Iαaþ f

�ðtÞ ¼ 1

ΓðαÞZ t

a

ðt � sÞα−1f ðsÞds; t > a;

where a∈ℝ and Γ is the Gamma function.

Definition 2.2 (See [13]). The left-sided Riemann–Liouville fractional derivative of orderα∈ ðn− 1; n� of function f : ½a;∞Þ→ℝ, is defined by

AJMS26,1/2

108

�Dα

aþ f�ðtÞ ¼ 1

Γðn� αÞ�d

dt

�n Z t

a

ðt � sÞn−α−1f ðsÞds; t > a;

where n ¼ ½α� þ 1; ½α� denotes the integer part of α.Remark 2.1. If f is an abstract function with values in X, then the integrals appearing inDefinitions 2.1 and 2.2 are taken in Bochner’s sense.

Definition 2.3 (See [7]). The left-sided Hilfer fractional derivative of order 0 < α < 1 andtype 0≤ β≤ 1, of function f ðtÞ is defined by

�Dα; β

aþ f�ðtÞ ¼

�Iβð1−αÞaþ D

�Ið1−βÞð1−αÞaþ

��ðtÞ;

where D :¼ ddt:

Remark 2.2 (See [7]). From Definition 2.3, we observe that:

(i) the operator Dα; βaþ can be written as

Dα; βaþ ¼ I

βð1−αÞaþ DI

ð1−γÞaþ ¼ I

βð1−αÞaþ Dγ; γ ¼ αþ β � αβ;

(ii) The Hilfer fractional derivative can be regarded as an interpolator between theRiemann–Liouville derivative (β ¼ 0) and Caputo derivative (β ¼ 1) as

Dα; βaþ ¼

(DI

ð1−αÞaþ ¼ Dα

aþ ; if β ¼ 0;

Ið1−αÞaþ D ¼ CDα

aþ ; if β ¼ 1:

In the forthcoming analysis, we need the spaces:

Cα; β1−γ ½J ;X � ¼

�f ∈C1−γ½J ;X �;Dα; β

aþ f ∈C1−γ½J ;X ��;

and

Cγ1−γ½J ;X � ¼

�f ∈C1−γ½J ;X �;Dγ

aþ f ∈C1−γ½J ;X ��:

Since Dα; βaþ f ¼ I

βð1−αÞaþ Dγ f , it is obvious that Cγ

1−γ ½J ;X �⊂Cα; β1−γ ½J ;X �.

Now, we state some known results related to our work.

Lemma 2.1 (See [5]). Let β > 0 and α > 0. Then

Iαaþðt � aÞβ−1ðxÞ ¼ ΓðβÞ

Γðβ þ αÞðx� aÞβþα−1

and Dα

aþðt � aÞα−1ðxÞ ¼ 0; 0 < α < 1:

Lemma 2.2 (See [5]). If α > 0 and β > 0, and f ∈ L1ðJÞ for t ∈ ½a; b�, then the followingproperties hold: �

Iαaþ Iβaþ f�ðtÞ ¼ �I αþβ

aþ f�ðtÞ and

�Dα

aþ Iβaþ f�ðtÞ ¼ f ðtÞ:

In particular, if f ∈Cγ ½J ;X � or f ∈C½J ;X �, then the above properties hold for each t ∈ ða; b� ort ∈ ½a; b� respectively.

Hilferfractional


109

Lemma 2.3 (See [5]). If 0 < α < 1, 0≤ γ < 1 and that f ∈Cγ ½J ;X �, I 1−αaþ f ∈C1γ ½J ;X �, then

I αaþDαaþ f ðtÞ ¼ f ðtÞ �

�I 1−αaþ f

�ðaÞΓðαÞ ðt � aÞα−1; ∀ t ∈ J :

Lemma 2.4 (See [6]). If 0≤ γ < 1 and f ∈Cγ ½J ;X �, then�I αaþ f

�ðaÞ ¼ limt→aþ

I αaþ f ðtÞ ¼ 0; 0 ≤ γ < α:

Lemma 2.5 (See [6]). Let α > 0, β > 0 and γ ¼ αþ β− αβ. If f ∈Cγ1−γ ½J ;X �, then

I γaþDγaþ f ¼ I αaþ D

α; βaþ f ;Dγ

aþ Iαaþ f ¼ D

βð1−αÞaþ f :

Lemma 2.6 (See [6]). Let f ∈L1ðJÞ and Dβð1−αÞaþ f ∈L1ðJÞ exists, then

Dα; βaþ I αaþ f ¼ I

βð1−αÞaþ D

βð1−αÞaþ f :

Lemma 2.7 (Theorem 23, [6]). Let f : J 3ℝ→ℝ be a function such that f ∈C1−γ ½J ;ℝ� forany y∈C1−γ ½J ;ℝ�. Then y∈C

γ1−γ ½J ;ℝ� is a solution of the initial value problem:(

Dα; βaþ yðtÞ ¼ f ðt; yðtÞÞ; t ∈ J ¼ ða; b�; 0 < α < 1; 0≤ β≤ 1;

I 1−γaþ yðaþÞ ¼ ya; α ≤ γ ¼ αþ β � αβ;

if and only if y satisfies the following Volterra integral equation:

yðtÞ ¼ ya

ΓðγÞðt � aÞγ−1 þ 1

ΓðαÞZ t

a

ðt � sÞα−1f ðs; yðsÞÞds:

Next we obtain the integral solution of the problem (1.1) by using Lemma 2.7.

Lemma 2.8. Let f : J 3X 3X →X be a function such that f ∈C1−γ ½J ;X � for anyy∈C1−γ ½J ;X �. Then y∈C

γ1−γ ½J ;X � is a solution of the problem (1.1) if and only if y satisfies

the following integral equation

yðtÞ ¼ w

uþ v

ðt � aÞγ−1ΓðγÞ � v

uþ v

ðt � aÞγ−1ΓðγÞ

1

Γð1� γ þ αÞ

3

Z b

a

ðb� sÞα−γf ðs; yðsÞ; ðSyÞðsÞÞds

þ 1

ΓðαÞZ t

a

ðt � sÞα−1f ðs; yðsÞ; ðSyÞðsÞÞds:

(2.1)

Proof. In view of Lemma 2.7, the solution of (1.1) can be written as

yðtÞ ¼ I 1−γaþ yðaþÞΓðγÞ ðt � aÞγ−1 þ 1

ΓðαÞZ t

a

ðt � sÞα−1f ðs; yðsÞ; ðSyÞðsÞÞds: (2.2)

Applying I1−γaþ on both sides of (2.2) and taking the limit t→ b−, we obtain

I 1−γaþ yðb−Þ ¼ I 1−γaþ yðaþÞ þ 1

Γð1� γ þ αÞZ b

a

ðb� sÞα−γf ðs; yðsÞ; ðSyÞðsÞÞds: (2.3)

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110

In a similar manner, we find that

I 1−γaþ yðaþÞ ¼ 1

1þ v

u

�w

u� v

u

1


a

ðb� sÞα−γf ðs; yðsÞ; ðSyÞðsÞÞds�

¼ 1

uþ v

�w� v

1


a

ðb� sÞα−γf ðs; yðsÞ; ðSyÞðsÞÞds�:

(2.4)

Submitting (2.4) into (2.2), we obtain

yðtÞ ¼ ðt � aÞγ−1ΓðγÞ

1

uþ v

�w� v

1


a

ðb� sÞα−γf ðs; yðsÞ; ðSyÞðsÞÞds�

þ 1

ΓðαÞZ t

a

ðt � sÞα−1f ðs; yðsÞ; ðSyÞðsÞÞds;

¼ w

uþ v


uþ v


1

Γð1� γ þ αÞ

3

Z b

a


þ 1

ΓðαÞZ t

a


Conversely, applying I1−γaþ on both sides of (2.1) and using Lemmas 2.1 and 2.2, we get

I 1−γaþ yðtÞ ¼ w

uþ v� v

uþ v

1


a


þ I1−βð1−αÞaþ f ðt; yðtÞ; ðSyÞðtÞÞ: (2.5)

Next, taking the limit t→ aþ of (2.5) and using Lemma 2.4, with 1− γ < 1− βð1− αÞ, weobtain

I 1−γaþ yðaþÞ ¼ w

uþ v� v

uþ v

1


a

ðb� sÞα−γf ðs; yðsÞ; ðSyÞðsÞÞds: (2.6)

Now, taking the limit t→ b− of (2.5), we get

I 1−γaþ yðb−Þ ¼ w

uþ v� v

uþ v

1


a


þ I1−βð1−αÞaþ f ðb; yðbÞ; ðSyÞðbÞÞ: (2.7)

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111

From (2.6) and (2.7), we find that

uI 1−γaþ yðaþÞ þ vI 1−γaþ yðb−Þ

¼ uw

uþ v� uv

uþ v

1


a


þ vw

uþ v� v2

uþ v

1


a

ðb� sÞα−γf ðs; yðsÞ; ðSyÞðsÞÞds:

þ vI1−βð1−αÞaþ f ðb; yðbÞ; ðSyÞðbÞÞ

¼ wðuþ vÞuþ v

� vðuþ vÞuþ v

I 1−γþαaþ f ðb; yðbÞ; ðSyÞðbÞÞ

þ vI1−βð1−αÞaþ f ðb; yðbÞ; ðSyÞðbÞÞ

¼ w;

which shows that the boundary condition I 1−γaþ ½uyðaþÞ þ vyðb−Þ� ¼ w is satisfied.Next, applying D

γaþ on both sides of (2.1) and using Lemmas 2.1 and 2.5, we have

DγaþyðtÞ ¼ D

βð1−αÞaþ f ðt; yðtÞ; ðSyÞðtÞÞ: (2.8)

Since y∈Cγ1−γ ½J ;X � and by definition of Cγ

1−γ ½J ;X �, we have Dγaþy∈C1−γ ½J ;X �, therefore,

Dβð1−αÞaþ f ¼ DI

1−βð1−αÞaþ f ∈C1−γ ½J ;X �. For f ∈C1−γ ½J ;X �, it is clear that I 1−βð1−αÞaþ f ∈C1−γ ½J ;X �.

Hence f and I1−βð1−αÞaþ f satisfy the hypothesis of Lemma 2.3.

Now, applying Iβð1−αÞaþ on both sides of (2.8), and using Lemma 2.3, we get

Dα; βaþ yðtÞ ¼ f ðt; yðtÞ; ðSyÞðtÞÞ � I

1−βð1−αÞaþ f ða; yðaÞ; ðSyÞðaÞÞ

Γðβð1� αÞÞ ðt � aÞβð1−αÞ−1:

By Lemma 2.4, we have I1−βð1−αÞaþ f ða; yðaÞ; ðSyÞðaÞÞ ¼ 0. Therefore, we have D

α; βaþ yðtÞ ¼

f ðt; yðtÞ; ðSyÞðtÞÞ. This completes the proof. ,Next, we recall definition of noncompactness measure of HausdorffΨð$Þon each bounded

subset Ω of Banach space X defined by

ΨðΩÞ ¼ inf fr > 0;Ω can be covered by finite number of balls with radii rg:

Lemma 2.9 ([3]). For all nonempty subsets A;B⊂X, the Hausdorff measure ofnoncompactness Ψð$Þ satisfies the following properties:

(1) A is precompact if and only if ΨðAÞ ¼ 0;

(2) ΨðAÞ ¼ ΨðAÞ ¼ ΨðconvAÞ, where Aand convAdenote the closure and convex hull ofA respectively;

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112

(3) ΨðAÞ≤ΨðBÞ when A⊆B;

(4) ΨðAþ BÞ≤ΨðAÞ þ ΨðBÞ, where Aþ B ¼ faþ b; a∈ A; b ∈ Bg;(5) ΨðA∪BÞ≤maxfΨðAÞ;ΨðBÞg;(6) ΨðλAÞ ¼ jλjΨðAÞ for any λ∈ℝ;

(7) Ψðfxg∪AÞ≤ΨðAÞ for any x∈X.

Lemma 2.10 ([3]). If B⊆Cð½a; b�;XÞ is bounded and equicontinuous, then ΨðBðtÞÞis continuous for t ∈ ½a; b� and Ψ ðBÞ ¼ supfΨðBðtÞÞ; t ∈ ½a; b�g, where BðtÞ ¼fxðtÞ; x ∈Bg⊆X.

Lemma 2.11 ([16]). If fung∞n¼1 is a sequence of Bochner integrable functions from J into XwithkunðtÞk≤ μðtÞ for almost all t ∈ J and every n≥ 1, where μ∈L1ðJ ;RÞ, then the functionΨðtÞ ¼ ΨðfunðtÞ : n≥ 1gÞ belongs to L1ðJ ;RÞ with

Ψ

��Z t

0

unðsÞds : n≥ 1

��≤ 2

Z t

0

ΨðsÞds:

In order to prove the existence of solutions for our problemwith lesser number of constraints,we will introduce another type of measure of noncompactness as follows.

Let Φ denote the measure of noncompactness in the Banach space C½J ;X � defined by

ΦðΩÞ ¼ maxE∈ΔðΩÞ

ðδðEÞ;modcðEÞÞ; (2.9)

for all bounded subsets Ω of C½J ;X �, where ΔðΩÞ is the set of countable subsets of Ω, δ isthe real measure of noncompactness given by

δðEÞ ¼ supt∈½0;b�

e−LtΨðEðtÞÞ;

with EðtÞ ¼ fxðtÞ : x∈Eg; t ∈ J, L is a suitably chosen constant and modcðEÞ is themodulus of equicontinuity of the function set E defined as

modcðEÞ ¼ limδ→0

supx∈ E

maxjt2−t1j≤δ

kxðt2Þ � xðt1Þk:

Observe thatΦ is well defined [9] (i.e., E0 ∈ΔðΩÞwhich attends the maximum in (2.9)) and isnonsingular, monotone and regular measure of noncompactness.

Lemma 2.12 (M€onch fixed point theorem, [15]). Let D be a closed convex subset of a Banachspace X with 0∈D. Suppose that F : D→X is a continuous map satisfying the M€onch’scondition (if M⊆D is countable and M⊆ convðf0g∪FðMÞÞ, then M is compact), then F has afixed point in D.

Hilferfractional


113

3. Existence of solutionsLet us begin this section by introducing the hypotheses needed to prove the existence ofsolutions for the problem at hand.

(H1) The function f : J 3X 3X →X satisfies (i) f ð$; x; yÞ : J →X is measurable for allx; y∈X and (ii) f ðt; $; $Þ : X 3X →X is continuous for a.e t ∈ J.

(H2) There exists a constant N > 0 such that

kf ðt; y; SyÞk ≤ Nð1þ ζkykÞ;

for each t ∈ J and all y∈X.

(H3) There exist constants m1;m2 > 0 such that

Ψðf ðt; x; yÞÞ ≤ m1ΨðxÞ þm2Ψ ðyÞ;

for bounded sets x; y⊂X, a.e t ∈ J.Now, we are ready to present the existence result for the problem (1.1), which is based onM€onch fixed point theorem.

Theorem 3.1. Suppose that f : J 3X 3X →X is such that f ð$; yð$Þ; Syð$ÞÞ∈Cβð1−αÞ1−γ ½J ;X �

for any y∈C1−γ ½J ;X � and satisfies the hypotheses (H1)-(H3). Then the Hilfer problem (1.1) has

at least one solution in Cγ1−γ ½J ;X �⊂C

α; β1−γ ½J ;X �, provided that

Qd1

ΓðγÞjvj

juþ vjNζ

Γð1� γ þ αÞðb� aÞαBðγ; α� γ þ 1Þ þ Nζ

ΓðαÞðb� aÞαBðγ; αÞ < 1:

Proof. Introduce the operator Q : C1−γ ½J ;X �→C1−γ ½J ;X � defined by

ðQyÞðtÞ ¼ w

uþ v


uþ v


1

Γð1� γ þ αÞ

3

Z b

a


þ 1

ΓðαÞZ t

a


(3.1)

Notice that the solutions of problem (1.1) are the fixed points of the operator Q. Define abounded closed convex set Br :¼ fy ∈ C1−γ ½J ;X � : kykC1−γ ≤ r; t ∈ Jgwith r ≥ ω

1�. ð. < 1Þand

ω :¼ 1

ΓðγÞjwj

juþ vj þNðb� aÞα−γþ1

Γðαþ 1Þ þ 1

ΓðγÞjvj

juþ vjNðb� aÞα−γþ1

Γð2� γ þ αÞ :

In order to satisfy the hypotheses of the M€onch fixed point theorem, we split the proof intofour steps.

Step 1. The operator Qmaps the set Br into itself.

AJMS26,1/2

114

By the assumption (H2), we have

kðQyÞðtÞðt � aÞ1−γk

¼�� 1

ΓðγÞw

uþ v� 1

ΓðγÞv

uþ v

1


a


þ ðt � aÞ1−γΓðαÞ

Z t

a

ðt � sÞα−1f ðs; yðsÞ; ðSyÞðsÞÞds��≤ 1

ΓðγÞjwj

juþ vj þ1

ΓðγÞjvj

juþ vj1

Γð1� γ þ αÞ

3

Z b

a

ðb� sÞα−γkf ðs; yðsÞ; ðSyÞðsÞÞkdsþ ðt � aÞ1−γΓðαÞ

Z t

a

ðt � sÞα−1kf ðs; yðsÞ; ðSyÞðsÞÞkds

≤1

ΓðγÞjwj

juþ vj þ1

ΓðγÞjvj

juþ vj1


a

ðb� sÞα−γNð1þ ζkyðsÞkÞds

þ ðt � aÞ1−γΓðαÞ

Z t

a

ðt � sÞα−1Nð1þ ζkyðsÞkÞds≤ 1

ΓðγÞjwj

juþ vj

þ 1

ΓðγÞjvj

juþ vjN


a

ðb� sÞα−γdsþ 1

ΓðγÞjvj

juþ vjN


a

ðb� sÞα−γζkyðsÞkds

þ Nðt � aÞ1−γΓðαÞ

Z t

a

ðt � sÞα−1dsþ Nðt � aÞ1−γΓðαÞ

Z t

a

ðt � sÞα−1ζkyðsÞkds

≤1

ΓðγÞjwj

juþ vj þ1

ΓðγÞjvj

juþ vjN

Γð1� γ þ αÞðb� aÞα−γþ1

ðα� γ þ 1Þ

þ 1

ΓðγÞjvj

juþ vjNζ


a

ðb� sÞα−γðs� aÞγ−1kykc1�γdsþ Nðt � aÞ1−γ

ΓðαÞðt � aÞα

α

þ Nζðt � aÞ1−γΓðαÞ

Z t

a

ðt � sÞα−1ðs� aÞγ−1kykC1�γds≤

1

ΓðγÞjwj

juþ vj þ1

ΓðγÞjvj


Γð2� γ þ αÞ

þ 1

ΓðγÞjvj

juþ vjNζr

Γð1� γ þ αÞðb� aÞαBðγ; α� γ þ 1Þ þ Nðb� aÞα−γþ1

Γðαþ 1Þ þ Nζr

ΓðαÞðt � aÞαBðγ; αÞ

≤1

ΓðγÞjwj

juþ vj þNðb� aÞα−γþ1

Γðαþ 1Þ þ 1

ΓðγÞjvj


Γð2� γ þ αÞ

þ

1

ΓðγÞjvj

juþ vjNζ

Γð1� γ þ αÞðb� aÞαBðγ; α� γ þ 1Þ þ Nζ

ΓðαÞðb� aÞαBðγ; αÞ�r;

where we used the factZ t

a

ðt � sÞα−1kyðsÞkds≤�Z t

a

ðt � sÞα−1ðs� aÞγ−1ds�kykC1�γ

¼ ðt � aÞαþγ−1Bðγ; αÞkykC1�γ

In consequence, we get kQykC1− γ≤ωþ . r≤ r, that is, QBr ⊂Br. Thus Q : Br →Br.

Step 2. The operator Q is continuous.

Suppose that fyng is a sequence such that yn → y inBr as n→∞. Since f satisfies (H1), for eacht ∈ J, we get

Hilferfractional


115

��ððQynÞðtÞ � ðQyÞðtÞÞðt � aÞ1−γ��

≤1

ΓðγÞjvj

juþ vj1

Γð1� γ þ αÞ3Z b

a

ðb� sÞα−γkf ðs; ynðsÞ; ðSynÞðsÞÞ � f ðs; yðsÞ; ðSyÞðsÞÞkds

þðt � aÞ1−γΓðαÞ

Z t

a

ðt � sÞα−1kf ðs; ynðsÞ; ðSynÞðsÞÞ � f ðs; yðsÞ; ðSyÞðsÞÞkds

≤1

ΓðγÞjvj

juþ vjðb� aÞαBðγ; α� γ þ 1Þ

Γð1� γ þ αÞ3 k f ð$; ynð$Þ; ðSynÞð$ÞÞ � f ð$; yð$Þ; ðSyÞð$ÞÞkC1�γ

þðt � aÞαΓðαÞ Bðγ; αÞk f ð$; ynð$Þ; ðSynÞð$ÞÞ � f ð$; yð$Þ; ðSyÞð$ÞÞkC1�γ

:

By (H1) and using the Lebesgue dominated convergence theorem, we have

kðQyn � QyÞkC1�γ→ 0 as n→∞;

which implies that the operator Q is continuous on Br.

Step 3. The operator Q is equicontinuous.

For any a < t1 < t2 < b and y∈Br, we get

��ðt2 � aÞ1−γðQyÞðt2Þ � ðt1 � aÞ1−γðQyÞðt1Þ��

≤1

ΓðαÞ��ðt2 � aÞ1−γ

Z t2

a

ðt2 � sÞα−1f ðs; yðsÞ; ðSyÞðsÞÞds

�ðt1 � aÞ1−γZ t1

a

ðt1 � sÞα−1f ðs; yðsÞ; ðSyÞðsÞÞds��

≤kfkC1�γ

ΓðαÞ��ðt2 � aÞ1−γ

Z t2

a

ðt2 � sÞα−1ðs� aÞγ−1ds

�ðt1 � aÞ1−γZ t1

a

ðt1 � sÞα−1ðs� aÞγ−1ds��

≤kfkC1�γ

ΓðαÞ Bðγ; αÞ��ðt2 � aÞ1−γðt2 � aÞαþγ−1 � ðt1 � aÞ1−γðt1 � aÞαþγ−1��

≤kfkC1�γ

ΓðαÞ Bðγ; αÞkðt2 � aÞα � ðt1 � aÞαk;

which tends to zero as t2 → t1, independent of y∈Br. Thus we conclude that QðBrÞ isequicontinuous, that is, modcðQðBrÞÞ ¼ 0.

AJMS26,1/2

116

Step 4. The M€onch condition is satisfied.

Suppose that D⊂Br is a countable set and D⊆ convðf0g∪QðDÞÞ. In order to show that D isprecompact, it is enough to obtain that ΦðDÞ ¼ ð0; 0Þ. Since ΦðQðDÞÞ is maximum, letfxng∞n¼1 ⊆QðDÞ be a countable set attaining its maximum. Then, there exists a setfyng∞n¼1 ⊆D such that xn ¼ ðQynÞðtÞ for all t ∈ J ; n≥ 1.

Now, using (H3) together with Lemmas 2.9–2.11, we obtain

Ψ�fxng∞n¼1

� ¼ Ψ�fðQynÞðtÞg∞n¼1

�

≤2jvj

juþ vjðt � aÞγ−1

ΓðγÞ1

Γð1� γ þ αÞ 3Z b

a

ðb� sÞα−γΨ�f ðs; fynðsÞg∞n¼1; ðSfynðsÞg∞n¼1

��Þds

þ 2

ΓðαÞZ t

a

ðt � sÞα−1Ψ�f ðs; fynðsÞg∞n¼1; ðSfynðsÞg∞n¼1

��Þds

≤2jvj


ΓðγÞ1


a

ðb� sÞα−γ�m1Ψ�fynðsÞg∞n¼1

�þm2ΨððSfynðsÞg∞n¼1

��Þds

þ 2

ΓðαÞZ t

a

ðt � sÞα−1�m1ΨðfynðsÞg∞n¼1

�þm2Ψ��SfynðsÞg∞n¼1

��Þds

≤2jvj


ΓðγÞ1


a

ðb� sÞα−γ�m1 supt∈½a;b�

ΨðfynðtÞg∞n¼1

�

þ 2m2ζ supt∈½a;b�

Ψ�fynðtÞg∞n¼1

��dsþ 2

ΓðαÞZ t

a

ðt � sÞα−1�m1 supt∈½a;b�

ΨðfynðtÞg∞n¼1

�

þ 2m2ζ supt∈½a;b�

Ψ�fynðtÞg∞n¼1

��ds ≤

2jvjjuþ vj


1


a

ðb� sÞα−γ

3 eLs�m1 sup

t∈½a;b�e−LtΨðfynðtÞg∞n¼1

�þ 2m2ζ supt∈½a;b�

e−LtΨ�fynðtÞg∞n¼1

��ds

þ 2

ΓðαÞZ b

a

ðt � sÞα−1 3 eLsðm1 supt∈½a;b�

e−LtΨðfynðtÞg∞n¼1Þ þ 2m2ζ supt∈½a;b�

e−LtΨðfynðtÞg∞n¼1ÞÞds

≤2jvj


ΓðγÞδ�fyng∞n¼1

�Γð1� γ þ αÞ

Z b

a

ðb� sÞα−γeLsðm1 þ 2m2ζÞds

þ 2δ�fyng∞n¼1

�ΓðαÞ

Z t

a

ðt � sÞα−1eLsðm1 þ 2m2ζÞds

≤

2jvj


ΓðγÞ1


a


þ 2

ΓðαÞZ t

a

ðt � sÞα−1eLsðm1 þ 2m2ζÞds�δ�fyng∞n¼1

�:

Hilferfractional


117

Hence

δ�fxng∞n¼1

�

≤ supt∈½a;b�

e−Lt

"2jvj


ΓðγÞ1


a


þ 2

ΓðαÞZ t

a

ðt � sÞα−1eLsðm1 þ 2m2ζÞds#δ�fyng∞n¼1

�:

Fixing a suitable constant 0 < L0< 1 given by

L0 ¼ sup

t∈½a;b�e−Lt

"2jvj


ΓðγÞ1

Γð1� γ þ αÞ

3

Z b

a


þ 2

ΓðαÞZ t

a

ðt � sÞα−1eLsðm1 þ 2m2ζÞds#:;

we get δðfxng∞n¼1Þ≤L0δðfyng∞n¼1Þ. Thus

δ�fyng∞n¼1

�≤ δðDÞ≤ δðconvðf0g∪QðDÞÞÞ ¼ δ

�fxng∞n¼1

�≤L

0δ�fyng∞n¼1

�;

which implies that δðfyng∞n¼1Þ ¼ 0 and hence δðfxng∞n¼1Þ ¼ 0.Now, according to the Step 3, we have found an equicontinuous set fxng∞n¼1 on J. Hence

ΦðDÞ≤Φðconvðf0g∪QðDÞÞÞ≤ΦðQðDÞÞ, whereΦðQðDÞÞ ¼ Φðfxng∞n¼1Þ ¼ ð0; 0Þ. Therefore,Dis precompact. Hence, by Lemma 2.12, there is a fixed point yof operatorQ, which is a solutionof the problem (1.1) in C1−γ ½J ;X �.

Next, we show that such a solution is indeed inCγ1−γ ½J ;X �. By applyingDγ

aþ on both sides of(2.1), we get

DγaþyðtÞ ¼ D

βð1−αÞaþ f ðt; yðtÞ; ðSyÞðtÞÞ:

Since f ðt; yðtÞ; ðSyÞðtÞÞ∈Cβð1−αÞ1−γ ½J ;X �, it follows by definition of the space C

βð1−αÞ1−γ ½J ;X � that

DγaþyðtÞ∈C1−γ ½J ;X �, which implies that y∈C

γ1−γ ½J ;X �. ,

4. e−Approximate solution

Definition 4.1. A function z∈Cγ1−γ ½J ;X � satisfying the Hilfer fractional integrodifferential

inequality ��Dα; βaþ zðtÞ � f ðt; zðtÞ; ðSzÞðtÞÞ�� ≤ e; t ∈ J ;

and

I 1−γaþ ½uzðaþÞ þ vzðb−Þ� ¼ w;

is called an e−approximate solutions of Hilfer fractional integrodifferential equation (1.1).

AJMS26,1/2

118

Lemma 4.1 (See [22]). For β > 0, let vðtÞ be a nonnegative function locally integrable on0 < t < T (some T ≤ þ∞) and gðtÞ be a nonnegative, nondecreasing continuous functiondefined on 0 < t < T with gðtÞ≤M (constant) and uðtÞ be a nonnegative and locally integrablefunction on 0 < t < T such that

uðtÞ≤ vðtÞ þ gðtÞZ t

0

ðt � sÞβ−1uðsÞds; 0 < t < T:

Then

uðtÞ≤ vðtÞ þZ t

0

"X∞n¼1

ðgðtÞΓðβÞÞnΓðnβÞ ðt � sÞnβ−1vðsÞ

#ds; 0 < t < T:

Theorem 4.1. Suppose that the function f : J 3X 3X →X satisfies the condition:

kf ðt; y1; x1Þ � f ðt; y2; x2Þk≤ n1ky1 � y2k þ n2kx1 � x2k;for each t ∈ J and all y1; y2; x1; x2 ∈X, where n1; n2 > 0 are constants. Let zi ∈Cγ

1−γ½J ;X �;i ¼ 1; 2, be an e−approximate solution of the following Hilfer fractional integrodifferentialequation (

Dα; βaþ ziðtÞ ¼ f ðt; ziðtÞ; ðSziÞðtÞÞ; t ∈ J ; 0 < α < 1; 0≤ β≤ 1;

I 1−γaþ ½uziðaþÞ þ vziðb−Þ� ¼ wi; α≤ γ ¼ αþ β � αβ; i ¼ 1; 2:(4.1)

Then

kz1 � z2kC1�γ≤ Z−1

3

"ðe1 þ e2Þ

ðb� aÞα−γþ1

Γðαþ 1Þ þX∞n¼1

ðn1 þ ζn2Þn 1

Γððnþ 1Þαþ 1Þðb� aÞðnþ1Þα−γþ1

!

þ jw1 � w2jjuþ vj

1

ΓðγÞ þX∞n¼1

ðn1 þ ζn2Þn 1

Γðnαþ γÞðb� aÞnα!#

;

(4.2)

where

Z ¼ 1� jvj

juþ vjðn1 þ ζn2Þ

Γðα� γ þ 1Þðb� aÞαBðγ; α� γ þ 1Þ

3

(1

ΓðγÞ þX∞n¼1

ðn1 þ ζn2Þn 1

Γðnαþ γÞðb� aÞnα)!

≠ 0:

(4.3)

Proof. Let zi ∈Cγ1−γ ½J ;X �; ði ¼ 1; 2Þ be an e−approximate solution of problem (4.1). Then

I1−γaþ ½uziðaþÞ þ vziðb−Þ� ¼ wi and��Dα; β

aþ ziðtÞ � f ðt; ziðtÞ; ðSziÞðtÞÞ��≤ ei; i ¼ 1; 2; t ∈ J : (4.4)

Hilferfractional


119

Applying Iαaþ on both sides of the above inequality and using Lemma 2.3, we get

I αaþei ≥ Iαaþ��Dα; β

aþ ziðtÞ � f ðt; ziðtÞ; ðSziÞðtÞÞ��≥

��ziðtÞ � wi

uþ v


þ v

uþ v

ðt � aÞγ−1ΓðγÞ Iα−γþ1

aþ f ðb; ziðbÞ; ðSziÞðbÞÞ � I αaþ f ðt; ziðtÞ; ðSziÞðtÞÞ��;

which implies that

ei

Γðαþ 1Þðt � aÞα ≥��ziðtÞ � wi

uþ v

ðt � aÞγ−1ΓðγÞ þ v

uþ v

ðt � aÞγ−1ΓðγÞ I α−γþ1

aþ f ðb; ziðbÞ; ðSziÞðbÞÞ

� Iαaþ f ðt; ziðtÞ; ðSziÞðtÞÞ��; i ¼ 1; 2:

Using jxj− jyj≤ jx− yj≤ jxj þ jyj in the above inequality yields

ðe1 þ e2ÞΓðαþ 1Þðt � aÞα

≥

��z1ðtÞ � w1

uþ v


uþ v

ðt � aÞγ−1ΓðγÞ I

α−γþ1aþ f ðb; z1ðbÞ; ðSz1ÞðbÞÞ � I αaþ f ðt; z1ðtÞ; ðSz1ÞðtÞÞ

��

þ��z2ðtÞ � w2

uþ v


uþ v


aþ f ðb; z2ðbÞ; ðSz2ÞðbÞÞ � I αaþ f ðt; z2ðtÞ; ðSz2ÞðtÞÞ��

≥

�� z1ðtÞ � w1

uþ v


uþ v


aþ f ðb; z1ðbÞ; ðSz1ÞðbÞÞ

� I αaþ f ðt; z1ðtÞ; ðSz1ÞðtÞÞ� � z2ðtÞ � w2

uþ v


þ v

uþ v


aþ f ðb; z2ðbÞ; ðSz2ÞðbÞÞ � I αaþ f ðt; z2ðtÞ; ðSz2ÞðtÞÞ��

≥

��ðz1ðtÞ � z2ðtÞÞ � ðw1 � w2Þ

uþ v


þ v

uþ v


aþ ½f ðb; z1ðbÞ; ðSz1ÞðbÞÞ � f ðb; z2ðbÞ; ðSz2ÞðbÞÞ�

� I αaþ ½f ðt; z1ðtÞ; ðSz1ÞðtÞÞ � f ðt; z2ðtÞ; ðSz2ÞðtÞÞ��

≥ kðz1ðtÞ � z2ðtÞÞk ��ðw1 � w2Þ

uþ v


��

þ�� v

uþ v


aþ ½f ðb; z1ðbÞ; ðSz1ÞðbÞÞ � f ðb; z2ðbÞ; ðSz2ÞðbÞÞ��

� ��I αaþ ½f ðt; z1ðtÞ; ðSz1ÞðtÞÞ � f ðt; z2ðtÞ; ðSz2ÞðtÞÞ��:

In consequence, we have

AJMS26,1/2

120

kðz1ðtÞ � z2ðtÞÞk

≤ðe1 þ e2ÞΓðαþ 1Þðt � aÞα þ

��ðw1 � w2Þuþ v


��

�� jujjuþ vj


aþ ½ f ðb; z1ðbÞ; ðSz1ÞðbÞÞ � f ðb; z2ðbÞ; ðSz2ÞðbÞÞ��

þkI αaþ ½f ðt; z1ðtÞ; ðSz1ÞðtÞÞ � f ðt; z2ðtÞ; ðSz2ÞðtÞÞ�k

≤ðe1 þ e2ÞΓðαþ 1Þðt � aÞα þ jw1 � w2j

uþ v


þ jvjjuþ vj


��Iα−γþ1aþ ½f ðb; z1ðbÞ; ðSz1ÞðbÞÞ � f ðb; z2ðbÞ; ðSz2ÞðbÞÞ�

��þkI αaþ ½f ðt; z1ðtÞ; ðSz1ÞðtÞÞ � f ðt; z2ðtÞ; ðSz2ÞðtÞÞ�k



ΓðγÞ

þ jvjjuþ vj


ðn1 þ ζn2ÞΓðα� γ þ 1Þ

Z b

a

ðb� sÞα−γkz1ðsÞ � z2ðsÞkds

þðn1 þ ζn2ÞΓðαÞ

Z t

a

ðt � sÞα−γkz1ðsÞ � z2ðsÞkds



ΓðγÞ

þ jvjjuþ vj


ðn1 þ ζn2ÞΓðα� γ þ 1Þðb� aÞαBðγ; α� γ þ 1Þkz1 � z2kC1�γ

þðn1 þ ζn2ÞΓðαÞ

Z t

a

ðt � sÞα−1kz1ðsÞ � z2ðsÞkds:

Using Lemma 4.1 with uðtÞ ¼ kðz1ðtÞ− z2ðtÞÞk, gðtÞ ¼ ðn1þζn2ÞΓðαÞ and vðtÞ ¼ ðe1 þ e2Þ

Γðαþ 1Þ ðt − aÞαþjw1 −w2jjuþvj

ðt − aÞγ−1ΓðγÞ þ jvj

juþvjðt − aÞγ−1ΓðγÞ

ðn1þζn2ÞΓðα− γþ1Þðb− aÞαBðγ; α− γ þ 1Þkz1−z2kC1− γ

, we get

Hilferfractional


121

kðz1ðtÞ � z2ðtÞÞk≤ ðe1 þ e2ÞΓðαþ 1Þðt � aÞα þ jw1 � w2j


ΓðγÞ

þ jvjjuþ vj



þZ t

a

X∞n¼1

ðn1 þ ζn2ÞnΓðnαÞ ðt � sÞnα−1

�ðe1 þ e2ÞΓðαþ 1Þðs� aÞα þ jw1 � w2j

juþ vjðs� aÞγ−1

ΓðγÞ

þ jvjjuþ vj

ðs� aÞγ−1ΓðγÞ


�ds



ΓðγÞ þ jvjjuþ vj



þ ðe1 þ e2ÞΓðαþ 1Þ

X∞n¼1

ðn1 þ ζn2ÞnI nαaþ ðt � aÞα

þ jw1 � w2jΓðγÞjuþ vj

X∞n¼1

ðn1 þ ζn2ÞnI nαaþ ðt � aÞγ−1

þjvjkz1 � z2kC1�γ

ΓðγÞjuþ vjðn1 þ ζn2Þ

Γðα� γ þ 1Þðb� aÞαBðγ; α� γ þ 1Þ

3X∞n¼1

ðn1 þ ζn2ÞnI nαaþ ðt � aÞγ−1 ≤ ðe1 þ e2ÞΓðαþ 1Þðt � aÞα þ jw1 � w2j


ΓðγÞ

þ jvjjuþ vj



þ ðe1 þ e2ÞΓðαþ 1Þ

X∞n¼1

ðn1 þ ζn2Þn Γðαþ 1ÞΓððnþ 1Þαþ 1Þðt � aÞðnþ1Þα

þ jw1 � w2jΓðγÞjuþ vj

X∞n¼1

ðn1 þ ζn2Þn ΓðγÞΓðnαþ γÞðt � aÞnαþγ−1 þ

jvjkz1 � z2kC1�γ

ΓðγÞjuþ vjðn1 þ ζn2Þ

Γðα� γ þ 1Þðb� aÞαBðγ; α� γ þ 1Þ3X∞n¼1

ðn1 þ ζn2Þn ΓðγÞΓðnαþ γÞðt � aÞnαþγ−1

¼ ðe1 þ e2Þ ðt � aÞαΓðαþ 1Þ þ

X∞n¼1

ðn1 þ ζn2Þn 1

Γððnþ 1Þαþ 1Þðt � aÞðnþ1Þα!


ðt � aÞγ−1

ΓðγÞ þX∞n¼1

ðn1 þ ζn2Þn 1

Γðnαþ γÞðt � aÞnαþγ−1

!

þ jvjjuþ vj


3

ðt � aÞγ−1

ΓðγÞ þX∞n¼1

ðn1 þ ζn2Þn 1

Γðnαþ γÞðt � aÞnαþγ−1

!:

AJMS26,1/2

122

Hence, for each t ∈ J, we have

ðt � aÞ1−γkðz1ðtÞ � z2ðtÞÞk

≤ ðe1 þ e2Þ ðt � aÞα−γþ1


ðn1 þ ζn2Þn 1

Γððnþ 1Þαþ 1Þðt � aÞðnþ1Þα−γþ1

!


1

ΓðγÞ þX∞n¼1

ðn1 þ ζn2Þn 1

Γðnαþ γÞðt � aÞnα!

þ jvjjuþ vj


3

1

ΓðγÞ þX∞n¼1

ðn1 þ ζn2Þn 1

Γðnαþ γÞðt � aÞnα!:

Thus

kz1 � z2kC1�γ

≤ ðe1 þ e2Þ ðb� aÞα−γþ1


ðn1 þ ζn2Þn 1


!


1

ΓðγÞ þX∞n¼1

ðn1 þ ζn2Þn 1

Γðnαþ γÞðb� aÞnα!

þ jvjjuþ vj


3

1

ΓðγÞ þX∞n¼1

ðn1 þ ζn2Þn 1

Γðnαþ γÞðb� aÞnα!;

which, together with (4.3), yields

kz1 � z2kC1�γ≤ Z−1½ðe1 þ e2Þ

3

ðb� aÞα−γþ1


ðn1 þ ζn2Þn 1


!


1

ΓðγÞ þX∞n¼1

ðn1 þ ζn2Þn 1

Γðnαþ γÞðb� aÞnα!#

: ,

(4.5)

Remark 4.1. If e1 ¼ e2 ¼ 0 in the inequality (4.4), then z1; z2 are solutions of the problem(1.1) in the space Cγ

1−γ ½J ;X � and the inequality (4.5) takes the form

kz1 � z2kC1�γ≤ Z−1jw1 � w2j

juþ vj

1

ΓðγÞ þX∞n¼1

ðn1 þ ζn2Þn 1

Γðnαþ γÞðb� aÞnα!;

Hilferfractional


123

which provides the information with respect to continuous dependence on the solution of theproblem (1.1). In addition, if w1 ¼ w2 we get kz1−z2kC1− γ

¼ 0;which proves the uniqueness ofsolutions of the system (1.1).

Remark 4.2. One can note that our results for the Hilfer fractional integrodifferentialequation (1.1) correspond to initial boundary value problem for u ¼ 1; v ¼ 0, terminalboundary value problem for u ¼ 0; v ¼ 1 and anti-periodic problem for u ¼ 1; v ¼ 1;w ¼ 0.

Remark 4.3. If β ¼ 1, then Eq. (1.1) reduces to the Caputo fractional integrodifferentialequation with boundary conditions as in [12].

References

[1] S. Abbas, M. Benchohra, J. Lazreg, Y. Zhou, Yong, A survey on Hadamard and Hilfer fractionaldifferential equations: Analysis and stability, Chaos Solitons Fractals 102 (2017) 47–71.

[2] B. Ahmad, A. Alsaedi, S.K. Ntouyas, J. Tariboon, Hadamard-Type Fractional DifferentialEquations Inclusions and Inequalities, Springer, Cham, 2017.

[3] J. Banas, K. Goebel, Measure of Noncompactness in Banach Spaces: Lecture Notes in Pure andApplied Mathematics, Dekker, New York, 1980.

[4] V.M. Bulavatsky, Closed form of the solutions of some boundary-value problems for anomalousdiffusion equation with Hilfer’s generalized derivative, Cybernet. Systems Anal. 50 (2014)570–577.

[5] K. Diethelm, The analysis of fractional differential equations, in: An Application-OrientedExposition Using Differential Operators of Caputo Type, in: Lecture Notes in Mathematics,Vol. 2004, Springer-Verlag, Berlin, 2010.

[6] K.M. Furati, M.D. Kassim, N.E. Tatar, Existence and uniqueness for a problem involving Hilferfractional derivative, Comput. Math. Appl. 64 (2012) 1616–1626.

[7] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc.,River Edge, NJ, Singapore, 2000.

[8] R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differentialequations with generalized Riemann–Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12(2009) 299–318.

[9] M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing multivalued maps and semilineardifferential inclusions in Banach spaces, in: De Gruyter Series in Nonlinear Analysis andApplications, Vol. 7, Walter de Gruyter & Co., Berlin, 2001.

[10] R. Kamocki, A new representation formula for the Hilfer fractional derivative and its application,J. Comput. Appl. Math. 308 (2016) 39–45.

[11] R. Kamocki, C. Obczynski, On fractional Cauchy-type problems containing Hilfer’s derivative,Electron. J. Qual. Theory Differ. Equ. (2016) 1–12.

[12] K. Karthikeyan, J.J. Trujillo, Existenes and uniqueness results for fractional integrodifferentialequations with boundary value conditions, Commun. Nonlinear Sci. Numer. Simul. 17 (2012)4037–4043.

[13] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional DifferentialEquations, Elsevier B.V., Amsterdam, 2006.

[14] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, JohnWiley & Sons, Inc., New York, 1993.

[15] H. M€onch, Boundary value problems for nonlinear ordinary differential equations of second orderin Banach spaces, Nonlinear Anal. 4 (1980) 985–999.

[16] D. O’Regan, R. Precup, Existence criteria for integral equations in Banach spaces, J. Inequal. Appl.6 (2001) 77–97.

AJMS26,1/2

124

[17] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[18] S.K. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory andApplications, Gordon and Breach Science, Switzerland, 1993.

[19] S.T.M. Thabet, M.B. Dhakne, On abstract fractional integro-differential equations via measure ofnoncompactness, Adv. Fixed Point Theory 6 (2016) 175–193.

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Corresponding authorSabri T.M. Thabet can be contacted at: [email protected]


Hilferfractional


125


Remarks on the critical nonlinearhigh-order heat equation

Tarek SaanouniQassim University, Buraidah, Saudi Arabia and

LR03ES04 Partial Differential Equations and Applications,Faculty of Science of Tunis, University of Tunis El Manar, Tunis, Tunisia

AbstractThe initial value problem for a semi-linear high-order heat equation is investigated. In the focusing case, globalwell-posedness and exponential decay are obtained. In the focusing sign, global and non global existence ofsolutions are discussed via the potential well method.

Keywords Nonlinear high-order heat equation, Global existence, Decay, Blow-up

Paper type Orginal Article

1. IntroductionConsider the Cauchy problem for a high-order nonlinear heat equation�

u: þ ð�ΔÞkuþ cu ¼ ejujp−1u;

ujt¼0j ¼ u0:(1.1)

Higher-order semi-linear and quasilinear diffusion operators occur in applications in thinfilm theory, non-linear diffusion and lubrication theory, flame and wave propagation, andphase transition at critical Lifschitz points and bistable systems (e.g., the Kuramoto–Sivashinsky equation and the extended Fisher–Kolmogorov equation). See models andreferences [16].

Here and hereafter k > 1, cef0; 1g, e ¼ ±1, u :¼ uðt; xÞ is a real-valued function of the

variables ðt; xÞ∈ℝ3ℝn for some integer ne

�2k; 2kð1þkÞ

k− 1

�. The non-linearity satisfies

k≤ p≤ p* :¼ pc − 1 :¼ nþ2kn− 2k. The k- Laplacian operator stands for

ð−ΔÞk :¼ ð−ΔÞ½−Δ�k−1; ð−ΔÞ0 :¼ I :

Remarks on thehigh-order heat

equation

127

JEL Classification — 35K55© Tarek Saanouni. Published in the Arab Journal of Mathematical Sciences. Published by Emerald

Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0)license. Anyone may reproduce, distribute, translate and create derivative works of this article (for bothcommercial and non-commercial purposes), subject to full attribution to the original publication andauthors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

The publisher wishes to inform readers that the article “Remarks on the critical nonlinear high-orderheat equation” was originally published by the previous publisher of the Arab Journal of MathematicalSciences and the pagination of this article has been subsequently changed. There has been no change tothe content of the article. This change was necessary for the journal to transition from the previouspublisher to the new one. The publisher sincerely apologises for any inconvenience caused. To accessand cite this article, please use Saanouni, T. (2019), “Remarks on the critical nonlinear high-order heatequation”, Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 127-152. The original publicationdate for this paper was 15/03/2019.



Received 28 August 2018Accepted 7 March 2019


Vol. 26 No. 1/2, 2020pp. 127-152


DOI 10.1016/j.ajmsc.2019.03.002

The energy space Cð½0; T�; HkðℝnÞÞ is naturally adapted to study the high-order heatproblem (1.1) using, with a minimal regularity, the following energy identity

vtEcðtÞ :¼ vtE

cðuðtÞÞ

:¼ vt

�Zℝn

�1

2

��∇kuðtÞ��2 þ c

2juðtÞj2 � e

1þ pjuðtÞj1þp

�dx

�

¼ −

Zℝn

ju: ðt; xÞj2dx

If e ¼ −1, the energy is positive and (1.1) is said to be defocusing. For e ¼ 1, the energy nolonger allows a control of theHk norm of an eventual solution. In such a case, (1.1) is focusing.

In the classical case k ¼ 1, Eq. (1.1) has been extensively studied in the scale of Lebesgue

spaces LqðℝnÞ. The critical index qc :¼ nðp− 1Þ2 gives the following three different regimes.

(1) Sub-critical case q > qc ≥ 1:Weissler [18] proved local well-posedness in Cð½0; TÞ;LqðℝnÞÞ∩L∞

locð�0; T�;L∞ðℝnÞÞ. Then Brezis–Cazenave [3] showed unconditionaluniqueness.

(2) Critical case q ¼ qc: There are two cases

(a) qc > pþ 1: local well-posedness holds [3,18];

(b) q ¼ qc ¼ pþ 1: Weissler [19] proved a conditional well-posedness.

(3) Super-critical case q < qc: There is no solution in any reasonable weak sense[3,18,19]. Moreover, uniqueness is lost [10] for the initial data u0 ¼ 0 and for1þ 1

n< p < nþ2

n− 2:

See [11] for exponential type non-linearity in two space dimensions.This manuscript seems to be one of few works treating well-posedness issues of the

nonlinear high-order heat equation in the energy space [2,8,9,17].The purpose of this paper is two-fold. First, global well-posedness and exponential

decay are established in the defocusing case. Second, in the focusing sign, global andnon global existence of solutions are discussed via potential-well method. Comparingwith the classical case, we need to operate with various modification due to the high-order Laplacian.

The rest of the paper is organized as follows. Section 2 is devoted to the main results andsome tools needed in the sequel. Section 3 deals with local well-posedness of (1.1). Section 4contains a proof of global existence of solutions in the critical case with small data. Section 5deals with the associated stationary problem. Section 6 is about global and non globalexistence of solutions with data in some stable sets in the spirit of Payne and Sattinger [15]. Inthe last one, the existence of infinitely many non global solutions near the ground state isproved.

We mention that C will be used to denote a constant which may vary from line to line.A(B means that A≤CB for some absolute constant C. For simplicity, denoteR$dx :¼ Rℝn$dx; Lp :¼ LpðℝnÞ is the Lebesgue space endowed with the norm

k$kp :¼ k$kLp and k$k :¼ k$k2. The classical Sobolev space is Hk;p :¼ ðI −ΔÞ−k2 Lp and

Hk :¼ Hk;2 is the energy space. Using Plancherel Theorem, the following norms areequivalent

kukHk :¼�Z

ℝn

�1þ jξj2kjbuðξÞj2dξ

�12

’kuk2 þ ��∇ku

��2�12

:

AJMS26,1/2

128

We denote the real numbers

p* :¼ 1þ 4k

n; p* :¼ pc � 1 :¼ nþ 2k

n� 2k

and we assume here and hereafter that

c ¼ 1� δp*

p ¼�

01

ifif

p ¼ p*;p≠ p*:

Finally, if T > 0 and X is an abstract functional space, we denote CTðXÞ :¼ Cð½0; T�; XÞ;LpTðXÞ :¼ Lpð½0; T�; XÞ and Xrd the set of radial elements in X, moreover for an eventual

solution to (1.1), we denote T* > 0 its lifespan.

2. Background and main resultsIn this section we give the main results and some technical tools needed in the sequel.

2.1 Main resultsResults proved in this paper are listed in what follows.

First, we deal with local well-posedness of the heat problem (1.1) in the energy space.

Theorem 2.1. Take k > 1, n∈ ð2k; 2kð1þkÞk− 1 Þ,1 < p ≤ p* and u0 ∈Hk. Then, there exist

an admissible pair ðq; rÞ in the meaning of Definition 2.8 and a unique maximal solution to(1.1),

u∈Lqð0;T*Þ; Hk;r

�:

Moreover,

(1) u∈Cð½0; T*Þ; HkÞ;(2) EðtÞ ¼ Eð0Þ− R t0 Rℝn j _uðs; xÞj2dx ds, for any t ∈ ½0; T*Þ;(3) if p < p*, then

(a) u is unique in Cð½0; T*Þ; HkÞ;(b) if T* < ∞, then lim sup

T*

kuðtÞkHk ¼ ∞ and

kuðtÞkHk ≥C

ðT* � tÞ 1p−1

−n−2k4k

;

(c) if e ¼ −1, then T* ¼ ∞ and there exists γ > 0 such that

kuðtÞkHk ¼ O�e−γt; when t→∞:

In the critical case, for small data, there exists a global solution to (1.1).

Theorem 2.2. Take k > 1, n∈ ð2k; 2kð1þkÞk− 1 Þ and p ¼ p*. Then, there exists e0 > 0 such that

if u0 ∈ _Hksatisfies ku0k _H

k ≤ e0, the problem (1.1) possesses a unique global solution

u∈Cðℝþ; _HkÞ, satisfying the decay


equation

129

limt→þ∞

kuðtÞkLp ¼ 0; for all 2 < p <2n

n� 2k:

Second, we are interested on the focusing case. Using the potential well method due to Payne–Sattinger [15], we discuss global and non global existence of solutions to (1.1), when the databelongs to some stable sets. Denote the quantities

μ :¼ maxf2αþ ðn� 2kÞβ; 2αþ nβg; ~μ :¼ minf2αþ ðn� 2kÞβ; 2αþ nβgand the set

A :¼ ðα; βÞ∈ℝ*þ 3ℝ s: t ~μ > 0 and αðp� 1Þ þ 2kβ > 0

�:

The following quantity will be called constraint

Kcα;βðvÞ ¼

1

2

Z �ð2αþ ðn� 2kÞβÞ��∇kv

��2 þ ð2αþ nβÞcjvj2 � 2

�αþ nβ

1þ p

�jvj1þp

�dx:

Take the minimizing problem under constraint

mcα;β :¼ inf

0≠v∈Hkrd

nEcðvÞ; s: t Kc

α;βðvÞ ¼ 0o:

For easy notation, set

mα;β :¼ m1α;β; ∈E :¼ E1 and Kα;β :¼ K1

α;β:

Definition 2.3. We call a ground state to (1.1) any solution to

−ð−ΔÞkf� cfþ jfjp−1f ¼ 0; 0≠f∈Hkrd; mα;β ¼ EðfÞ: (2.2)

The existence of ground state is claimed.

Theorem 2.4. Take k > 0, n≥ 2, 1 < p≤ p* and ðα; βÞ∈A. So, there exists a ground statesolution to (2.2). Moreover, mc :¼ mc

α;β is nonzero and independent of ðα; βÞ.Denote the spaces

Ac;þα;β :¼

nf∈Hk; s: t EcðfÞ < mc

α;β and Kcα;βðfÞ≥ 0

o;

Ac;−α;β :¼

nf∈Hk; s: t EðfÞ < mc

α;β and Kcα;βðfÞ < 0

o;

Aþα;β :¼ A1;þ

α;β ; A−

α;β :¼ A1;−α;β :

Let us discuss global and non global existence of solutions to the heat problem (1.1).

Theorem 2.5. Take k > 1, n∈ ð2k; 2kð1þkÞk− 1 Þ, 1 < p ≤ p* and ðα; βÞ∈A,e ¼ 1 and u∈C

ð½0;T*Þ; HkÞ be a maximal solution to (1.1). Then,

(1) if p < p* and u0 ∈Aþα;β, then T* ¼ ∞ and uðtÞ∈Aþ

α;β for any time t ≥ 0. Moreover,for small ku0k, there exists γ > 0 such that

kuðtÞk _Hk ¼ O

�e−γt; when t→∞;

AJMS26,1/2

130

(2) if u0 ∈Ac;−α;β, then u blows-up in finite time.

The last result concerns instability by blow-up for stationary solutions to the heatproblem (1.1). Indeed, near ground state, there exist infinitely many data giving non globalsolutions.

Theorem 2.6. Take k > 1, n∈ ð2k; 2kð1þkÞk− 1 Þ, e ¼ 1and p* < p ≤ p*. Let f be a ground state

solution to (2.2). Then, for any ε > 0, there exists u0 ∈Hk such that ku0−fkHk < ε and themaximal solution to (1.1) is not global.

2.2 ToolsLet us collect some classical estimates needed forward this manuscript. We start with sometechnical results about the high-order heat equation. Some useful properties of the free heatkernel are gathered in what follows.

Proposition 2.7. Denoting the free operator associated to the high-order heat equation

TkðtÞf :¼ e−tð−ΔÞk

:¼ F−1ðe−tj:j2kÞ*f :¼ KkðtÞ*f;yields

(1) e−tð−ΔÞk

u0 þ eR t0 e

−ðt−sÞð−ΔÞk jujp−1u ds is the solution to the problem (1.1);

(2) TkTβ ¼ Tkþβ T*k ¼ Tk:

Let us recall the so-called Strichartz estimate [20].

Definition 2.8. A couple of real numbers ðq; rÞ is said to be admissible if

q; r≥ 2 and2k

q¼ n

�1

2� 1

r

�:

Proposition 2.9. Let n≥ 2, k > 0, u0 ∈L2 and ðq; rÞ, ðq; rÞ two admissible pairs. Then,there exists C :¼ Cq;~q such that

kukLqtðLrÞ ≤C

ku0k þ

�� _uþ ð−ΔÞku��L~q0t

L~r0��

:

Proof. Compute

ðKkðtÞÞðxÞ ¼ F−1e−tj _j

2k�ðxÞ

¼ 1

tn2k

F−1e−j _j

2k�� x

t12k

�

¼ 1

tn2k

K

�1

tn2k

�;

where K ∈ ðL1 ∩L∞ÞðℝnÞ (see [7]). Thus,kTkðtÞfk(kfk; ��TkðtÞT*

k ðsÞf��∞(

1

jt � sj n2kkfk1:

The proof is finished via Theorem 1.2 in [12]. -Using the above computation via Young inequality, the following smoothing effect yields.


equation

131

Lemma 2.10. There exists a positive constant C such that for all 1≤ r≤ q≤∞, we have

kTkðtÞwkLq ≤C

tN2kð1r−1

qÞkwkLr ; ∀t > 0; ∀w e∈LrðℝN Þ: (2.3)

The following Sobolev injections [1,13] give a meaning to the energy and severalcomputations done in this note.

Lemma 2.11. Let n≥ 2, k > 0 and p∈ ð1; ∞Þ. Then,(1) Wk;pðℝnÞ↪LqðℝnÞwhenever 1 < p < q < ∞; and 1

p≤ 1

qþ k

n;

(2) WkðℝnÞ↪LqðℝnÞ for any q∈ ½2; 2nn− 2k�; n > 2k

(3) HkrdðℝnÞ↪↪LqðℝnÞ for any q∈ ð2; 2n

n− 2kÞ; n≥ 2k.

The following Gagliardo–Nirenberg inequality is useful throughout the manuscript [14].

Lemma 2.12. Let n≥ 2, k > 0 and p; q; r∈ ð1; ∞Þ. Then,k$kp(

��∇k$��θrk$k1−θq ;

for 1p¼ θð1

r− k

nÞ þ 1− θ

qsuch that θ∈ ½0; 1�.

In the critical case, recall some properties of the best constant of Sobolev injection [5,6].

Proposition 2.13. Take n≥ 2 and 0 < 2k < n. Then,

C*n;k :¼ inf

0≠u∈ _Hk

kuk2pck∇kuk2

¼ 1

22kπk

Γn2� k�

Γn2þ k� ΓðnÞ2kn

Γn2

�2kn

:

Moreover, u is such a minimizer if and only if there exist c∈ℝ, μ > 0 and x0 ∈ℝn such that

uðxÞ ¼ c�μ2 þ jx� x0j2

−n−2k2 :

Let us give an abstract result.

Lemma 2.14. Let T > 0 and X ∈Cð½0; T�; ℝþÞ such that

X ≤ aþ bX θ on ½0; T�;where a b > 0; θ > 1; a <

�1− 1

θ

�ðθbÞ −1

θ−1 and Xð0Þ≤ ðθbÞ −1θ−1. Then

X ≤θ

θ � 1a on ½0; T�:

Proof. The function f ðxÞ :¼ bxθ − xþ a is decreasing on ½0; ðbθÞ 11−θ� and increasing on

½ðbθÞ 11−θ; ∞Þ. The assumptions imply that f ððbθÞ 1

1−θÞ < 0 and f ð θθ− 1 aÞ≤ 0. As f ðXðtÞÞ≥ 0,

f ð0Þ > 0 and Xð0Þ≤ ðbθÞ 11−θ, we conclude the result by a continuity argument. -

We close this subsection with a classical result about ordinary differential equations.

Proposition 2.15. Let ε > 0. There is no real function G∈C2ðℝþÞ satisfyingGð0Þ > 0; G

0 ð0Þ > 0 and GG00 � ð1þ εÞðG0 Þ2 ≥ 0 on ℝþ:

AJMS26,1/2

132

Proof. Assume the existence of such a function. Then ðG−ð1þεÞG0 Þ0 ≥ 0 and

G0

G1þε≥

G0 ð0Þ

G1þεð0Þ > 0:

Integrating on ð0; TÞ the previous inequality, yields

0 <1

GεðTÞ≤1

Gεð0Þ � εG

0 ð0ÞG1þεð0ÞT;

which implies that T < 1ε

Gð0ÞG0 ð0Þ. This is a contradiction, which achieves the proof. -

3. Local well-posednessThis section is devoted to proving Theorem 2.1 about local well-posedness of the high-orderheat problem (1.1). The result follows by a standard fixed point argument. Take theadmissible couple ðq; rÞ :¼ ð 4ð1þpÞ

ðp− 1Þðnk− 2Þ;

pþ11þk

n ðp− 1ÞÞ. Let us start with an intermediary result.

Lemma 3.1. Take u0 ∈Hk. There exist T > 0 and a unique u∈LqTðHk;rÞ solution to (1).

Proof. For R;T > 0 consider the space

XT;R :¼nu∈Lq

T

Hk;r

�s: t kuk

Lq

TðHk;rÞ ≤Ro

endowed with the complete distance

dðu; vÞ :¼ ku� vkLqTðLrÞ:

Take the function

~v :¼ fðvÞ :¼ e−tð−ΔÞk

u0 þZ t

0

e−ðt−sÞð−ΔÞk��vjp−1vds:

We prove that f is a contraction of XT;R, for some positive T;R.Let u; v∈XT;R and w :¼ u− v. Then, using the equality

1

r0 ¼ ðp� 1Þ

�1

r� k

n

�þ 1

r;

we get by Sobolev injection

��w��vjp�1 þ ��ujp�1��r0(kwkr

0@kvkp−1rn

n−krþ kukp−1rn

n−kr

1A

(kwkrkvkp−1

Hk;r þ kukp−1Hk;r

�:

Since p≤ p*, there exists α > 0 such that α ¼ ∞ if and only if p ¼ p* and

1

α:¼ 1� 1þ p

q:


equation

133

Thanks to Strichartz estimate

kWkLqðI ; LrÞ(��wðjvjp�1 þ jujp�1Þ��

Lq0 ðI ;Lr0 Þ

(Τ1αkwkLqðI ;LrÞ½kvkp−1

LqðI ;Lrnr−kÞ

þ kukp−1LqðI ;L

rnr−kÞ

�

(Τ1αkwkLqðI ; LrÞ½kvkp−1LqðI ;Hk;rÞ þ kukp−1

LqðI ;Hk;rÞ�

(Τ1αRp−1kwkLqðI ;LrÞ:

(3.4)

Applying the previous inequality for v ¼ 0, yields

kukLqðI ; LrÞ(��e�tð�ΔÞku0

��LqðI ;LrÞ

þ T1αRp−1kukLqðI ;LrÞ

≤Cku0k þ CT1αRp:

Write now, for jαj ¼ k, ��∇k~u��LqðI ;LrÞ(k ~u0k _H

k þ��∇kðupÞ��

Lq0 ðI ;Lr0 Þ

(k ~u0k _Hk þ ðIÞ

Denoting PjðαÞ :¼ fαi ∈ ðN*Þj such thatPj

i¼1αi ¼ αg, we get

ðIÞ(Xkj¼1

XPjðαÞ

��up�jYji¼1

vαiu

��Lq

0 ðI ; Lr0 Þ:

Take the real numbers

1

a0:¼ 1

r� k

n;1

ai:¼ 1

r� k� jαij

n:

Then

p� j

a0þXj

i¼1

1

ai¼ 1

r0 :

With H€older inequality,

ðIÞ(Xkj¼1

XPjðαÞ

��up�jYji¼1

vαi u

��Lq

0 ðI ;Lr0 Þ

(T1α

Xkj¼1

XPjðαÞ

kukp−jLqðI ; La0 ÞYji¼1

kvαi ukLqðI ; Lai Þ

(T1α

Xkj¼1

XPjðαÞ

kukp−jLqðI ; L

rnn−rkÞ

Yji¼1

kukLqðI ; _Hαi ;ai Þ:

AJMS26,1/2

134

Taking account of Sobolev embedding

ðIÞ(T1α

Xkj¼1

XPjðαÞ

kukp−jLqðI ; _Hk;rÞ

Yji¼1

kukLqðI ; _Hk;rÞ

(T1α

Xkj¼1

XPjðαÞ

kukp−jLqðI ; _Hk;rÞ

kukLqðI ; _Hk;rÞ

(T1αkukp

LqðI ; _Hk;rÞ

(T1αRp:

Then

k~ukLqðI ;Hk;rÞ ≤Cku0kHk þ CT

1αRp: (3.5)

If p < p*, 1α > 0, so choosing R :¼ 2Cku0kHk and T > 0 small enough, it follows that f is acontraction ofXT;R. If p ¼ pc using previous computation with the fact that whenT vanishes,��e−tð−ΔÞku0��

Lq

TðHk;rÞ

→ 0, it follows that f is a contraction of XT;R for small time. Thanks to

Picard fixed point theorem, existence of a solution of (1.1) is proved. For uniqueness of such asolution, it is sufficient to apply (3.4) and use a translation argument. -

Lemma 3.2. Take u0 ∈Hk and u∈LqTðHk;rÞ be a solution of (1.1). Then, u∈CTðHkÞ∩Lq1

T

ðHk;r1Þ for any admissible couple ðq1; r1Þ.Proof. Take 0 < t1; t2 < T, by Strichartz estimate via the integral formula

kuðt1Þ � uðt2ÞkHk(

��Z t2

t1

e�ðt�sÞð�ΔÞkðjujp�1uÞds

��L∞ððt1 ; t2Þ;HkÞ

(kupkLq

0 ððt1 ; t2Þ;Hk;r0 Þ

(ðt1 � t2Þ1αkukp

Lqððt1 ; t2Þ; _Hk;rÞ:

This completes the proof. -Let us prove unconditional uniqueness in the sub-critical case. Take σ :¼ 1þ p and an

admissible couple ða; σÞ. With Strichartz estimate

k~wkLaðI ;LσÞ(��w�jvjp�1 þ jujp�1��

La0 ðI ;Lσ0 Þ

(T1−2akwkLaðI ; LσÞ

hkvkp−1L∞ðI ;LσÞ þ kukp−1L∞ðI ;Lσ Þ

i

(T1−2akwkLaðI ; LσÞ

hkvkp−1

L∞ðI ;HkÞ þ kukp−1L∞ðI ;HkÞ

i

(T1−2a Rp−1kwkLaðI ;LσÞ:


equation

135

The sub-critical condition implies that σ < 1þ pc, which gives a < 2. Then, unconditionaluniqueness is established via the last inequality.

Now, for t ∈ ð0; T*Þ, taking account of (3.5), if there exists R > 0 such that

CkuðtÞkHk þ CðT � tÞ1α Rp≤R;

then, T < T*. Thus, for any R > 0,

CkuðtÞkHk þ CðT* � tÞ1α Rp≤R;

Choosing R :¼ 2CkuðtÞkHk, it follows that

ðT* � tÞ1αkuðtÞkp−1Hk ≥C:

Let us prove that the maximal solution of (1.1) is global in the sub-critical defocusing case.The global existence is a consequence of the energy decay and previous calculations. Letu∈Cð½0; T*Þ; HkÞ be the unique maximal solution of (1.1). We prove that u is global. Bycontradiction, suppose that T* < ∞. Consider for 0 < s < T*, the problem

ðPsÞ�

_vþ ð�ΔÞkvþ vþ jvjp−1v ¼ 0;vðs; :Þ ¼ uðs; :Þ:

Using the same arguments of local existence, we can find a real τ > 0 and a solution v to ðPsÞon Cð½s; sþ τ�; HkÞ. Thanks to the energy decay, we see that τ does not depend on s. Thus, ifwe let s be close to T* such that T* < sþ τ, this fact contradicts the maximality of T*.

Let us prove that u∈Cðℝþ; HkÞ, the global solution to (1.1) for c ¼ −e ¼ 1 and1 < p < p* satisfies an exponential decay in the energy space.

Denoting the quantity KðuðtÞÞ :¼ kuðtÞk2Hk

Rℝn juðtÞj1þp

dx, yields

EðuðtÞÞ≤KðuðtÞÞ≤ ðpþ 1ÞEðuðtÞÞ:On the other hand, for T > 0,Z T

t

KðuðsÞÞds ¼ 1

2

�kuðtÞk2 � kuðTÞk2

≤1

2kuðtÞk2

≤EðuðtÞÞ:So, Z T

t

EðuðsÞÞds(Z T

t

KðuðsÞÞds(EðuðtÞÞ:

Thus, for some positive real number T0 > 0,

yðtÞ :¼Z ∞

t

EðuðsÞÞds(EðuðtÞÞ≤� T0y

0 ðtÞ

AJMS26,1/2

136

This implies that, for t ≥T0,

yðtÞ≤ yðT0Þe1−tT0 ≤T0EðuðT0ÞÞe1−

tT0 :

Taking account of the monotonicity of the energy, for large T > 0,Z T

t

EðuðsÞÞds≥Z tþT0

t

EðuðsÞÞds≥T0Eðuðt þ T0ÞÞ:

Then,

Eðuðt þ T0ÞÞ≤EðuðT0ÞÞe1−tT0 :

Finally,

kuðt þ T0Þk2Hk(Eðuðt þ T0ÞÞ≤EðuðT0ÞÞe1−tT0 :

The proof is finished.

4. Global well-posedness in the critical caseThis section is devoted to prove Theorem 2.2 about global well-posedness of the critical high-order heat type equation (1.1). Denote the norms

kukZ ðIÞ :¼ kukL2p

* ðI ;L2p* Þ;

kukMðIÞ :¼��∇ku

��L2p

* ðI ;L2nðnþ2kÞn2þ4k2 Þ

;

kukW ðIÞ :¼ k∇ukL2p

* ðI ;L2nðnþ2kÞn2þ4k2 Þ

;

kukNðIÞ :¼ k∇ukL2ðI ;L

2nnþ2kÞ

:

Let us start with an intermediary result.

Lemma 4.1. The following continuous injection holds.

kukW ðIÞ↪kukZðIÞ:

Proof. Write

kuk2p* ¼ ku2ð1− 1pcÞk

pc

2p*

Pc

(ku2ð1− 1pcÞk

pc

2p*

_H1

(k∇uu2ð1� 1pcÞ�1k

pc

2p*

(

�ku2ð1� 1

pcÞ�1k 2p*

2ð1� 1pcÞ�1

k∇uk 2p*

pc�2ð1� 1pcÞ

� pc

2p*

(kukpc−2

2p*

2p*k∇uk

pc

2p*

2p*

pc−2ð1− 1pcÞ:


equation

137

Then

kukZðIÞ(kkukpc−2

2p*

2p*k∇uk

pc

2p*

2p*

pc−2ð1− 1pcÞkL2p

* ðIÞ

(kukpc−2

2p*

Z ðIÞk∇ukpc

2p*

L2p* ðI ;L

2p*

pc−2ð1− 1pcÞÞ

(k∇ukL2p

* ðI ; L2p*

pc�2ð1� 1pcÞÞ: ▪

Proposition 4.2. Take the critical case p :¼ p* and I an interval containing zero. Thereexists δ > 0 such that for any u0 ∈Hk satisfying

ke−tð−ΔÞku0kW ðIÞ < δ;

there exists a unique solution u∈CðI ; HkÞ to (1.1). Moreover,

kukW ðIÞ ≤ 2δ; kukMðIÞ þ kukL∞ðI ;HkÞ ≤C

�ku0kHk þ δp*: (4.6)

Proof. First, we establish the existence of a local solution to (1.1) by a fixed point argument.For M :¼ Cku0kHk, T > 0 and I :¼ ð0; TÞ, take the set

XM ; δ :¼ fv∈MðIÞ; kvkW ðIÞ ≤ 2δ; kvkL2ð2kþnÞ

n ðI ;L2ð2kþnÞn Þ

≤ 2Mg

endowed with the complete distance

dðu; vÞ :¼ ku� vkL2ð2kþnÞ

n ðI ;L 2ð2kþnÞn Þ:

Take the function

~v :¼ fðvÞ :¼ e−tð−ΔÞk

u0 þZ t

0

e−ðt−sÞð−ΔÞk jvjpc−2vds:

Let us prove that for some positive M ; δ; f is a contraction of XM ; δ.We establish that XM ;δ is stable by f for some small positiveM ; δ. Let v∈XM ;δ. Compute,

using Strichartz and H€older inequalities

k~vkL2ð2kþnÞ


(ku0k þ kvp*kL2ð2kþnÞ4kþn ðI ;L

2ð2kþnÞ4kþn Þ

(ku0k þ kvkL2ð2kþnÞ

n ðI ;L2ð2kþnÞ

n Þ

��vpc�2��L2kþn2k ðI ;L

2kþn2k Þ



kvkpc−2L2p

* ðI ;L2p* Þ



kvkpc−2Z ðIÞ

≤Mð1þ δpc−2Þ

AJMS26,1/2

138

On the other hand

k~vkW ðIÞ(ke�itðΔÞku0kW ðIÞ þ kvjvjpc�2kNðIÞ

(M þ k∇vvpc�2kL2ðI ;L

2n2kþnÞ

(M þ kvkpc−2Z ðIÞ kvkW ðIÞ

(M þ δp*:

Always using Strichartz estimate

k~vkMðIÞ(��∇ku0

��þ ��∇kðvjvjpc�2Þ��L2ðI ;L

2nnþ2kÞ

(ku0k _Hkþ��∇kðvjvjpc�2Þ��

L2ðI ;L2n

nþ2kÞ:

Using Faa-di bruno [4] identities, we get

∇kðvp*Þ ¼

Xki¼1

vp*−iXks¼1

XPE ðνÞ

ν!Ykj¼1

�vl jvkj

kj!ðlj!Þkj

where in PEðνÞ, we havePk

j¼1kj ¼ i,Pk

j¼1kjlj ¼ νand jνj ¼ k. Then, it is sufficient to estimatethe term

kvp*�iYkj¼1

ðvl jvÞkjkL2ðI ;L

2nnþ2kÞ

:

Taking the choice

αj :¼ 2p*

kj;

1

βj¼ kj

�jljjnþ 1

2p*

�;

it follows that

1

2¼ p* � i

2p*þXkj¼1

1

αj

¼ 1

2� i

2p*þXkj¼1

1

αj

;

1

2þ k

n¼ nþ 2k

2n¼ p* � i

2p*þXkj¼1

1

βj¼ 1

2� i

2p*þXkj¼1

1

βj:

Thus, with H€older inequality

kvp*�iYkj¼1

ðvl jvÞkjkL2ðI ;L

2nnþ2kÞ

≤ kvkp*−iZðIÞYkj¼1

kvl jvkkjLkjαjðI ;L

kjβj Þ :

With Sobolev injection, yields

Wk;2nðnþ2kÞn2þ4k2 ↪W

k−nð n2þ4k2

2nðnþ2kÞ−1

kjβjÞ; kjβj

↪W jljj;kjβj :

This implies that


equation

139

k~vkMðIÞ(ku0k _Hk þXki¼1

kvkp*−iZ ðIÞYkj¼1

kvl jvkkjLkjαjðI ;L

kjβj Þ

(ku0k _Hk þXki¼1

kvkp*−iZðIÞkvkiMðIÞ:

This finishes the stability of XM ;δ. Now, let u; v∈XM ;δ and w :¼ u− v. Then

dðu; vÞ(��wðvpc�2 þ upc�2Þ��L2ð2kþnÞ4kþn ðI ;L

2ð2kþnÞ4kþn Þ

(kwkL2ð2kþnÞ

n ðI ;L2ð2kþnÞ

n Þ½��vpc�2

��L2kþn2k ðI ;L

2kþn2k Þ

þ ��upc�2��L2kþn2k ðI ;L

2kþn2k Þ

�

(½kvkpc−2Z ðIÞ þ kukpc−2Z ðIÞ �dðu; vÞ:Then, using Lemma 4.1, we get

dðu; vÞ(δpc−2dðu; vÞ:This proves the contraction via taking small δ; M > 0. -

Now, let us prove global existence.By Strichartz estimate, if u exists on ½0; t0� and satisfies ku0k _H

k small enough, we can use

(4.6) to extend uon ½t0; t0 þ 1�. Hence, in order to prove global well-posedness, it is sufficient toprove that ku0k _H

k remains small on the whole ½0; T*Þ. Let a positive time t < T*. With the

decay of energy and Sobolev injection, yields

2EðuðtÞÞ ¼ ��∇ku0��2 þ 2μ

pc

Zju0jpcdx

(��∇ku0

��2 þ ��∇ku0��pc :

Then,

��∇kuðtÞ��2 ¼ 2EðuðtÞÞ þ 2

pc

ZjuðtÞjpcdx

(��∇ku0

��2 þ ��∇ku0��2pc þ ��∇kuðtÞ��pc :

The proof is closed via Lemma 2.14.Let us finish this section by proving the decay of solutions. Using the previous

proposition, it follows that

u∈MðℝþÞ∩W ðℝþÞ:Using previous computation and denoting vðtÞ :¼ Tkð−tÞuðtÞ, we get for t; t 0 → þ∞,

kvðtÞ � vðt0 Þk _Hk(

Z t0

t

Tkð�sÞ�jujpc�2uds��

_Hk

(Xki¼1

kukp*−iZ ðt; t0 Þkuk

i

Mðt;t0 Þ → 0:

Finally, taking account of Sobolev embeddings and denoting f :¼ limt→þ∞

vðtÞ in _Hk, yields

AJMS26,1/2

140

kuðtÞkp ≤ kuðtÞ � TkðtÞfkp þ kTkðtÞfkp(kuðtÞ � TkðtÞfk _H

k þ kTkðtÞfkp(kvðtÞ � fk _H

k þ kTkðtÞfkp:

Thanks to the smoothing effect (2.3), the decay is proved.

5. Existence of a ground stateThe goal of this section is to prove that the elliptic problem

−ð−ΔÞkf� cfþ jfjp−1f ¼ 0; f∈Hkrd

has a ground state in the meaning that it has a nontrivial positive radial solution whichminimizes of the energy when Kα;β vanishes. Let us define the quantities

fλ:¼ eαλfðe−βλ:Þ;

Lα;βEðfÞ :¼ vλðEðfλÞÞjλ¼0 :¼ Kα;βðfÞ;

Hα;β :¼1� Lα;β

μ

�E:

With a direct calculation

Kα;βðvÞ ¼ 1

2

Z �ð2αþ ðn� 2kÞβÞ��∇kv

��2 þ ð2αþ nβÞjvj2 � 2

�αþ nβ

1þ p

�jvj1þp

�dx;

Hα;βðvÞ ¼ 1

2

�1� 2αþ ðn� 2kÞβ

μ

��∇kv��2 þ 1

2

�1� 2αþ nβ

μ

�kvk2

þ��

αþ nβ

pþ 1

�1

μ� 1

1þ p

� Z ��vj1þpdx:

Denote the quadratic part and the nonlinear parts of Kα;β,

KQα;βðvÞ :¼

Zℝn

hαþ

n2� k�β��∇kv

��2 þ αþ n

2β�jvj2idx; KN :¼ K � KQ:

Remark 5.1. Note that,

(1) in this section ðα; βÞ∈A;(2) the proof of Theorem 2.2 is based on several Lemmas;

(3) in this section, we write, for easy notation, K ¼ Kα;β;KQ ¼ KQ

α;β;KN ¼

KNα;β;L ¼ Lα;β and H ¼ Hα;β.

Lemma 5.2. We have

(1) mðLHðfÞ; HðfÞÞ > 0, for all 0≠f∈Hk;

(2) λ↦HðfλÞ is increasing.


equation

141

Proof. Compute

LHðfÞ ¼ L1�L

μ

�EðfÞ

¼ −ðL � ~μÞðL � μÞEðfÞμ

þ ~μ1�L

μ

�EðfÞ

¼ −ðL � ~μÞðL � μÞEðfÞμ

þ ~μHðfÞ:

Now, since ðL− ð2αþ βðn− 2kÞÞÞ��∇kf��2 ¼ ðL− ð2αþ nβÞÞkfk2 ¼ 0, we have ðL− ~μÞ−

ðL− μÞkfk2Hk ¼ 0. Moreover Lð��fj1þpÞ ¼ ðαð1þ pÞ þ nβÞ��fj1þp, so because ðα; βÞ∈A,

LHðfÞ≥ 1

μðL� ~μÞðL� μÞ

Z jfj1þp

1þ pdx

¼ αðp� 1Þðαðp� 1Þ þ 2kβÞμð1þ pÞ

Z ��fj1þpdx

> 0:

The first point of the Lemma follows. The last point is a consequence of the equalityvλHðfλÞ ¼ LHðfλÞ. -

The next intermediate result is the following.

Lemma 5.3. Let ðfnÞ be a bounded sequence of Hk − f0g such that limnKQðfnÞ ¼ 0. Then,

there exists n0 ∈ℕ such that KðfnÞ > 0 for all n≥ n0.

Proof. Since ðα; βÞ∈A, and KQðfnÞ vanishes at infinity, by Sobolev injection, we have

KN ðfnÞ(kfnk1þp

1þp(kfnk1þp

Hk ¼ okfnk2Hk

�:

Then KðfÞ ’ KQðfnÞ > 0. The proof is achieved. -The last auxiliary result of this section reads as follows.

Lemma 5.4.

mα;β ¼ inf0≠f∈Hk

rd

fHðfÞ; s:t KðfÞ≤ 0g: (5.7)

Proof. Let m1 be the right hand side, then it is sufficient to prove that m≤m1. Take f∈Hk

such that KðfÞ < 0 then by Lemma 5.3, the fact that limx→−∞

KQðfλÞ ¼ 0 and λ↦HðfλÞ isincreasing, there exists λ < 0 such that

KðfλÞ ¼ 0;HðfλÞ≤HðfÞ: (5.8)

The proof is closed. -

Proof of Theorem 2.4

(1) sub-critical case. Let ðfnÞ be a minimizing sequence, namely

0≠fn ∈Hkrd;KðfnÞ ¼ 0 and lim

nHðfnÞ ¼ lim

nEðfnÞ ¼ m:

AJMS26,1/2

142

� First step: ðfnÞ is bounded in Hk. First case β≥ 0. Then

kfnk2_Hk(HðfnÞ→m:

So ðfnÞ is bounded in _Hk. Assume that lim sup

nkfnk ¼ ∞. Then

kfnk2(KQðfnÞ¼ −KN ðfnÞ

(kfnk1þp

1þp

(kfnk1þp−nðp−1Þ2k

��∇kfn

��nðp−1Þ2k

(kfnk1þp−nðp−1Þ2k :

This contradiction achieves this case. Second case β < 0. Using the fact thatαðp− 1Þ þ 2kβ > 0 and Kα;βðfnÞ ¼ 0,

2μHðfnÞ ¼ −2kβkfnk2 þ1

1þ pðαðp� 1Þ þ 2kβÞ

Zjfj1þp

dx

≥1

1þ pðαðp� 1Þ þ 2kβÞ

Zjfj1þp

dx

≥ kfnk2Hk :

Then, ðfnÞ is bounded in Hk.

� Second step: m > 0.

Taking account of the compact injection of the radial Sobolev space Hkrd on the Lebesgue

space Lp for any 2 < p < pc, we take

fn →f in Hk and fn →f in Lp; ∀p∈ ð2; pcÞ:Assume that f ¼ 0, since ðfnÞ is bounded in Hk, we have

KN ðfnÞ(kfnk1þp

1þp → 0:

By Lemma 5.3, KðfnÞ > 0 for large nwhich is absurd. So

f≠ 0:

With lower semi continuity of Hk norm, we have KðfÞ≤ 0 and HðfÞ≤m. Using (8), we canassume that KðfÞ ¼ 0 and EðfÞ ¼ HðfÞ≤m. So that f is a minimizer satisfying0≠f∈Hk

rd, KðfÞ ¼ 0 and EðfÞ ¼ HðfÞ ¼ m. Thus

m ¼ HðfÞ > 0:

� f is a solution to (2).


equation

143

Now, there is a Lagrange multiplier η∈ℝ such that E0 ðfÞ ¼ ηK

0 ðfÞ. Recall thatLðfÞ :¼ ðvλfλ

α;βÞjλ¼0 and LEðfÞ :¼ ðvλEðfλα;βÞÞjλ¼0. Compute

0 ¼ KðfÞ ¼ LEðfÞ ¼ hE 0 ðfÞ;LðfÞi¼ ηhK 0 ðfÞ;LðfÞi¼ ηLKðfÞ ¼ ηL2EðfÞ:

With a previous computation

�ðL � μÞðL � ~μÞEðfÞ ¼ kp� 1

pþ 1ðkðp� 1Þ þ 2kβÞ

Z ��fj1þpdx

¼ −L2EðfÞ � ~μμEðfÞ> 0:

Thus η ¼ 0 and E0 ðfÞ ¼ 0. So, f is a ground state and m is independent of α; β.

(2) Critical case. Define the mass less action

K0α;βðfÞ :¼ Lα;βE

0ðfÞ

¼ 1

2ð2αþ ðN � 2kÞβÞ��∇kf

��2 ��αþ Nβ

pc

�kfkpcpc

¼�αþ Nβ

pc

��∇kf��2 � kfkpcpc

�

and the operator

H 0α;βðfÞ :¼

�E0 � 1

αpc þ NβK0

α;β

�ðfÞ

¼ k

N

��∇kf��2:

Let m0α;β :¼ mα;β for p ¼ p* and the real number

d0α;β :¼ inf0≠f∈Hk

nH 0

α;βðfÞ s: t K0α;βðfÞ < 0

o:

Claim. m0α;β ¼ d0α;β.

Since K0α;β ¼ 0 implies that E0 ¼ H 0

α;β, it follows that m0α;β ≥ d0α;β. Conversely, take

0≠f∈Hk such that K0α;βðfÞ < 0. Thus, when 0 < λ→ 0, we get

K0α;βðλfÞ ¼

1

2ð2αþ ðN � 2kÞβÞλ2��∇kf

��2 ��αþ Nβ

pc

�λpckfkpcpc

’ 1

2ð2αþ ðN � 2kÞβÞλ2��∇kf

��2 > 0:

AJMS26,1/2

144

So, there exists λ∈ ð0; 1Þ satisfying K0α;βðλfÞ ¼ 0 and

m0α;β ≤H 0

α;βðλfÞ ¼ λ2H 0α;βðfÞ≤H 0

α;βðfÞ:

Thus, m0α;β ≤ d0α;β.

So m0α;β ¼ d0α;β. Because of the definitions of K0

α;β and H 0α;β, it is clear that m0

α;β is

independent of ðα; βÞ and

m :¼ m0α;β ¼ inf

0≠f∈Hk

�k

N

��∇kf��2 s: t

��∇kf��2 < kfkpcpc

�:

Taking the scaling λf,

m ¼ inf0≠f∈Hk

rd

�k

Nλ2��∇kf

��2 s: t λ2−pc��∇kf

��2 < kfkpcpc�

¼ inf0≠f∈Hk

rd

8>><>>:

k

N

��∇kf��2

kfkpcpck∇kfk2

! 22−pc

9>>=>>;

¼ k

Ninf

0≠f∈Hkrd

8>><>>:��∇kf

��kfkpc

�Nk

9>>=>>;

¼ k

NðC*Þ−N

α :

Here, C* denotes the best constant of the Sobolev injection

kfkpc ≤C*��∇kf

��;is known [16] to be attained by the following explicit Q∈ _H

k,

QðxÞ :¼ a

ð1þ jxj2ÞN2−k

which solves the mass less equation

ð−ΔÞkQ ¼ Q*

6. Invariant sets and applicationsThis section is devoted to establish Theorem 2.5. The proof is based on two auxiliary results.

Lemma 6.1. The sets Ac;þα;β and Ac;−

α;β are independent of the couple ðα; βÞ.Proof. Take ðα; βÞ and ðα0

; β0 Þ inA. By Theorem 2.4, the union A

c;þα;β∪A

c;−α;β is independent of

ðα; βÞ. So, it is sufficient to prove that Ac;þα;β is independent of ðα; βÞ. If EcðvÞ < m and

Kcα;βðvÞ ¼ 0, then v ¼ 0. So, Ac;þ

α;β is open. The rescaling vλ :¼ eαλvðe−βλ:Þ implies that a

neighborhood of zero is in Ac;þα;β . Moreover, this rescaling with λ→ 0 gives that Ac;þ

α;β is

contracted to zero and so it is connected. Now, write


equation

145

Ac;þα; β ¼ Ac;þ

α; β ∩Ac;þ

α0 ; β0;∪Ac;−

α0 ; β0

�¼Ac;þ

α; β; ∩Ac;þα0 ; β0

�∪Ac;þ

α; β;∪Ac;−

α0 ; β0

�:

Since by the definition, Ac;−α;β is open and 0∈Ac;þ

α;β ∩Ac;þα0 ;β0

, using a connectivity argument, wehave Ac;þ

α;β ¼ Ac;þα0 ;β0

. The proof is ended. -

Lemma 6.2. The sets Ac;þα;β and A

c;−α;β are invariant under the flow of (1.1).

Proof. Take ðα; βÞ∈A. Let u0 ∈Ac;þα;β and u∈CT*ðHkÞ be the maximal solution of (1.1). The

proof follows with contradiction. Assume that for some time t0 ∈ ð0;T*Þ, uðt0Þ∉Ac;þα;β and

uðtÞ∈Ac;þα;β for all t ∈ ð0; t0Þ. Since the energy is decreasing and Eðuðt0ÞÞ < m, then, with a

continuity argument, there exists a positive time t1 ∈ ð0; t0Þ such that Kα;βðuðt1ÞÞ ¼ 0. Thiscontradicts the definition of m and finishes the proof in this case. The proof is similar toAc;þα;β . -

(1) Proof of the first part of Theorem 2.5. Using the two previous Lemmas via atranslation argument, we can assume that uðtÞ∈Aþ

1;1 for any t ∈ ½0;T*Þ. Takingaccount of the definition of m, we get

m > EðuðtÞÞ

> EðuðtÞÞ � 1

2þ nK1;1ðuðtÞÞ

¼ α2þ n

��∇kuðtÞ��2 þ p� 1

ð1þ pÞð2þ nÞkuðtÞk1þp

1þp:

This implies, via decay of the equality

vt�kuðtÞk2 ¼ 2K1;0ðuðtÞÞ < 0;

that

sup½0;T* �

kuðtÞkHk < ∞:

Then, u is global.Now, we prove an exponential decay. For small ku0k, since supt kuðtÞkHk(1, we get using

Gagliardo–Nirenberg inequality in Lemma 2.12,

K1;0ðuðtÞÞ ¼ kuðtÞk2Hk �Zℝn

juðtÞj1þpdx

≥ kuðtÞk2 þ kuðtÞk2_Hk � CkuðtÞkpþ1−nðp−1Þ2k kuðtÞk

nðp−1Þ2k

_Hk

≥ kuðtÞk2 þ kuðtÞk2_Hkð1� Cku0kpþ1−nðp−1Þ2k kuðtÞk

nðp−1Þ2k

_Hk Þ

≥ CkuðtÞk2_Hk

≥ CEðuðtÞÞ:

AJMS26,1/2

146

On the other hand

EðuðtÞÞ ¼ 1

2kuðtÞk2Hk � 1

1þ p

Zℝn

juðtÞj1þpdx

¼ 1

2kuðtÞk2Hk � 1

1þ p

kuðtÞk2Hk � K1;0ðuðtÞÞ

�

¼�1

2� 1

1þ p

�kuðtÞk2Hk þ 1

1þ pK1;0ðuðtÞÞ

≥CmaxnK1;0ðuðtÞÞ; kuðtÞk2Hk

o:

Moreover, for T > 0, Z T

t

K1;0ðuðsÞÞds ¼ 1

2

�kuðtÞk2 � kuðTÞk2

≤1

2kuðtÞk2

≤C EðuðtÞÞ:So, Z T

t

EðuðsÞÞds(Z T

t

K1;0ðuðsÞÞds(EðuðtÞÞ:

Thus, for some positive real number T0 > 0,

yðtÞ :¼Z ∞

t

EðuðsÞÞds(EðuðtÞÞ≤� T0y

0 ðtÞThis implies that, for t ≥T0,

yðtÞ≤ yðT0Þe1−tT0 ≤T0EðuðT0ÞÞe1−

tT0 :

Taking account of the monotonicity of the energy, for large T > 0,Z T

t

EðuðsÞÞds≥Z tþT0

t

EðuðsÞÞds≥T0Eðuðt þ T0ÞÞ:

Then,

Eðuðt þ T0ÞÞ≤EðuðT0ÞÞe1−tT0 :

Finally,

kuðt þ T0Þk2Hk(Eðuðt þ T0ÞÞ≤EðuðT0ÞÞe1−tT0 :

The proof is finished.


equation

147

(2) Proof of the second part of Theorem 2.4. Using the two previous Lemmas via atranslation argument, we can assume that uðtÞ∈Ac;−

1; λ for any t ∈ ½0;T*Þ and anyλ > 0. Take the real function

LðtÞ :¼ 1

2

Z t

0

kuðsÞk2ds; t ∈ ½0; T*Þ:

Using Eq. (1.1), a direct computation gives

L00 ðtÞ ¼

Zℝn

_uudx ¼ −kuðtÞk2_Hk � ckuðtÞk2 þZℝn

juj1þpdx:

We discuss two cases.

(a) First case: Ecðu0Þ > 0. For any λ > 0,

H1; λðuÞ ¼ 1

2þ Nλ

�kλ��∇ku

��2 þ p� 1

pþ 1

Zℝn

jujpþ1dx

�> m:

Thus, for any ε > 0,

L00 ¼ ε

��∇ku��2 � ð1þ εÞ��∇ku

��2 � ckuðtÞk2 þZℝn

��ujpþ1dx

>εk

��2

λþ n

�m� 1

λ

p� 1

pþ 1

Zℝn

��ujpþ1dx

�

� 2ð1þ εÞ�Ecðu0Þ þ 1

2ð1þ pÞZ ��ujpþ1

dx

�

þ 2ð1þ εÞZ t

0

k _uðsÞk2dsþZℝn

��ujpþ1dx

>

�εk

�2

λþ n

�m� 2ð1þ εÞEcðu0Þ

�þ�1� 1þ ε

1þ p� εðp� 1Þkλðpþ 1Þ

�

3

Zℝn

��ujpþ1dxþ 2ð1þ εÞ

Z t

0

k _uðsÞk2ds

:¼ ðIÞ þ ðIIÞpþ 1

Zℝn

��ujpþ1dxþ 2ð1þ εÞ

Z t

0

k _uðsÞk2ds:

Taking λ :¼ aε and γ :¼ m−Ecðu0Þ, we get

ðIÞ ¼ 2γð1þ εÞ þm

�2

ka� 2þ ε

��2þ N

k

��

¼ ε�2γ � 2mþ Nm

k

�þ 2m

�1

ka� 1

�þ 2γ:

AJMS26,1/2

148

On the other hand,

ðIIÞ ¼ pþ 1� ð1þ εÞ � p� 1

ka

¼ ðp� 1Þ�1� 1

ka

�þ 1� ε:

The choice 1kp− 1p− ε < a < 1

k, via ε > 0 near to zero implies that the terms ðIÞ and ðIIÞ are non

negative. Thus,

L00> 2ð1þ εÞ

Z t

0

ku _uðsÞk2ds:

Thanks to Cauchy–Schwarz inequality, it follows that

LL00> ð1þ εÞk _uk2

L2t ðL2Þkuk

2

L2t ðL2Þ

> ð1þ εÞku _uk2L1t ðL1Þ

> ð1þ εÞL02:

Indeed, if LðtÞ ¼ 0 for some positive time, we get u0 ¼ Eðu0Þ ¼ 0, which is a contradiction.Thus

ðL−εÞ0 0 ¼ −εL−ε−2�L

00L� ð1þ εÞðL00 Þ2��0:

Taking account of Proposition 2.15, for some finite time T > 0,

lim supt→T

Z T

0

kuðsÞk2ds ¼ ∞:

Thus, T* < ∞ and u is not global. This ends the proof.

(b) Second case: Ecðu0Þ≤ 0. Compute

L00 ¼ −kuk2_Hk � ckuk2 þ

Zℝn

��ujpþ1dx

≥ ð2þ εÞ�Z

ℝn

jujpþ1

pþ 1dx� 1

2kuk2_Hk

c

2kuk2

�

≥� ð2þ εÞEcðuÞ:

So, thanks to the identity _EcðuÞ ¼ −k _uk2, we get

L00≥ ð2þ εÞ

k _uk2

L2t ðL2Þ � Ecðu0Þ

�: (6.10)

Now, the proof goes by contradiction assuming that T* ¼ ∞.

Claim 1. There exists t1 > 0 such thatR t10 k _uðsÞk2ds > 0.

Indeed, otherwise uðtÞ ¼ u0 almost everywhere and solves the elliptic stationary equationð−ΔÞkuþ cu ¼ ��ujp−1u. Therefore, kuk2_Hk þ Ckuk2 ¼ R

ℝn

��ujpþ1dx and


equation

149

ku0k2_Hk þ cku0k2 � 2

pþ 1

Zℝn

��u0jpþ1dx ¼

�1� 2

pþ 1

�Zℝn

��u0jpþ1dx ¼ 2Eðu0Þ≤ 0:

Then, u0 ¼ 0 which contradicts the fact that K0;1ðu0Þ < 0.

Claim 2. For any 0 < α < 1, there exists tα > 0 such that

ðL0 � L0 ð0ÞÞ2 ≥ αL

02; on ðtα;∞Þ:The claim immediately follows from the first one and (6.10) observing that

limt→∞

LðtÞ ¼ limt→∞

L0 ðtÞ ¼ þ∞:

Claim 3. One can choose α ¼ αðεÞ such that

LL00≥ ð1þ αÞL02; on ðtα;∞Þ:

Indeed, we have

LL00≥

2þ ε2

kuk2L2t ðL2Þk _uk

2

L2t ðL2Þ

≥2þ ε2

ku _uk2L1t ðL1Þ

≥2þ ε2

ðL0 � L0 ð0ÞÞ2

≥ð2þ εÞα

2L

02;

where we used (6.10) in the first estimate, Cauchy–Schwarz inequality in the second andClaim 2 in the last one. Now choosing α such that 1 < ð2þεÞα

2:¼ 1þ ε, we get

LL00> ð1þ εÞL02; for large time:

Thanks to Proposition 2.15, this ordinary differential inequality blows up in finite time andcontradicts our assumption that the solution is global. This ends the proof.

7. Strong instabilityThis section is devoted to prove Theorem 2.5 about strong instability of stationary solutionsto (1.1). Take here and hereafter c ¼ e ¼ 1. Denote the scaling uλ :¼ λ

N2uðλ:Þ. Let us write an

auxiliary result.

Lemma 7.1. Let u∈Hk such that K1;−2nðuÞ≤ 0. Then, there exists λ0 ≤ 1 such that

(1) K1;−2nðuλoÞ ¼ 0;

(2) λ0 ¼ 1 if and only if K1;−2nðuÞ ¼ 0;

(3) vvλEðuλÞ > 0 for λ∈ ð0; λ0Þ and v

vλEðuλÞ < 0 for λ∈ ðλ0;∞Þ;(4) λ→EðuλÞ is concave on ðλ0;∞Þ;(5) v

vλEðuλÞ ¼ N2λK1;−2

nðuλÞ.

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150

Proof. With direct computations, we have

K1;−2nðuλÞ ¼ 2kλ2k

N

��∇ku��2 �

�1� 2

1þ p

�λN2ðp−1Þ

Zℝn

��uj1þpdx;

vλEðuλÞ ¼ N

2λK1;−2

nðuλÞ;

which proves ð5Þ. Now

K1;−2nðuλÞ ¼ 2kλ2k

N

24��∇ku

��2 � N

k

�1

2� 1

1þ p

�λN2ðp−1Þ−2k

Zℝn

��uj1þpdx

35:

Amonotonicity argument via the inequality p < p* closes the proof of ð1Þ; ð2Þand ð3Þ. For ð4Þ,it is sufficient to compute using ð3Þ. -Lemma 7.2. Let f be a ground state solution of (2.2), λ > 1 a real number close to one anduλ ∈Cð½0;T*Þ;HkÞ be the solution to (1.1) with data fλ. Then, for any t ∈ ð0;T*Þ,

EðuλðtÞÞ < EðfÞ and K1;−2nðuλðtÞÞ < 0:

Proof. By Lemma 7.1, we have

EðfλÞ < EðfÞ and K1;−2nðfλÞ < 0:

Moreover, thanks to the decay of energy, it follows that for any t > 0,

EðuλðtÞÞ≤EðfλðtÞÞ < EðfÞ:Then K1;−2

nðuλðtÞÞ≠ 0 because f is a ground state. Finally K1;−2

nðuλðtÞÞ< 0 with a continuity

argument. -Now, we are ready to prove the instability result.Take uλ ∈CT*ðHkÞ the maximal solution to (1.1) with data fλ, where λ > 1 is close to one

and f is a ground state solution to (2.2). With the previous Lemma, we get

uλðtÞ∈A−

1;−2n; for any t ∈ ð0;T*Þ:

Then, using Theorem 2.5, it follows that

lim supt→T*

kuλðtÞkkH ¼ ∞:

The proof is finished via the fact that

limλ→1

kfλ � fkHk ¼ 0:

References

[1] D.R. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2] F. Bernis, A. Friedman, Higher order nonlinear degenerate parabolic equations, J. DifferentialEquations 83 (1990) 179–206.

[3] H. Brezis, T. Cazenave, A nonlinear heat equation with singular initial data, J. d’Anal. Math. 68(1996) 73–90.

[4] G.M. Constantine, T.H. Savitis, A multivariate Faa Di Bruno formula with applications, Trans.Amer. Math. Soc. 348 (2) (1996) 503–520.


equation

151

[5] A. Cotsiolis, N.K. Tavoularis, Best constants for Sobolev inequalities for higher order fractionalderivatives, J. Math. Anal. Appl. 295 (2004) 225–236.

[6] J. Davila, M.D. Pino, Y. Sire, Non degeneracy of the bubble in the critical case for non localequations, Proc. Amer. Math. Soc. 141 (2013) 3865–3870.

[7] S.D. Eidel’man, Parabolic systems, in: Translated from the Russian by Scripta Tech- nica, North-Holland Publishing, London, Amsterdam, 1969.

[8] V.A. Galaktionov, Critical global asymptotics in high-order semilinear parabolic equations, Int. J.Math. Math. Sci. 60 (2003) 3809–3825.

[9] V.A. Galaktionov, S.I. Pohozaev, Existence and blow-up for higher-order semilinear parabolicequations: Majorizing order-preserving operators, Indiana Univ. Math. J. 51 (6) (2002) 1321–1338.

[10] A. Haraux, F.B. Weissler, Non uniqueness for a semilinear initial value problem, Indiana Univ.Math. J. 31 (1982) 167–189.

[11] S. Ibrahim, M. Majdoub, R. Jrad, T. Saanouni, Local well posedness of a 2D semilinear heatequation, Bull. Belg. Math. Soc. Simon Stevin 21 (3) (2014) 535–551.

[12] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998) 955–980.

[13] P.L. Lions, Symetrie et compacit�e dans les espaces de Sobolev, J. Funct. Anal. 49 (1982) 315–334.

[14] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super Pisa Cl. Sci. 13 (1955)116–162.

[15] L.E. Payne, D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, IsraelJ. Math. 22 (3–4) (1975) 273–303.

[16] L.A. Peletier, W.C. Troy, Spatial Patterns. Higher Order Models in Physics and Mechanics, in:Progress in Nonlinear Differential Equations and their Appli- cations, vol. 45, Birkhuser Boston,Massachusetts, 2001.

[17] T. Saanouni, Global well-posedness and finite time blow-up of some heat type equations, Proc.Edinb. Math. Soc. 60 (2017) 481–497.

[18] F.B. Weissler, Local existence and nonexistence for a semilinear parabolic equation in Lp, IndianaUniv. Math. J. 29 (1980) 79–102.

[19] F.B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,Israel J. Math. 38 (1981) 29–40.

[20] Z. Zhai, Strichartz type estimates for fractional heat equations, J. Math. Anal. Appl. 356 (2009)642–658.

Corresponding authorTarek Saanouni can be contacted at: [email protected]


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Approximative K-atomicdecompositions and frames in

Banach spacesShah Jahan

Department of Mathematics, Ramjas College, University of Delhi, Delhi, India

AbstractL. Gǎvruta (2012) introduced a special kind of frames, named K-frames, where K is an operator, in Hilbertspaces, which is significant in frame theory and has many applications. In this paper, first of all, we haveintroduced the notion of approximative K-atomic decomposition in Banach spaces. We gave twocharacterizations regarding the existence of approximative K-atomic decompositions in Banach spaces.Also some results on the existence of approximative K-atomic decompositions are obtained. We discussseveral methods to construct approximative K-atomic decomposition for Banach Spaces. Further,approximative Xd-frame and approximative Xd-Bessel sequence are introduced and studied. Twonecessary conditions are given under which an approximative Xd-Bessel sequence and approximativeXd-frame give rise to a bounded operator with respect to which there is an approximative K-atomicdecomposition. Example and counter example are provided to support our concept. Finally, a possibleapplication is given.

Keywords Frames, K-frames, Atomic decomposition, K-atomic decomposition, Xd-Bessel sequence,

Xd-frames


1. Introduction and preliminariesFourier transform has been amajor tool in analysis for over a century. It has a serious lackingfor signal analysis in which it hides its phase information concerning themoment of emissionand duration of a signal. What actually needed was a localized time frequency representationwhich has this information encoded in it. In 1946, Dennis Gabor [14] filled this gap andformulated a fundamental approach to signal decomposition in terms of elementary signals.On the basis of this development, in 1952, Duffin and Schaeffer [10] introduced frames forHilbert spaces to study some deep problems in non-harmonic Fourier series. In fact, theyabstracted the fundamental notion of Gabor for studying signal processing. Let H be a real

K-atomicdecompositionsand frames inBanach spaces

153

JEL Classification — 42A38, 46B15, 42C15, 42C30© Shah Jahan. Published in Arab Journal of Mathematical Sciences. Published by Emerald


The author would like to thank referees whose reports led to an improvement in the presentation ofthis manuscript.

The publisher wishes to inform readers that the article “Approximative K-atomic decompositionsand frames in Banach spaces”was originally published by the previous publisher of theArab Journal ofMathematical Sciences and the pagination of this article has been subsequently changed. There has beenno change to the content of the article. This change was necessary for the journal to transition from theprevious publisher to the new one. The publisher sincerely apologises for any inconvenience caused. Toaccess and cite this article, please use Jahan, S. (2019), “Approximative K-atomic decompositions andframes in Banach spaces”, Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 153-166. Theoriginal publication date for this paper was 08/04/2019.



Received 17 November 2018Revised 9 February 2019Accepted 29 March 2019


Vol. 26 No. 1/2, 2020pp. 153-166


DOI 10.1016/j.ajmsc.2019.03.003

(or complex) separable Hilbert space with inner product h:; :i. A countable sequence ffkg⊂His called a frame for the Hilbert spaceH, if there exist positive constants A;B > 0 such that

Ajjf jj2H ≤X∞n¼1

jhf ; fnij2 ≤ Bjjf jj2H; for all f ∈H (1.1)

The positive numbers A and B are called the lower and upper frame bounds of the frame,respectively. These bounds are not unique. The inequality in (1.1) is called the frameinequality of the frame. If ffng is a frame for H then the following operators are associatedwith it.

(a) Pre-frame operator T : l2ðℕÞ→H is defined as Tfcng∞n¼1 ¼P∞

k¼1cnfn; fcng∞n¼1 ∈

l2ðℕÞ.(b) Analysis operator T* : H→ l2ðℕÞ;T*f ¼ fhf ; fkig∞k¼1 f ∈H.

(c) Frame operator S ¼ TT* ¼: H→H; Sf ¼P∞

k¼1hf ; fkifk; f ∈H. The frameoperator S is bounded, linear and invertible on H. Thus, a frame for H allows eachvector inH to be written as a linear combination of the elements in the frame, but thelinear independence between the elements is not required; i.e for each vector f ∈Hwehave,

f ¼ SS−1f ¼X∞k¼1

hf ; fkifk:

For more details related to frames and Riesz bases in Hilbert spaces, one may refer to [4,6].These ideas did not generate much interest outside of non-harmonic Fourier series and signalprocessing for more than three decades until Daubechies et al. [9] reintroduced frames. Afterthis landmark paper the theory of frames begin to be studied widely and found manyapplications to wavelet and Gabor transforms in which frames played an important role.Feichtinger and Gr€ocheing [12] extended the idea of Hilbert frames to Banach spaces andcalled it atomic decomposition. A more general concept called Banach frame was introducedby Gr€ocheing [18] and were further studied in [22,33]. Banach frames were developed for thetheory of frames in the context of Gabor and Wavelet analysis. Christensen and Heil [7]studied some perturbation results for Banach frames and atomic decompositions.

In particular, frames which are widely used in sampling theory in [2] amount to theconstruction of Banach frames consisting of reproducing kernels for a large class of shiftinvariant spaces. Aldroubi et al. [1] used Banach frames in various irregular samplingproblems. Eldar and Forney [11] used tight frames for quantummeasurement. Gr€ochenig [19]emphasized that localization of a frame is a necessary condition for its extension to a Banachframe for the associated Banach spaces. He also observed that localized frames are universalBanach frames for the associated family of Banach spaces. Fornasier [13] studied Banachframes for α-modulation spaces. In fact, he gave a Banach frame characterization for theα-modulation spaces. Shah et al. [21] defined and studied Banach frames to a new geometricnotation; in fact they gave a sufficient condition and a necessary condition for a coneassociated with a Banach frame to be a generating cone.

Casazza et al. [5] studied X d-frames and Xd-Bessel sequences in Banach spaces. Stoeva[30] gave some perturbation results for Xd-frames and atomic decompositions. Kaushik andSharma [23] studied approximative atomic decompositions in Banach spaces. For furtherstudies related to approximative frame one may refer [20,24,28]. Gavruta [15], introduced andstudied atomic system for an operatorK and the notion ofK-frame in a Hilbert space, see also[16]. Frames for operators in Banach spaces were further studied in [8,17,25]. Xiao et al. [32]

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discussed relationship betweenK-frames and ordinary frames in Hilbert spaces. Poumai andJahan [26] introduced K-atomic decompositions in Banach spaces.

Outline of the paper. In this paper, we have introduced the notion of approximativeK-atomic decomposition in Banach spaces. We gave two characterizations regarding theexistence of approximativeK-atomic decompositions in Banach spaces. Also some results onthe existence of approximative K-atomic decompositions are obtained. We discuss severalmethods to construct approximative K-atomic decomposition for Banach Spaces. Further,approximativeXd-frame and approximativeXd-Bessel sequence are introduced and studied.Two necessary conditions are given under which an approximative Xd-Bessel sequence andapproximative Xd-frame give rise to bounded operators with respect to which there is anapproximative K-atomic decomposition. Example and counter example are provided tosupport our concept of approximative K-atomic decomposition. Finally, we gave a possibleapplication of our work.

Next we give some basic notations. Throughout this paper, X will denote a separableBanach space over the scalar field K(ℝ or ℂ), X* the dual space of X, Xd a BK-space andLðX ;YÞwill denote the space of all bounded linear operators fromX intoY. ForT ∈LðXÞ,T*

denotes the adjoint of T, π : X →X** is the natural canonical projection from X onto X **.Also Ty denote the pseudo inverse of the operator T. Note that TTy f ¼ f for all f ∈RðKÞ.Throughout RðKÞ is closed.

A sequence space S is called a BK-space if it is a Banach space and the co-ordinatefunctionals are continuous on S. That is the relations xn ¼ fαðnÞj g, x ¼ fαjg∈S,limn→∞xn ¼ x imply limn→∞α

ðnÞj ¼ αjðj ¼ 1; 2; 3; . . .Þ.

Definition 1.1. ([18]). Let X be a Banach space and Xd be a BK-space. A sequenceðxn; fnÞðfxng⊂X ; ffng⊂X*Þ is called an atomic decomposition forX with respect toXd if thefollowing statements hold:

(a) ffnðxÞg∈Xd, for all x∈X.

(b) There exist constants A and Bwith 0 < A≤B < ∞ such that

AjjxjjX ≤ jjffnðxÞgjj ≤ BjjxjjX ; for all x∈X

(c) x ¼P∞

n¼1 fnðxÞxn, for all x∈X.

Next, we state some lemmas which we will use in the subsequent results.

Lemma 1.2. ([31,33]). Let X, Y be Banach spaces and T : X →Y be a bounded linearoperator. Then, the following conditions are equivalent:

(a) There exist two continuous projection operators P : X →X and Q: Y→Y such that

PðXÞ ¼ kerT and QðYÞ ¼ TðXÞ: (1.2)

(b) T has a pseudo inverse operator Ty.

If two continuous projection operators P : X →X andQ : Y→Y satisfy (1.2), then there exists apseudo inverse operatorTy of T such that TyT ¼ IX −P and TTy ¼ Q, where IX is the identityoperator on X.

Lemma 1.3. ([3,27]). Let X be a Banach space. If T ∈LðXÞ has a generalized inverseS ∈ LðXÞ, then TS, ST are projections and TSðXÞ ¼ TðXÞ and STðXÞ ¼ SðXÞ.Lemma 1.4. ([23,29]). Let X be a Banach space and ffng⊂X* be a sequence such thatfx∈X : fnðxÞ ¼ 0; for all n∈ℕg ¼ f0g. Then X is linearly isometric to the Banachspace X d ¼ fffnðxÞg : x∈Xg, where the norm is given by jjffnðxÞgjjX d ¼ jjxjjX, x∈X.


155

2. Main resultsPoumai and Jahan [26] defined and studied K-atomic decomposition as a generalization ofK-frames in Banach spaces. Here we shall extend this study further and introduce the concept ofapproximativeK-atomic decomposition in Banach spaces and obtain new and interesting results.We start this section with the following definition of approximative K-atomic decomposition:

Definition 2.1. Let X be a Banach Space and Xd be a BK-space, fxng⊂X ;

fhn;igi¼1;2;3;...;mnn∈ℕ⊂X*, where fmng is an increasing sequence of positive integer and

K ∈LðXÞ. A pair ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is called an approximative K-atomic

decomposition for X with respect to X d, if the following statements hold:

(a) fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ∈Xd, for all x∈X.


AkKðxÞkX ≤��hn;iðxÞ�i¼1;2;3;...;mn

n∈ℕ

��Xd

≤ BjjxjjX ; for all x∈X :

(c) limn→∞

Pmn

i¼1hn;iðxÞxi converges for all x∈X and

KðxÞ ¼ limn→∞

Xmn

i¼1

hn;iðxÞxi:

The constants A and B are called lower and upper bounds of the approximative K-atomicdecomposition ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕ

Þ.Observation. If ðfxng; ffngÞ is aK-atomic decomposition for X with respect to Xd, then forhn;i ¼ fi; i ¼ 1; 2; . . . ; n; n∈ℕ, ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕ

Þ is an approximative K-atomic

decomposition for X with respect to some associated Banach space X d.

Remark 2.2. Let ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞbe an approximativeK-atomic decomposition

for X with respect to X d with bounds A and B.

(I). IfK ¼ IX , then ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative atomic decomposition

for X with respect to X d with bounds A and B.

(II). If K is invertible, then ðK−1ðfxngÞ; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative atomic

decomposition for X with respect to Xd.

In the following example, we show the existence of approximative K-atomic decompositionfor a Banach space X with respect to an associated BK space Xd.

Example 2.3. Let X be a Banach Space. Let fxng⊆X, fhn;igi¼1;2;3;...;mnn∈ℕ⊆ X* such that

limn→∞

Pmn

i¼1hn;iðxÞxi converges for all x∈X and xn ≠ 0, for all n∈N. Also, letXd ¼ ffhn;igi¼1;2;3;...;mnn∈ℕ

j limn→∞

Pmn

i¼1hn;ixi convergesg. Then Xd is a BK-space with

norm jjfhn;igi¼1;2;3;...;mnn∈ℕjjX d ¼ sup1≤n<∞jj

Pni¼1hn;ixijj. Define an operator as T : X d →X

as Tfhn;igi¼1;2;3;...;mnn∈ℕ¼ limn→∞

Pmn

i¼1hn;ixi and define S : X →Xd as SðxÞ ¼ fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ; x∈X. Take K ¼ TS. Then K : X →X is such that KðxÞ ¼ T SðxÞ ¼limn→∞

Pmn

i¼1 hn;iðxÞxi, for all x∈X ; i ¼ 1; 2; . . . ; n; n∈ℕ. Clearly, fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ∈X d and

kKðxÞkX ¼ limn→∞

��Xmn

i¼1

hn;iðxÞxi��≤ sup

1≤n<∞

��Xnk¼1

hkðxÞxk��

¼ ��hn;iðxÞ�i¼1;2;3;...;mnn∈ℕ

��Xd

≤CkxkX ; for all x∈X ;

where C ¼ sup1≤n<∞kSnk and SnðxÞ ¼ limn→∞

Pmn

i¼1hn;iðxÞxi.

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156

Hence, ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximativeK-atomic decomposition forX with

respect to Xd.In the following result, we give the characterization regarding the existence of

approximative K-atomic decompositions in Banach spaces.

Theorem 2.4. Let K ∈LðXÞ with K ≠ 0. Then a Banach spaceX has an approximativeK-atomic decomposition if and only if there exists a sequence fvig⊂BðXÞ of finite rankendomorphism such that KðxÞ ¼Pn

i¼1viðxÞ; x∈X.

Proof. Let fxng⊂X and fhn;igi¼1;2;3;...;mnn∈ℕ⊂X*, where fmng is an increasing sequence of

positive integer such that ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative K-atomic

decomposition for X with respect to X d. Define

SnðxÞ ¼Xmn

i¼1

hn;iðxÞxi; for all x∈X ; n∈ℕ:

Then for each n∈ℕ and x∈X, SnðxÞ is a well defined continuous linear mapping on X suchthat limn→∞SnðxÞ ¼ x; x∈X. Also by uniform boundedness principle we have sup1≤n≤∞kSnðxÞk < ∞. Assume that v1 ¼ S1, v2n ¼ v2nþ1 ¼ 1

2 ðSnþ1 − SnÞ, n∈ℕ. Now, we compute

limn→∞

Xni¼1

viðxÞ ¼ limn→∞

�S1ðxÞ þ 1

2ðS2ðxÞ � S1ðxÞÞ þ 1

2ðS2ðxÞ � S1ðxÞÞ þ 1

2ðS3ðxÞ � S2ðxÞÞ

þ 1

2ðS3ðxÞ � S2ðxÞÞ þ � � �

�

¼ limn→∞

SnðxÞ

¼ KðxÞ; for all x∈X ;K ∈ LðXÞ:

Therefore, limn→∞

Pni¼1viðxÞ ¼ KðxÞ.

Conversely assume that there exists a sequence of finite rank endomorphism fSng⊂LðXÞsuch that limn→∞SnðxÞ ¼ KðxÞ; x∈X. Then, each SnðxÞ is of a finite rank, there exist asequence fyn;igmn

i¼mn−1þ1 ⊂X and a total sequence of row finite matrix of functionals

fgn;igmn

i¼mn−1þ1 ⊂X* such that

SnðxÞ ¼Xmn

i¼mn−1þ1

gn;iðxÞyn;i; for all x∈X ; n∈ℕ:

Define sequences fxng⊂X and fhn;igi¼1;2;3;...;mnn∈ℕ⊂X *, where fmng is an increasing

sequence of positive integers, by

xi ¼ yn;i; i ¼ mn−1 þ 1; . . . ;mn; n ¼ 1; 2; 3:::

and

hn;i ¼�0; for i ¼ 1; 2; . . . ;mn−1

gn;i; for i ¼ mn−1 þ 1; . . . ;mn:

Then xn ≠ 0, so for each x∈X and n∈ℕ, we get

limn→∞

Xmn

i¼1

hn;iðxÞxi ¼ limn→∞

SnðxÞ ¼ KðxÞ: (2.3)


157

Let x∈X be such that hn;iðxÞ ¼ 0; for all i ¼ 1; 2; . . . ;mn; n∈ℕ. Then by Eq. (2.3) KðxÞ ¼ 0.Thus by Lemma 1.4 there exists an associated Banach space X d ¼ ffhn;igi¼1;2;3;...;mnn∈ℕ

; x∈Xgwith norm given by

��fhn;igi¼1;2;3;...;mnn∈ℕ

��X d ¼ kxkX ; for all x∈X. Hence ðfhn;igi¼1;2;3;...;mnn∈ℕ; fxngÞ is

an approximative K-atomic decomposition for X with respect to X d. ,Next, we give an example of an approximativeK-atomic decomposition forX which is not

an approximative atomic decomposition for X.

Example 2.5. Let X ¼ c0 and Xd ¼ l∞. Let fxng⊂X be the sequence of standard unit

vectors in X and fhn;igi¼1;2;3;...;mnn∈ℕ⊆ X* be such that for x ¼ fαng ∈X ; hn;1ðxÞ ¼ 0;

hn;2ðxÞ ¼ α2; . . . ; hn;iðxÞ ¼ αn; . . .. It is clear that limn→∞

Pmn

i¼1hn;iðxÞxi converges for x∈X.DefineK : X →X byKðxÞ ¼ limn→∞

Pmn

i¼1hn;iðxÞxi; x∈X. Then fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ∈X d

is such that ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximativeK-atomic decomposition forX with respect

to X d. But ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is not an approximative atomic decomposition for X.

Next, we give various methods for the construction of approximative K-atomicdecompositions for X.

Theorem 2.6. Let ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ be an approximative atomic decomposition for

X with respect to Xd with bounds A and B. Let K ∈LðXÞ with K ≠ 0. Then ðfKxng;fhn;igi¼1;2;3;...;mnn∈ℕ

Þ is an approximative K-atomic decomposition forX with respect toX d with

bounds AkKk and B.

Proof. ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative atomic decomposition forX with respect

to Xd with bounds A and B. So for each x∈X, we have x ¼ limn→∞

Pmn

i¼1hn;iðxÞxi. This impliesKðxÞ ¼ limn→∞

Pmn

i¼1hn;iðxÞKðxiÞ. Also, we have kKðxÞkX ≤ kKkkxkX , for all x∈X. This gives

A

kKkkKðxÞkX ≤��hn;iðxÞ�i¼1;2;3;...;mn

n∈ℕ

��Xd

≤BkxkX ; for all x∈X : ,

Theorem 2.7. Let ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ be an approximative atomic decomposition for

X with respect to Xd with bounds A and B. Let K ∈LðXÞ with K ≠ 0. Then ðfxng;fK*hn;igi¼1;2;3;...;mnn∈ℕ

Þ is an approximative K-atomic decomposition for X with respect to X d

with bounds A and BkKk.Proof. Construction of proof is similar to Theorem 2.6. ,

Theorem2.8.Let ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞbe an approximativeK-atomic decomposition for

X with respect to Xd with bounds A and B and let T ∈LðXÞ with T ≠ 0. Then�fTxng; fhn;ig

i¼1;2;3;...;mnn∈ℕ

is an approximative T K-atomic decomposition forX with respect toXd with bounds

AkTk and B.

Proof. Can be easily proved with the help of Theorem 2.6. ,

Theorem 2.9. Let ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ be an approximative K-atomic decomposition for X

with respect to Xd with bounds A and B and let T ∈LðXÞ with kTk≠ 0. Thenðfxng; fT*hn;igi¼1;2;3;...;mnn∈ℕ

Þ is an approximative KT-atomic decomposition for X with

respect to Xd with bounds A and BkTk.Proof. One can easily prove. ,

Theorem 2.10. If ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ be an approximative K-atomic decomposition for X

with respect to Xd and K has pseudo inverse Ky, then there exists ðfgn;igi¼1;2;3;...;mnn∈ℕ⊆X*Þ

such that (fxng; fgn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative K-atomic decomposition for X with

respect to Xd with bounds A and BkKk2.

AJMS26,1/2

158

Proof. Since ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative K-atomic decomposition for X

with respect to X d, then for each x∈X we have

AkKðxÞkX ≤��hn;iðxÞ�i¼1;2;3;...;mn

n∈ℕ

��Xd

≤BkxkX ; x∈X :

Also, for each x∈X, we have

KðxÞ ¼ KðKyKðxÞÞ ¼ limn→∞

Xmn

i¼1

hn;iðKyKðxÞÞxi

¼ limn→∞

Xmn

i¼1

ððKyKÞ*ðhn;iÞðxÞÞxi:

For each n∈ℕ, define gn;i ¼ ðKyKÞ*ðhn;iÞ; i¼1;2;3;...;mnn∈ℕ. Then

kKðxÞkX ¼ kKðKyKðxÞÞkX ≤1

A

��hn;iðKyKðxÞÞ��Xd¼ 1

A

��gn;iðxÞ��Xd; x∈X

and ��gn;iðxÞ��Xd¼ ��hn;iðKyKðxÞÞ��Xd

≤BkKykkKkkxkX ; x∈X :

Hence, we conclude that ðfxng; fgn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative K-atomic

decomposition for X with respect to Xd. ,

3. Approximative Xd-frameCasazza et al. [5] defined and studied Xd-Bessel sequences and Xd-frames in Banach spaces.Later on Stoeva [30] studied perturbation of Xd-Bessel sequences, Xd-frames, atomicdecomposition and Xd-Riesz bases in separable Banach spaces. We have generalized thisconcept and defined approximative Xd-Bessel sequences and approximative Xd-frames inBanach spaces. We begin this section with the following definitions:

Definition 3.1.A sequence fhn;igi¼1;2;3;...;mnn∈ℕ⊆X*, where fmng is an increasing sequence

of positive integers, is called an approximative Xd-frame for X if

(a) fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ∈Xd, for all x∈X.


AkxkX ≤��hn;iðxÞ�i¼1;2;3;...;mn

n∈ℕ

��Xd

≤BkxkX ; for all x∈X : (3.4)

The constants A and B are called approximative Xd-frame bounds. If at least (a) and theupper bound condition in (3.4) are satisfied, then fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ

is called anapproximative Xd-Bessel sequence for X.

One may note that if ffng is an Xd-frame for X, then for fhn;ig ¼ fi; i ¼ 1; 2; 3; . . . ; n;n∈ℕ, fhn;igi¼1;2;3;...;mnn∈ℕ

is an approximative X d-frame for X. Also, note that if ffng is anXd-Bessel sequence for X, then for fhn;ig ¼ fi; i ¼ 1; 2; 3; . . . ; n; n∈ℕ, fhn;igi¼1;2;3;...;mnn∈ℕ

is an approximative X d-Bessel sequence for X.In the next two results, we give necessary conditions under which an approximative

Xd-frame gives rise to a bounded operatorKwith respect to which there is an approximativeK-atomic decomposition for X.


159

Theorem 3.2. Let fhn;igi¼1;2;3;...;mnn∈ℕ⊆ X* be an approximative X d-frame for X with

bounds Aand B. Letfxng⊆X with sup1≤n<∞kxnk < ∞ and let limn→∞

Pmn

i¼1

hn;iðxÞ < ∞, for

all x∈X. Then there exists an operator K ∈LðXÞ such that ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an

approximative K-atomic decomposition for X with respect to Xd.

Proof. Since fhn;igi¼1;2;3;...;mnn∈ℕ⊆ X * is an approximative X d-frame for X with

sup1≤n<∞kxnk < ∞ and limn→∞

Pmn

i¼1

hn;iðxÞ < ∞. Then, by Theorem 2.4, we have

limn→∞

Pmn

i¼1hn;iðxÞxi exist for all x∈X ; n∈ℕ.Define K : X →X by KðxÞ ¼ limn→∞

Pmn

i¼1hn;iðxÞxi; x∈X. Then K is a bounded linearoperator such that

kKðxÞkX ≤ sup1≤n<∞

��Xmn

i¼1

hn;iðxÞxi��X≤CkxkX ;

where C ¼ sup1≤n<∞

Pmn

i¼1hn;iðxÞxi. ThusA

CkKðxÞkX ≤

��hn;iðxÞ��Xd≤BkxkX ; for all x∈X :

Hence, ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative K-atomic decomposition for X with

respect to Xd with bounds ACand B. ,

Theorem 3.3. Let fhn;igi¼1;2;3;...;mnn∈ℕ⊆ X* be an approximativeX d-frame with bounds A, B

and let fxng⊆X. Let T : X d →X given by Tðfhn;igi¼1;2;3;...;mnn∈ℕÞ ¼ limn→∞

Pmn

i¼1hn;ixi be a

well defined operator. Then, there exists a linear operator K ∈LðXÞ such thatðfxng; fhn;igi¼1;2;3;...;mnn∈ℕ

Þ is an approximative K-atomic decomposition for X with respect to Xd.

Proof. Define U : X →Xd by UðxÞ ¼ fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ; x∈X. Then U is well defined

and kUk≤B. Take K ¼ TU. Then KðxÞ ¼ limn→∞

Pmn

i¼1hn;iðxÞxi; x∈X. Therefore, byuniform boundedness principle, we have

kKðxÞkX ≤ sup1≤n<∞

��Xmn

i¼1

hn;iðxÞxi��X≤CkxkX ; x∈X ;

where C ¼ sup1≤n<∞

��Pmn

i¼1hn;iðxÞxi��X. Thus, we have

A

CkKðxÞk≤ ��hn;iðxÞ�i¼1;2;3;...;mn

n∈ℕ

��≤Bkxk; for all x∈X :

Hence ðfxng; fhn;iðxÞgi¼1;2;3;...;mnn∈ℕÞ is an approximativeK-atomic decomposition forX with

respect to Xd with bounds ACand B. ,

Next, we give the existence of an approximative K-atomic decomposition from anapproximative X d-Bessel sequence.

Theorem3.4. LetX be a reflexive Banach space andXd be a BK-space which has a sequence ofcanonical unit vectors feng as a basis. Let fhn;igi¼1;2;3;...;mnn∈ℕ

⊆X* be an approximativeX d-Besselsequence with bound B and let fxng ⊆ X. If fhðxnÞg∈ ðX dÞ* for all h∈X*, then there exists abounded linear operator K ∈LðXÞ such that ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕ

Þ is an approximativeK-atomic decomposition for X with respect to Xd.

Proof. Clearly U : X →Xd given by UðxÞ ¼ fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ; x∈X is well defined.

Define amap R : X*→ ðXdÞ* byRðhÞ ¼ fhðxnÞg; x∈X. Then, its adjointR* : ðXdÞ** →X **

is given by R*ðejÞðhÞ ¼ ejðRðhÞÞ ¼ hðxjÞ. Let T ¼ ðR*ÞjXd and fhn;igi¼1;2;3;...;mnn∈ℕ∈X d.

Then

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160

T��

hn;iðxÞ�i¼1;2;3;...;mn

n∈ℕ

�¼ lim

n→∞

Xmn

i¼1

hn;iTðeiÞ ¼ limn→∞

Xmn

i¼1

hn;ixi:

But fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ; ∈Xd. So Tðfhn;iðxÞgi¼1;2;3;...;mnn∈ℕ

Þ ¼ limn→∞

Pmn

i¼1hn;iðxÞxi. TakeK ¼ TU. Then K ∈ LðXÞ and KðxÞ ¼ limn→∞

Pmn

i¼1hn;iðxÞxi. Moreover, T is a bounded linear

operator such that kKðxÞk≤ kTk��fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ

��. Hence1

kTk kKðxÞk≤ ��hn;iðxÞ�i¼1;2;3;...;mnn∈ℕ

��≤Bkxk; x∈X,

Next, we construct an approximative K*-atomic decomposition for X* from a givenapproximative K-atomic decomposition for X.

Theorem 3.5. Let Xd be a BK-space with dual ðXdÞ* and let Xd andðXdÞ* have sequences ofcanonical unit vectors feng and fvng respectively as bases. Let ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕ

Þbe anapproximative K-atomic decomposition for X with respect to X d. Let S : X d →X given bySðfhn;igi¼1;2;3;...;mnn∈ℕ

Þ ¼ limn→∞

Pmn

i¼1hn;ixi be a well definedmapping. Then, ðfhn;igi¼1;2;3;...;mnn∈ℕ; πðxnÞÞ

is an approximative K*-atomic decomposition for X * with respect to ðXdÞ*.Proof. Since ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕ

Þ is an approximative K-atomic decomposition for Xwith respect to X d, so for each x∈X, KðxÞ ¼ limn→∞

Pmn

i¼1hn;iðxÞxi. Thus hðKðxÞÞ ¼limn→∞

Pmn

i¼1hn;iðxÞhðxiÞ. Therefore, by Theorem 2.4 we have limn→∞

Pmn

i¼1hðxiÞhn;i exists forall h∈X*. Also, for x∈X, we compute

ðK*ðhÞÞðxÞ ¼ h

limn→∞

Xmn

i¼1

hn;iðxÞxi!

¼ limn→∞

Xmn

i¼1

hðxiÞhn;iðxÞ:

This gives K*ðhÞ ¼ limn→∞

Pmn

i¼1hðxiÞhn;i, for h∈X*. Note that S*ðhÞðejÞ ¼ hðSðejÞÞ ¼hðxjÞ; h∈X*. So, S*ðhÞ ¼ fhðxnÞg and fhðxnÞg ¼ fhðSðenÞÞg∈ ðXdÞ*; h∈X *. Also

kfhðxnÞgkðXdÞ* ¼ kS*ðhÞk≤ kSkkhkX* ; h∈X *:

Define R : X →X d by RðxÞ ¼ fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ; x∈X. Then, R*ðvjÞðxÞ ¼ vjðRðxÞÞ ¼

hj;iðxÞ; x∈X. So, R*ðvjÞ ¼ hj;i, for all j∈ℕ and for fgn;iðxÞgi¼1;2;3;...;mnn∈ℕ∈ ðXdÞ* we have

R*��

gn;iðxÞ�i¼1;2;3;...;mn

n∈ℕ

�¼ R*

limn→∞

Xmn

i¼1

gn;iðxÞvi!

¼ limn→∞

Xmn

i¼1

gn;iðxÞhn;i:

Therefore, we have

R*S*ðhÞ ¼ R*ðfhðxiÞgÞ ¼ limn→∞

Xmn

i¼1

hðxiÞhn;i; h∈X *:

Note that, K* ¼ R*S* and so

kK*ðhÞkX* ¼ kR*S*ðhÞkX* ≤ kR*kkfhðxnÞgkðXdÞ*; h∈X *:

This gives

1

kR*kkK*ðhÞkX* ≤ kfhðxnÞgkðXdÞ* ≤ kSkkhkX* ; h∈X *: (3.5)


161

Hence, ðfhn;igi¼1;2;3;...;mnn∈ℕ; πðxnÞÞ is an approximative K*-atomic decomposition for X * with

respect to ðX dÞ*. ,Next, we give the following result characterizing the class of approximative K-atomic

decompositions.

Theorem 3.6. Let ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ be an approximative K-atomic decomposition for X

with respect to X d with bounds A and B. Let T : Xd →X given by Tðfhn;igi¼1;2;3;...;mnn∈ℕÞ

¼ limn→∞

Pmn

i¼1hn;ixi is well defined for fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ∈ Xd and let U : X →X d be the

mapping given by UðxÞ ¼ fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ. If K is invertible, then the following statements are

equivalent.

(a) T is the pseudo inverse of U.

(b) ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative atomic decomposition for X with

respect to Xd.

(c) T is a linear extension of U−1 : UðXÞ→X.

(d) UðXÞ is a complemented subspace of Xd.

(e) KerT is a complemented subspace of X d and T is surjective.

Proof. ðaÞ0ðbÞ By hypothesis, fx∈X : hn;iðxÞ ¼ 0; for all n∈ℕg ¼ f0g. So, KerU ¼ f0g. Since T is the pseudo inverse of U, by Lemma 1.2 there exists a continuousprojection operator θ : X →X such that TU ¼ IX − θ and kerU ¼ θðXÞ. Thus, for eachx∈X, we have

TUðxÞ ¼ ðIX � θÞðxÞ ¼ x; x∈X :

Hence, for every x∈X, limn→∞

Pmn

i¼1hn;iðxÞxi ¼ x.ðbÞ0ðaÞ For x∈X, we have

UTUðxÞ ¼ UT��


n∈ℕ

�¼ U

limn→∞

Xmn

i¼1

hn;iðxÞxi!

¼ UðxÞ:

Hence, UTU ¼ U.ðcÞ0ðbÞ If T is a linear extension of U−1 : UðXÞ→X, then TU : X →X is the identity

map on X. So, TUðxÞ ¼ x and limn→∞

Pmn

i¼1hn;iðxÞxi ¼ x.ðcÞ0ðaÞ Obvious, since UTU ¼ U IX ¼ U.ðdÞ0ðbÞ Suppose Xd ¼ UðXÞ⊕G, where G is a closed subspace of Xd. Let P be a

projection of Xd onto UðXÞ along G.Then, Pðfhn;igi¼1;2;3;...;mnn∈ℕ

Þ ¼ fgn;iðlimn→∞

Pmn

i¼1hn;ixiÞg, for all fhn;igi¼1;2;3;...;mnn∈ℕ∈X d. Therefore

U−1+P��

hn;i�i¼1;2;3;...;mn

n∈ℕ

�¼ U−1

(gn;i

limn→∞

Xmn

i¼1

hn;ixi

!)

¼ limn→∞

Xmn

i¼1

hn;ixi ¼ T��

hn;i�i¼1;2;3;...;mn

n∈ℕ

�; for all

�hn;i�∈X d:

This gives, T ¼ U−1+P and

T��


n∈ℕ

�¼ U−1+P

��hn;iðxÞ

�i¼1;2;3;...;mn

n∈ℕ

�

¼ U−1��


n∈ℕ

�:

Hence, x ¼ limn→∞

Pmn

i¼1hn;iðxÞxi, for all x∈X.

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162

(b)0(d) Obvious.(e)0(b) Let Xd ¼ kerT⊕M, where M is a closed subspace of Xd. Take

Y ¼ kerT⊕UðXÞ. Let Q : X d →M be a projection from Xd onto M along kerT. DefineL : Xd →Y by LðαÞ ¼ ðα−QðαÞ;UTðαÞÞ, for α ¼ fhn;igi¼1;2;3;...;mnn∈ℕ

∈Xd. Let LðαÞ ¼ 0.This gives QðαÞ ¼ α. So α∈M. Let UTðαÞ ¼ 0. Then

U

limn→∞

Xmn

i¼1

hn;ixi

!¼(gn;i

limn→∞

Xmn

i¼1

hn;ixi

!)¼ 0; for n∈ℕ:

This gives limn→∞

Pmn

i¼1hn;ixi ¼ 0 and so, α∈ kerT. Thus, α∈ kerT ∩M ¼ f0g. Hence, L isone–one.

Let ðα0;UðxÞÞ∈ ker T⊕UðXÞ, for α0 ∈ kerU and UðxÞ∈UðXÞ.Since, T is onto, for each x∈X, there exists β∈Xd such that TðβÞ ¼ x and this gives

UTðβÞ ¼ UðxÞ. Take α ¼ α0 þ QðβÞ. Then QðαÞ ¼ Qðα0Þ þ Q2ðβÞ ¼ QðβÞ and α0 ¼ α−QðαÞ. Also, we have

UTðαÞ ¼ UTðα� α0Þ ¼ UTðQðβÞÞ ¼ UTðβÞ ¼ UðxÞ: (3.6)

Thus LðαÞ ¼ ðα0;UTðxÞÞ and L is an isomorphism from Xd onto Y. So, there is a projectionP ¼ UT : Xd →UðXÞ onto UðXÞ along kerT. This gives

U−1+P ¼ T and U−1+P��


n∈ℕ

�¼ T

��hn;iðxÞ

�i¼1;2;3;...;mn

n∈ℕ

�:

Finally, we compute

U−1��


n∈ℕ

�¼ lim

n→∞

Xmn

i¼1

hn;iðxÞxi and x ¼ limn→∞

Xmn

i¼1

hn;iðxÞxi:

Therefore, ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative atomic decomposition for X with

respect to Xd.(b)0(e) Obvious. ,In the following result, we prove a duality type approximative K-atomic decomposition

for X.

Theorem 3.7. Let Xd be a reflexive BK-space with its dual ðXdÞ* and let sequences of

canonical unit vectors feng and fvng be bases for Xd andðX dÞ*, respectively. Let

ðfhn;igi¼1;2;3;...;mnn∈ℕ; πðxnÞÞ be an approximative K-atomic decomposition for X * with respect to

ðX dÞ*. If S : ðXdÞ* →X* given by SðfdigÞ ¼ limn→∞

Pmn

i¼1dihn;i is well defined for fdig∈X*d,

then there exists a linear operator L∈LðXÞ such that ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an

approximative L-atomic decomposition for X with respect to Xd.

Proof. Since ðfhn;igi¼1;2;3;...;mnn∈ℕ; πðxnÞÞ is an approximative K-atomic decomposition for

X* with respect to ðX dÞ*. For h∈X *, we have KðhÞ ¼ limn→∞

Pmn

i¼1hðxiÞhn;i. Also, byTheorem 2.4 we have limn→∞

Pmn

i¼1hn;iðxÞxi exist, for all x∈X. Define L : X →X byLðxÞ ¼ limn→∞

Pmn

i¼1hn;iðxÞxi; x∈X. Note that SðvnÞ ¼ hn;i;i¼1;2;3;...;mnn∈ℕ and for x∈X, the

linear bounded operator S* : X **→ ðXdÞ** satisfies

S*ðπðxÞÞðvnÞ ¼ πðxÞSðvnÞ ¼�hn;iðxÞ

�i¼1;2;3;...;mn

n∈ℕ

:


163

So, fhn;iðxÞgi¼1;2;3;...;mnn∈ℕis identified with S*ðπðxÞÞ∈ ðX dÞ** ¼ X d. Further, we have

��hn;iðxÞ�i¼1;2;3;...;mnn∈ℕ

��Xd

¼ kS*ðπðxÞÞkXd≤ kSkkxkX ; x∈X : (3.7)

Letting U ¼ S* jX , we have UðxÞ ¼ fhn;iðxÞgi¼1;2;3;...;mnn∈ℕand kUk≤ kSk.

Define R : X*→ ðXdÞ* by Rðf Þ ¼ fhðxnÞg; h∈X *. Then

R*ðejÞðhÞ ¼ ejðRðhÞÞ ¼ hðxjÞ; h∈X *:

So, R*ðejÞ ¼ xj; for all j∈ℕ. Take T ¼ ðR*ÞjXd. Then, forfhn;igi¼1;2;3;...;mnn∈ℕ

∈Xd wecompute

T��

hn;i�i¼1;2;3;...;mn

n∈ℕ

�¼ Tðhn;ieiÞ ¼ lim

n→∞

Xmn

i¼1

hn;iTðeiÞ ¼ limn→∞

Xmn

i¼1

hn;ixi:

Thus, TUðxÞ ¼ limn→∞

Pmn

i¼1hn;iðxÞxi, for all x∈X and this gives TU ¼ L on X. Therefore,1

kTkkLðxÞkX ≤��fhn;iðxÞgi¼1;2;3;...;mnn∈ℕ

��Xd. Then

1

kTkkLðxÞkX ≤��hn;iðxÞ�i¼1;2;3;...;mn

n∈ℕ

��Xd

≤ kSkkxkX :

Hence, ðfxng; fhn;igi¼1;2;3;...;mnn∈ℕÞ is an approximative L-atomic decomposition for X with

respect to Xd. ,

4. Possible applicationOne of the most important devices in modern world is digital camera. In our notation a digitalpicture is a two-dimensional sequence, fhnmg. So, it can be seen either as an infinite lengthsequence with a finite number of non-zeros samples; that is fhnmg; n;m∈ℤ, or as a sequencewith domain n∈ f0; 1; 2; . . . ;N − 1g,m∈ f0; 1; 2; . . . ;M − 1g, can be expressed as a matrix:

h ¼24 h0;0 h0;1; : : :; hM−1

h1;0 h1;1; : : :; hM−1

hN−1;0 hN−1;1; : : :; hN−1;M−1;

35

where each elements hnm is called a pixel and the image has NM pixels. In real life for hn;m torepresent colour image, it must have more than one component, usually, red, green and bluecomponents are used (RGB colour space).

References

[1] A. Aldroubi, A. Baskakov, I. Krishtal, Slanted matrices, Banach frames and sampling, J. Funct.Anal. 255 (7) (2008) 1667–1691.

[2] A. Aldroubi, K.A. Gr€ocheing, Nonuniform sampling and reconstruction in shift – invariantspaces, SIAM Rev. 43 (4) (2001) 585–620.

[3] S.R. Caradus, Generalized Inverses and Operator Theory, Queen’s Papers in Pure and AppliedMathematics, Vol. 50, Queen’s University, Kingston, Ont, 1978.

[4] P.G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (2000) 129–201.

[5] P.G. Casazza, O. Christensen, D.T. Stoeva, Frame expansions in separable Banach spaces, J. Math.Anal. Appl. 307 (2) (2005) 710–723.

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[6] O. Christensen, An introduction to frames and Riesz bases, Applied and Numerical HarmonicAnalysis, Birkhuser Boston, Inc., Boston, MA, 2003.

[7] O. Christensen, C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr.185 (1997) 33–47.

[8] B. Dastourian, M. Jafanda, Frames for operators in Banach spaces via semi-inner products, Int. J.Wavelets Multiresolut. Inf. Process. 14 (3) (2016).

[9] I. Daubechies, A. Grossmann, Y. Meyer, Painless non orthogonal expansions, J. Math. Phys. 27(1986) 1271–1283.

[10] R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72(1952) 341–366.

[11] Y.C. Eldar, G.D. Forney, Optimal tight frames and quantum measurement, IEEE Trans. Inform.Theory 48 (2002) 599–610.

[12] H.G. Feichtinger, K.A. Gr€ocheing, Unified approach to atomic decompositions via integrablegroup representations, Function Spaces and Applications, Springer, Berlin, Heidelberg, 1988,pp. 52–73.

[13] M. Fornasier, Banach frames for α-modulation spaces, Appl. Comput. Harmon. Anal. 22 (2) (2007)157–175.

[14] D. Gabor, Theory.of. communication, Theory of communication Part 1: The analysis ofinformation, J. Inst. Electr. Eng. III Radio Commun. Eng. 93 (26) (1946) 429–441.

[15] L. Gǎvruta, Frames for operators, Appl. Comput. Harmon. Anal. 32 (1) (2012) 139–144.

[16] L. Gǎvruta, Atomic decompositions for operators in reproducing Hilbert spaces, Math. Rep. 17(67) (2015) 303–314. 3.

[17] R. Geddavalasa, P.S. Johnson, Frames for operators in Banach spaces, Acta Math. Vietnam 42(2017) 665–673.

[18] K.A. Gr€ocheing, Describing functions: atomic decompositions versus frames, Monatsh. Math. 112(1) (1991) 1–42.

[19] K.A. Gr€ocheing, Localization of frames, Banach frames, and the invertibility of the frameoperator, J. Fourier Anal. Appl. 10 (2) (2004) 105–132.

[20] S. Jahan, V. Kumar, S.K. Kaushik, On the existence of non-ninear frames, Arch. Math. (BRNO) 53(2017) 101–109.

[21] S. Jahan, V. Kumar, C. Shekhar, Cone associated with frames in Banach spaces, Palestine J. Math.7 (2) (2018) 641–649.

[22] S.K. Kaushik, Some results concerning frames in Banach spaces, Tamkang J. Math. 38 (3) (2007)267–276.

[23] S.K. Kaushik, S.K. Sharma, On approximative atomic decompositions in Banach spaces, Commun.Math. Appl. 3 (3) (2012) 293–301.

[24] S.K. Kaushik, Shalu Sharma, Generalized Schauder frames, Arch. Math. (BRNO) Tomus 50 (2014)39–49.

[25] K.T. Poumai, S. Jahan, Atomic systems for operators, Int. J. Wavelets Multiresolut. Inf. Process. 16(5) (2018).

[26] K.T. Poumai, S. Jahan, On K-atomic decompositions in Banach Spaces, Electron. J. Math. Anal.Appl. 6 (1) (2018) 183–197.

[27] C. Schmoeger, Partial Isometries on Banach Spaces, Mathematisches Institut I, Universit€atKarlsruhe, 2005.

[28] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces,Springer - Verlag, New York, Heidelberg, Berlin, 1970.

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[30] D.T. Stoeva, Perturbation of frames in banach spaces, Asian-Eur. J. Math. 5 (1) (2012) 15.

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[32] X. Xiao, Y. Zhu, L. Gǎvruta, Some properties of K-frames in Hilbert spaces, Result Math. 63 (2013)1243–1255.

[33] Y.C. Zhu, S.Y. Wang, The stability of Banach frames in Banach spaces, Acta Math. Sin. (Engl.Ser.) 26 (12) (2009) 2369–2376.

Corresponding authorShah Jahan can be contacted at: [email protected]


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Existence of self-similar solutionsof the two-dimensional

Navier–Stokes equation fornon-Newtonian fluids

Dongming WeiDepartment of Mathematics, Nazarbayev University, Astana, Kazakhstan, and

Samer Al-AshhabDepartment of Mathematics and Statistics,

Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

AbstractThe reduced problem of the Navier–Stokes and the continuity equations, in two-dimensional Cartesiancoordinates with Eulerian description, for incompressible non-Newtonian fluids, is considered. TheLadyzhenskaya model, with a non-linear velocity dependent stress tensor is adopted, and leads to thegoverning equation of interest. The reduction is based on a self-similar transformation as demonstrated inexisting literature, for two spatial variables and one time variable, resulting in an ODE defined on a semi-infinite domain. In our search for classical solutions, existence and uniquenesswill be determined depending onthe signs of two parameters with physical interpretation in the equation. Illustrations are included to highlightsome of the main results.

Keywords Non-linear boundary value problem, Singular, Self-similar transformation, Existence, Uniqueness


1. IntroductionThe study of non-Newtonian fluids, both mathematically and physically, has gained muchimportance during the last few decades due to their many applications in industry and indescribing physical phenomena. The basic physical theory, and itsmathematical formulationcan be found in [1,8,18]. Many researchers studied non-Newtonian fluids from a numerical orcomputational point of view, in some instances accompanied with certain techniques or

Existence ofNavier–Stokes

equation

167

JEL Classification — 34B40, 76A05© Dongming Wei and Samer Al-Ashhab. Published in Arab Journal of Mathematical Sciences.


Dongming Wei is supported by the Kazakhstan Ministry of Education Grant # AP05134166.The publisher wishes to inform readers that the article “Existence of self-similar solutions of the two-

dimensional Navier–Stokes equation for non-Newtonian fluids” was originally published by theprevious publisher of the Arab Journal of Mathematical Sciences and the pagination of this article hasbeen subsequently changed. There has been no change to the content of the article. This change wasnecessary for the journal to transition from the previous publisher to the new one. The publishersincerely apologises for any inconvenience caused. To access and cite this article, please use Wei, D.,Al-Ashhab, S. (2019), “Existence of self-similar solutions of the two-dimensional Navier–Stokes equationfor non-Newtonian fluids”, Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 167-178. Theoriginal publication date for this paper was 20/04/2019.



Received 23 October 2018Revised 12 March 2019Accepted 6 April 2019


Vol. 26 No. 1/2, 2020pp. 167-178


DOI 10.1016/j.ajmsc.2019.04.001

transformations to elucidate investigating the problem [6,9]. Other studies involved existenceand uniqueness of solutions to problems involving non-Newtonian fluids [10,11,20,21]. Manytimes, it is found that solutions for Newtonian and non-Newtonian flows are not unique[7,13,15,17]. In some instances or special cases, exact solutions were established, see forexample [12]. Our interest in this paper is in a Ladyzhenskaya type non-Newtonian fluid [16],where self-similar transformations of the Navier–Stokes equations, for non-Newtonianincompressible fluids, lead to an ODE with dependence on one similarity variable. Navier–Stokes equations in two dimensions, for incompressible non-Newtonian fluids, consist of asystem of PDEs with two spatial variables, and a time variable. However, a two-dimensionalgeneralization of the well-known self-similar Ansatz reduces the PDE system into an ODE.This resulting ODE was used for example in [4], to study the compressible NewtonianNavier–Stokes equations. Symmetry reductions analysis can also be applied to obtain somesolutions, as was done in [14], and as was done for three dimensions in [19].

Recently in [3], the authors considered a self-similar transformation to obtain analyticsolutions of the two-dimensional Navier–Stokes equations, with Eulerian description, for a non-Newtonian fluid. However, it remains to investigate existence and uniqueness of solutions forthat particular reduced Navier–Stokes equation, with suitable boundary conditions. A similarproblem was studied in [5], but where the parameters were tied together via certain relations,and where the authors used a different approach to investigate the problem.

We shall discuss existence (or non-existence) and uniqueness of solutions for the resultingNavier–Stokes reduced problem. In Section 2, we introduce the problem with a briefderivation including the main ideas leading to the governing equation of interest. The mainresults are then derived in Section 3, where we discuss separate cases depending on the signof two parameters: the flow behavior index (mathematically an exponent r) and the leadingcoefficient k in the governing equation.

2. The problemConsider the Ladyzhenskaya model of non-Newtonian fluid dynamics, with the followingformulation (c.f. [16]):

ρvui

vtþ ρuj

vui

vxj¼ −

vp

vxi

þ vΓij

vxjþ ρFi (1)

vuj

vxj¼ 0 (2)

where the Einstein summation convention is assumed on the j index. The parameters ρ;u; pandF represent the density, the two dimensional velocity field, the pressure, and the externalforce, respectively. On the other hand, observe that Γij is defined via:

Γij ¼ ðμ0 þ μ1jEð∇uÞjrÞEijð∇uÞ (3)

where μ0; μ1 and r represent the dynamical viscosity, the consistency index, and the flowbehavior index, respectively, and where

Eijð∇uÞ ¼ 1

2

�vui

vxjþ vuj

vxi

�(4)

is the Newtonian linear stress tensor. Observe that x represents the two dimensionalCartesian coordinates, say x ¼ ðx; yÞ. Now, setting the external force to zero F ¼ 0,observing that in two dimensions:

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168

jEj ¼�u2x þ v2y þ

1

2

�u2y þ v2x

��1=2

;

(where u and v are the components of u) and letting:

L ¼ μ0 þ μ1jEjr;simplifies the formulation, using compact notation, to the following equations:

ux þ vy ¼ 0; (5)

ut þ uux þ vuy ¼ −px

ρþ Lxux þ Luxx þ Ly

2ðuy þ vxÞ þ L

2ðuyy þ vxyÞ; (6)

vt þ uvx þ vvy ¼ −py

ρþ Lyvy þ Lvyy þ Lx

2ðuy þ vxÞ þ L

2ðvxx þ uxyÞ: (7)

The following transformation (8) (self-similar Ansatz, c.f. [3]) leads to solutions of physicalinterest, and shall further simplify the problem consisting of the 3 3 3 PDE system (5)–(7)given above. Namely, this transformation is given by:

u ¼ t−αf ðηÞ; v ¼ t−βgðηÞ; p ¼ t−γhðηÞ; η ¼ t−δðxþ yÞ (8)

where η is called a similarity variable. The functions f ; g, and h are referred to as shapefunctions. We shall consider μ0 ¼ 0; μ1 ≠ 0, and we note that the details of the entirederivation and simplification process can be found in the references, c.f. [2,3] and thereferences therein. We choose to skip those details since our main interest is in the resultingODE for f below. However, we do point out that through the simplification process, the shapefunctions are assumed to have interrelations relating them to one another, while the followingrelations are obtained for the above exponents:

α ¼ β ¼ ð1þ rÞ=2; δ ¼ ð1� rÞ=2; γ ¼ r þ 1: (9)

Solutions of physical relevance and interest will require all exponents in (9) to be positive,from which we must have: −1 < r < 1. It is noted that in similar power-law problems, apower-law index n is used and is related to r mathematically via r ¼ n− 1. In this respect,−1 < r < 0 corresponds to pseudo-plastic or shear-thinning fluid, while 0 < r < 1corresponds to a shear-thickening fluid. (Since r > 1 has been eliminated, the fluid ofinterest here maybe considered as a restricted Ostwald–de Waele-type fluid.) The followingODE is the reduced and simplified equation that is of our interest, and it is the followingreduced Navier–Stokes equation:

2rþ1ð1þ rÞμ1 f00 j f 0 jr−1f 0 þ ð1� rÞηf 0 þ ð1þ rÞf ¼ 0: (10)

Observe that this ODE is for f , while g and h are related to f via certain relations as can befound in the references. Due to the conditions we shall consider, see (12), we shall supposef0≤ 0. (Observe that if f

0reaches zero at some point, say f

0 ðη0Þ ¼ 0, then the equation maybecome inconsistent in case f ðη0Þ≠ 0 for r > 0, or it may become undefined if r < 0.) Byfurther assuming

k ¼ 2rþ1ð1þ rÞμ1;we obtain the equivalent equation (11). Before proceedingwith the analysis, however, observethat if f

0 ðη0Þ ¼ 0 while f ðη0Þ≠ 0, for some η0 > 0, then Eq. (10) becomes inconsistent for


equation

169

positive r. The solution assumes a point of termination at such instances. Solutions alsoassume a terminal point for negative values of rwhen f

0 ðη0Þ ¼ 0 as the first term in the ODEbecomes undefined. It is noted that practical values of k > 0were listed in [3], while k < 0 canbe found in the similar Rayleigh problem. So, now, consider:

−kf00 ð−f 0 Þr þ ð1� rÞηf 0 þ ð1þ rÞf ¼ 0 (11)

We shall make a few observations regarding (11). First, notice that if r ¼ 0 then we havethe equation −kf

00 þ ðηf Þ0 ¼ 0 which leads to a solution: −kf0 þ ηf ¼ c and therefore

f ðηÞ ¼ f ð0Þeη2=2k þ f0 ð0Þeη2=2k R η

0 e−u2=2kdu. This solution approaches zero for k < 0 as

η→∞, and consequently it is an explicit illustration of the existence of a solution whenr ¼ 0; k < 0, which satisfies (12).

Additionally, observe that it is not possible to have f → c≠ 0 as η→∞, for some constantc≠ 0, unless f reaches c at some finite η. To establish this, let gðηÞ ¼ f ðηÞ− c so thatf ðηÞ ¼ cþ gðηÞ, then we must have gðηÞ→ 0 as η→∞, and therefore −kg

00 ð−g 0 Þrþð1− rÞηg 0 ¼ −ð1þ rÞðcþ gÞ, which upon integration would imply that:

kð−g 0 ðηÞÞrþ1

r þ 1¼ −ð1þ rÞcη� ð1� rÞηgðηÞ � 2r

Z η

0

gðuÞduþ K;

where K ¼ kð−g0 ð0ÞÞrþ1

rþ1is a constant. Now, since r > − 1 and the first term on the right-hand

side would make that side of the equation diverge and become unbounded as η→∞, thiswould in turn imply that the equation does not balance, or otherwise g

0 ðηÞ has to take oninfinite values as η→∞, which is a contradiction. It is very important to emphasize here thatit will be shown that solutions do exist where f reaches c≠ 0 at a terminal point in finite η:f ðη0Þ ¼ c≠ 0; f

0 ðη0Þ ¼ 0 for some η0 > 0, as is also shown in numerical illustrations in [3] forr < 0. The boundary conditions for an equation such as (11) are typically given at 0 and at∞.The boundary conditions of interest to us take the form:

f ð0Þ ¼ a; f ð∞Þ ¼ 0 (12)

where a > 0.

3. Existence of solutionsTo establish existence of solutions, a shooting method is utilized where the condition atinfinity is replaced by an initial condition f

0 ð0Þ: we shall first show that Eq. (11) subject tof ð0Þ ¼ a (the first of the two conditions in (12)) has solutions for which f

0 ðη0Þ ¼ 0 at somefinite η0 < ∞ and where f ðη0Þ ¼ b > 0 (such solutions terminate at η0 as discussed above)for some appropriate choice of f

0 ð0Þ. We shall also show that it has solutions that extend toinfinite ηwhile crossing the horizontal axis at some point.

Observe that subtracting 2rf from both sides of Eq. (11) yields the following:

−kf00 ð−f 0 Þr þ ð1− rÞηf 0 þ ð1− rÞf ¼ −2rf , where now observe that the left-hand side is

an exact derivative. Now integrating from 0 to η and using a dummy variable of integration,say t, we obtain

ð−f 0 ðηÞÞrþ1 ¼ ð−f 0 ð0ÞÞrþ1 � ðr þ 1Þk

�ð1� rÞηf ðηÞ þ 2r

Z η

0

f ðtÞdt�: (13)

To begin with, let us consider the case r > 0; k > 0:Theorem 1.There exists a unique solution to (11) subject to (12) for r > 0; k > 0, and

where f ðηÞ > 0 for all η > 0.

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Proof. To begin with, we show that for some appropriate choice of the initial conditionf0 ð0Þ < 0one obtains a solution that terminates at some finite η0 where f

0 ðη0Þ ¼ 0; f ðη0Þ > 0.Observe that (11) implies that f

00 ð0Þ > 0. We further assume f00> 0 on the entire interval

ð0; η0Þwhich will be verified at the end of the proof, and with f00> 0 we must have:

ð−f 0 ðηÞÞrþ1< ð−f 0 ð0ÞÞrþ1 � ðr þ 1Þ

k

�0þ 2r

Z η

0

ðf ð0Þ þ f0 ð0ÞtÞdt

�;

and therefore

ð−f 0 ðηÞÞrþ1< ð−f 0 ð0ÞÞrþ1 � 2rðr þ 1Þ

k

�f ð0Þηþ f

0 ð0Þη2�2�:Taking ð−f 0 ð0ÞÞrþ1

< rðrþ1Þk

f ð0Þand j f 0 ð0Þj < f ð0Þ (whichever yields a smaller j f 0 ð0Þj, recallthat f

0 ð0Þ is negative) would in fact show that for η ¼ 1 we have ð−f 0 ð1ÞÞrþ1< 0, but by

assumption this last quantity should be non-negative (due to f0< 0). This contradiction

shows that f0 ¼ 0 at some finite η0 < 1. Finally one checks that with the additional condition

j f 0 ð0Þj < ðrþ1Þ2

f ð0Þwe have f 00> 0 and f > 0 for all η < 1, so that the above arguments hold

(note that this strong condition for j f 0 ð0Þj establishes our point here, but it might be relaxedsignificantly once a particular solution is determined).

On the other hand, it can be shown that for large enough j f 0 ð0Þjwe obtain a solution for

which f0 ðηÞ < 0 for all η > 0, and where f ðηÞ < 0 for all η > η0, for some η0 > 0 (i.e. a

solution that crosses the η-axis). Now observe that for f0< 0 it follows from Eq. (11) that

−kf00 ð−f 0 Þr ¼ −ð1− rÞf 0

− ð1þ rÞf > − ð1þ rÞf , which can be integrated to obtain

ð−f 0 ðηÞÞrþ1> ð−f 0 ð0ÞÞrþ1 � ðr þ 1Þ

k

Z η

0

f ðtÞdt; (14)

from which we have

ð−f 0 ðηÞÞrþ1> ð−f 0 ð0ÞÞrþ1 � ðr þ 1Þ

kf ð0Þη; (15)

by choosing f0 ð0Þ to be large enough in absolute value such that

ð−f 0 ð0ÞÞrþ1> ðf ð0ÞÞrþ1 þ ðr þ 1Þ

kf ð0Þ (16)

then it is guaranteed from (15) and (16) that ð−f 0 ðηÞÞrþ1> ðf ð0ÞÞrþ1

for all 0 < η < 1, andtherefore f

0 ðηÞ < − f ð0Þ < 0 for all 0 < η < 1, which in turn guarantees the existence ofsome η0 < 1 such that f ðη0Þ ¼ f ð0Þ þ R η0

0f0 ðtÞdt ¼ 0. Once we have f ðη0Þ ¼ 0 with

f0 ðη0Þ < 0, then Eq. (11) will show that this solution will satisfy: f ðηÞ < 0; f

0 ðηÞ < 0 for allη > η0. (We note that the same argument can be used for−1 < r < 0since the exponent r þ 1is positive for this range of r, as will be needed for later proofs.)

Now to show existence of solutions: given the above results, suppose that y1 is a solutionthat terminates at some finite η1 where y

01ðη1Þ ¼ 0 and y1ðη1Þ ¼ e > 0. One can find

another solution that terminates at y2ðη2Þ ¼ e=2 for some η2, i.e., y2ðη2Þ ¼ e=2; y02ðη2Þ ¼ 0.

It is not difficult to prove this last mathematical statement, following similar analysis asabove, coupled with the continuity with respect to initial conditions (on the interval ð0; η1Þ).We, however, leave out some of the obvious details.

In fact, a general assumption that there is aminimum value for a solution f > 0 where f0

reaches zero so the solution terminates (at say η1, i.e. f ðη1Þ ¼ emin > 0; f0 ðη1Þ ¼ 0, and where


equation

171

no solution with smaller f -values will terminate), leads to a contradiction for the case

r > 0; k > 0. Since then, one can still take a slightly larger j f 0 ð0Þj so that f ðη1Þdecreases veryslightly, while the new j f 0 ðη1Þj is very small so that f ðηÞwill still have to decrease for η > η1.But on the other hand, f

00 ðηÞwould be large enough for η > η1, and will approach infinity fastsince r > 0, see (11). The new solution will then terminate with a smaller f > 0at say η2 > η1.

We still need to prove that there exists a solution that will not reach f ¼ 0 at finite η,i:e:, we need to show that f → 0 with f > 0 for all η > 0.

So now with y2ðη2Þ ¼ e=2 as above, observe that if we let δ2 ¼ ð−y02ð0ÞÞrþ1

, where y2ð0Þis the initial condition corresponding to the solution y2, which is extended to, and terminates

at η2, then Eq. (13) yields the following: δ2 ¼ ðrþ1Þk

ðð1− rÞη2�e2

�þ 2r

R η20 y2ðtÞdtÞ since

ð−y02ðη2ÞÞrþ1 ¼ 0. Similarly δ1 ¼ ðrþ1Þ

kðð1− rÞη1eþ 2r

R η10 y1ðtÞdtÞ, where δ1 ¼ ð−y01ð0ÞÞ

rþ1,

and y1 is the solution extending to η1 with y1ðη1Þ ¼ e; y01ðη1Þ ¼ 0. Therefore

δ2 � δ1 ¼ ðr þ 1Þk

�eð1� rÞ

�η22� η1

�þ 2r

Z η1

0

ðy2ðtÞ � y1ðtÞÞdt þ 2r

Z η2

η1

y2ðtÞdt�:

Observe that the last two terms in parentheses on the right-hand side of the equation abovesatisfy:

2r

Z η1

0

ðy2ðtÞ � y1ðtÞÞdt þ 2r

Z η2

η1

y2ðtÞdt < 3e

2ðη2 � η1Þ;

since the first integral is negative, and the second integral is smaller than the trapezoidal areaunder the line extending between ðη1; eÞ and ðη2; e=2Þ. This area is equal to 3e

4ðη2 − η1Þ, and

after multiplying this area by 2r and recalling that 0 < r < 1, the desired result is obtained.Now, note that δ2 − δ1 > 0 so we can deduce that eð1− rÞðη2

2− η1Þ þ 3e

2ðη2 − η1Þ > 0, and

therefore η2η1> 5− 2r

4− r¼ K > 1, for 0 < r < 1. In this manner, it can be shown that the solution

can be extended to η ¼ ∞ since we can go step by step to y ¼ e=2n; n ¼ 1; 2; 3; . . ., and reachη > Knη1, where K ¼ 5− 2r

4− r> 1 as given above.

To verify that f00stays negative for the new solution y2 one can check that

f000 ¼ −f

0 ðηf 00 ð1− rÞ2þ2f0 Þ þ rð1 þ rÞf f 00

kð−f 0 Þrþ1 . So, on the one hand, if y2ðη1Þ goes significantly below e,

with y02ðη1Þ relatively small in absolute value so that y

002ðη1Þ is large, and f

00approaches

infinity quickly, then it is obvious that f00stays positive (from (11)). On the other hand, if y2ðη1Þ

goes slightly below e, say to e0, with y02ðη1Þ becoming relatively large in absolute value, then

keep δ2 small, or close enough to δ1, so that y02ðη1Þ ¼ −e0ð1 þ rÞ

η1ð1− rÞ þ e0for some very small e

0that

will yield y002ðη1Þ ¼ −2y

02ðη1Þ

η1ð1− rÞ2 from (11). Observe now that the above expression for f000is

positive at η1 (with both terms in the numerator being positive) and will stay positive with f00

increasing, and f0increasing (becoming closer to zero). The fact that now y

002ðη1Þ is relatively

very small and using the above expression for f000, shows that by the point where we get to a

terminal point with y02 ¼ 0 and y

002 becoming unbounded, it must be that y2 is significantly

smaller than e, and where we leave out some of the details. The process can be repeated to

eventually get to a solution where y2ðη2Þ ¼ e=2 and where y002 > 0 is guaranteed on the

maximal interval of continuation for y2. Observe that this also reinforces our earlierdiscussion on the existence of y2 reaching e=2 and terminating.

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To establish uniqueness, suppose that f ðηÞ is a solution that satisfies (11) subject to (12).Define

FðηÞ ¼ ðr þ 1Þk

�ð1� rÞηf ðηÞ þ 2r

Z η

0

f ðtÞdt�;

and note thatFðηÞ is an increasing function such that in the limit we have: FðηÞ→ ð−f 0 ð0ÞÞrþ1

as η→∞, and where f0 ð0Þ is the initial condition corresponding to the given solution f .

Suppose that gðηÞ is another solution with g0 ð0Þ≠ f

0 ð0Þ, say ð−g 0 ð0ÞÞrþ1 ¼ ð−f 0 ð0ÞÞrþ1 þ ewith e≠ 0. Take e > 0: the solution g will then satisfy gðηÞ < f ðηÞ, for all η > 0, so that:

GðηÞ ¼ ðr þ 1Þk

ðð1� rÞηgðηÞ þ 2r

Z η

0

gðtÞdtÞ≤ FðηÞ; (17)

and where GðηÞ→ ð−g 0 ð0ÞÞrþ1 ¼ ð−f 0 ð0ÞÞrþ1 þ e as η→∞, which follows from ourassumption that g is another solution that satisfies (12). But then we would have

Gð∞Þ > ð−f 0 ð0ÞÞrþ1 ¼ Fð∞Þ, and this last inequality requires GðηÞ > FðηÞ for large η,which is a contradiction (it contradicts (17)). This completes the proof.

Figure 1 shows a typical solution to the Navier–Stokes equation (11) illustrating the aboveresult. Another result can readily be obtained here for r > 0; k < 0:

Proposition 2. There exists no solution to (11) subject to (12) for r > 0; k < 0 and wheref ðηÞ≥ 0 for all η > 0.

Proof. Under the hypotheses of the preceding theorem where f ðηÞ > 0 for all η, Eq. (13)will show that ð−f 0 ðηÞÞrþ1 > ð−f 0 ð0ÞÞrþ1 > 0. This implies that it is not possible to havef → 0 as η→∞. Nor is it possible to have a solution that reaches zero equilibrium at finite η:f0 ðηÞ ¼ 0 when f ðηÞ ¼ 0, for the same reason.In fact, solutions where r > 0; k < 0, will cross the axis, and will eventually terminate at

some point where f0 ðη0Þ ¼ 0; f ðη0Þ < 0, for some finite η0. This can be illustrated with the

aid of numerical integrators. (See Figure 2.)

Figure 1.A typical solution tothe Navier–Stokes

equation (11)with r > 0; k > 0.


equation

173

3.1 The case r < 0; k < 0As for the case where r < 0; k > 0, we begin by showing that a solution exists where

f0 ðη0Þ ¼ 0 at some finite η0 > 0: observe that with f

0< 0; f

00> 0 we have f ðηÞ > f ð0Þ

þ f0 ð0Þη, so that Eq. (13) yields:

ð−f 0 ðηÞÞrþ1< ð−f 0 ð0ÞÞrþ1 � ðr þ 1Þ

kðð1� rÞðf ð0Þ þ f

0 ð0ÞηÞηþ 2rf ð0ÞηÞ

< ð−f 0 ð0ÞÞrþ1 � ðr þ 1Þk

��1þ rÞf ð0Þηþ ð1� rÞf 0 ð0Þη2Þ:

Choose f0ð0Þ small enough in absolute value so that:

f ð0Þ > k

ð1þ rÞ2ð−f0 ð0ÞÞrþ1 � ð1� rÞ

ð1þ rÞ f0 ð0Þ:

This choicewill show that a solution exists such that for some η0 < 1, we have f0 ðη0Þ ¼ 0, and

the solution terminates. It can readily be verified that f0< 0; f

00> 0, within the interval of the

given solution, so that the above arguments stay valid.On the other hand, there exists a solution which crosses the axis at some finite η. This can

be established using the same arguments in the proof of the preceding theorem, as was statedearlier. However, observe that since k > 0 and f

0< 0, we must have

ð1� rÞηf 0 þ ð1þ rÞf > 0 (18)

in order to avoid any inflection point (with f > 0, and since the solution will cross the axisonce it has an inflection point, as the curvaturewill continue to be negative once it is negative).

Observe, now, that inequality (18) implies f0

f> −

ð1þrÞð1− rÞη, and therefore f > cη−

ð1þrÞð1−rÞ, where c is a

constant, and −ð1þrÞð1− rÞ < 0 for −1 < r < 1. Now, if f ¼ ηp where p > −

ð1þrÞð1− rÞ, then the above

inequality for f holds, but inequality (13) will have a divergent term on the right-hand side,and therefore f

0will reach zero in finite time say η1, with f ðη1Þ > 0, so that conditions (12)

will not be satisfied. On the other hand, if we let f ðηÞ ¼ cη−ð1þrÞð1−rÞ þ gðηÞ, with 0 < gðηÞ < ηq

(of order q less than p ¼ −ð1þrÞð1− rÞ, q is real and q < p) then the above inequality still holds, but

Figure 2.A typical solution toEq. (11) with r > 0;k < 0. It crossesthe axis.

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again a contradiction occurs upon substituting into (11), where we are led again to obtainingan inflection point. Therefore:

Theorem 3. There exists no solution to (11) subject to (12) if r < 0, and k > 0.

The dynamics here is the following: Solutions exist where f0reaches zero at some η0 > 0,

and f ðηÞ ¼ b for all η0 < η < ∞, for some large enough b > 0. However, there exists a certain

value for b > 0where further reduction of the initial condition f0ð0Þ (increase in absolute value

of the gradient) shall yield a solution that crosses the horizontal axis (f0ðηÞdoes not reach zero

but rather stays negative). This happens since the decay of solutions (changes in f and f0)

becomes extremely slow with f00proportional to ðð1− rÞη f 0 þ ð1þ rÞf Þðf 0 Þ−r (namely

observe the factor ðf 0 Þ−r with f0≈ 0 and where now r < 0), allowing the non-autonomous

term ð1− rÞηf 0with the presence of η, to exceed the last term ð1þ rÞf , of the governing

equation (11). This leads to a change in curvature, and therefore solutions will cross the axis,and will not satisfy f ð∞Þ ¼ 0 from (12). This is verified by numerical integrators, and isillustrated in Figure 3: In particular the two upper curves reach a pointwhere (11) is undefined

with f0 ¼ 0. Such solutions reach a terminal point, that they cannot be extended beyond. The

solution in the bottom illustrates that there is a minimum for f with those terminal points,after which solutions change curvature, and eventually will cross the axis.

3.2 The case r < 0; k < 0Unlike some of the previous cases, observe that in this case the governing equation (11)

implies that f00ð0Þ < 0. In fact, the curvature stays negative for some interval say ð0; η0Þ, until

f ðηÞdrops in value while f 0ðηÞbecomesmore negative (see (11)). Then f00 ðηÞbecomes positive,

and it can readily be established that f00ðηÞ stays positive, on the infinite interval, if j f 0 ð0Þj is

large enough. Additionally, if the solution crosses the horizontal axis then f00 ðηÞwill continue

to be positive in this case of k < 0, and in fact if the solution does cross the axis it will

eventually terminate with f0 ¼ 0: once the solution attains a negative value, say f0, then we

have f00 ð−f 0 Þr > ð1þ rÞ f0=k, so that −ð−f 0 ðηÞÞrþ1

≥ ðð1þ rÞ2f0=kÞðη− η0Þ− ð−f 0 ðη0ÞÞrþ1,

which implies that f0 ðηÞwill reach zero at finite η. With the existence of solutions that cross

the axis and then reach f0 ¼ 0, as stated by the remarks given above, another result is needed:

Figure 3.A set of solutions to Eq.(11) with r < 0; k > 0.They do not satisfy

(12): There is aminimum for f wheref0reaches zero and (11)becomes undefined(a terminal point),

beyond whichsolutions changecurvature with

f0 ðηÞ < 0 on the entire

solution domain.


equation

175

Lemma4.Two different solutions of (11)with the same initial f(0), but two different initial

gradients f01ð0Þ≠ f

02ð0Þ, do not intersect for any η > 0. Furthermore, if f

02ð0Þ < f

01ð0Þ with

f2ð0Þ≤ f1ð0Þ, then f02ðηÞ < f

01ðηÞ for all η > 0.

Proof. Given a solution with say f01ð0Þ, take another solution with f

02ð0Þ < f

01ð0Þ, and

where f2ð0Þ ¼ f1ð0Þ. The two solutions will be different in, at least a small interval say ð0; η0Þ,and f2 < f1 on that interval. If the two solutions intersect, then ηf ðηÞwould be the same for f1and f2 at the point of intersection, and therefore the right-hand side of (13) would be larger forthe solution f2. This, in turn, implies that f2ðηÞ is larger than f1ðηÞ in absolute value, so that

f02ðηÞ < f

01ðηÞ at the point of intersection, and now this is a contradiction (which in fact can

also be illustrated geometrically, as well as analytically).Now, using the continuity with respect to initial conditions, it can be concluded that the

solution f2 with the larger initial absolute gradient j f 02ð0Þj > j f 0

1ð0Þjwill always have a largerj f 0

2ðηÞj, at all η > 0where f01ðηÞ < 0 (i.e. avoiding a situationwhere f

0ðηÞ ¼ 0). Otherwise, at an

η where f02ðηÞ ¼ f

01ðηÞ, let us say that e > 0 represents the difference between the two

solutions: f2ðηÞ ¼ f1ðηÞ− e. Then, observe that we would have f002 ðηÞ > f

001 ðηÞ, where f

002 ðηÞ is

larger precisely by the amount eð1þ rÞð−f 0 Þ−r=k (see (11)). Nowwe can take esmall enough sothat the two solutions would intersect at some point, say at ηþ Δη (an argument here can bemade, for example, using a Taylor series expansion). This contradicts the first result in the

lemma, proven above. Now, note that the possibility f02ðηÞ > f

01ðηÞ would imply that

f02ðη0Þ ¼ f

01ðη0Þat some 0 < η0 < η, since f

02ð0Þ < f

01ð0Þ. Therefore, the obtained contradiction

would still eliminate this last possibility. This result can be generalized using similararguments for f2ð0Þ < f1ð0Þ.

With solutions that reach f0 ¼ 0; f ¼ constant < 0, and the above lemma, we may

“construct” a solution that reaches zero equilibrium (f ¼ 0) at finite η: given a solution thatreaches equilibrium at a constant f ¼ c < 0, take another solution with a smaller j f 0 ð0Þj sothat it reaches a terminal point f ¼ d > c, at a smaller value of η (with f

0 ðηÞ ¼ 0). (This is aconsequence of the preceding lemma.) Proceed in this fashion to find a solution that reacheszero at finite η (See Figure 4). Another way to view this is the following: we have solutions thatcross the horizontal axis at η0 with a negative f

0 ðη0Þ, so that taking another solution with asmaller j f 0 ð0Þj leads to a less negative f 0 ðη0Þat η0, and with f ðη0Þ > 0. If the change in f

0 ð0Þ issmall enough, the new solutionwill then cross the axis, but at a larger ηandwith a smaller j f 0 j

Figure 4.A typical solution toEq. (11) withr < 0; k < 0. It reacheszero equilibrium atfinite η ( ≈ 30 inthis particular figure).

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(at the point of crossing). This process can be continued until the required solution is reached.So this solution is established here, mathematically, as a limiting case.

Remark. Observe that the two different views above involve the same set of solutions.Theorem 5. Solutions to (11) subject to (12) exist for r < 0; k < 0, and where f ðηÞ≥ 0 for

all η > 0.In fact, analysis of Eq. (13) suggests that other solutions may exist but where f ðηÞ > 0 for

all η > 0, and with possibly an infinite number of points where the solution changescurvature. In such a case, the quantity ηf ðηÞdoes not approach zero due to balancing positiveand negative terms in (13), which cannot approach zero. Furthermore, it can be easily checkedthat any solution of (11), with r < 0; k < 0; f ð0Þ > 0, and any choice of f

0 ð0Þ < 0, will satisfyf0 ðηÞ < 0 for all η > 0 as long as f ðηÞ > 0, and cannot approach an equilibrium f ¼ c > 0.

4. ConclusionsWe studied a reduced problem from the Navier–Stokes and the continuity equations in two-dimensional Cartesian coordinates, with Eulerian description, for incompressible non-Newtonian fluids. We have shown the existence of positive solutions to the reduced ODE,f ≥ 0, f

0≤ 0, and where f ð∞Þ ¼ 0. Such solutions exist if rk > 0. Those solutions may not be

unique if the flow behavior index r < 0. On the other hand, positive solutions do not exist ifrk < 0. Additionally, a solution exists and has been explicitly expressed when r ¼ 0; k < 0.

References

[1] G. Astarita, G. Marrucci, Principles of Non–Newtonian Fluid Mechanics, McGraw-Hill,Malabar, 1974.

[2] I.F. Barna, Self-similar solutions of three-dimensional Navier–Stokes equation, Commun. Theor.Phys. 56 (4) (2011) 745–750, http://dx.doi.org/10.1088/0253-6102/56/4/25.

[3] I. Barna, G. Bognar, K. Hriczo, Self-similar analytic solution of the two-dimensional Navier–Stokesequation with a non-Newtonian type of viscosity, Math. Model. Anal. 21 (1) (2016) 83–94, http://dx.doi.org/10.3846/13926292.2016.1136901.

[4] I.F. Barna, L. Matyas, Analytic solutions for the three-dimensional compressible Navier–Stokesequation, Fluid Dyn. Res. 46 (5), 2014, http://dx.doi.org/10.1088/0169-5983/46/5/055508.

[5] N. Bedjaoui, M. Guedda, Z. Hammouch, Similarity Solutions of the Rayleigh problem for Ostwald-de Wael electrically conducting fluids, Anal. Appl. 9 (2) (2011) 135–159.

[6] G. Bogn�ar, Similarity solution of a boundary layer flow for non-Newtonian fluids, Int. J. NonlinearSci. Numer. Simul. 10 (2010) 1555–1566.

[7] G. Bogn�ar, On similarity solutions of boundary layer problems with upstream moving wall innon-Newtonian power-law fluids, IMA J. Appl. Math. 77 (2012) 546–562.

[8] G. Bohme, Non–Newtonian fluid mechanics, in:North-Holland Series in Applied Mathematics andMechanics, Amsterdam, 1987.

[9] J.P. Denier, P. Dabrowski, On the boundary-layer equations for power–law fluids, Proc. R. Soc. A460 (2004) 3143–3158.

[10] W. Gao, J. Wang, Similarity solutions to the power-law generalized Newtonian fluid, J. Comput.Appl. Math. 222 (2008) 381–391.

[11] M. Guedda, Z. Hammouch, Similarity flow solutions of a non-Newtonian power-law fluid flow, Int.J. Nonlinear Sci. 6 (3) (2008) 255–264.

[12] M. Guedda, R. Kersner, Non-Newtonian pseudoplastic fluids: Analytical results and exactsolutions, Int. J. Non-Linear Mech. 4 (7) (2011) 949–957.

[13] Z. Hammouch, Multiple solutions of steady MHD flow of dilatant fluids, Eur. J. Pure Appl. Math. 1(2) (2008) 11–20.


equation

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[14] X. Hu, Z. Dong, F. Huang, Y. Chen, Symmetry reductions and exact solutions of the (2þ1)-dimensional Navier–Stokes equations, Z. Nat.forsch. A 65 (2010) 504–510.

[15] M.Y. Hussaini, W.D. Lakin, Existence and non-uniqueness of similarity solutions of a boundary-layer problem, Q. J. Mech. Appl. Math. 39 (1986) 177–191.

[16] O.A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, AmericanMathematical Society, Providence, RI, 1970.

[17] J.H. Merkin, On dual solutions occurring in mixed convection in a porous medium, J. Eng. Math.20 (1985) 171–179.

[18] H. Schlichting, Boundary Layer Theory, McGraw-Hill Press, New York, 1979.

[19] L. Sedov, Similarity and Dimensional Methods in Mechanics, CRC Press, Boca Raton, 1993.

[20] D. Wei, S. Al-Ashhab, Similarity solutions for a non-newtonian power-law fluid flow, Appl. Math.Mech. (English Ed.) 35 (2014) 1155–1166, http://dx.doi.org/10.1007/s10483-014-1854-6.

[21] L. Zheng, X. Zhang, J. He, Existence and estimate of positive solutions to a nonlinear singularboundary value problem in the theory of dilatant non-Newtonian fluids, Math. Comput. Modelling45 (2007) 387–393.

Corresponding authorSamer Al-Ashhab can be contacted at: [email protected]


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Coupled fixed points and coupledbest proximity points for cyclic

�Ciri�c type operatorsAdrian Magdas

Faculty of Mathematics and Computer Science, Babes Bolyai University,Cluj-Napoca, Romania

AbstractThe purpose of this paper is to study the coupled fixed point problem and the coupled best proximityproblem for single-valued and multi-valued contraction type operators defined on cyclic representations ofthe space. The approach is based on fixed point results for appropriate operators generated by the initialproblems.

KeywordsMetric space, Single-valued operator, Multi-valued operator, Fixed point, Coupled fixed point, Best

proximity point, Coupled best proximity point, Generalized contraction, Data dependence, Ulam–Hyers

stability, Well-posedness


1. IntroductionOne of the most important metrical fixed point theorem, Banach contraction principle, hasbeen generalized in several directions, see for example [1]. The concept of coupled fixed pointwas introduced by Guo and Lakshmikantham (see [2]). A new research direction for thetheory of coupled fixed points was developed by many authors (see [3–9]) using contractivetype conditions.

Definition 1.1 ([10]). Let X be a nonempty set. A pair ðx; yÞ∈X 3X is called coupled fixedpoint of the operator F : X 3X →X if Fðx; yÞ ¼ x and Fðy; xÞ ¼ y. If Fðx; xÞ ¼ x then x iscalled a strong coupled fixed point of F (or, in several papers, a fixed point of F).

Another generalization of the Banach principle was given by Kirk, Srinivasan andVeeramani using the concept of cyclic operators.

Coupled fixedpoints of cyclictype operators

179

JEL Classification — 41A50, 47H09, 47H10, 57H25© Adrian Magdas. Published in the Arab Journal of Mathematical Sciences. Published by Emerald


The author is thankful to the referees for their useful suggestions.Declaration of Competing Interest: No author associated with this paper has disclosed any potential

or pertinent conflicts which may be perceived to have impending conflict with this work.The publisher wishes to inform readers that the article “Coupled fixed points and coupled best

proximity points for cyclic �Ciri�c type operators” was originally published by the previous publisher ofthe Arab Journal of Mathematical Sciences and the pagination of this article has been subsequentlychanged. There has been no change to the content of the article. This change was necessary for thejournal to transition from the previous publisher to the new one. The publisher sincerely apologises forany inconvenience caused. To access and cite this article, please use Magdas, A. (2019), “Coupled fixedpoints and coupled best proximity points for cyclic �Ciri�c type operators”, Arab Journal of MathematicalSciences, Vol. 26 No. 1/2, pp. 179-196. The original publication date for this paper was 22/05/2019.



Received 14 December 2018Revised 15 May 2019

Accepted 16 May 2019


Vol. 26 No. 1/2, 2020pp. 179-196


DOI 10.1108/j.ajmsc.2019.05.002

Definition 1.2 ([11]). Let A and B be nonempty subsets of a given set X. An operatorT : A∪B→A∪B is called cyclic if TðAÞ⊆B and TðBÞ⊆A.

More recently, Choudbury and Maity formulated the following definition.

Definition 1.3 ([12]). Let A and B be nonempty subsets of a given set X. An operatorF : X 3X →X having the property that for any x ∈ A and y ∈ B, Fðx; yÞ ∈ B andFðy; xÞ ∈ A, is called a cyclic operator with respect to A and B.

Definition 1.4 ([13]). Let A and B be nonempty subsets of a metric space ðX ; dÞ.An operator F : X 3X →X is called a cyclic �Ciri�c operator with respect to A and B if F iscyclic with respect to A and B and for some constant q∈ ð0; 1Þ, F satisfies the followingcondition:

dðFðx; yÞ;Fðu; vÞÞ ≤ q$Mðx; v; y; uÞ;

where x; v ∈ A, y; u ∈ B, and

Mðx; v; y; uÞ ¼ max

�dðx; uÞ; 1

2dðu;Fðx; yÞÞ; 1

2dðx;Fðu; vÞÞ;

1

2½dðx;Fðx; yÞÞ þ dðu;Fðu; vÞÞ�

�:

Theorem 1.1 ([13]). Let A and B be nonempty closed subsets of a complete metric spaceðX ; dÞ, F : X 3X →X a cyclic �Ciri�c type operator with respect to A and B, with A∩B≠ 0= .Then F has a strong coupled fixed point in A∩B.

The first aim of this paper is to generalize the above theorem, weakening the contractivecondition and excluding the condition A∩B≠ 0=. We prove the uniqueness of the strongcoupled fixed point andwe provide an iterative method for approximating the strong coupledfixed point.

We also present coupled fixed point and coupled best proximity point results for cycliccoupled �Ciri�c-type multivalued operators.

On the other hand, some qualitative properties of the coupled fixed point set, such as datadependence, generalized Ulam–Hyers stability and well-posedness are studied.

Our approach is based on the following idea: we transform the coupled fixed point/ bestproximity point problem into a fixed point/ best proximity point problem for an appropriateoperator defined on a cartesian product of the spaces. In this way, many coupled fixed point/best proximity point results can be obtained using classical fixed point/ best proximity pointtheorems.

2. PreliminariesThe standard notations and terminologies in nonlinear analysis will be used throughoutthis paper.

Let ðX ; dÞ be a metric space. We denote:

PðXÞ :¼ fY ⊆X jY is nonemptyg;PbðXÞ :¼ fY ∈PðXÞ jY is boundedg;PclðXÞ :¼ fy∈PðXÞ jY is closedg;PcpðXÞ :¼ fY ∈PðXÞ jY is compactg:

Let us define the following (generalized) functionals used in this paper:

• The gap functional

D : PðXÞ3PðXÞ→ℝþ; DðA;BÞ ¼ inffdða; bÞ j a∈A; b∈Bg;

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180

• The generalized excess functional

ρ : PðXÞ3PðXÞ→ℝþ ∪ fþ∞g; ρðA;BÞ ¼ supfDða;BÞ j a∈Ag;• The generalized Pompeiu–Hausdorff functional

H : PðXÞ3PðXÞ→ℝþ ∪ fþ∞g;HðA;BÞ ¼ maxfρðA;BÞ; ρðB;AÞg:There are several conditions upon the comparison function that have been considered in

literature. In this paper we shall refer only to:

Definition 2.1 ([14]).A function w : ℝþ →ℝþ is called a comparison function if it satisfies:

(i) w is increasing;

(ii) ðwnðtÞÞn∈ℕ converges to 0 as n→∞, for all t ∈ℝþ.

If the condition (ii) is replaced by the condition:

(iii)P∞

k¼0wkðtÞ < ∞, for any t > 0, then w is called a strong comparison function.

Lemma 2.1 ([1]). If w : ℝþ →ℝþ is a comparison function, then wðtÞ < t , for any t > 0 ,wð0Þ ¼ 0 and w is continuous at 0.

Lemma 2.2 ([14]). If w : ℝþ →ℝþ is a strong comparison function, then the following hold:

(i) w is a comparison function;

(ii) the function s : ℝþ →ℝþ , defined by

sðtÞ ¼X∞k¼0

wkðtÞ;

is increasing and continuous at 0.

Example 2.1 ([15]). (1) w : ℝþ →ℝþ, wðtÞ ¼ at, where a∈ ½0; 1Þ, is a strong comparisonfunction;

(2) w : ℝþ →ℝþ, wðtÞ ¼ 12 t, for t ∈ ½0; 1� and wðtÞ ¼ t − 1

2, for t > 1, is a strongcomparison function;

(3) w : ℝþ →ℝþ, wðtÞ ¼ at þ 12 ½t�, where a∈ ð0; 12Þ, is a strong comparison function;

(4) w : ℝþ →ℝþ, wðtÞ ¼ t1þt

, is a comparison function, but is not a strong comparisonfunction.

For more examples and considerations on comparison functions see [1] and thereferences therein.

3. Coupled fixed points of cyclic �Ciri�c type single valued operatorsIn this section we present some coupled fixed point results for cyclic �Ciri�c type operators oncomplete metric spaces.

We introduce now the following new concept.

Definition 3.1 Let ðX ; dÞ be a metric space, A;B∈PclðXÞ, Y ¼ A∪B and w: Rþ →Rþ astrong comparison function. An operator F : Y 3Y →Y is called a cyclic coupledw-contraction of �Ciri�c type if the following statements hold:

(i) F is cyclic with respect to A and B;


181

(ii) dðFðx; yÞ;Fðu; vÞÞ≤wðMðx; v; y; uÞÞ; (3.1)

for any x; v∈A and y; u∈B, where

Mðx; v; y; uÞ ¼ max

�dðx; uÞ; dðv; yÞ; dðx;Fðx; yÞÞ; dðu;Fðu; vÞÞ; dðv;Fðv; uÞÞ;

dðy;Fðy; xÞÞ; 12½dðx;Fðu; vÞÞ þ dðu;Fðx; yÞÞ�;

1

2½dðy;Fðv; uÞÞ þ dðv;Fðy; xÞÞ�

�:

The following theorem (which is a particular case of Theorem 3.2 in [16]) will be used toprove our results presented in this section.

Theorem 3.1 ([16]). Let ðX ; dÞ be a complete metric space, A;B∈PclðXÞ , w : ℝþ →ℝþ be astrong comparison function and f : A∪B→A∪B be an operator such that f ðAÞ⊆B andf ðBÞ⊆A . If f is a cyclic w -contraction of �Ciri�c type, that is

dðf ðxÞ; f ðyÞÞ≤w

�max

�dðx; yÞ; dðx; f ðxÞÞ; dðy; f ðyÞÞ;

1

2½dðx; f ðyÞÞ þ dðy; f ðxÞÞ�

��;

for any x∈A and y∈B , then the following statements hold:(1) f has a unique fixed point x* ∈A∩B and the Picard iteration fxngn≥0 defined by

xn ¼ f ðxn−1Þ, n≥ 1 , converges to x* for any starting point x0 ∈A∪B;(2) the following estimates hold:

dðxn; x*Þ ≤ sðwnðdðx0; x1ÞÞÞ; n≥ 1;dðxn; x*Þ ≤ sðdðxn; xnþ1ÞÞ; n≥ 1;

(3) for any x∈A∪B, dðx; x*Þ≤ sðdðx; f ðxÞÞÞ , where s is given by Lemma 2.2.The main result of this section is the following theorem.

Theorem 3.2. Let ðX ; dÞ be a complete metric space, A;B∈PclðXÞ, Y ¼ A∪B andF : Y3Y →Y a cyclic coupled w -contraction of �Ciri�c type. Then:

(1) F has a unique strong coupled fixed point x* ∈A∩B;

(2) for any ðx0; y0Þ∈A3B, there exists a sequence fðxn; ynÞgn∈ℕ⊂X 3X defined by�xn ¼ Fðyn−1; xn−1Þyn ¼ Fðxn−1; yn−1Þ ; n ≥ 1;

that converges to ðx; xÞ;(3) the following estimates hold:

maxfdðxn; x*Þ; dðyn; x*Þg ≤ sðwnðmaxfdðx0;Fðx0; y0ÞÞ; dðy0;Fðy0; x0ÞÞgÞÞ; n≥ 1;maxfdðxn; x*Þ; dðyn; x*Þg ≤ sðmaxfdðxn; xnþ1Þ; dðyn; ynþ1ÞgÞ; n≥ 1;

(4) for any x; y∈Y, dðx; x*Þ ≤ sðmaxfdðx;Fðx; yÞÞ; dðy;Fðy; xÞÞgÞ , where s is given byLemma 2.2.

Proof. ð1Þ−ð2Þ Changing the roles between x and v and similarly for y and u, the inequality(3.1) becomes:

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dðFðv; uÞ;Fðy; xÞÞ≤wðMðv; x; u; yÞÞ; for x; v∈A and y; u∈B: (3.2)

Obviously, Mðx; v; y; uÞ ¼ Mðv; x; u; yÞ. From the inequalities (3.1) and (3.2) we obtain

maxfdðFðx; yÞ;Fðu; vÞÞ; dðFðy; xÞ;Fðv; uÞÞg≤wðMðx; v; y; uÞÞ: (3.3)

For z ¼ ðx; yÞ∈A3B, w ¼ ðu; vÞ∈B3A, denote

d*ðz;wÞ ¼ maxfdðx; uÞ; dðy; vÞg: (3.4)

Then ðX 3X ; d*Þ is a complete metric space.Let T : Y 3Y →Y 3Y be defined by Tðx; yÞ ¼ ðFðx; yÞ;Fðy; xÞÞ. We have:

1

2½d*ðz;TðwÞÞ þ d*ðw;TðzÞÞ� ¼ 1

2maxfdðx;Fðu; vÞÞ; dðy;Fðv; uÞÞg

þ 1

2maxfdðu;Fðx; yÞÞ; dðv;Fðy; xÞÞg

≥max

�1

2½dðx;Fðu; vÞÞ þ dðu;Fðx; yÞÞ�;1

2½dðy;Fðv; uÞÞ þ dðv;Fðy; xÞÞ�

�:

Using the above relation, from (3.3) we get

d*ðTðzÞ;TðwÞÞ≤w

�max

�d*ðz;wÞ; d*ðz;TðzÞÞ; d*ðw;TðwÞÞ;1

2½d*ðz;TðwÞÞ þ d*ðw;TðzÞÞ�

��;

(3.5)

for any z∈A3B, w∈B3A.Because FðA3BÞ⊆B and FðB3AÞ⊆A, we have

TðA3BÞ⊆B3A and TðB3AÞ⊆A3B: (3.6)

(3.5) and (3.6) means that the operator T is a cyclic w-contraction of �Ciri�c type. ApplyingTheorem 3.1, there exists a unique z* ¼ ðx*; y*Þ∈ ðA3BÞ∩ðB3AÞsuch thatTðz*Þ ¼ z* andthe Picard iteration zn ¼ Tðzn−1Þ converges to z* for any starting point z0 ∈Y . So�

Fðx*; y*Þ ¼ x*

Fðy*; x*Þ ¼ y*(3.7)

where x*; y* ∈A∩B.From unicity of the pair ðx*; y*Þ and the symmetry with respect to x* and y* of the

system (3.7) we conclude x* ¼ y*.Then F has a unique strong coupled fixed point x* ∈A∩B and for any starting point

ðx0; y0Þ∈A3B there exists a sequence fðxn; ynÞgn∈ℕ⊂Y 3Y with�xn ¼ Fðyn−1; xn−1Þyn ¼ Fðxn−1; yn−1Þ ; n ≥ 1

that converges to ðx*; x*Þ.


183

(3) By the second conclusion of Theorem 3.1,

d*ðzn; ðx*; x*ÞÞ ≤ sðwnðd*ðz0; z1ÞÞÞand

d*ðzn; ðx*; x*ÞÞ ≤ sðd*ðzn; znþ1ÞÞ; n≥ 1:Hence

maxfdðxn; x*Þ; dðyn; x*Þg ≤ sðwnðmaxfdðx0;Fðx0; y0ÞÞ; dðy0;Fðy0; x0ÞÞgÞÞmaxfdðxn; x*Þ; dðyn; x*Þg ≤ sðmaxfdðxn; xnþ1Þ; dðyn; ynþ1gÞÞ; n≥ 1:

(4) Using (3) from Theorem 3.1, for any ðx; yÞ∈Y 3Y ,

d*ððx; yÞ; ðx*; x*ÞÞ ≤ sðd*ððx; yÞ;Tðx; yÞÞÞ:Hence

maxfdðx; x*Þ; dðy; x*Þg ≤ sðmaxfdðx;Fðx; yÞÞ; dðy;Fðy; xÞÞgÞ: ,

Example 3.1. Let X ¼ ℝ; dðx; yÞ ¼ jx− yj; for any x; y∈ℝ, A ¼ ½0; 2�, B ¼ ½0; 1�, Y ¼A∪B, F : Y 3Y →Y , Fðx; yÞ ¼ xþ3y

9 .It is easy to verify that F is cyclic with respect to A and B.For any x, v∈A and y, u∈B

dðFðx; yÞ;Fðu; vÞÞ ¼ jxþ 3y

9� uþ 3v

9j

¼ jx� u

9þ y� v

3j

≤ j19ðx� uÞ þ 10

27ðy� vÞj

¼ 1

3jy� vþ 3u

9þ yþ 3x

9� vj

≤1

3ðjy� Fðv; uÞj þ jv� Fðy; xÞjÞ

≤2

3$1

2½dðy;Fðv; uÞÞ þ dðv;Fðy; xÞÞ�:

Then F is a cyclic coupled w-contraction of �Ciri�c type, where wðtÞ ¼ 23$t.

The hypotheses of Theorem 3.2 are satisfied, so by Theorem 3.2, F has a unique strongcoupled fixed point x* ∈A∩B. By calculation we get:

Fðx*; x*Þ ¼ x*5x* ¼ 0:

Our next theorem gives the well-posedness property for the coupled fixed point problem.For the concept of well-posedness for the fixed point problems see [17].

Theorem 3.3. Let F : Y 3Y →Y be as in Theorem 3.2. Then the coupled fixed pointproblem is well posed, that is, if there exists a sequence fðan; bnÞg n∈ℕ⊂Y 3Y such that�

dðan;Fðan; bnÞÞ→ 0dðbn;Fðbn; anÞÞ→ 0

as n→∞;

then an → x* and bn → x*, as n→∞.

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Proof. Using the inequality

dðx; x*Þ ≤ sðmaxfdðx;Fðx; yÞÞ; dðy;Fðy; xÞÞgÞfrom Theorem 3.2 for x :¼ an and next for x :¼ bn, we have:�

dðan; x*Þ ≤ sðmaxfdðan;Fðan; bnÞÞ; dðbn;Fðbn; anÞÞgÞdðbn; x*Þ ≤ sðmaxfdðbn; Fðbn; anÞÞ; dðan;Fðan; bnÞÞgÞ ; n∈ℕ;

and letting n→∞we obtain �dðan; x*Þ→ 0dðbn; x*Þ→ 0

; n→∞:

For the data dependence problem we have the following result.

Theorem 3.4. Let F : Y 3Y →Y be as in Theorem 3.2. Let G: Y 3Y →Y be such that:

(i) G has at least one strong coupled fixed point x*G;

(ii) there exists η > 0 such that

dðFðx; xÞ;Gðx; xÞÞ≤ η; for any x∈Y :

Then dðx*F ; x*GÞ≤ sðηÞ , where x*F is the unique strong coupled fixed point of F and

sðtÞ ¼X∞k¼0

wkðtÞ; t ∈ℝþ:

Proof. By letting x :¼ x*G and y :¼ x*G in the inequality

dðx; x*Þ≤ sðmaxfdðx;Fðx; yÞÞ; dðy;Fðy; xÞÞgÞ;we have

dðx*G; x*FÞ ≤ sðdðx*G;Fðx*G; x*GÞÞÞ ¼ sðdðGðx*G; x*GÞ;Fðx*G; x*GÞÞÞ;and using the monotonicity of swe obtain

dðx*F ; x*GÞ≤ sðηÞ:

Theorem 3.5. Let F : Y 3Y →Y be as in Theorem 3.2 and Fn: Y 3Y →Y , n∈N , besuch that:

(i) for each n∈ℕ there exists a strong coupled fixed point x*n of Fn ;

(ii) fFngn∈ℕ converges uniformly to F.Then x*n → x* as n→∞ , where x* is the unique strong coupled fixed point of F.

Proof. The sequence fFngn∈ℕ converges uniformly to F. Then there exist ηn ∈ℝþ, n∈ℕsuch that ηn → 0 as n→∞ and

dðFnðx; yÞ;Fðx; yÞÞ≤ ηn for any ðx; yÞ∈Y3Y :

Using Theorem 3.3 for G :¼ Fn, n∈N, we have

dðxn; x*Þ≤ sðηnÞ as n→∞:


185

We will discuss Ulam–Hyers stability for the coupled fixed point problem correspondingto a cyclic operator.

Definition 3.2. Let ðX ; dÞ be a metric space, Y ∈PðXÞ and F : Y 3Y →Y be an operator.The coupled fixed point problem �

Fðx; yÞ ¼ x

Fðy; xÞ ¼ y; x; y∈Y (3.8)

is called generalized Ulam–Hyers stable if there existsψ : ℝþ →ℝþ increasing, continuous at0 and ψð0Þ ¼ 0 such that for any ε1 > 0; ε2 > 0 and for any solution ðx; yÞ∈Y 3Y of thesystem �

dðx;Fðx; yÞÞ≤ ε1dðy;Fðy; xÞÞ≤ ε2

there exists a solution ðx*; y*Þ of the coupled fixed point problem such that�dðx; x*Þ≤ψðεÞdðy; y*Þ≤ψðεÞ ; where ε ¼ maxfε1; ε2g:

In particular, if x* ¼ y*, then we have generalized Ulam–Hyers stability for the strongcoupled fixed point problem Fðx; xÞ ¼ x; x∈Y .

Theorem 3.6. Suppose that all the hypotheses of Theorem 3.2 hold. Then the coupled fixedpoint problem (3.8) is generalized Ulam–Hyers stable.

Proof. By Theorem 3.2 we have a unique x* ∈Y such that Fðx*; x*Þ ¼ x*.Let ε1 > 0; ε2 > 0 and ð~x;~yÞ∈Y 3Y such that�

dð~x;Fð~x;~yÞÞ≤ ε1dð~y;Fð~y;~xÞÞ≤ ε2:

We know that

dðx; x*Þ≤ sðmaxfdðx;Fðx; yÞÞ; dðy;Fðy; xÞÞgÞ; ∀ðx; yÞ∈Y 3Y :

Then for �x :¼ ~xy :¼ ~y

�

and next for �x :¼ ~yy :¼ ~x

�

using the monotonicity of s, we obtain that

maxfdð~x; x*Þ; dð~y; x*Þg≤ sðmaxfdð~x;Fð~x;~yÞÞ; dð~y;Fð~y;~xÞÞgÞ≤ sðmaxfε1; ε2gÞ:As a conclusion, the coupled fixed point problem (3.8) is generalized Ulam–Hyers stablewith ψ ¼ s.

4. Coupled fixed points and coupled best proximity points of cyclic �Ciri�c typemultivalued operatorsThe purpose of this section is to consider the above problems in the multi-valued setting. Wepresent first a new concept of cyclic multi-valued operator.

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Definition 4.1. Let ðX ; dÞ be a metric space, A;B∈PðXÞ, Y ¼ A∪B and w : ℝþ →ℝþ astrong comparison function. A multivalued operator F : Y 3Y →PðY Þ is called a cycliccoupled w-contraction of �Ciri�c type multivalued operator if the following statements hold:

(i) F is cyclic with respect to A and B, that is

FðA3BÞ⊆B and FðB3AÞ⊆A;

(ii)

HðFðx; yÞ;Fðu; vÞÞ≤wð ~Mðx; v; y; uÞÞ; for any x; v∈A; y; u∈B (4.1)

where

~Mðx; v; y; uÞ ¼ max

�dðx; uÞ; dðv; yÞ;Dðx;Fðx; yÞÞ;Dðu;Fðu; vÞÞ;Dðv;Fðv; uÞÞ;

Dðy;Fðy; xÞÞ; 12½Dðx;Fðu; vÞÞ þ Dðu;Fðx; yÞÞ�; 1

2½Dðy;Fðv; uÞÞ þ Dðv;Fðy; xÞÞ�

�:

Definition 4.2. Let ðX ; dÞ be a metric space. Then Y ∈PðXÞ is called proximinal if for anyx∈X, there exists y∈Y such that

dðx; yÞ ¼ Dðx;Y Þ:We denote Pprox ¼ fy∈PðXÞ jY is proximinalg.Remark 4.1. Let ðX ; dÞ be a metric space. Then

PcpðXÞ⊂PproxðXÞ⊂PclðXÞ:

Remark 4.2. Every closed convex subset of a uniformly Banach space is proximinal,see [18].

For details concerning the above notions see [1,19] and [20].The following theorem (which is a particular case of Theorem 2.7 in [21]) will be used to

prove the first result in this section.

Theorem 4.1. ([21]). Let ðX ; dÞ be a complete metric space, A;B∈PclðXÞ andT : A∪B→PproxðA∪BÞ a multivalued cyclic w -contraction of �Ciri�c type, that is:

(i) TðAÞ ⊆ B and TðBÞ⊆A;

(ii) there exists a strong comparison function w : ℝþ →ℝþ such that

HðTðxÞ;TðyÞÞ≤w

�max

�dðx; yÞ;Dðx;TðxÞÞ;Dðy;TðyÞÞ;

1

2½Dðx;TðyÞÞ þ Dðy;TðxÞÞ�

��;

for any x∈A and y∈B .Then the following statements hold:

(1) there exists x* ∈A∩B such that x* ∈Tðx*Þ;(2) for any x∈A and y∈TðxÞ , there exists a sequence ðxnÞn∈ℕ with x0 ¼ x , x1 ¼ y and

xn ∈Tðxn−1Þ, n≥ 1 , that converges to a fixed point x* ∈A∩B of T.

The following lemma presents a well-known result (see for example [22]).


187

Lemma 4.1. Let ðX ; dÞ be a metric space, d* the metric defined on X 3X by (3,4) and D* thegap functional, respectively H* the generalized Pompeiu–Hausdorff functional generated by d* .Then for any a; b∈X and any A;B;C;D∈PproxðXÞ, the following statements hold:

(1) D*ðða; bÞ;C3DÞ ¼ maxðDða;CÞ;Dðb;DÞÞ;(2) D*ðA3B;C3DÞ ¼ maxðDðA;CÞ;DðB;DÞÞ;(3) H *ðA3B;C3DÞ ¼ maxfHðA;CÞ;HðB;DÞg;(4) D*ðA3B;B3AÞ ¼ DðA;BÞ.

Proof. (1)þ(2) Since the sets C and D are proximinal then there exists c0 ∈C; d0 ∈D such thatDða;CÞ ¼ dða; c0Þ and Dðb;DÞ ¼ dðb; d0Þ.

Then

D*ðða; bÞ;C3DÞ ¼ inffd*ðða; bÞ; ðc; dÞÞjc∈C; d∈Dg¼ inffmaxfdða; cÞ; dðb; dÞgjc∈C; d∈Dg¼ maxfdða; c0Þ; dðb; d0Þg:

Similarly, we can prove (2).(3) H*ðA3B;C3DÞ ¼max

�supða;bÞ∈A3BfD*ðða; bÞ;C3DÞg; supðc;dÞ∈C3DfD*ððc; dÞ;A3BÞgg:

Using statement (1), we have

H *ðA3B;C3DÞ ¼ max�supða;bÞ∈A3BfDða;CÞ;Dðb;DÞg; supðc;dÞ∈C3DfDðc;AÞ;Dðd;BÞgg

¼ maxfHðA;CÞ;HðB;DÞg(4) We use statement (2) for C ¼ A;D ¼ B.

Lemma 4.2. Let ðX ; dÞ be a metric space, d* the metric defined on X 3X by (3.4) . If amultivalued operator F : X 3X →PðXÞ takes proximinal values with respect to d then themultivalued operator T : X 3X →PðX 3XÞ, Tðx; yÞ ¼ ðFðx; yÞ;Fðy; xÞÞ takes proximinalvalues with respect to d*.

Proof. For any pair ða; bÞ∈X 3X ;Fða; bÞ is a proximinal set, which means that for anyx∈X, there exists c∈Fða; bÞ such that

dðx; cÞ ¼ Dðx;Fða; bÞÞ:In a similar way, for any y∈X, there exists d∈Fðb; aÞ such that

dðy; dÞ ¼ Dðy;Fðb; aÞÞ:Then for any ðx; yÞ∈X 3X, there exists ðc; dÞ∈Tða; bÞ such that

d*ððx; yÞ; ðc; dÞÞ ¼ maxfdðx; cÞ; dðy; dÞg¼ maxfDðx;Fða; bÞÞ;Dðy;Fðb; aÞÞg¼ D*ððx; yÞ;Tða; bÞÞ:

The first result in this section is the following theorem.

Theorem 4.2. Let ðX ; dÞ be a complete metric space, A;B∈PclðXÞ, Y ¼ A∪B andF : Y 3Y →PproxðY Þ a cyclic coupled w-contraction of �Ciri�c type multivalued operator.

Then the following statements hold:

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(1) there exist x*; y* ∈A∩B such that

x* ∈Fðx*; y*Þ; y* ∈Fðy*; x*Þ;(that is the pair ðx*; y*Þ is a coupled fixed point of F );

(2) for each ða; bÞ∈A3Bthere exists a sequence ðan; bnÞn∈ℕ* ∈Y 3Y with a0 ¼ a, b0 ¼ band

an ∈Fðbn−1; an−1Þ; bn ∈Fðan−1; bn−1Þ for n ≥ 1

that converges to a coupled fixed point ðx*; y*Þ∈A∩B of F .

Proof. It is easy to observe that

~Mðx; v; y; uÞ ¼ ~Mðv; x; u; yÞ; for any x; v∈A; y; u∈B:

If we change the roles between x and v and similarly for y and u, then the inequality (4.1)becomes

HðFðv; uÞ;Fðy; xÞÞ≤wð ~Mðx; v; y; uÞÞ: (4.2)

From (4.1) and (4.2) we obtain

maxfHðFðx; yÞ;Fðu; vÞÞ;HðFðy; xÞ;Fðv; uÞÞg≤wð ~Mðx; v; y; uÞÞ:

Let T : Y 3Y →PðY 3Y Þ, Tðx; yÞ ¼ ðFðx; yÞ;Fðy; xÞÞ.We consider onY 3Y themetric d* defined by (3.4), using the same functionalsD* andH *

as in Lemma 4.1.For z ¼ ðx; yÞ∈A3B, w ¼ ðu; vÞ∈B3A, using Lemma 4.1,

H *ðTðzÞ;TðwÞÞ ¼ H *ððFðx; yÞ;Fðy; xÞÞ; ðFðu; vÞ;Fðv; uÞÞÞ¼ maxfHðFðx; yÞ;Fðu; vÞÞ;HðFðy; xÞ;Fðv; uÞÞg≤wð ~Mðx; v; y; uÞÞ:

(4.3)

By Lemma 4.1,

D*ðz;TðzÞÞ ¼ maxfDðx;Fðx; yÞÞ;Dðy;Fðy; xÞÞg;D*ðw;TðwÞÞ ¼ maxfDðu;Fðu; vÞÞ;Dðv;Fðv; uÞÞg;

1

2½D*ðw;TðzÞÞ þ D*ðz;TðwÞÞ� ¼ 1

2½maxfDðu;Fðx; yÞÞ;Dðv; Fðy; xÞÞg

þmaxfDðx;Fðu; vÞÞ;Dðy;Fðv; uÞÞg�

≥max

�1

2½Dðu;Fðx; yÞÞ þ Dðx;Fðu; vÞÞ�;

1

2½Dðv;Fðy; xÞÞ þ Dðy;Fðv; uÞÞ�

�:

Using the monotonicity of w, (4.3) becomes


189

H *ðTðzÞ;TðwÞÞ≤wðmax

�d*ðz;wÞ;D*ðz;TðzÞÞ;D*ðw;TðwÞÞ;1

2½D*ðw;TðzÞÞ þ D*ðz;TðwÞÞ�

��; for any z∈A3B;

w∈B3A;

and because T satisfies the cyclic condition

TðA3BÞ ¼ ðFðA3BÞ;FðB3AÞÞ⊆B3A;TðB3AÞ⊆A3B;

where A3B;B3A∈PclðY 3Y Þ, we conclude that T is a multivalued cyclic w-contractionof �Ciri�c type.

By Lemma 4.2, the property of the operator F to have proximinal values is transferred tothe operator T, so we are in the conditions of Theorem 4.1.

Then there exists ðx*; y*Þ∈ ðA3BÞ∩ðB3AÞ such that ðx*; y*Þ∈ ðFðx*; y*Þ;Fðy*; x*ÞÞand for each ða; bÞ∈A3B there exists a sequence ðan; bnÞn∈ℕ∈Y 3Y with a0 ¼ a, b0 ¼ band

ðan; bnÞ∈ ðFðbn−1; an−1Þ;Fðan−1; bn−1ÞÞ; n≥ 1

that converges to ðx; yÞ.Hereinafter we define and study the generalized Ulam–Hyers stability of the following

coupled fixed point problem.

Definition 4.3. Let ðX ; dÞbe ametric space,Y ∈PðXÞ,F : Y 3Y →PðY Þbe amultivaluedoperator. By definition, the coupled fixed point problem�

x∈Fðx; yÞy∈Fðy; xÞ ; x; y∈Y (4.4)

is said to be generalized Ulam–Hyers stable if there exists an increasing functionψ : ℝþ →ℝþ, continuous at 0, with ψð0Þ ¼ 0 such that for each ε > 0 and for eachsolution ðx; yÞ∈Y 3Y of the inequality

maxfDðx;Fðx; yÞÞ;Dðy;Fðy; xÞÞg≤ ε;

there exists a solution ðx*; y*Þ∈Y3Y of the coupled fixed point problem such that

maxfdðx; x*Þ; dðy; y*Þg≤ψðεÞ:Our stability result is a consequence of the following theorem.

Theorem 4.3 ([21]). Let T : Y →PproxðY Þ be as in Theorem 4.2, ε > 0 and x∈Y be suchthat Dðx;TðxÞÞ≤ ε . Then there exists x* a fixed point of T such that dðx; xÞ≤ sðεÞ, where s isgiven by Lemma 2.2.

Theorem4.4. If all the hypotheses of Theorem 4.2 hold, then the coupled fixed point problem(4.4) is generalized Ulam–Hyers stable.

Proof. Let any ε > 0 and let ðx; yÞ∈Y 3Y such that�Dðx;Fðx; yÞÞ≤ εDðy;Fðy; xÞÞ≤ ε:

As before, we consider T : Y 3Y →PðY 3Y Þ,Tðx; yÞ ¼ ðFðx; yÞ;Fðy; xÞÞ:

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For z ¼ ðx; yÞ,D*ðz;TðzÞÞ ¼ maxfDðx;Fðx; yÞÞ;Dðy;Fðy; xÞÞg≤ ε:

Applying Theorem 4.3, there exists a fixed point z* ¼ ðx*; y*Þ of T such that d*ðz; z*Þ≤ sðεÞ,that is there exists a solution ðx*; y*Þ of the coupled fixed point problem (4.4) such that

maxfdðx; x*Þ; dðy; y*Þg≤ sðεÞ: ,

In the last part of this section we will consider the following best proximity problem for acyclic coupled multivalued operator:

If ðX ; dÞ is a metric space, A;B∈PðXÞ, Y ¼ A∪B, F : Y 3Y →PðY Þ is a coupledmultivalued operator satisfying the cyclic condition FðA3BÞ⊆B, FðB3AÞ⊆A, then weare interested in finding ðx*; y*Þ∈A3B such that

Dðx*;Fðx*; y*ÞÞ ¼ Dðy*;Fðy*; x*ÞÞ ¼ DðA;BÞ: (4.5)

ðx*; y*Þ is said to be a coupled best proximity point of F.Notice that, in particular, if A∩B≠ 0= then ðx*; y*Þ is a coupled fixed point of F.

Definition 4.4. Let ðX ; dÞ be a metric space, A;B∈PðXÞ, Y ¼ A∪B. A multivaluedoperator F : Y 3Y →PðY Þ is called a cyclic coupled �Ciri�c type multivalued operator if:

(i) FðA3BÞ⊆B and FðB3AÞ⊆A;

(ii) there exists a comparison function w : ℝþ →ℝþ such that

HðFðx; yÞ;Fðu; vÞÞ≤wð ~Mðx; v; y; uÞ � DðA;BÞÞ þ DðA;BÞ;

for any x; v∈A, y; u∈B.In 2009, Suzuki, Kikkawa and Vetro introduced the following property.

Definition 4.5. [23] LetAandBbe nonempty subsets of a metric space ðX ; dÞ. Then ðA;BÞis said to satisfy the property UC if for ðxnÞn∈ℕ and ðznÞn∈ℕ sequences in A and ðynÞn∈ℕ asequence in B such that dðxn; ynÞ→DðA;BÞ and dðzn; ynÞ→DðA;BÞ as n→∞, thendðxn; znÞ→ 0 as n→∞.

Example 4.1. [24] [23] (1) Any pair of nonempty subsets ðA;BÞ of a metric space ðX ; dÞwith DðA;BÞ ¼ 0 satisfies the property UC;

(2) Any pair of nonempty subsets ðA;BÞ of a uniformly convex Banach space with Aconvex satisfies the property UC.

Lemma 4.3. Let Aand B be nonempty subsets of a metric space ðX ; dÞ , and d* be the metricdefined on X 3X by (3.4). If ðA;BÞ and ðB;AÞ satisfy the property UC with respect to d thenðA3B;B3AÞ satisfy the property UC with respect to d.

Proof. We denote D*ðA3B;B3AÞ ¼ DðA;BÞ ¼ D. Let xn ¼ ðan; bnÞ; zn ¼ ða0n; b

0nÞ∈

A3B; yn ¼ ðβn; αnÞ∈B3A such that d*ðxn; ynÞ→D and d*ðzn; ynÞ→D as n→∞.Then

maxfdðan; βnÞ; dðbn; αnÞg→D and

max�dða0

n; βnÞ; dðb0n; αnÞ

�→D as n→∞:

It is obvious that dðan; βnÞ→D; dða0n; βnÞ→D and because ðA;BÞ satisfies the property

UC we get dðan; a0nÞ→ 0.


191

From dðbn; αnÞ→D; dðb0n; αnÞ→D as n→∞ and using ðB;AÞ satisfies the property UC

we get dðbn; b0nÞ→ 0.

Finally,

d*ðxn; znÞ ¼ max�dðan; a0

nÞ; dðbn; b0nÞ�→ 0 as n→∞:

We recall the following result.

Theorem 4.5 ([25]). Let ðX ; dÞ be a complete metric space, A∈PclðXÞ;B∈PðXÞ such thatðA;BÞ satisfies the property UC. Let T : A∪B→PproxðXÞ be a multivalued �Ciri�c type cyclicoperator that is:

(i) TðAÞ⊆B and TðBÞ⊆A;

(ii) there exists a comparison function w : ℝþ →ℝþ such that

HðTðxÞ;TðyÞÞ≤wðMðx; yÞ � DðA;BÞÞ þ DðA;BÞ; where

Mðx; yÞ ¼ max

�dðx; yÞ;Dðx;TðxÞÞ;Dðy;TðyÞÞ; 1

2½Dðx;TðyÞÞ þ Dðy;TðxÞÞ�

�:

Then the following statements hold:

(1) T has a best proximity point x*A ∈A ;

(2) there exists a sequence ðxnÞn∈ℕwith x0 ∈A, and xnþ1 ∈TðxnÞ , n≥ 0 , such that ðx2nÞn∈ℕconverges to x*A.

The next result is a consequence of the above theorem.

Theorem 4.6. Let ðX ; dÞ be a complete metric space, A;B∈PclðXÞ such that ðA;BÞ andðB;AÞ satisfy the property UC, and Y ¼ A∪B . If F : Y 3Y →PproxðY Þ is a cyclic coupled�Ciri�c type multivalued operator, then the following statements hold:

(i) F has a coupled best proximity point ðx*; y*Þ∈A3B ;

(ii) there exist two sequences ðxnÞn∈ℕ , ðynÞn∈ℕ with

ðx0; y0Þ∈A3B; xnþ1 ∈Fðxn; ynÞ; ynþ1 ∈Fðyn; xnÞ;such that ððx2n; y2nÞÞn∈ℕ converges to ðx*; y*Þ .

Proof. Considering again on Y 3Y the metric d* defined by (3.4), in a similar manner as inTheorem 4.2, we obtain that the operator T : Y 3Y →PðY 3Y Þ,

Tðx; yÞ ¼ ðFðx; yÞ;Fðy; xÞÞ:is a multivalued �Ciri�c type cyclic operator which takes proximinal values.

Using Lemma 4.1, the pair ðA3B;B3AÞ satisfies the property UC with respect to d*.Consequently, we are in the conditions of Theorem 4.5, so T has a best proximity point

ðx*; y*Þ∈A3B and there exists a sequence ðxn; ynÞn∈ℕ with ðx0; y0Þ∈A3B andðxnþ1; ynþ1Þ∈Tðxn; ynÞ such that ðx2n; y2nÞn∈ℕ converges to ðx*; y*Þwith respect to d*.

5. An application to a system of integral equationsWe apply the results given by Theorem 3.2 to study the existence and the uniqueness ofsolutions of the following system of integral equations:

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8>><>>:

xðtÞ ¼Z b

a

Gðt; sÞf ðs; xðsÞ; yðsÞÞds

yðtÞ ¼Z b

a

Gðt; sÞf ðs; yðsÞ; xðsÞÞds; t ∈ ½a; b� (5.1)

where a; b∈ℝ, a < b, G∈Cð½a; b�3½a; b�; ½0;∞ÞÞ;f ∈Cð½a; b�3ℝ3ℝ;ℝÞ:

Theorem 5.1. We suppose that:

(i) there exist α; β∈Cð½a; b�;ℝÞ , with αðtÞ≤ βðtÞ , for any t ∈ ½a; b� , such that8>><>>:

αðtÞ≤Z b

a

Gðt; sÞf ðs; βðsÞ; αðsÞÞds

βðtÞ≥Z b

a

Gðt; sÞf ðs; αðsÞ; βðsÞÞdsfor any t ∈ ½a; b�; (5.2)

(ii) there exists a strong comparison function w : ℝþ →ℝþ such that

jf ðs; u1; u2Þ � f ðs; v1; v2Þj≤wðmaxfju1 � v1j; ju2 � v2jgÞ;for any s∈ ½a; b� and u1; u2; v1; v2 ∈ℝ;

(iii) supt∈½a;b�R b

aGðt; sÞds≤ 1 ;

(iv) f ðs; $; yÞ is monotone decreasing for any s∈ ½a; b� and any y∈ℝ;

(v) f ðs; x; $Þ is monotone increasing for any s∈ ½a; b� and any x∈ℝ .Then the system (5.1) has a unique solution ðx*; x*Þ∈Cð½a; b�;ℝ2Þ , with α≤ x* ≤ β .

Proof. Let us consider

X :¼ Cð½a; b�;ℝÞ; and the Chebyshev norm jxj∞¼ maxt∈½a;b�jxðtÞj:

Then ðX ; j$j∞Þ is a Banach space. We consider the following closed subsets of X:

A ¼ fx∈X j x≤ βg;B ¼ fx∈X jx≥ αg;

Y ¼ A∪B and the operator F : Y 3Y →Y ,

Fðx; yÞðtÞ :¼Z b

a

Gðt; sÞf ðs; xðsÞ; yðsÞÞds:

The system (5.1) is equivalent to �Fðx; yÞ ¼ x

Fðy; xÞ ¼ y; x; y∈Y :

We will prove that F is cyclic with respect to A and B, that is

FðA3BÞ⊆B and FðB3AÞ⊆A:

Let x∈A and y∈B0xðsÞ≤ βðsÞ; yðsÞ≥ αðsÞ; ∀s∈ ½a; b�.


193

Using the monotonicity of f we have

Gðt; sÞf ðs; xðsÞ; yðsÞÞ≥Gðt; sÞf ðs; βðsÞ; αðsÞÞ;and from (i), by integration, Z b

a

Gðt; sÞf ðs; xðsÞ; yðsÞÞds≥ αðtÞ;

which means thatFðx; yÞðtÞ≥ αðtÞ; ∀t ∈ ½a; b�0Fðx; yÞ∈B:

So FðA3BÞ⊆B. In a similar way we have FðB3AÞ⊆A.Using the conditions (ii) and (iii), and the monotonicity of w, for any x; v∈A and y; u∈B,

we have

j f ðs; xðsÞ; yðsÞÞ � f ðs; uðsÞ; vðsÞÞj≤wðmaxs∈½a;b�

fjxðsÞ � uðsÞj; jyðsÞ � vðsÞjgÞ

≤wðmaxfjx� uj∞; jy� vj

∞gÞ0

jFðx; yÞðtÞ � Fðu; vÞðtÞj≤Z b

a

Gðt; sÞjf ðs; xðsÞ; yðsÞÞ � f ðs; uðsÞ; vðsÞÞjds

≤wðmaxfjx� uj∞; jy� vj

∞gÞZ b

a

Gðt; sÞds≤wðmaxfjx� uj

∞; jy� vj

∞gÞ; ∀t ∈ ½a; b�:

We have

jFðx; yÞ � Fðu; vÞj∞≤wðmaxfjx� uj

∞; jy� vj

∞gÞ for any x; v∈A and y; u∈B;

so the operator F is a cyclic coupled w-contraction of �Ciri�c type.All the conditions of Theorem 3.2 are satisfied, so T has a unique strong coupled fixed

point ðx*; x*Þ∈A∩B; with αðtÞ≤ x*ðtÞ≤ βðtÞ; for any t ∈ ½a; b�.Definition 5.1. The system (5.1) is said to be generalized Ulam–Hyers stable if there existsψ : ℝþ →ℝþ increasing, continuous at 0 and ψð0Þ ¼ 0 such that for any ε1 > 0; ε2 > 0 andfor any solution ðx; yÞ∈Cð½a; b�;ℝ2Þ, of the system8>><

>>:jxðtÞ �

Z b

a

Gðt; sÞf ðs; xðsÞ; yðsÞÞdsj≤ ε1

jyðtÞ �Z b

a

Gðt; sÞf ðs; yðsÞ; xðsÞÞdsj≤ ε2

there exists a solution ðx*; y*Þ∈Cð½a; b�;ℝ2Þ of the system (5.1) such that for any t ∈ ½a; b�,� jxðtÞ � x*ðtÞj≤ψðεÞjyðtÞ � y*ðtÞj≤ψðεÞ ; where ε ¼ maxðε1; ε2Þ:

Theorem 5.2. Suppose that the hypotheses of Theorem 5.1 hold. Then the system (5.1) isgeneralized Ulam–Hyers stable.

Proof. By Theorem 5.1, the system (5.1) has a unique solution ðx*; x*Þ∈Cð½a; b�;ℝ2Þ, withα≤ x* ≤ β. Applying Theorem 3.6 to the operator F : Y 3Y →Y ,

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Fðx; yÞðtÞ :¼Z b

a

Gðt; sÞf ðs; xðsÞ; yðsÞÞds;

in the same setting as in the proof of Theorem 5.1, we get the conclusion.

References

[1] I.A. Rus, A. Petrusel, G. Petrusel, Fixed Point Theory, Cluj University Press, Cluj-Napoca, 2008.

[2] D. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications,Nonlinear Anal. Theory Methods Appl. 11 (1987) 623–632.

[3] J.G. Kadwin, M. Marudai, Fixed point and best proximity point results for generalised cycliccoupled mappings, Thai J. Math. 14 (2) (2016) 431–441.

[4] V. Lakshmikantham, L. �Ciri�c, Coupled fixed point theorems for nonlinear contractions in partiallyordered metric spaces, Nonlinear Anal. 70 (2009) 4341–4349.

[5] A. Petrusel, G. Petrusel, B. Samet, J.-C. Yao, Coupled fixed point theorems for symmetriccontractions in b-metric spaces with applications to operator equation systems, Fixed PointTheory 17 (2) (2016) 457–476.

[6] A. Petrusel, G. Petrusel, B. Samet, A study of the coupled fixed point problem for operatorssatisfying a max-symmetric condition in b-metric spaces with applications to a boundary valueproblem, Miskolc Math. Notes 17 (1) (2016) 501–516.

[7] B. Samet, C. Vetro, Coupled fixed point, F-invariant set and fixed point of N-order, Ann. Funct.Anal. 1 (2010) 46–56.

[8] B. Samet, C. Vetro, Coupled fixed point theorems for multi-valued nonlinear contractionmappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 4260–4268.

[9] B. Samet, E. Karapinar, H. Aydi, V.C. Raji�c, Discussion on some coupled fixed point theorems,2013:50, 12 pp. (2013).

[10] T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces andapplications, Nonlinear Anal. 65 (2006) 1379–1393.

[11] W.A. Kirk, P.S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclicalcontractive conditions, Fixed Point Theory 4 (1) (2003) 79–89.

[12] B.S. Choudbury, P. Maity, Cyclic coupled fixed point result using Kannan type contractions, J.Oper. 2014 (2014) 876749, 5.

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[14] I.A. Rus, M.A. Serban, Some generalizations of a Cauchy lemma and applications, in: Topics inMathematics, Computer Science and Philosophy (St. Cobzas Ed.), Cluj University Press, 2008,pp. 173–181.

[15] I.A. Rus, Generalized Contractions and Applications, Cluj Univ. Press, 2001.

[16] A. Magdas, Fixed point theorems for generalized contractions defined on cyclic representations, J.Nonlinear Sci. Appl. 8 (2015) 1257–1264.

[17] S. Reich, Zaslavski, Well-posedness of fixed point problems, Far East J. Math. Sci. III (2001)393–401.

[18] J. Fletcher, W.B. Moors, Chebyshev Sets, J. Aust. Math. Soc. 98 (2015) 161–231.

[19] G. Petrusel, Cyclic representations and periodic points, Stud. Univ. Babes-Bolyai Math. 50 (3)(2005) 107–112.

[20] S.P. Singh, B. Watson, P. Srivastava, Fixed Point Theory and Best Approximation: The KKM-Map Principle, Kluwer Academic Publishers, Dordrecht, 1997.


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[21] A. Magdas, A fixed point theorem for �Ciri�c type multivalued operators satisfying a cyclicalcondition, J. Nonlinear Convex Anal. 17 (6) (2015) 1109–1116.

[22] O. Mlesnite, A. Petrusel, Existence and Ulam-Hyers stability results for multivalued coincidenceproblems, Filomat 26 (5) (2012) 965–976.

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[24] A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal.Appl. 323 (2) (2006) 1001–1006.

[25] A. Magdas, Best proximity problems for �Ciri�c type multivalued operators satisfying a cycliccondition, Stud. Univ. Babes -Bolyai Math. 62 (3) (2017) 395–405.

Corresponding authorAdrian Magdas and can be contacted at: [email protected]


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Nonlinear Jordan centralizer ofstrictly upper triangular matrices

Driss Aiat Hadj AhmedCentre R�egional des Metiers d’Education et de Formation (CRMEF),

Tangier, Morocco

AbstractLet F be a field of zero characteristic, let NnðFÞ denote the algebra of n3n strictly upper triangular matriceswith entries in F , and let f : NnðFÞ→NnðFÞ be a nonlinear Jordan centralizer of NnðFÞ; that is, a mapsatisfying that f ðXY þ YXÞ ¼ Xf ðY Þ þ f ðY ÞX, for all X ; Y ∈NnðFÞ. We prove that f ðXÞ ¼ λX þ ηðXÞwhere λ∈F and η is a map from NnðFÞ into its center ZðNnðFÞÞ satisfying that ηðXY þ YXÞ ¼ 0 for everyX ;Yin NnðFÞ.Keywords Jordan centralizer, Strictly upper triangular matrices, Commuting map


1. IntroductionConsider a ring R. An additive mapping T : R→R is called a left (respectively right)centralizer if TðabÞ ¼ TðaÞb ðrespectivelyTðabÞ ¼ aTðbÞÞ for all a; b∈R. The map T iscalled a centralizer if it is a left and a right centralizer. The characterization of centralizers onalgebras or rings has been a widely discussed subject in various areas of mathematics.

In [11] Zalar proved the following interesting result: if R is a 2 -torsion free semiprime ringand T is an additive mapping such that Tða2Þ ¼ TðaÞa ðorTða2Þ ¼ aTðaÞÞ, then T is acentralizer. Vukman [10] considered additive maps satisfying similar conditions, namely2Tða2Þ ¼ TðaÞaþ aTðaÞ for any a∈R, and showed that if R is a 2 -torsion free semiprimering then T is also a centralizer. Since then, the centralizers have been intensivelyinvestigated by many mathematicians (see, e.g., [2–5,7]).

Let R be a ring. An additive map f : R→R, is called a Jordan centralizer of R if

∀x; y ∈ Rf ðxyþ yxÞ ¼ xf ðyÞ þ f ðyÞx: (1)

NonlinearJordan

centralizer

197

©Driss Aiat Hadj Ahmed. Published in theArab Journal ofMathematical Sciences. Published byEmeraldPublishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license.Anyone may reproduce, distribute, translate and create derivative works of this article (for bothcommercial and non-commercial purposes), subject to full attribution to the original publication andauthors. The full terms of this licensemay be seen at http://creativecommons.org/licences/by/4.0/legalcode

The author would like to thank the referee for providing useful suggestions which served to improvethis paper.

Declaration of Competing Interest:No author associated with this paper has disclosed any potential orpertinent conflicts which may be perceived to have impending conflict with this work. For full disclosurestatements refer to https://doi.org/10.1016/j.ajmsc.2019.08.002.

The publisher wishes to inform readers that the article “Nonlinear Jordan centralizer of strictly uppertriangularmatrices”was originally published by theprevious publisher of theArab Journal ofMathematicalSciences and the pagination of this article has been subsequently changed. There has been no change to thecontent of the article. This change was necessary for the journal to transition from the previous publisher tothe newone. The publisher sincerely apologises for any inconvenience caused. To access and cite this article,please use Hadj Ahmed, D. A. (2019), “Nonlinear Jordan centralizer of strictly upper triangular matrices”,Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 197-201. The original publication date for thispaper was 07/09/2019.



Received 17 December 2018Revised 24 August 2019

Accepted 25 August 2019


Vol. 26 No. 1/2, 2020pp. 197-201


DOI 10.1016/j.ajmsc.2019.08.002

Recently, Ghomanjani and Bahmani [8] dealt with the structure of Lie centralizers of trivialextension algebras, whereas Fo�sner and Jing [6] studied Lie centralizers of triangular rings.

The inspiration of this paper comes from the articles [1,4,6] in which the authors deal withthe Lie centralizermaps of triangular algebras and rings. In this note wewill consider nonlinearJordan centralizers on strictly upper triangular matrices over a field of zero characteristic.

Throughout this article, F is a field of zero characteristic. Let MnðFÞ and NnðFÞ denotethe algebra of all n3 nmatrices and the algebra of all n3 n strictly upper triangular matricesover F , respectively. We use diagða1; a2; . . . ; anÞ to represent a diagonal matrix withdiagonal ða1; a2; . . . ; anÞ where ai ∈F . The set of all n3 n diagonal matrices over F is

denoted by DnðFÞ. Let In be the identity inMnðFÞ; J ¼ Pn−1i¼1 Ei;iþ1 and fEij : 1≤ i; j ≤ ng

the canonical basis of MnðFÞ, where Eij is the matrix with 1 in the ði; jÞ position and zeroselsewhere. By CNnðFÞðXÞwe will denote the centralizer of the element X in the ring NnðFÞ.

The notation f : NnðFÞ→NnðFÞ means a nonlinear map satisfying ∀X ; Y ∈ NnðFÞ :f ðXY þ Y XÞ ¼ X f ðY Þ þ f ðY ÞX.

Notice that it is easy to check that the ZðNnðFÞÞ ¼ FE1n.The main result in this paper is the following:

Theorem 1. Let F be a field of zero characteristic. If f : NnðFÞ→NnðFÞ is a nonlinearJordan centralizer then there exists λ∈F and a map η : NnðFÞ→ZðNnðFÞÞ satisfyingηðXY þ Y XÞ ¼ 0 for every X ; Y in NnðFÞ such that f ðXÞ ¼ λX þ ηðXÞ for all X in NnðFÞ.

2. Proof of the main resultLet us start with some basic properties of Lie centralizers.

Lemma 2. Let f be a nonlinear Jordan centralizer of NnðFÞ. Then(1) f ð0Þ ¼ 0;

(2) For every X ; Y ∈ NnðFÞ, we have f ðXY þ Y XÞ ¼ Yf ðXÞ þ f ðXÞY.Proof. To prove (1) it suffices to notice that

f ð0Þ ¼ 0f ð0Þ þ f ð0Þ0 ¼ 0:

(2) Observe that if f ðXY þ YXÞ ¼ Yf ðXÞ þ f ðXÞY , Interchanging X and Y in the aboveidentity, we have f ðXY þ YXÞ ¼ Yf ðXÞ þ f ðXÞY . -

Lemma 3. Let f be a nonlinear Jordan centralizer of NnðFÞ. Then(1) f ðPn−1

i¼1 ai Ei;iþ1Þ ¼Pn−1

i¼1 bi Ei;iþ1;

(2) There exists λ∈F such that f ðJÞ ¼ λJ.

Proof. Let D ¼ Pni¼1 αi Ei;i ∈DnðFÞ, As F is infinite, we can find a set fαi ∈F=1≤ i≤ ng

whose elements satisfy conditions: αi þ αiþ1 ¼ 1 for 1≤ i ≤ n− 1and αi þ αj ≠ 1 for j≠ i þ 1.

(1) ConsiderA∈MnðFÞ. It is well known thatDAþ AD ¼ A if and only ifA ¼ Pni¼1 ai Ei;iþ1:

Hence, if A ¼ Pn−1i¼1 ai Ei;iþ1; , we have A ¼ DAþ AD. Thus f ðAÞ ¼ f ðDAþ ADÞ ¼

Df ðAÞ þ f ðAÞD. Therefore f ðAÞ ¼ Pn−1i¼1 bi Ei;iþ1:

(2) As in (1), let N ¼ Pn−1i¼1 ð−1Þi Ei;iþ1 ∈ NnðFÞ, consider A ¼ Pn−1

i¼1 ai Ei;iþ1: for someai ∈F . Then NAþ AN ¼ 0 if and only if A ¼ aJ for some a∈F .

Indeed, f ðJÞ ¼ Pn−1i¼1 ai Ei;iþ1:by (1). Thus, 0 ¼ f ð0Þ ¼ f ðNAþ ANÞ ¼ Nf ðAÞ þ f ðAÞN.

Hence, there exists λ∈F such that f ðJÞ ¼ λJ. -

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We will need the following lemma.

Lemma 4 (Lemma 2.1, [9]). Suppose thatF is an arbitrary field. If G; H ∈UTnðFÞ are suchthat gi;iþ1 ¼ hi;iþ1 ≠ 0 for all 1≤ i≤ n− 1, then G and H are conjugated in UTnðFÞ.

Here UTnðFÞ is the multiplicative group of n3 n upper triangular matrices with only 1’sin the main diagonal. From the lemma above we obtain the following corollary.

Corollary 5. Let F be a field. For every A ¼ P1≤i<j≤n aij Eij, where ai;iþ1 ≠ 0 for all

1≤ i≤ n− 1, there exists B∈TnðFÞ such that B−1AB ¼ J and TnðFÞ is the ring of uppertriangular matrices.

Proof. Let A be a matrix in NnðFÞ of the mentioned form. Then In þ A is a unitriangularmatrix. Let us notice first that there exists B1 ∈DnðFÞ such that ðB−1

1 AB1Þi;iþ1 ¼ 1 for alli∈ℕ. We can construct B1 ∈DnðFÞ recursively by:

ðB1Þ11 ¼ 1; ðB1Þiþ1;iþ1 ¼ ðB1Þii$ðAi;iþ1Þ−1for i≥ 1:

Consider the matrix In þ B−11 AB ∈ UTnðFÞ. The unitriangular matrices In þ J and

In þ B−11 AB fulfill the condition in Lemma 4. Hence, there exists B2 ∈UTnðFÞ such that

In þ J ¼ B−12 ðIn þ B−1

1 AB1ÞB2. Then J ¼ B−12 ðB−1

1 AB1ÞB2. Taking B ¼ B1B2 ∈TnðFÞ, weget J ¼ B−1AB as wanted. -

Lemma 6. Let A ¼ Pi<j aij Eij be a matrix in NnðFÞ with ai;iþ1 ≠ 0 for every

i ¼ 1; . . . ; n− 1. Then there exists λA ∈F such that f ðAÞ ¼ λAA.

Proof. Since A ¼ P1≤i<j≤n aij Eij, where ai;iþ1 ≠ 0, there exists T ∈TnðFÞ such that

T AT−1 ¼ J by the previous corollary. Define h : NnðFÞ→ NnðFÞ by hðXÞ ¼ T f ðT−1XTÞT−1. Then h is a nonlinear Jordan centralizer map. Indeed, ∀X ; Y ∈ NnðFÞ, we have:

hðXY þ Y XÞ ¼ T f�T−1ðXY þ Y XÞTÞT−1

¼ T f�T−1ðXY þ Y XÞTÞT−1

¼ T f�T−1XT T−1Y T þ T−1Y T T−1XT

�T−1

¼ T f��T−1XT

��T−1Y T

�þ �T−1Y T

��T−1XT

��T−1

¼ T��T−1XT

�f�T−1Y T

�þ f�T−1Y T

��T−1XT

��T−1

¼ XT f�T−1Y T

�T−1 þ T f

�T−1Y T

�T−1X

¼ XhðY Þ þ hðY ÞX

Hence, hðJÞ ¼ λAJ by lemme 2.2. Then

T f ðAÞT−1 ¼ T f�T−1ðT AT−1

�TÞT−1 ¼ hðJÞ ¼ λAJ ¼ λAT AT−1:

Multiplying the left and right sides by T−1 and T respectively yields f ðAÞ ¼ λAA. -Now we wish to extend Lemma 2.3 to all elements of NnðFÞ. In order to do this, let us

introduce the following set:

S ¼ �B ¼ ðbijÞ∈NnðFÞ : bi;iþ1 ≠ 0 ∀ i ¼ 1; . . . ; n� 1

�:

This set has an important property that is established below.

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Lemma 7. Let F be a field. Every element of NnðFÞ can be written as a sum of at most twoelements of S.Proof. If ai;iþ1 ≠ 0 for all i ¼ 1; . . . ; n− 1, thenAbelongs toS, so there is nothing to prove. IfA is not in S, then we can define B1 and B2 as follows:

ðB1Þij ¼�ai;iþ1 � bi if j ¼ i þ 1aij if j > i þ 1;

ðB2Þij ¼�bi if j ¼ i þ 10 otherwise;

where bi is an element in F different from ai;iþ1. It is easy to see that B1; B2 are in S, andA ¼ B1 þ B2, so we wanted. -

Lemma 8. Let F be a field. For arbitrary elements A;B of NnðFÞ, there exists λA;B ∈F suchthat

f ðAþ BÞ ¼ f ðAÞ þ f ðBÞ þ λA;B E1n:

Proof. For any A; B; X of NnðFÞ, we havef ððAþ BÞX þ XðAþ BÞÞ ¼ Xf ðAþ BÞ þ f ðAþ BÞX

¼ Xf ðAþ BÞ þ f ðAþ BÞX¼ Af ðXÞ þ f ðXÞAþ Bf ðXÞ þ f ðXÞB¼ f ðAX þ XAÞ þ f ðBX þ XBÞ¼ Xf ðAÞ þ f ðAÞX þ Xf ðBÞ þ f ðBÞX

hence

Xðf ðAÞ þ f ðBÞ � f ðAþ BÞÞ ¼ ðf ðAþ BÞ � f ðBÞ � f ðAÞÞXwhich implies that ðf ðAþ BÞ – f ðAÞ – f ðBÞÞ2 ∈ZðNnðFÞÞ. Thus, there exists λA;B ∈F suchthat f ðAþ BÞ ¼ f ðAÞ þ f ðBÞ þ λA;B E1n. -

Now we can prove the main theorem.

Proof of Theorem 1. For every X ∈NnðFÞ there exists a A; B∈S such that X ¼ Aþ B.First take A; B∈S such that ABþ BA≠ 0. Then, by Lemma 2.3, f ðAÞ ¼ λAA;

f ðBÞ ¼ λB B for some λA; λB ∈F . Since f is nonlinear Jordan centralizer map, thefollowing holds:

f ðABþ BAÞ ¼ Af ðBÞ þ f ðBÞA ¼ B f ðAÞ þ f ðAÞBwe must have λA ¼ λB.

Consider now A andB from S such thatABþ BA ¼ 0. Then there exists C ∈S such thatthe pairs C andA; C andB, C are AC þ C A ≠ 0 and BC þ CB ≠ 0, so we have λA ¼ λCand λB ¼ λC.

Thus, there exists λ∈F , η : NnðFÞ→ZðNnðFÞÞ nonlinear Jordan centralizer map suchthat f ðXÞ ¼ λX þ ηðXÞ for all X ∈NnðFÞ.

we have

f ðXY þ Y XÞ ¼ λðXY þ Y XÞ þ ηðXY þ Y XÞ¼ Xf ðY Þ þ f ðY ÞX¼ XðλY þ ηðY ÞÞ þ ðλY þ ηðY ÞÞX¼ λðXY þ Y XÞ þ XηðY Þ þ ηðY ÞX

we obtain that ηðXY þ Y XÞ ¼ XηðY Þ þ ηðY ÞX for all X ;Y ∈NnðFÞ.

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Now we use Lemma 2.5 we get f ðXÞ ¼ λX þ ηðXÞ for all X ∈NnðFÞ, whereη : NnðFÞ→ZðNnðFÞÞ is a nonlinear Jordan centralizer map and ηðXÞ ¼ 0 for allX ∈S: ,

References

[1] J. Bounds, Commuting maps over the ring of strictly upper triangular matrices, Linear AlgebraAppl. 507 (2016) 132–136.

[2] M. Bre�sar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993) 385–394.

[3] M. Bre�sar, Commuting traces of biadditive mappings, commutativity-preserving mappings andLie mappings, Trans. Amer. Math. Soc. 335 (1993) 525–546.

[4] W.-S. Cheung, Commuting maps of triangular algebras, J. Lond. Math. Soc. 63 (2) (2001) 117–127.

[5] D. Eremita, Commuting traces of upper triangular matrix rings, Aequationes Math. 91 (2017)563–578.

[6] A. Fo�sner, W. Jing, Lie centralizers on triangular rings and nest algebras, Adv. Oper. Theory,in press.

[7] W. Franca, Commuting maps on some subsets of matrices that are not closed under addition,Linear Algebra Appl. 437 (2012) 388–391.

[8] F. Ghomanjani, M.A. Bahmani, A note on Lie centralizer maps, Palest. J. Math. 7 (2) (2018) 468–471.

[9] R. Słowik, Expressing infinite matrices as products of involutions, Linear Algebra Appl. 438(2013) 399–404.

[10] J. Vukman, An identity related to centralizers in semiprime rings, Comment. Math. Univ. Carolin.40 (3) (1999) 447–456.

[11] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin. 32 (4) (1991) 609–614.

Further Reading

[1] D. Aiat Hadj Ahmed, R. Slowik, M-commuting maps of the rings of infinite triangular and strictlytriangular matrices, (in preparation).

[2] M. Bre�sar, Centralizing mappings on von Neumann algebra, Proc. Amer. Math. Soc. 111 (1991)501–510.

[3] L. Chen, J.H. Zhang, Nonlinear Lie derivation on upper triangular matrix algebras, LinearMultilinear Algebra 56 (2008) 725–730.

[4] Ghahramani, Characterizing Jordan maps on triangular rings through commutative zero products,H. Mediterr. J. Math. 15 (2018) 38.

[5] T.K. Lee, Derivations and centralizing mappings in prime rings, Taiwanese J. Math. 1 (1997)333–342.

[6] T.K. Lee, T.C. Lee, Commuting additive mappings in semiprime rings, Bull. Inst. Math. Acad.Sinica 24 (1996) 259–268.

[7] L. Liu, On Jordan centralizers of triangular algebras, Banach J. Math. Anal. 10 (2) (2016) 223–234.

Corresponding authorDriss Aiat Hadj Ahmed can be contacted at: [email protected]


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Subcommuting and comparableiterative roots of order

preserving homeomorphismsVeerapazham Murugan

Department of Mathematical and Computational Sciences,National Institute of Technology Karnataka, Surathkal, Mangalore, India, and

Murugan Suresh KumarDepartment of Mathematics, The Gandhigram Rural Institute, Gandhigram, India

AbstractIt is known that the iterative roots of continuous functions are not necessarily unique, if it exist. In this note, byintroducing the set of points of coincidence, we study the iterative roots of order preserving homeomorphisms.In particular, we prove a characterization of identical iterative roots of an order preserving homeomorphismusing the points of coincidence of functions.

Keywords Iterative roots, Homeomorphisms, Commuting functions, Subcommuting functions,

Comparable functions


1. IntroductionGiven a function F : X →X and a positive integer n, if there is a function f : X →X such that

f nðxÞ ¼ FðxÞ; for all x∈X (1)

(where f n is n times composition of f ) then f is called an n th iterative root or fractional iterate oforder n of F. The problem of finding the iterative root of functions was initiated in the classicalworks of Charles Babbage [1]. The iterative roots of continuous monotone and piecewisemonotone function was developed in the works of B€odewadt [2], Łojasiewicz [7], Kuczma [4],Zhang [6,12] and many others. For a detailed study of recent results on iterative roots ofcontinuous piecewise monotone functions can be found in the survey paper by Zdun andSolarz [11].

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JEL Classification — 39B12, 39B22© Veerapazham Murugan and Murugan Suresh Kumar. Published inArab Journal of Mathematical

Sciences. Published by Emerald Publishing Limited. This article is published under the CreativeCommons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and createderivative works of this article (for both commercial and non-commercial purposes), subject to fullattribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

Declaration of Competing Interest: The authors declare that they have no known competing financialinterests or personal relationships that could have appeared to influence the work reported in this paper.

The publisher wishes to inform readers that the article “Subcommuting and comparable iterativeroots of order preserving homeomorphisms” was originally published by the previous publisher of theArab Journal ofMathematical Sciences and the pagination of this article has been subsequently changed.There has been no change to the content of the article. This change was necessary for the journal totransition from the previous publisher to the new one. The publisher sincerely apologises for anyinconvenience caused. To access and cite this article, please use Murugan, V., Kumar, M.S. (2019),“Subcommuting and comparable iterative roots of order preserving homeomorphisms”,Arab Journal ofMathematical Sciences, Vol. 26 No. 1/2, pp. 203-210. The original publication date for this paper was31/10/2019.



Received 15 November 2018Revised 24 October 2019

Accepted 25 October 2019


Vol. 26 No. 1/2, 2020pp. 203-210


DOI 10.1016/j.ajmsc.2019.10.003

For the class of strictly increasing continuous functions, we have the following result.Theorem 1.1 ([5]). Let I ⊆ ℝ be any interval. Then every strictly increasing continuous

function F from I into itself possesses a strictly increasing continuous iterative roots of order n, forall n∈ℕ.

Theorem 1.1 guarantees the existence of strictly increasing continuous iterative roots of astrictly increasing continuous functions. Moreover, this strictly increasing continuous nthorder iterative root depends on arbitrary strictly increasing homeomorphisms (see Theorem11.2.2 [5]), and hence its iterative roots are not necessarily unique. In fact, every strictlyincreasing continuous function, other than identity, possesses infinitely many strictlyincreasing continuous nth order iterative roots.

In fact, uniqueness of iterative roots of a special class ofmonotonic functionswas conjecturedby B€odewadt [2] and answered in negative by Smajdor [9]. Motivated by B€odewadt, supposef and g are two iterative roots of order n of a strictly increasing homeomorphism F (i.e.f n ¼ gn ¼ F ), it is reasonable to ask under what condition f and g are identically equal?It is known that, if f n ¼ gn ¼ F and f ; g commutes each other (i.e. f g ¼ g f ) then f must be

equal to g (see [10]). In this article, we further investigate this problem. We give some sufficientconditions, using the set of points of coincidence of two functions. Also, for given orderpreserving homeomorphism from an interval onto itself, by generalizing the result in [10], wecharacterize the conditions of identical iterative roots of an order preserving homeomorphism.

2. Set of points of coincidenceThroughout our discussion we fix I ¼ ða; bÞ, where −∞≤ a≤ b≤∞, and HðIÞ denotes theset of all order preserving homeomorphisms from I onto itself. Here after we always assumeall the functions are in the class HðIÞ unless otherwise stated.

Let f and g be two order preserving homeomorphisms from the interval I onto J ⊆ I. Wesay f and g are comparable, if either f ðxÞ≤ gðxÞ or gðxÞ≤ f ðxÞ for all x∈ I, and if theinequalities are strict then we say f and g are strictly comparable.

Proposition 2.1. If f and g are two strictly comparable order preservinghomeomorphisms from I onto J ⊆ I, then f n and gn are strictly comparable order preservinghomeomorphisms, for all n∈ℕ. In addition to that, if J ¼ I then f −n and g−n are also strictlycomparable order preserving homeomorphisms, for all n∈ℕ.

Proof. First we prove the result for positive integers using induction on n. Assumef ðxÞ < gðxÞ for all x∈ I. Suppose there exists t ∈ f ðIÞ such that f 2ðtÞ≥ g2ðtÞ. Sincef ðtÞ < gðtÞwe have f ðf ðtÞÞ < f ðgðtÞÞ. Therefore

gðgðtÞÞ≤ f ðf ðtÞÞ < f ðgðtÞÞ:i:e:; ðf � gÞðgðtÞÞ > 0:

Since ðf – gÞðtÞ < 0, by intermediate value theorem there exists c∈ ðt; gðtÞÞ such thatf ðcÞ ¼ gðcÞ, which is a contradiction. Hence f 2 < g2 on ða; bÞ.

Assume f kðxÞ < gkðxÞ for all x∈ ða; bÞ and 1≤ k≤ n – 1. Suppose there is a t ∈ f n – 1ðIÞsuch that

f nðtÞ≥ gnðtÞ:Since f n−1ðtÞ < gn−1ðtÞwe have f nðtÞ < f ðgn−1ðtÞÞ. Therefore

gnðtÞ≤ f nðtÞ < f ðgn−1ðtÞÞ:i:e:; ðf � gÞðgn−1ðtÞÞ > 0:

Since ðf – gÞðtÞ < 0, by intermediate value theorem there exists c∈ ðt; gn−1ðtÞÞ such thatf ðcÞ ¼ gðcÞ, which is a contradiction. Hence f n < gn on ða; bÞ.

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Now, we prove the result for negative integers by assuming J ¼ I. First we prove iff ðxÞ < gðxÞ for all x∈ I, then g−1ðxÞ < f −1ðxÞ for all x∈ I. Suppose there is a t ∈ I such thatg−1ðtÞ≥ f −1ðtÞ. If g−1ðtÞ ¼ f −1ðtÞ then there exists x∈ I such that f ðxÞ ¼ gðxÞ ¼ t, which isnot possible as f ðxÞ < gðxÞ for all x∈ I. Therefore g−1ðtÞ > f −1ðtÞ. But this implies

f ðg−1ðtÞÞ > t > gðf −1ðtÞÞ:

Since f ðg−1ðtÞÞ < gðg−1ðtÞÞ we have t > f ðg−1ðtÞÞ > t, which is a contradiction. Thusg−1ðxÞ < f −1ðxÞ for all x∈ I. Therefore, as above g−nðxÞ < f −nðxÞ for all x∈ I and for alln∈ℕ. ,

For any two functions f and g, we denote the set of points of coincidence of f and g byZðf ; gÞ. i.e., Zðf ; gÞ ¼ fx∈ I j f ðxÞ ¼ gðxÞg.

Theorem 2.2. If Zðf ; gÞ is a finite set, then f n ≠ gn for all n∈ℤnf0g.Proof. If Zðf ; gÞ is empty, then either f ðxÞ < gðxÞ or gðxÞ < f ðxÞ for all x∈ I. Therefore

by Proposition 2.1, gnðxÞ≠ f nðxÞ for all x∈ I and for all n∈ℤnf0g.On the other hand, if Zðf ; gÞ is non empty, we argue as follows:If f and g do not have a common fixed point, then there exists t ∈ I such that f ðtÞ ¼ t but

gðtÞ≠ t. Without loss of generality, let gðtÞ < t. Therefore gnðtÞ < t but f nðtÞ ¼ t which inturn implies f n ≠ gn for all n∈ℤ.

If f and g have common fixed points, then the set fx∈ I j f ðxÞ ¼ gðxÞ ¼ xg must befinite. Let αi where 1≤ i≤ k be the common fixed points of f and g with

a ¼ α0 < α1 < � � � < αk < αkþ1 ¼ b:

Now, to prove our result it is enough to prove f n ≠ gn on ðαi; αiþ1Þ for some i. Since on eachðαi; αiþ1Þ both the functions f and g are self maps and has no fixed points, we may assume fand g do not have fixed points in I.

Case 1. x < f ðxÞ and gðxÞ < x for all x∈ I.Since gðxÞ < x < f ðxÞ for all x∈ ða; bÞ, for any positive integer n, gnðxÞ < x < f nðxÞ.

Moreover for any positive integer n, f −nðxÞ < x < g−nðxÞ for all x∈ ða; bÞ asf −1ðxÞ < x < g−1ðxÞ for all x∈ ða; bÞ. Hence for any n∈ℤnf0g, f n ≠ gn.

Case 2. x < f ðxÞ and x < gðxÞ for all x∈ I.Step 1: We prove the result for positive integers.Let α ¼ maxfx∈ ða; bÞj f ðxÞ ¼ gðxÞg, then f ðαÞ ¼ gðαÞ and f ðxÞ≠ gðxÞ for all x∈ ðα; bÞ.

Without loss of generality assume f ðxÞ < gðxÞ for all x∈ ðα; bÞ. To prove f n ≠ gn on I, weprove f n < gn on ðα; bÞ for all n∈ℕ.

Since f and g are self maps on ðα; bÞ, By Proposition 2.1, f nðxÞ < gnðxÞ for all x∈ ðα; bÞ.Step 2: We prove the result for negative integers.Let β ¼ minfx∈ ða; bÞj f ðxÞ ¼ gðxÞg, then f ðβÞ ¼ gðβÞ and f ðxÞ≠ gðxÞ for all x∈ ða; βÞ.

We may assume f ðxÞ < gðxÞ for all x∈ ða; βÞ. Sincex < f ðxÞ < gðxÞ for all x∈ ða; βÞ; (2)

replacing x by g−1ðxÞ in Eq. (2) we get g−1ðxÞ < f ðg−1ðxÞÞ < x for all x∈ ða; g−1ðβÞÞ. Inparticular,

g−1ðxÞ < f −1ðxÞ for all x∈ ða; g−1ðβÞÞ: (3)

To prove f −n ≠ g−n on I, we prove g−n < f −n on ða; g−1ðβÞÞ for all n∈ℕ. Since, both f −1 andg−1 are self maps on ða; g−1ðβÞÞ, by Proposition 2.1, g−nðxÞ < f −nðxÞ for all x∈ ða; g−1ðβÞÞ.

Moreover the cases f ðxÞ < x and x < gðxÞ for all x∈ I and f ðxÞ < x and gðxÞ < x for allx∈ I are similar to case 1 and case 2. ,

Lemma 2.3. If fg ¼ gf , then f ngm ¼ gmf n for all n;m∈ℤ .

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Proof. First we prove f ng ¼ g f n for all n∈ℤ. As f and g commute, we see that

f 2g ¼ f ðfgÞ ¼ f ðgf Þ ¼ ðgf Þf ¼ gf 2:

Assume f kg ¼ gf k for all 1≤ k≤ n− 1. Again, by using induction hypothesis and f and gcommute, we see that

gf n ¼ ðgf n−1Þf ¼ ðf n−1gÞf ¼ f n−1ðgf Þ ¼ f n−1ðfgÞ ¼ f ng:

Therefore f ng ¼ g f n for all n∈ℕ. Since g f ¼ f g, pre-multiplying by f −1 we get f −1g f ¼ g.Now, post multiply by f −1 to get f −1g ¼ g f −1. Hence by repeating the above process we getf −ng ¼ g f −n. Therefore f −ng ¼ g f −n for all n∈ℤ.

Since f ng ¼ g f n for each n∈ℤ, again by above argument, we have f ngm ¼ gmf n forall m∈ℤ. ,

Proposition 2.4. If x∈ Zðf ; gÞ and f g ¼ g f , then f nðxÞ; gnðxÞ∈ Zðf ; gÞ for all n∈ℤ.Proof. For x∈ Zðf ; gÞ, we have f ðf ðxÞÞ ¼ f ðgðxÞÞ ¼ gðf ðxÞÞ. Therefore f ðxÞ∈ Zðf ; gÞ.

By repeating the above process we see that f nðxÞ∈ Zðf ; gÞ for all n∈ℕ. Now, by applyingLemma 2.3, we see that

f ðf −1ðxÞÞ ¼ f −1ðf ðxÞÞ ¼ f −1ðgðxÞÞ ¼ gðf −1ðxÞÞ:Therefore f −1ðxÞ∈ Zðf ; gÞ. Hence, by above argument, f −nðxÞ∈ Zðf ; gÞ for all n∈ℕ. i.e.,f nðxÞ∈ Zðf ; gÞ for all n∈ℤ. Similarly gnðxÞ∈ Zðf ; gÞ for all n∈ℤ. ,

Theorem 2.5. If f g ¼ g f , then Zðf ; gÞ ¼ Zðf n; gnÞ for all n∈ℤnf0gProof. Step 1:We prove Zðf ; gÞ ¼ Zðf n; gnÞ for all n∈ℕ using induction on n. First we

prove Zðf ; gÞ ¼ Zðf 2; g2Þ.For x∈ Zðf ; gÞ, we have

f 2ðxÞ ¼ f ðf ðxÞÞ ¼ f ðgðxÞÞ ¼ gðf ðxÞÞ ¼ gðgðxÞÞ ¼ g2ðxÞ:Let x∈ Zðf 2; g2Þ. If f ðxÞ≠ gðxÞ, without loss of generality, say f ðxÞ < gðxÞ then

f 2ðxÞ < f ðgðxÞÞ ¼ gðf ðxÞÞ ¼ g2ðxÞwhich is not possible. Therefore Zðf ; gÞ ¼ Zðf 2; g2Þ.

Assume Zðf ; gÞ ¼ Zðf k; gkÞ for 2≤ k≤ n− 1. Therefore, by applying Proposition 2.4, forx∈ Zðf ; gÞ, we have

f nðxÞ ¼ f n−1ðf ðxÞÞ ¼ f n−1ðgðxÞÞ ¼ gn−1ðgðxÞÞ:This shows that Zðf ; gÞ ⊆ Zðf n; gnÞ. Suppose x∈ Zðf n; gnÞ with f ðxÞ < gðxÞ. Then, byapplying Lemma 2.3

f nðxÞ < f n−1ðgðxÞÞ ¼ gn−1ðf ðxÞÞ ¼ gðgn−1ðxÞÞ ¼ gnðxÞ;which is not possible. Therefore Zðf n; gnÞ ⊆ Zðf ; gÞ. This completes the proof of step 1.

Step 2: We prove Zðf ; gÞ ¼ Zðf −n; g−nÞ for all n∈ℕ.It is clear from Step 1 that, Zðf −1; g−1Þ ¼ Zðf −n; g−nÞ for all n∈ℕ. Therefore to prove

Step 2, it is enough to prove Zðf ; gÞ ¼ Zðf −1; g−1Þ.Let x∈ Zðf ; gÞ. Suppose f −1ðxÞ < g−1ðxÞ. Then, by applying Lemma 2.3 we see that,

x < f ðg−1ðxÞÞ ¼ g−1ðf ðxÞÞ ¼ g−1ðgðxÞÞ ¼ x:

which is not possible. On the other hand, if g−1ðxÞ < f −1ðxÞ thenx < gðf −1ðxÞÞ ¼ f −1ðgðxÞÞ ¼ f −1ðf ðxÞÞ ¼ x;

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again a contradiction. Therefore f −1ðxÞ ¼ g−1ðxÞ whenever f ðxÞ ¼ gðxÞ, i.e. Zðf ; gÞ⊆Zðf −1; g−1Þ. Now by replacing f and g by f −1 and g−1 respectively, we getZðf ; gÞ ¼ Zðf −1; g−1Þ. ,

Corollary 2.6. If f ; g satisfy f g ¼ g f and f n ¼ gn for some n∈ Z then f ¼ g.Proof. Since fg ¼ gf , by Theorem 2.5, we have Zðf n; gnÞ ¼ Zðf ; gÞ. But Zðf n; gnÞ ¼ I as

f n ¼ gn. Therefore f ¼ g on I. ,Theorem2.7. Let f ; g ∈HðIÞwithout fixed points such that fg ¼ gf . Suppose Zðf n; gnÞ is

an interval for some n∈ℤ , then f ¼ g on I.Proof. Since fg ¼ gf , by Theorem 2.5, Zðf ; gÞ ¼ Zðf n; gnÞ. Without loss of generality, let

α∈ Zðf ; gÞ such that α < f ðαÞ. Also by Proposition 2.4, f ðαÞ∈ Zðf ; gÞ. Since f mðαÞ→ b andf −mðαÞ→ a as m→∞. Therefore

I ¼ ða; bÞ ¼[m∈ℤ

½f mðαÞ; f mþ1ðαÞ�:

Let y∈ ½f mðαÞ; f mþ1ðαÞ� be arbitrary. Then there is an element x∈ ½α; f ðαÞ� such thaty ¼ f nðxÞ. Since f ¼ g on ½α; f ðαÞ�we have y ¼ f mðxÞ ¼ gmðxÞ. Therefore, by Lemma 2.3,

f ðyÞ ¼ f ðgmðxÞÞ ¼ gmðf ðxÞÞ ¼ gmðgðxÞÞ ¼ gðgmðxÞÞ ¼ gðyÞ:This completes the proof. ,

3. Subcommuting and comparable iterative rootsDefinition 3.1 ([3]). Let f and g be two order preserving homeomorphisms on I. We say fsubcommutes with g if f ðgðxÞÞ≤ gðf ðxÞÞ, for all x∈ I .

Note that every commuting functions are subcommuting, but the converse is notnecessarily true. For example, consider the functions f ; g : ð0;∞Þ→ ð0;∞Þ by f ðxÞ ¼ 2x andgðxÞ ¼ x2. Clearly f subcommutes with g as f ðgðxÞÞ ¼ 2x2 ≤ gðf ðxÞÞ ¼ 4x2 for all x∈ ð0;∞Þ.But f ðgðxÞÞ ¼ 2x2 ≠ gðf ðxÞÞ ¼ 4x2 for all x∈ ð0;∞Þ.

Let F : I → I be an order preserving homeomorphism. We prove that it is not possible tohave different iterative roots of F which are either comparable or subcommuting.

Theorem 3.2. Let F ∈HðIÞ. Suppose f ; g ∈HðIÞ satisfy f n ¼ gn ¼ F for some n∈ℤ.Then the following are equivalent.

1. f subcommutes with g.

2. f and g are comparable.

3. f ¼ g.

Proof. 3 implies 1 and 2 are trivial.(103) In view of Corollary 2.6, it is enough to prove that f g ¼ g f on I.Suppose f gðxÞ < g f ðxÞ for some x. Then

gnþ1ðxÞ ¼ gnðgðxÞÞ¼ f nðgðxÞÞ¼ f n−1ðf ðgðxÞÞÞ< f n−1ðgðf ðxÞÞÞ≤ f n−2ðgðf 2ðxÞÞÞ...

≤ gðf nðxÞÞ¼ gnþ1ðxÞ:

i.e., gnþ1ðxÞ < gnþ1ðxÞ, a contradiction. Hence f g ¼ g f . Therefore by Corollary 2.6, f ¼ g on I.

Subcommutingand


207

(203) Assume f ≤ g. If possible, let f ðtÞ≠ gðtÞ for some t ∈ I, therefore f ðtÞ < gðtÞ. Sincef n ¼ gn, we have

gnðtÞ ¼ f nðtÞ < f n−1ðgðtÞÞ≤ gðf n−2ðgðtÞÞÞ;

where the last inequality holds since f ≤ g. But then gn−1ðtÞ < f n−2ðgðtÞÞ as g−1 is an order-preserving homeomorphisms. Now

gn−1ðtÞ < f n−2ðgðtÞÞ≤ gðf n−3ðgðtÞÞÞ;

since f ≤ g. This implies gn−2ðtÞ < f n−3ðgðtÞÞ, since g−1 is an order-preservinghomeomorphisms. Continuing this process up to ðn− 2Þ times we get

gðgðtÞÞ < f ðgðtÞÞ;

a contradiction to our assumption. Therefore f ¼ g on I. ,Part of a theorem due to McShane [8] is observed below.Corollary 3.3 ([8]). The only order preserving iterative root of any order of the identity

function on ℝ is the identity function.Proof. Clearly, identity function is an iterative root of any order of the identity function, it

follows from Theorem 3.2, that any order preserving homeomorphism whose iteration isidentity becomes identity, as the identity function subcommutes (also commutes, so Corollary2.6 also applicable) with any function. ,

Further, if f ∈HðIÞ such that f nðxÞ ¼ x for all x∈ I but f is not the identity, then thereexists an interval ðα; βÞ such that either f ðxÞ < x or f ðxÞ > x for all x∈ ðα; βÞ andf ððα; βÞÞ ¼ ðα; βÞ. Since f nðxÞ ¼ x for all x∈ ðα; βÞ and f is comparable with identity, byTheorem 3.2 f ðxÞ ¼ x on ðα; βÞ, which is a contradiction. This forces that identity is the onlyorder preserving homeomorphism of the identity function.

From Theorem 3.2, we can conclude that the non-commuting, non-comparable iterativeroots of an order preserving homeomorphism are all different. We provide an illustrativeexample. The construction given in this example is based on Theorem 11.2.2 in [5].

Example 1. Consider the order preserving homeomorphism F : ½0; 1�→ ½0; 1�defined by

FðxÞ ¼

8>>>>>>><>>>>>>>:

4x if x∈

�0;1

8

�

4

3xþ 1

3if x∈

�1

8;1

4

�

4

9xþ 5

9if x∈

�1

4; 1

�:

In order to construct iterative roots of this function, first we define a sequence of disjointintervals whose union is ½0; 1� and on each interval we define homeomorphism which servesas an iterative root of order 2 of F.

To start with, let x0 ¼ 18 and x1 ¼ 1

4. Define x2k :¼ Fðx2k−2Þ; x2kþ1 :¼ Fðx2k−1Þ for allk∈ℕ and x−ð2kþ1Þ :¼ F−1ðx−ð2k−1ÞÞ; x−2k :¼ F−1ðx−ð2k−2ÞÞ for all k∈ℕ ∪f0g. Note that

x2 ¼ Fðx0Þ ¼ 12; x3 ¼ Fðx1Þ ¼ 2

3; x4 ¼ Fðx2Þ ¼ 12 ð49Þ þ 5

9; x5 ¼ Fðx3Þ ¼ 23 ð49Þ þ 5

9 , in general

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x2k ¼ 1

2

�4

9

�k−1

þ 5

9

Xk−2i¼0

�4

9

�i

; x2kþ1 ¼ 2

3

�4

9

�k−1

þ 5

9

Xk−2i¼0

�4

9

�i

∀ k∈ℕ;

and x−1 ¼ F−1ðx1Þ ¼ 14ð14Þ; x−2 ¼ F−1ðx0Þ ¼ 1

8ð14Þ; x−3 ¼ F−1ðx−1Þ ¼ 1

4ð14Þ2; x−4 ¼ F−1ðx−2Þ ¼ 1

8ð14Þ2, in

generalx−ð2kþ1Þ ¼ 1

4

�1

4

�kþ1

; x−2k ¼ 1

8

�1

4

�k

∀ k∈ℕ ∪ f0g:

Define Ik ¼ ½xk; xkþ1� for k∈ℤ. Since x2k → 1; x2kþ1 → 1; x−2k → 0, x−ð2kþ1Þ → 0 as k→∞ wehave∪k∈ℤIk

¼ ½0; 1�. Letf0 : I0 → I1 be thehomeomorphismdefinedby㱦0ðxÞ ¼ 2x for allx∈ I0.

Now, define fk : Ik → Ikþ1 by fkðxÞ ¼ F+f−1k−1ðxÞ for all x∈ Ik and k∈ℕ, also define

f−k : I−k → I−ðk−1Þ by f−kðxÞ ¼ f−1−ðk−1Þ+FðxÞ for all x∈ Ik and k∈ℕ. Consider the

homeomorphism f : ½0; 1�→ ½0; 1� defined by f ðxÞ ¼ fkðxÞ if x∈ Ik for all k∈ℤ. By calculationwe can show that

f ðxÞ ¼

8>><>>:

2x if x∈

�0;1

4

�

2

3xþ 1

3if x∈

�1

4; 1

�:

and f 2ðxÞ ¼ FðxÞ ∀ x∈ ½0; 1�. Now we construct another order preserving homeomorphism gwhich do not subcommute and not comparable with f but g2 ¼ F. For this, let ψ 0: I0 → I1 be thehomeomorphism defined by

ψ 0ðxÞ ¼

8>><>>:

xþ 1

8if x∈

�1

8;3

16

�

3x� 1

4if x∈

�3

16;1

4

�:

Now, define ψ k : Ik → Ikþ1 by ψ kðxÞ ¼ F+ψ−1k−1ðxÞ for all x∈ Ik and k∈ℕ, also define

ψ−k : I−k → I−ðk−1Þ by ψ−kðxÞ ¼ ψ−1−ðk−1Þ+FðxÞ for all x∈ Ik and k∈ℕ. Then the homeomorphism

g : ½0; 1�→ ½0; 1� defined by gðxÞ ¼ ψ kðxÞ if x∈ Ik for all k∈ℤ satisfies g2ðxÞ ¼ FðxÞ for allx∈ ½0; 1�. Since,

ψ 1ðxÞ ¼ F+ψ−10 ðxÞ ¼

8>>>><>>>>:

4

3xþ 1

6if x∈

�1

4;5

16

�

4

9xþ 4

9if x∈

�5

16;1

2

�;

and

ψ 2ðxÞ ¼ F+ψ−11 ðxÞ ¼

8>>>><>>>>:

1

3xþ 1

2if x∈

�1

2;7

12

�

xþ 1

9if x∈

�7

12;2

3

�;

Subcommutingand


209

we observe that

f

�g

�3

16

��¼ f

�ψ 0

�3

16

��¼ f

�5

16

�¼ 13

24< g

�f

�3

16

��¼ ψ 1

�3

8

�¼ 11

18;

and

g

�f

�13

32

��¼ ψ 2

�29

48

�¼ 103

144< f

�g

�13

32

��¼ f

�ψ 1

�13

32

��¼ f

�45

72

�¼ 27

36:

Moreover, gð 316Þ ¼ 516 < f ð 316Þ ¼ 3

8 and f ð 516Þ ¼ 1324 < gð 516Þ ¼ 7

12 . Thus we have two orderpreserving homeomorphisms f and g such that they are neither comparable norsubcommuting but f 2 ¼ g2 ¼ F and f ≠ g.

References

[1] C. Babbage, Essay towards the calculus of functions I, Philos. Trans. R. Soc. Lond. 105 (1815)389–423.

[2] U.T. B€odewadt, Zur iteration reeller funktionen, Math. Z. 49 (1944) 497–516.

[3] D. Głazowska, J. Matkowski, Subcommuting and commuting real homographic functions, J.Difference Equ. Appl. 22 (2016) 177–187.

[4] M. Kuczma, On the functional equation fn(x) 5 g(x), Ann. Polon. Math. 11 (1961) 161–175.

[5] M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, in: Encyclopedia of Mathematicsand its Applications, Vol. 32, Cambridge University Press, Cambridge, 1990.

[6] L. Liu, W. Jarczyk, L. Li, W. Zhang, Iterative roots of piecewise monotonic functions ofnonmonotonicity height not less than 2, Nonlinear Anal. 75 (1) (2012) 286–303.

[7] S. Łojasiewicz, Solution g�en�erale de l’�equation fonctionelle f(f($$$f(x)$$$)) 5 g(x), Ann. Soc. Polon.Math. 24 (1951) 88–91.

[8] N. Mcshane, On the periodicity of homeomorphisms of the real line, Amer. Math. Monthly 6 (1961)562–563.

[9] A. Smajdor, On some special iteration groups, Fund. Math. 82 (1973) 67–74.

[10] M.C. Zdun, Note on commutable functions, Aequationes Math. 36 (1988) 153–164.

[11] M.C. Zdun, P. Solarz, Recent results on iteration theory: Iteration groups and semigroups in thereal case, Aequationes Math. 87 (2014) 201–245.

[12] W. Zhang, P.M. functions, PM functions their characteristic intervals and iterative roots, Ann.Polon. Math. 65 (1997) 119–128.

Corresponding authorMurugan Suresh Kumar can be contacted at: [email protected]


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Strong consistency of akernel-based rule for spatially

dependent dataAhmad Younso

Department of Mathematical Statistics, Faculty of Sciences, Damascus University,Syrian Arab Republic, and

Ziad Kanaya and Nour AzhariDepartment of Mathematics, Faculty of Sciences, Tishreen University,

Syrian Arab Republic

AbstractWeconsider the kernel-based classifier proposed byYounso (2017). This nonparametric classifier allows for theclassification of missing spatially dependent data. The weak consistency of the classifier has been studied byYounso (2017). The purpose of this paper is to establish strong consistency of this classifier under mildconditions. The classifier is discussed in a multi-class case. The results are illustrated with simulation studiesand real applications.

Keywords Bayes rule, Kernel rule, Random field, Bandwidth, Strong consistency


1. IntroductionIn many applications one needs to classify spatial data that have been collected incompletely.The classification of incomplete-data problem, in which certain features are missing fromparticular feature vectors, exists in a wide range of fields, including image labeling, computervision and others. For example, in the remote sensing technology, because of the internalmalfunction of satellite sensors and poor atmospheric conditions such as thick cloud, theacquired remote sensing images often suffer from missing information at certain pixels andone wants to classify these pixels using the information in the nearest identified pixels. Manyexisting classification algorithms assume either certain parametric distributions for the dataor certain forms of separating curves or surfaces. These parametric classifiers are suboptimal

Strongconsistency ofa kernel-based

rule

211

© Ahmad Younso, Ziad Kanaya and Nour Azhari. Published in the Arab Journal of MathematicalSciences. Published by Emerald Publishing Limited. This article is published under the CreativeCommons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and createderivative works of this article (for both commercial and non-commercial purposes), subject to fullattribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

The author would like to thank the anonymous referees whose valuable comments led to animproved version of the paper.

The publisher wishes to inform readers that the article “Strong consistency of a kernel-based rulefor spatially dependent data”was originally published by the previous publisher of theArab Journal ofMathematical Sciences and the pagination of this article has been subsequently changed. There hasbeen no change to the content of the article. This change was necessary for the journal to transitionfrom the previous publisher to the new one. The publisher sincerely apologises for any inconveniencecaused. To access and cite this article, please use “Younso, A., Kanaya, Z., Azhari, N. (2019), “Strongconsistency of a kernel-based rule for spatially dependent data”, Arab Journal of MathematicalSciences, Vol. 26 No. 1/2, pp. 211-225. The original publication date for this paper was 13/11/2019.



Received 1 September 2018Revised 27 October 2019

Accepted 28 October 2019


Vol. 26 No. 1/2, 2020pp. 211-225


DOI 10.1016/j.ajmsc.2019.10.004

and of limited use in practical applications where little information about the underlyingdistributions is available a priori. In comparison, nonparametric classifiers are usually moreflexible in accommodating different data structures, and are hence more desirable. [21] hasproposed a nonparametric approach allowing to include contextual features for classifyingmissing spatial data and has investigated the consistency of the classifier under mildconditions. In nonparametric spatial estimation, the existing works concern mainly theestimation of a probability density and regression functions, see the key references: [2–4,15]and [14]. More recently, [5] has proposed a kernel spatial density estimator allowing for theanalysis of spatial clustering. In this work, we establish strong consistency of the classifierproposed by [21] and then, we check its performance with simulation studies andapplications. We consider a strictly stationary random field fðXi;YiÞgi∈ℤN defined on someprobability space ðΩ;F ; ℙÞ and taking values inℝd3f0; . . . ;Mg, for some integerM ≥ 1. Inthe problem of classification, for each i∈ℤN ,Xi is a vector of features andYi is the label (class)ofXi. A point i ¼ ði1; . . . ; iN Þ∈ ℤN will be referred to as a site. Forn ¼ ðn1; . . . ; nN Þ∈ ðℕ*ÞN ,we define the rectangular region In by In ¼ fi∈ ℤN : 1 ≤ ik ≤ nk; ∀k ¼ 1; . . . ;Ng. We willwrite n→∞ if mink¼1;...; N nk →∞. Define bn ¼ n13 � � �3 nN ¼ cardðInÞ and assume thatthe random field is observed on a subset Sn ⊂In with In −Sn is a bounded set for bn largeenough. When processing a particular site, its features are not used at all, but only thefeatures of its neighbors will be considered. In other words, wewish to predict the labelYj of anew site j based only on observations in a vicinity, say νj ⊂Sn, where the set νj is notcontaining j. Let νj ¼ jþ ν, where ν⊂ℤN is a fixed bounded set of sites not containing 0withcardðνÞ ¼ l (l is also the cardinal of each νj). We assume that XðjÞ ¼ fXi: i∈ νjg is a randomvector taking values inℝ

~dwith ~d ¼ ld, and that the components ofXðjÞ are ordered accordingto an arbitrary order on indices, for example the lexicographic order. The pair ðXðjÞ;YjÞmaybe completely described by μ, the probability measure for XðjÞ, and ηðxÞ, the regression of Yj

on XðjÞ ¼ x. Assume that for each i∈ℤN , ðXðiÞ;YiÞ has the same distribution as the pairðXð1Þ;Y1Þ. We will create a classifier g : ℝ

~d→ f0; . . . ;Mg mapping XðjÞ into the predicted

label of Xj. The error rate, or risk, of a rule g is LðgÞ ¼ ℙfgðXðjÞÞ≠Yjg. This is minimized bythe rule

g*ðxÞ ¼ arg max0≤k≤M

ℙðYj ¼ kjXðjÞ ¼ xÞ; (1.1)

whose error rateL* ¼ Lðg*Þ is called the Bayes-optimal risk and g*ðxÞ is called the Bayes rule.Clearly, g*ðxÞpredicts the labelYj of the site jusing only x, the value ofXðjÞ, while the featuresvector Xj does not affect the classification procedure at all. This means that g*ðxÞwell workevent if Xj is completely missing. Unfortunately, we cannot use (1.1) directly because itdepends on the distribution of ðXðjÞ;YjÞ which is generally unknown. So, we takeJ n ¼ fi ∈ Sn : νi ⊂Sng and we use the training data Dn ¼ fðXi;YiÞ : i∈J ng toconstruct a classifier gnðxÞ. We consider the classifier gnðxÞ obtained by extending theclassifier of [21] to the multi-class case as follows:

gnðxÞ ¼ arg max0≤k≤M

Xi∈J n

1fYi¼kgK

�x� XðiÞ

bn

�: (1.2)

where 1A denotes the indicator of the set A, the kernelK : ℝ~d→ℝþ is a density function on

ℝ~d, and bn is a sequence of bandwidths tending to zero as n tends to infinity. In one hand, the

sum in (1.2) is taken over J n instead of Sn just to ensure that XðiÞ always exists and that thesums make sense. On the other hand, for each new site j∉Sn, the classifier gnðxÞ predictsthe missing label Yj independently of its features vector Xj which does not belong neither tothe training sample Dn nor to the components set of XðjÞ. Consequently, gnðxÞmay classify jeven if its own features vector Xj is completely missing and that makes our method exhibit

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good performance in comparison with the classical spatial Markovian model. [6] proposes anonparametric approach to extend the result of [2] to the non-Markovian case by using twokernels in the estimator in order to control both the distance between observations and thatbetween spatial locations without using a specific vicinity for the non-observed site. Thislatter approach may be developed to classify spatial data but it does not work when onewants to classify sites with missing or incomplete features. Let Ln ¼ ℙfgnðXðjÞÞ≠YjjDng bethe error probability of gnðxÞ. Generally, we cannot hope to design a classifier that achieve theBayes error probability L* but it is possible that the limit behavior of Ln compares favorablyto L*. This idea is encapsulated in the notion of consistency.

Definition 1.1. The classifier gnðxÞ is called weakly consistent if

ELn →L*as n→∞

and strongly consistent if

Ln →L* as n→∞ with probability one:

The classifier is called universally (weakly or strongly) consistent if it is (weakly or strongly)consistent for all distribution of ðX1;Y1Þ.Remark 1.1. Since Ln is bounded, the weak consistency of Ln is equivalent to theconvergence of Ln towards L

* in probability whichmeans that strong consistency implies theweak consistency.In this paper, we investigate the strong consistency of gnðxÞ under some mild mixingconditions.

2. Notation and general hypothesesLet ðΩ;F ; ℙÞ be a probability space and let A and B be two sub σ-fields of F . The α-mixingcoefficient between A and B is defined by

α ¼ αðA;BÞ ¼ supA∈A; B∈B

jℙðA ∩ BÞ � ℙðAÞℙðBÞj

and the β-mixing coefficient is defined by

β ¼ βðA;BÞ ¼ EfsupA∈A

jℙðAjBÞ � ℙðAÞjg:

Let ðZiÞi∈ℤN be a random field on (Ω;F ; ℙÞ and taking values in some space (Ω0;F0

).

Definition 2.1. The random field ðZiÞi∈ℤN is called strongly mixing if there existsχ : ℝ→ℝþ with χðtÞa0 as t→∞, and for any E;E

0⊂ ℤN with finite cardinals,

αðBðEÞ;BðE 0 ÞÞ ≤ χðdistðE;E 0 ÞÞ;where distðE;E 0 Þ denotes the Euclidean distance between E and E

0.

The α-mixing condition is one of the most popular mixing conditions. This condition issatisfied by many spatial models. Examples can be found in [17,19] and [11].

Definition2.2. The random field ðZiÞi∈ℤN is called β-mixing if there existsw : ℝ→ℝþwithwðtÞa0 as t→∞, and for any E;E

0⊂ℤN with finite cardinals,

βðBðEÞ;BðE 0 ÞÞ≤wðdistðE;E 0 ÞÞ:Linear processes or more generally Markov chains may be β-mixing (see [9]). Similar mixingcoefficient is used by [2] to establish some asymptotic properties of the kernel regression


rule

213

estimator in the spatial case. The two mixing coefficients α and β are related by theinequality 2α ≤ β (see [18]). It means that any β-mixing random field is a strongly mixing one.

Now, we need some regularity assumptions.

Assumption 1. K is a regular kernel, that is, there exist δ > 0 and c > 0 such that

c1Bð0;δÞ ≤ KðxÞ for all x∈ℝ~d and

Rℝ

~d Sup u∈vþBð0;δÞ KðuÞdv < ∞, where Bðx; δÞ is the

closed ball of radius δ > 0 and center at x.

Assumption 2. For each i, XðiÞ has a density f with respect to Lebesgue measure and foreach i≠ j with νi ∩ νj ¼ f, ðXðiÞ;XðjÞÞ has a density fi;j such that sup

u;v∈ℝ~d j fi;jðu; vÞ−

f ðuÞf ðvÞj≤C, for some C > 0.

Assumption 3. The random field fðXi;YiÞgi∈ℤN is β-mixing and there exists θ > 0 suchthat wðtÞ ¼ O ðt−θÞfor all t ∈ ℝ*

þ.Assumption 1 is used by [8] and [7] in the i.i.d. case. It may be satisfied if KðxÞ ¼ ξðkxkÞ

where ξ is a non-negative and decreasing function on ½0;þ∞� and k:k is the Euclidean norm.Hence, the Gaussian kernel is regular. Assumption 2, used by [21] to prove the weakconsistency, is similar to that used by [3]. It is satisfied for example if f and fi;j are uniformlybounded. Assumption 3means that the random field is arithmetically β-mixingwhich impliesthat it is also strongly mixing with αðBðEÞ;BðE 0 ÞÞ≤ w ðdistðE;E 0 ÞÞ since 2α ≤ β.

3. Preliminary lemmasThis section is a collection of technical lemmas which will be used to prove the strongconsistency result stated in Theorem 4.1. Let k:kr denote the Lr-norm for any real r ≥ 1. Thefollowing lemma is a direct consequence of the covariance inequality of Ibragimov [12] andthe inequality 2α ≤ β.

Lemma3.1. If r, sand t are strictly positive reals such that r−1 þ s−1 þ t−1 ¼ 1and Z1 and Z2are two ℝ-valued random variables such that kZ1ks < ∞ and kZ2kt < ∞, then

jcovðZ1; Z2Þj ≤ 2fβðσðZ1Þ; σðZ2ÞÞg1=rkZ1kskZ2kt;where σðZiÞ is the σ-field generated by Zi for i ¼ 1; 2.

For any sub σ-fields A and B of F , we denote by A ∨ B the σ-field generated by A ∪ B.The following coupling lemma of Berbee [1] will be needed to establish the asymptotic results.

Lemma3.2. Let Z be a random variable on (Ω;F ; ℙÞwith values in some Polish spaceΩ0and

M a sub σ-field ofF . Assume that there exists a random variable U uniformly distributed over½0; 1�, independent of σðZÞ ∨M. Then, there exists a random variable ~Z measurable withrespect to σðUÞ ∨ σðZÞ ∨ M, distributed as Z and independent of M, such that

ℙðZ ≠ ~ZÞ ¼ βðM; σðZÞÞ:

Remark 3.1. We recall that a Polish spaceΩ0is a topological space which is separable and

completely metrizable (see [13]) and that most of the familiar objects of study in analysisinvolve Polish spaces. For example, ℝd for each integer d ≥ 1, is Polish with the usualtopology and f0; 1; . . . ; ng, for all n∈ℕ, is Polish with discrete topology.We also recall that acountable product of Polish spaces is Polish.

The following covering lemma can be found in [8].

Lemma 3.3. Let K be a regular kernel on ℝ~d and bn be a sequence of bandwidths. Denote

KnðxÞ ¼ b−~d

n Kðx=bnÞ. Then, for any probability measure μ,

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supu∈ℝ

~d

Zℝ~d

Knðx� uÞEKnðx� Xð1ÞÞμðdxÞ < ρ;

for some ρ > 0 dependent only on K.The proof of the following lemma is in [4] (see also [21]).

Lemma 3.4. Let ζ ¼ −N − eþ ð1− γÞNa−1 for some 0 < a < 1=2, with γ and e being smallpositive numbers such that a−1 − ðN þ eÞð1− γÞ−1N−1 > 1. If Assumption 3 holds for someθ > 2N, then for any δ > 0, X

kik≥δkikζfwðkikÞg1−γ < ∞:

The proof of the following lemma follows from the reverse triangle inequality.

Lemma 3.5. For each i; j∈J n, distðνi; νjÞ≥maxfki− jk−~r; 0g, where ~r ¼ maxfki− jk;i; j∈ νg is the diameter of ν⊂ℤN .

4. Main resultThe weak consistency of the classifier (1.2) has been established by [21]. In this section westudy the strong consistency of (1.2). The following theorem states the strong consistencyunder mild conditions.

Theorem 4.1. Assume that Assumptions 1–3 hold for some θ > 2N. If bnb~dn →∞ as n→∞,then

Ln →L* asn→∞with probability one:

Remark 4.1. Note that the assumption on the bandwidth, using by [21] to prove the weakconsistency, is similar to the classical assumption used by [7] and [8] in the independent case.In addition, the condition on bn is minimal compared to that used by [4] and [3] since they havestudied the rate of uniform convergence for the estimators. However, the restrictiveconstraints on the bandwidth in [4] and [3] are related to θ and one has to let θ→∞ in order toattain the classical assumption.

5. Simulation study including comparison with the classical kernel ruleOur aim in this section is to look at how the classifier (1.2) behaves on simulated samples bycomparing it with the classical kernel rule. We use the R statistical programmingenvironment to run a simulation study for N ¼ 2. Let fðXði;jÞ;Yði;jÞÞg be the field of interestand suppose that the simulated data are observed on the area Iðn;nÞ ¼ fði; jÞ∈ℤ2 :1 ≤ i; j ≤ ng. Let

J ðn;nÞ ¼ Iðn;nÞn��

νði;jÞ ∪ fði; jÞg; ði; jÞ∈Mg∪ fð1; jÞ; ðk; 1Þ; ðn; lÞ; ðm; nÞ : 1 ≤ j; k; l;m ≤ ngg;

where M ¼ fð2k; 2lÞ; 1 ≤ k; l ≤ 10g is the set of non-observed sites which need to beclassified. In this particular case, the vicinity of any missing site ði; jÞ may be taken as inFigure 1.

It is important to note that the vicinity νði;jÞmay be designed depending on the location ofthemissing site (see some typical examples in Figure 2) and that samples with larger size givemore freedom to design vicinities.


rule

215

Figure 2 shows some examples of vicinities that can be usedwhen themissing sites are notcompletely surrounded by already labeled sites (located at the edges of Sn for example).

We suppose that the simulated fields have the covariance function

CðuÞ ¼ 4kuk−4:5 for eachu∈ℝ*2 :

We use the classifier (1.2) with KðxÞ ¼Q8i¼1 KiðxiÞ for x ¼ ðx1; . . . ; x8Þ∈ℝ8 where KiðxiÞ is

the standard Gaussian density (Gaussian kernel). We suppose that fXði;jÞ; 1 ≤ i; j ≤ ng areobservations of a Gaussian mixture model:

π0N�μ0; σ

20

�þ π1N�μ1; σ

21

�þ π2N�μ2; σ

22

�;

with μ0 < μ1 < μ2 and π1 þ π2 þ π3 ¼ 1. In order to illustrate the fact that our method worksfor multi-class, the data set fXði; jÞ; 1 ≤ i; j ≤ ng is partitioned in three clusters as follows:

class ðYði; jÞ ¼ 0Þ : Xði; jÞ < ðμ0 þ μ1Þ�2

class ðYði; jÞ ¼ 1Þ : ðμ0 þ μ1Þ�2 ≤ Xði; jÞ ≤ ðμ1 þ μ2Þ

�2

class ðYði; jÞ ¼ 2Þ : Xði; jÞ > ðμ1 þ μ2Þ�2:

For each n ¼ 50; 75; 100, we generate 100 samples on the region Iðn;nÞ with μ0 ¼ 5, μ1 ¼ 15,μ2 ¼ 25, π0 ¼ π1 ¼ π2 ¼ 1=3 and σ20 ¼ σ21 ¼ σ22 ¼ 4. In each replication, we use the classifier(1.2), constructed on the basis of the training data observed onJ ðn;nÞ, to re-predict the labels ofsites in the test set M. Figure 3 displays one replication for n ¼ 50.

Figure 2.Three typical vicinitiescorresponding to threemissing sites ði; jÞ indifferent locations.

Figure 1.The missing site ði; jÞand its vicinity νði;jÞwith boundary in greendashed lines.

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The optimal bandwidth bbopt is obtained by minimizing the cross-validation criterion on atraining sample and themisclassification error rate (E R) is evaluated based on the associatedtest sample. The average error rate (A E R) is obtained by averaging the error ratesassociated with the corresponding 100 test samples.

Table 1 shows that the estimated optimal bandwidth and the average error rate decreasewhen the training sample size increases. This means that the practical results in thesimulation study are in line with the theoretical results. Now, let us compare the average errorrate (A E R) resulting from application of the proposed classifier with that resulting fromapplication of the classical kernel rule.

5.1 Comparison with the classical kernel ruleThe classical kernel rule is given, for any unlabeled site jwith Xj ¼ x, by

~gnðxÞ ¼ argmax0≤k≤M

Xi∈In

1fYi¼kg~K

�x� Xi

hn

�:

where ~K : ℝd→ℝþ is a kernel on ℝd (the Gaussian kernel is considered here), and hn is a

sequence of bandwidths. In order for the classical kernel classifier to be usable in our case, wehave to adjust it slightly by taking the sum over In −M instead of In, i:e:, for each j∈Mwith Xj ¼ x,

n 50 75 100

bbopt 2.04 1.93 1.77AER 28.1% 21.2% 14.8%

Figure 3.The training sites are

colored in red (0), green(1) or blue (2) and the

sites to classifyare blank.

Table 1.Estimated optimalbandwidths and

average error ratescorresponding to theclassifier (1.2) with

samples ofdifferent sizes.


rule

217

~gnðxÞ ¼ arg max0≤k≤M

Xi∈In−M

1fYi¼kg~K

�x� Xi

hn

�:

From the theoretical point of view, this is justified by the fact that ~gn has the same asymptoticbehavior on In as on In −M since M is bounded. In this classical kernel method, weconsider knowing the features vectorXj of each element jofMandwe use x, the value ofXj, topredict its class while we needed only observations in nearby sites to predict the label of j bythe classifier (1.2). We apply the classical kernel classifier to re-classify the elements of Musing the same training samples generated above and taking into account all the replicationsfor each size n ¼ 50; 75; 100. Similar to what we have done in application of (1.2), the optimalbandwidth bhopt is chosen by minimizing the cross-validation criterion on a training sampleand the misclassification error rate (E R) is evaluated based on the associated test sample.Table 2 reports the average error rate (AER), obtained by averaging the error rates associatedwith the corresponding 100 test samples.

By comparing Tables 1 and 2, we observe that the corresponding error values in the twotables begin to be close as n increases. This supports the possibility of using the classifier (1.2)as an alternative to the classical kernel classifier when we have to classify sites with missingfeatures.

6. Application to a real dataAdigital image is nothing than data numbers indicating variation of red, greenand blue (RGB)at a particular location on a grid of pixels. An RGB color value is specified with:rgbðred; green; blueÞ. Each parameter ðred; green; blueÞdefines the intensity of the color as aninteger between 0 and 255. For example, rgbð0; 0; 255Þ is rendered as blue, because the blueparameter is set to its highest value 255 and the others are set to 0. One can divide RGB colorvalues by 255 in order to provide values in the interval ½0; 1�. Let us have an image of Eiffeltower with 100 missing pixels as in Figure 3.

We use the R package jpeg to convert a jpg image into 3-d array of numbers. The packagejpeg offers the read JPEGðÞ function which can read raster graphics (consisting of “pixelmatrices”) in jpg format intoR. It returns either a single matrix with gray values in ½0; 1�or 3-darray with the RGBvalues in ½0; 1�, say E. In our example of Figure 3, the dimensions of E are306 3 165 3 3. Thus, the elements of E½; j� represent the intensities of the color j, forj ¼ “red”; “green” or “blue”, at all pixels of the grid Ið306;165Þ. For example, the matrixE½55 : 60; 1 : 6; 1� displays the intensities of red in each pixel of the region:

fði; jÞ; 55 ≤ i ≤ 60; 1 ≤ j ≤ 6g:Le Xði; jÞ ¼ ðX ð1Þ

ði; jÞ; Xð2Þði; jÞ; X

ð3Þði; jÞÞ where X ðkÞ

ði; jÞ is the intensity of the color k at the pixel ði; jÞ.Since our purpose is to classify new sites with completely missing features, we set anarbitrary threshold of 0.4 and we define labels as follow:

Yði; jÞ ¼8<:

1; if min1≤k≤3

XðkÞði;jÞ > 0:4

0; otherwise:

n 25 50 80

bhopt 1.85 1.72 1.69AER 23.4% 18.7% 13.2%

Table 2.Estimated optimalbandwidths andaverage error ratescorresponding to theclassical kernelclassifier with samplesof different sizes.

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The set of 100 missing pixels is taken as a test set, say M. We use the classifier (1.2)(see (1.7) for the binary version) to classify each element ofM based on its eight-neighbors.The optimal bandwidth is evaluated by minimizing the cross-validation criterion on theknown sites where we get bbopt ≈ 0:72. The misclassification error rate (E R) is evaluated onMwhere we obtain E R ¼ 0:04 which indicates that there are only four misclassified casesout of 100 classified cases (see Figure 4).

Now let us use the support vector machine (S V M) classifier to re-classify the elements ofM. In this case we should suppose that the RGB value is known for each element ofM. Forimplementing support vector machine in R programming language, we use the packagee1071. According to this classifier, we get a misclassification error of E R ¼ 0:11 and thispermits to conclude that our kernel classifier in this example proceeds well compared to the(SVM) procedure.

7. Proof of Theorem 4.1Without loss of generality, we prove the theorem in the binary case where Yj takes values inf0; 1g since no additional argument is required to prove it in the multi-class case. However,the Bayes classifier (1.1) in the binary case is given by

g*ðxÞ ¼�0 if ℙfYj ¼ 0jXðjÞ ¼ xg ≥ ℙfYj ¼ 1jXðjÞ ¼ xg1 otherwise;

and the classifier (1.2) is given by

gnðxÞ ¼

8>><>>:

0 ifXi∈J n

1fYi¼0gK

�x� XðiÞ

bn

�≥Xi∈J n

1fYi¼1gK

�x� XðiÞ

bn

�

1 otherwise:

(7.1)

Figure 4.Digital image of Eiffeltower with 100 missing

pixels (blank pixels).


rule

219

Define

ηnðxÞ ¼P

i∈J nYiKnðx� XðiÞÞ

bnEKnðx� Xð1ÞÞ :

Consequently, the classifier (7.1) can be written as

gnðxÞ ¼

8><>:

0 if ηnðxÞ ≤

Pi∈J n

ð1� YiÞKnðx� XðiÞÞbnEKnðx� Xð1ÞÞ

1 otherwise:

By Theorem 2.3 in [7], the consistency will be proved if we show thatZℝ~d

j ηðxÞ � ηnðxÞjμðdxÞ→ 0 asn→∞with probability one: (7.2)

But

jηðxÞ � ηnðxÞj ≤ jηðxÞ � EηnðxÞj þ jηnðxÞ � EηnðxÞj; ∀x∈ℝ~d:

Hence, in order to prove (7.1), it suffices to show thatZℝ~d

jηðxÞ � EηnðxÞjμðdxÞ→ 0 as n→∞ (7.3)

and Zℝ~d

jηnðxÞ � EηnðxÞjμðdxÞ→ 0 as n→∞ with probability one: (7.4)

The proof of (7.3) is the same as in the i.i.d. case (see [7], pp. 156–157 ). So, it suffices to prove(7.4). To do that, wewill employ the blocking technique used in [4]. Let p ¼ pn ¼ ½bnγ � for some1=θ < γ < 1=ð2NÞ (where ½:� stands for the integer part). Without loss of generality, wesuppose that there exists a positive integer qk such that nk ¼ 2pqk for each k ¼ 1; . . . ;N. Let

Jq ¼ fj ¼ ðj1; . . . ; jN Þ∈ℕN : 0 ≤ jk ≤ qk � 1; ∀k ¼ 1; . . . ;N:

We define blocks as follow, for each j∈ Jq,

Sð1Þj ¼ fi∈ In : 2jkpþ 1 ≤ ik ≤ ð2jk þ 1Þp; k ¼ 1; . . . ;Ng

Sð2Þj ¼ fi∈In : 2jkpþ 1 ≤ ik ≤ ð2jk þ 1Þp; k ¼ 1; . . . ;N � 1

and ð2jN þ 1Þpþ 1 ≤ iN ≤ 2ðjN þ 1Þpg. . .

Sð2N−1Þj ¼ fi∈ In: ð2jk þ 1Þpþ 1 ≤ ik ≤ 2ðjk þ 1Þp; k ¼ 1; . . . ;N � 1

and 2jNpþ 1 ≤ iN ≤ ð2jN þ 1Þpg

Sð2NÞj ¼ fi∈ In : ð2jk þ 1Þpþ 1 ≤ ik ≤ 2ðjk þ 1Þp; k ¼ 1; . . . ;Ng:

As a consequence, we have In ¼ S2N

k¼1

Sj∈Jq

SðkÞj , and for each k ¼ 1; . . . ; 2N , cardðSðkÞ

j Þ ¼ pN

and distðSðkÞj ; S

ðkÞj0 Þ≥ p for any j≠ j

0. Let ΓðkÞ

j ¼ fi∈SðkÞj : νi ⊂Sng, for each k ¼ 1; . . . ; 2N

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220

and j∈Jq. Hence, for a fixed k, we have distðΓðkÞj ;Γk

j0 Þ≥ p for any j ≠ j, cardðΓðkÞ

j Þ≤cardðSðkÞ

j Þ ¼ pN and

J n ¼[2Nk¼1

[j∈Jq

ΓðkÞj : (7.5)

Let fðX*ðiÞ;Y

*i Þgi∈In−J n

be a set of independent and identically distributed random vectorssuch that they are independent of fðXðiÞ;YiÞgi∈J n

and ðX*ðiÞ;Y

*i Þ is identically distributed

with ðXð1Þ;Y1Þ. In order tomake sense to the blocking technique, we define randomvectors asfollow: for each i∈ In,

ðXðiÞ;YiÞ ¼( ðXðiÞ;YiÞ if νi ⊂ Sn�

X *ðiÞ;Y

*i

�if νi ⊄ Sn:

It is clear that fðXðiÞ;YiÞ; i∈J ng ¼ fðXðiÞ;YiÞ; i∈J ng and fðXðiÞ;YiÞ; i∈ΓðkÞj g ¼

fðXðiÞ;YiÞ; i∈ΓðkÞj g. Now, for a fixed k and each j∈Jq, letW

ðkÞj ¼ fðXðiÞ;YiÞ; i∈S

ðkÞj g be

a vector whose components are ordered according to a given order on indices. ApplyingLemma 3.2 together with the blocks decomposition introduced by [10] (see also [20]) on the

family of vectors fW ðkÞj ; j∈ Jqg, we can generate independent copies f ~W

ðkÞj ; j∈Jqgsuch that:

they are mutually independent, and for each j∈ Jq, ~WðkÞj ¼ fð ~XðiÞ; ~YiÞ; i∈S

ðkÞj g has the

same distribution as WðkÞj ¼ fð ~XðiÞ; ~YiÞ; i∈S

ðkÞj g. Furthermore, by Lemma 3.5, we have

PðW ðkÞj ≠ ~W

ðkÞj Þ≤wðp−~rÞ since p ≥~r for bn large enough. Thus, the two vectors ð ~XðiÞ; ~YðiÞÞ

and ð ~Xði0Þ; ~Yi0 Þare independent for each i∈SðkÞj and i0 ∈S

ðkÞj0 with j≠ j0. Now, for each i∈J n,

there exists j∈ Jq such that fðXðiÞ;YiÞ≠ ð ~XðiÞ; ~YiÞg ⊆ ðW ðkÞj ≠ ~W

ðkÞj Þ. Since ð ~XðiÞ; ~YiÞ

¼ ð ~XðiÞ; ~YiÞ for each i∈J n, denote ð ~XðiÞ; ~YiÞ ¼ ð~X ðiÞ; ~Y iÞ, for each i∈J n (or i∈ΓðkÞj ). As a

consequence

P

�ðXðiÞ;YiÞ≠ ð~X ðiÞ; ~Y iÞ

≤wðp� ~rÞ; for each i∈J n: (7.6)

By (7.5), we can write

Xi∈J n

~Y iKnðx� ~X ðiÞÞ ¼X2Nk¼1

Xj∈Jq

Xi∈Γ

ðkÞj

~Y iKnðx� ~X ðiÞÞ:

If we denote

~ηnðxÞ ¼

Pi∈J n

~Y iKnðx� ~X ðiÞÞbnEKnðx� Xð1ÞÞ and ~ηn;kðxÞ ¼

Pj∈Jq

Pi ∈Γ

ðkÞj

~Y iKnðx� ~X ðiÞÞ

bnEKnðx� Xð1ÞÞ ; (7.7)

then

~ηnðxÞ ¼X2Nk¼1

~ηn;kðxÞ: (7.8)


rule

221

Using Markov’s inequality and Lemma 3.3 together with (7.7), we have for any e > 0,

ℙ

��Zℝ~d

jηnðxÞ � EηnðxÞjμðdxÞ �Zℝ~d

j~ηnðxÞ � E~ηnðxÞjμðdxÞ�� > e

�

≤ e−1E

��Zℝ~d


j~ηnðxÞ � E~ηnðxÞjμðdxÞ��

≤ e−1E

�Zℝ~d

j~ηnðxÞ � ηnðxÞjμðdxÞ þ E

Zℝ~d

j~ηnðxÞ � ηnðxÞjμðdxÞ�

¼ 2e−1E

Zℝ~d

j~ηnðxÞ � ηnðxÞjμðdxÞ

¼ 2e−1E

Zℝ~d

��P

i∈J n

~Y iKnðx� ~X ðiÞÞbnEKnðx� Xð1ÞÞ �

Pi∈J n

YiKnðx� XðiÞÞbnEKnðx� Xð1ÞÞ

�� μðdxÞ

≤ 4e−1Xi∈J n

E1fð~X ðiÞ ;~Y iÞ ≠ ðXðiÞ ;YiÞgsupu∈ℝ

~d

Zℝ~d

Knðx� uÞbnEKnðx� Xð1ÞÞμðdxÞ

≤ 4ðebnÞ−1ρXi∈J n

E1fð~X ðiÞ ;~Y iÞ ≠ ðXðiÞ;YiÞg≤ 4e−1ρwðp� ~rÞ;

where ρ > 0 is the constant defined in Lemma 3.3. Since~r is bounded and p→∞as n→∞, sop−~r ≥ p=2 for bn large enough. Therefore, we get

ℙ

��Zℝ~d


j~ηnðxÞ � E~ηnðxÞjμðdxÞ�� > e

�

≤ 4e−1ρwðp=2Þ≤Ce−1 ρbn−γθ;

for some generic positive constant C > 0. Since γθ > 1, by Borel–Cantelli lemma, we haveZℝ~d


j~ηnðxÞ � E~ηnðxÞjμðdxÞ→ 0; (7.9)

with probability one. Now, we will show thatZℝ~d

j~ηnðxÞ � E~ηnðxÞjμðdxÞ→ 0with probability one: (7.10)

By (7.7) and (7.8), we haveZℝ~d

j~ηnðxÞ � E~ηnðxÞjμðdxÞ≤X2Nk¼1

Zℝ~d

��~ηn;kðxÞ � E~ηn;kðxÞ��μðdxÞ: (7.11)

Consequently, in order to establish (7.10), it is sufficient to show that

Zℝ~d

��~ηn;kðxÞ � E~ηn;kðxÞ��μðdxÞ→ 0 asn→∞with probability one ; (7.12)

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for each 1 ≤ k ≤ 2N . Without loss of generality, we show (7.12) for k ¼ 1. If the elements ofJq are enumerated in an arbitrary manner, we can write Jq ¼ f1; . . . ;mg with

m ¼ cardðJqÞ ¼QN

k¼1qk. Denote~Z j ¼ fð ~XðiÞ; ~YiÞ; i∈S

ð1Þj g, for each j ¼ 1; . . . ;m, where

the components of ~Z j are ordered according to an arbitrary order on indices. Recall that

ð ~XðiÞ; ~YiÞ ¼ ð~X ðiÞ; ~Y iÞ for i∈Γð1Þj and suppose that ð ~XðiÞ;YiÞ is replaced by ð0~d; 0Þ if i∉Γð1Þ

j

where 0~d ¼ ð0; . . . ; 0Þ∈ℝ~d. Hence, by the blocks decomposition, the random vectors

~Z 1; . . . ; ~Zm are independent. Let F : ððℝ~d3f0; 1gÞpN Þm →ℝ be a real function defined asfollows

Fð~Z 1; . . . ; ~ZmÞ ¼Zℝ~d

��Xmj¼1

Xi∈S

ð1Þj

�~YiKnðx� ~XðiÞÞbnEKnðx� Xð1ÞÞ �E~Y 1Knðx� ~X ð1ÞÞbnEKnðx� Xð1ÞÞ Þ

��μðdxÞ

¼Zℝ~d

��Xmj¼1

Xi∈Γ

ð1Þj

� ~Y iKnðx� ~X ðiÞÞbnEKnðx� Xð1ÞÞ �E~Y 1Knðx� ~X ð1ÞÞbnEKnðx� Xð1ÞÞ Þ

��μðdxÞ

¼Zℝ~d

j~ηn;1ðxÞ � E~ηn;1ðxÞjμðdxÞ:

For ~zj ≠ ~z0j where ~zj ¼ fð~xðiÞ;~yiÞ; i∈S

ð1Þj g;~z0j ¼ fð~x0

ðiÞ;~y0iÞ; i∈S

ð1Þj g∈ ðℝ~d3f0; 1gÞpN and

ð~xðiÞ;~yiÞ ¼ ð~x0ðiÞ;~y

0iÞ ¼ ð0~d; 0Þ for each i∉Γð1Þ

j , using Lemma 3.3, we have��Fð~Z 1; . . . ;~zj; . . . ; ~ZmÞ � F�~Z 1; . . . ;~z

0j; . . . ;

~Zm

��

≤

Zℝ~d

��Xi∈Γ

ð1Þj

~yiKnðx� ~xðiÞÞbnEKnðx� Xð1ÞÞ �Xi∈Γ

ð1Þj

~y0iKn

�x� ~x

0ðiÞ�

bnEKnðx� Xð1ÞÞ

��μðdxÞ

≤ 2pN supu∈ℝ

~d

Zℝ~d

Knðx� uÞbnEKnðx� Xð1ÞÞ μðdxÞ ≤ 2ρpNbn−1

:

Hence, since bn ¼ 2NpNm with m ¼QNk¼1qk, by McDiarmid’s inequality [16], we have for

every e > 0,

ℙðjFð~Z 1; . . . ; ~ZmÞ � EFð~Z 1:::; ~ZmÞj > eÞ≤ 2 exp

−2N−1e2bnρ2pN

!:

Since p ¼ ½bnγ � with 1=θ < γ < 1=ð2NÞ, then bn1−γN=logðbnÞ→∞ and Borel–Cantelli lemma

yields

Fð~Z 1; . . . ; ~ZmÞ � EFð~Z 1:::; ~ZmÞ→ 0 with probability one:

As a consequenceZℝ~d

��~ηn;1ðxÞ � E~ηn;1ðxÞ��μðdxÞ � E

Zℝ~d

��~ηn;1ðxÞ � E~ηn;1ðxÞ��μðdxÞ→ 0 (7.13)


rule

223

with probability one. In order to complete the proof of (7.12) for k ¼ 1, it remains to showthat

EFð~Z 1; . . . ; ~ZmÞ ¼ E

Zℝ~d

��~ηn;1ðxÞ � E~ηn;1ðxÞ��μðdxÞ→ 0: (7.14)

The proof of (7.14) can be achieved by the same arguments used by ([21], Section 5), inaddition to benefiting from Lemmas 3.1, 3.4 and 3.5. Combining (7.9), (7.10), (7.12)–(7.14),we get (7.4). Finally, (7.3) and (7.4) yield (7.2) and the proof is completed. ,

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[13] A.S. Kechris, Classical Descriptive Set Theory, Spriner-Verlag, New York, 1995.

[14] M.E. Machkouri, Asymptotic normality of the parzen–rosenblatt density estimator for stronglymixing random fields, J. Statist. Plann. Inference 14 (2011) 73–84.

[15] M.E. Machkouri, R. Stoica, Asymptotic normality of kernel estimates in a regression model forrandom fields, J. Nonparametr. Stat. 22 (2010) 366–377.

[16] C. McDiarmid, On the method of bounded differences, Surveys in combinatorics 1989, CambridgeUniversity Press, Cambridge, 1989, pp. 148–188.

[17] C.C. Nedearhouser, Convergence of blocks spins defined by a random fields, J. Stat. Phys. 22(1980) 673–684.

AJMS26,1/2

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[18] E. Rio, Th�eorie Asymptotique des Processus Al�eatoires Faiblement D�ependants. Math�ematiqueset Applications, Spriner, Berlin, 2000.

[19] M. Rosenblatt, Stationary Sequences and Random Fields, Birkh€auser, Boston, 1985.

[20] G. Viennet, Inequalities for absolutely sequence. Application to density estimation, Probab.Theory Related Fields 107 (4) (1967) 467–492.

[21] A. Younso, On the consistency of a new kernel rule for spatially dependent data, Statist. Probab.Lett. 131 (2017) 64–71.

Corresponding authorAhmad Younso can be contacted at: [email protected]



rule

225


Generators and number fields fortorsion points of a special

elliptic curveHasan Sankari and Mustafa Bojakli

Department of Mathematics, Faculty of Science, Tishreen University, Lattakia, Syria

AbstractLet E be an elliptic curve with Weierstrass form y2 ¼ x3 − px;where p is a prime number and let E½m� be itsm-torsion subgroup. Let p1 ¼ ðx1; y1Þ and p2 ¼ ðx2; y2Þ be a basis for E½m�, then we prove thatℚðE½m�Þ ¼ ℚðx1; x2; ξm; y1Þ in general. We also find all the generators and degrees of the extensionsℚðE½m�Þ=ℚ for m ¼ 3 and m ¼ 4.

Keywords Elliptic curves, Torsion points, Algebraic extensions


1. IntroductionLet E be an elliptic curve withWeierstrass form y2 ¼ x3 − px, where p is a prime number. Letm be a positive number, we denote by E½m� the m -torsion subgroup of E, by ℚðE½m�Þ thenumber field generated by the coordinates of them -torsion points of E, and byℚðEx½m�Þ thenumber field generated by the abscissas ofm -torsion points ofE. Mazur proves them -torsionsubgroup is isomorphic to one of 15 finite groups [5]. Let p1 ¼ ðx1; y1Þand p2 ¼ ðx2; y2Þbe twopoints in E forming a basis of E½m�, then ℚðE½m�Þ ¼ ℚðx1; x2; y1; y2Þ. By Artin’s primitiveelement theorem the extension ℚðx1; x2; y1; y2Þ=ℚ is monogeneous and we can find uniquegenerator forℚðx1; x2; y1; y2Þ=ℚby combining the above coordinates. As usual, we denote byμm the group ofmth roots of unity and by ξm one of its generators. By Weil pairing, we haveξm ∈ℚðE½m�Þ, so ℚðξmÞ ⊆ ℚðE½m�Þ for all m [5]. In [3] Paladino gives a family of ellipticcurves such that ℚðE½3�Þ ¼ ℚðξ3Þ and in [4] finds the number fields generated by the 4thtorsion points, degrees and Galois groups of an elliptic curve y2 ¼ ðx−αÞðx− βÞðx− γÞwhere α; β; γ ∈ℚ, and α ≠ β≠ γ. In [1] Bandini and Paladino determine the number fieldsgenerated by the 3-torsion points, degrees and Galois groups of an elliptic curve y2 ¼ x3 þ cwhere c∈ℚ*. In [2] the result of Brau and Jones says that the rational points on the modular

Fields of aspecial elliptic

curve

227

JEL Classification — 11G04, 12F05© Hasan Sankari and Mustafa Bojakli. Published in the Arab Journal of Mathematical Sciences.



The publisher wishes to inform readers that the article “Generators and number fields for torsion pointsof a special elliptic curve” was originally published by the previous publisher of the Arab Journal ofMathematical Sciences and the pagination of this article has been subsequently changed. There has been nochange to the content of the article. This change was necessary for the journal to transition from theprevious publisher to the new one. The publisher sincerely apologises for any inconvenience caused. Toaccess and cite this article, please use Sankari, H., Bojakli, M. (2019), “Generators and number fields fortorsion points of a special elliptic curve”,Arab Journal ofMathematical Sciences, Vol. 26 No. 1/2, pp. 227-231.The original publication date for this paper was 29/10/2019



Received 3 August 2019Revised 19 September 2019Accepted 21 October 2019


Vol. 26 No. 1/2, 2020pp. 227-231


DOI 10.1016/j.ajmsc.2019.10.002

curve of level 6 yield elliptic curve E satisfying the given containment. In the first part of thispaper we prove ξm ∈ℚðEx½m�Þ and ℚðE½m�Þ ¼ ℚðx1; x2; ξm; y1Þ for allm. In the second partof this paper we find the number fields of torsion points E½m� for casesm ¼ 3; 4, extensionsand degrees. These theorems have applications in local–global divisibility problem [4] andmodular curves [2].

2. Generators for ℚðE ½m�ÞLet p1 ¼ ðx1; y1Þand p2 ¼ ðx2; y2Þ form a basis ofE½m�. We haveℚðE½m�Þ ¼ ℚðx1; x2; y1; y2Þ.We will denote by L the field ℚðx1; x2Þ and by K the field ℚðE½m�Þ. Suppose ðx3; y3Þ be thecoordinates of the point p3 ¼ p1 þ p2 and ðx4; y4Þbe the coordinates of the point p4 ¼ p1 − p2.In next theorem we will prove ξm ∈ℚðEx½m�Þ for all m.Lemma 2.1. Let fP;Qg be a basis for E½m�. Then emðP;QÞ is a primitive mth root of unity.

Proof.We know that there are S;T ∈E½m� such that emðS;TÞ ¼ ξm, a primitivemth root ofunity. Write S ¼ aP þ bQ and T ¼ cP þ dQ. Then the antisymmetry properties of the Weilpairing imply that

ξm ¼ emðS;TÞ ¼ emðP;QÞad−bc:

Since emðP;QÞ is an mth root of unity and a power of it is a primitive mth root of unity, itfollows that emðP;QÞ is a primitive mth root of unity. ,

Theorem 2.2. Let fp1; p2g be a basis for E½m�, let p3 ¼ p1 þ p2 and p4 ¼ p1 − p2, and writepi ¼ ðxi; yiÞ. Then

ℚðξmÞ⊆ℚðx1; x2; x3; x4Þ⊆ℚðEx½m�Þ:

Proof. The second inclusion is by the definition ofℚðEx½m�Þ. For the first inclusion. Let σ bean automorphism of ℚðE½m�Þ that fixes ℚðx1; x2; x3; x4Þ. Then σðyiÞ ¼ ±yi since σðy2i Þ ¼ y2i .The equation

y1y2 ¼ ðx4 � x3Þðx1 � x2Þ24

shows that σðy1y2Þ ¼ y1y2. This means that either σðyiÞ ¼ yi for i ¼ 1; 2, or σðyiÞ ¼ −yi fori ¼ 1; 2. These mean that either σðpiÞ ¼ pi for i ¼ 1; 2, or σðpiÞ ¼ −pi for i ¼ 1; 2. In thefirst case,

emðp1; p2Þσ ¼ emðσðp1Þ; σðp2ÞÞ ¼ emðp1; p2Þ:In the second case,

emðp1; p2Þσ ¼ emðσðp1Þ; σðp2ÞÞ ¼ emð–p1;�p2Þ ¼ emðp1; p2Þ:

Since emðp1; p2Þ is a primitivemth root of unity, we find that ℚðξmÞ ⊆ ℚðx1; x2; x3; x4Þ. ,We know that ℚðx1; x2; y1; y2Þ ¼ ℚðx1; x2; y1; y1y2Þ. In next theorem we will prove that

ℚðE½m�Þ is equal to the field ℚðx1; x2Þ by adding ξm and y1.

Theorem 2.3. ℚðE½m�Þ ¼ ℚðx1; x2; ξm; y1Þ:Proof. We have ℚðx1; x2; ξm; y1; y2Þ ¼ ℚðE½m�Þ. If we do not have the equality in thetheorem, then y2 ∉ ℚðx1; x2; ξm; y1Þ. Since y22 is in this field, there is an automorphism σ suchthat σðy2Þ ¼ –y2 and σ is the identity on ℚðx1; x2; ξm; y1Þ. Then

AJMS26,1/2

228

emðp1; p2Þ ¼ emðp1; p2Þσ ¼ emðσðp1Þ; σðp2ÞÞ ¼ emðp1;�p2Þ ¼ emðp1; p2Þ−1:This implies that emðp1; p2Þ2 ¼ 1. Since emðp1; p2Þ is a primitive mth root of unity, we musthave m ¼ 2. But then y1 ¼ y2 ¼ 0, in which case the theorem is true. ,

3. Number fields ℚðE ½m�Þ for cases m ¼ 3, 4It is well known that the abscissas of the 3-torsion points of an elliptic curve y2 ¼ x3 – px arethe roots of the polynomial

w3 ¼ 3x4 � 6px2 � p2;

then the roots bx1; bx2; bx3; bx4 of w3 are:

bx1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip� 2pffiffiffi

3p

s; bx2 ¼ –

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip� 2pffiffiffi

3p

s; bx3 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipþ 2pffiffiffi

3p

s; bx4 ¼ –

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipþ 2pffiffiffi

3p

s:

In next theorems we will determine the field generated by 3 and 4 torsion points.

Theorem 3.1. Let E be an elliptic curve with Weierstrass form E : y2 ¼ x3 − px, where p is aprime number. Then

ℚðEx½3�Þ ¼ ℚ


3p

s; ξ3

!with ½ℚðEx½3�Þ : ℚ� ¼ 8;

ℚðE½3�Þ ¼ ℚ

0BB@

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p

ffiffiffi3

p � 3p

q3

vuut; ξ3

1CCA with ½ℚðE½3�Þ : ℚ� ¼ 16:

Proof. We have ℚðbx1; bx2; bx3; bx4Þ ¼ ℚðbx1; bx3Þ. On the other hand we have

bx1 bx3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�p� 2pffiffiffi

3p��

pþ 2pffiffiffi3

p�s

¼ffiffiffiffiffiffiffi–p2

3

r¼

ffiffiffiffiffi–3

pp

3;

so ℚðbx1; bx3Þ ¼ ℚðbx1; bx1 bx3Þ ¼ ℚðbx1; ξ3Þ ¼ ℚ

ffiffiffiffiffiffiffiffiffiffiffip – 2pffiffi

3p

q; ξ3

!.

We have"ℚ


3p

s; ξ3

!: ℚ

#¼"ℚ


3p

s; ξ3

!: ℚðξ3Þ

#½ℚðξ3Þ : ℚ�:

Put α ¼ffiffiffiffiffiffiffiffiffiffiffip – 2pffiffi

3p

q, then

f ðxÞ ¼ minðα;ℚðξ3ÞÞ ¼ 3α4 þ 6pα2 � p2 ¼ 0

is irreducible over ℚðξ3Þ, because the roots of f ðxÞ are bx1; bx2; bx3; bx4. They are irrational, soeither f ðxÞ is irreducible or it has a quadratic factor that has bx1 and some other bxi as roots.Since bx1 bx2 ∉ ℚðξ3Þ, the other root is not bx2. Suppose the other root is bx3 or bx4. Then (using bx3)

2p

3

�3±

ffiffiffiffiffiffi−3

p �¼ ðbx1 þ bx3Þ2


curve

229

is a square in ℚðξ3Þ. But its norm to ℚ is 16p2

3, which is not a square, so it cannot be a square.

Therefore, there is noquadratic factor and f ðxÞ is irreducible. So"ℚ

ffiffiffiffiffiffiffiffiffiffiffiffip− 2pffiffi

3p

q; ξ3

� �: ℚðξ3Þ

#¼ 4.

It is easy to verify that ½ℚðξ3Þ : ℚ� ¼ 2. Hence

½ℚðEx½3� : ℚÞ� ¼"ℚ


3p

s; ξ3

!: ℚ

#¼ 4 • 2 ¼ 8:

By Theorem 2.2 we proved that ℚðE½3�Þ ¼ ℚðbx1; bx2; ξ3; by1Þ ¼ ℚðbx1; ξ3; by1Þ, where bx1 ¼ – bx2.As by12 ¼ bx13 – pbx1, then

y1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibx31 � pbx1

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip� 2pffiffiffi

3p

s !3

� p


3p

s !vuut ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p

ffiffiffi3

p � 3p

q3

vuut

and ½ℚðbx1; ξ3; by1Þ : ℚðbx1; ξ3Þ� ¼ 2. We found in previous case that ½ℚðbx1; ξ3Þ : ℚ� ¼ 8.Hence

½ℚðE½3�Þ : ℚ� ¼ ½ℚðbx1; ξ3; by1Þ : ℚ� ¼ ½ℚðbx1; ξ3; by1Þ : ℚðbx1; ξ3Þ�½ℚðbx1; ξ3Þ : ℚ� ¼ 2 • 8 ¼ 16: ,

It is well known that the abscissas of the 4-torsion points of an elliptic curve y2 ¼ x3 – px arethe roots of the polynomial

w4 ¼ x6 � 5px4 � 5p2x2 þ p3;

then the roots bx1; bx2; bx3; bx4; bx5; bx6 of w4 are

bx1 ¼ iffiffiffip

p; bx2 ¼ þ ffiffiffi

pp þ ffiffiffiffiffi

2pp

; bx3 ¼ – iffiffiffip

p;

bx4 ¼ ffiffiffip

p �ffiffiffiffiffi2p

p; bx5 ¼ –

ffiffiffip

p þ ffiffiffiffiffi2p

p; bx6 ¼ –

ffiffiffip

p–ffiffiffiffiffi2p

p:

Theorem3.2. Let E be an elliptic curve withWeierstrass form y2 ¼ x3 – px, where p is a primenumber. Then

ℚðEx½4�Þ ¼�ℚði;

ffiffiffi2

p;ffiffiffip

p Þ with½ℚðEx½4�Þ : ℚ� ¼ 8 if p≠ 2;

ℚði;ffiffiffi2

pÞ with½ℚðEx½4�Þ : ℚ� ¼ 4 if p ¼ 2:

ℚðE½4�Þ ¼�ℚði;

ffiffiffi2

p;ffiffiffip4

p Þ with½ℚðE½4�Þ : ℚ� ¼ 16 if p≠ 2;

ℚði;ffiffiffi8

4p

Þ with½ℚðE½4�Þ : ℚ� ¼ 8 if p ¼ 2:

Proof. The points of exact order 4 of y2 ¼ x3 – px are ±p1;±p2;±p3;±p4;±p5;±p6, where

p1 ¼�iffiffiffip

p;�

ffiffiffiffip34

pþ i

ffiffiffiffip34

p �; p2 ¼

� ffiffiffip

p þffiffiffiffiffi2p

p; 2

ffiffiffiffip34

pþ

ffiffiffi2

p ffiffiffiffip34

p �;

p3 ¼�−i

ffiffiffip

p;�

ffiffiffiffip34

p� i

ffiffiffiffip34

p �; p4 ¼

� ffiffiffip


p;�2

ffiffiffiffip34

pþ

ffiffiffi2

p ffiffiffiffip34

p �;

p5 ¼ –ffiffiffip


p;2pffiffiffiffip34

p þ 2p

iffiffiffi2

p ffiffiffiffip34

p!; p6 ¼

–ffiffiffip

p–

ffiffiffiffiffi2p

p;2pffiffiffiffip34

p –2p

iffiffiffi2

p ffiffiffiffip34

p!:

We have:

AJMS26,1/2

230

ℚðEx½4�Þ ¼ ℚðbx1; bx2; bx3; bx4; bx5; bx6Þ¼ ℚ

�iffiffiffip

p;ffiffiffip


p;�i

ffiffiffip

p;ffiffiffip


p;� ffiffiffi

pp þ

ffiffiffiffiffi2p

p;�

ffiffiffi2

p�

ffiffiffiffiffi2p

p �¼ ℚ

�i;ffiffiffi2

p;ffiffiffip

p �

with ½ℚðEx½4�Þ : ℚ� ¼ 8 if p≠ 2 and ½ℚðEx½4�Þ : ℚ� ¼ 4 if p ¼ 2. ,Let fp1; p2g be a basis for E½4�, then

ℚðE½4�Þ ¼ ℚðbx1; bx2; by1; by2Þ¼ ℚ

�iffiffiffip

p;ffiffiffip


p;�

ffiffiffiffip34

pþ i

ffiffiffiffip34

p; 2

ffiffiffiffip34

pþ

ffiffiffi2

p ffiffiffiffip34

p �

¼ ℚ

�i;ffiffiffi2

p;ffiffiffiffip34

p �

with ½ℚðE½4�Þ : ℚ� ¼ 16 if p ≠ 2 and ½ℚðE½4�Þ : ℚ� ¼ ½ℚði; ffiffiffi84

p Þ� ¼ 8 if p ¼ 2. ,

References

[1] A. Bandini, L. Paladino, Number fields generated by the torsion points of an elliptic curve, J.Number Theory 169 (2016) 103–133.

[2] J. Brau, J. Jones, Elliptic curves with 2-torsion contained in the 3-torsion field, AMS 144 (2016)925–936.

[3] L. Paladino, Elliptic curves with ℚðE½3�Þ ¼ ℚðξ3Þand counterexamples to local global divisibilityby 9, J. Th�eor. Nombres Bordeaux 22 (2010) 138–160.

[4] L. Paladino, Local global divisibility by 4 in elliptic curves defined over ℚ, Ann. Mat. Pura Appl.189 (2010) 17–23.

[5] H. Silverman, The Arithematic of Elliptic Curves, Springer-Verlag, Heidelberg, 2009.

Corresponding authorMustafa Bojakli can be contacted at: [email protected]



curve

231


On the primeness of near-ringsKhalid H. Al-Shaalan

Department of Mathematics, College of Science, King Saud University,Riyadh, Saudi Arabia

AbstractIn this paper, we study the different kinds of the primeness on the class of near-rings and we give newcharacterizations for them. For that purpose, we introduce new concepts called set-divisors, ideal-divisors, etc.and we give equivalent statements for 3-primeness which make 3-primeness looks like the forms of the otherkinds of primeness. Also, we introduce a new different kind of primeness in near-rings called K-primenesswhich lies between 3-primeness and e-primeness. After that, we study different kinds of prime ideals in near-rings and find a connection between them and new concepts called set-attractors, ideal-attractors, etc. to makenew characterizations for them. Also, we introduce a new different kind of prime ideals in near-rings calledK-prime ideals.

Keywords Near-rings, Rings, Primeness, Prime ideals


1. IntroductionWe say that R is a right (left) near-ring if ðR;þÞ is a group, ðR; $Þ is a semigroup and Rsatisfies the right (left) distributive law. Throughout this paper, Rwill be a left near-ring. Wesay that R is an abelian near-ring if xþ y ¼ yþ x for all x; y∈R and we say that R is acommutative near-ring if xy ¼ yx for all x; y∈R. A zero-symmetric element is an elementx∈R satisfying 0x ¼ 0. A near-ring R is called a zero-symmetric near-ring, if 0x ¼ 0 for allx∈R. A constant element is an element y∈R satisfying zy ¼ y for all z∈R. An element x∈Ris called a right (left) zero divisor in R if there exists a non-zero element y∈R such that yx ¼ 0(xy ¼ 0). A zero divisor is either a right or a left zero divisor. By a near-ring without zerodivisors, wemean a near-ring without non-zero divisors of zero. IfAandBare two non-emptysubsets of R, then the product ABmeans the set fabja∈A; b∈Bg. We say that U is a right(left) R-subgroup of R, ifU is a subgroup of ðR;þÞ satisfiesUR⊆U (RU ⊆U). We say thatUis a two-sidedR-subgroup of R, ifU is both a right and a leftR-subgroup ofR. We say that I isa right (left) ideal of R, if I is a normal subgroup of ðR;þÞ satisfies ðr þ iÞs− rs∈ I for alli∈ I ; r; s∈R (RI ⊆ I). We say that I is an ideal of R if it is both a right and a left ideal of R. Wesay that U is a semigroup right (left) ideal of R, if U is a non-empty subset of R satisfiesUR⊆U (RU ⊆U). We say thatU is a semigroup ideal of R if it is both a semigroup right and

Primeness ofnear-rings

233

JEL Classification — 16D25, 16N60, 16Y30©Khalid H. Al-Shaalan. Published in theArab Journal ofMathematical Sciences. Published byEmerald

Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license.Anyonemay reproduce, distribute, translate and create derivativeworks of this article (for both commercialand non-commercial purposes), subject to full attribution to the original publication and authors. The fullterms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode


The publisher wishes to inform readers that the article “On the primeness of near-rings”was originallypublished by the previous publisher of the Arab Journal of Mathematical Sciences and the pagination ofthis article has been subsequently changed. There has been no change to the content of the article. Thischangewasnecessary for the journal to transition from thepreviouspublisher to the newone.Thepublishersincerely apologises for any inconvenience caused. To access and cite this article, please use Al-Shaalan,K. H. (2019), “On the primeness of near-rings”, Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2,pp. 233-243. The original publication date for this paper was 23/12/2019.



Received 11 October 2019Revised 16 December 2019



Vol. 26 No. 1/2, 2020pp. 233-243


DOI 10.1016/j.ajmsc.2019.12.004

left ideal of R (some authors call U a right (left, two-sided) R-subset of R [8]). For any groupðG;þÞ, MðGÞ denotes the near-ring of all maps from G to G with the two operations ofaddition and composition of maps. MoðGÞ is the zero-symmetric subnear-ring of MðGÞconsisting of all zero preserving maps from G to itself (and to make them left near-rings weshould write f ðgÞby gf , where f ∈MðGÞorMoðGÞand g ∈G ). A trivial zero-symmetric near-ring R is a zero-symmetric near-ring such that the multiplication on the group ðR;þÞ isdefined by xy ¼ y and 0y ¼ 0 for all x∈R− f0g; y∈R. A near-field N is a near-ring in whichðN − f0g; $Þ is a group. For further information about near-rings, see [8] and [9].

In near-rings, there are five well-known kinds of primeness. We say that: R is 0-prime (theusual primeness) if, for every two ideals I and J ofR, IJ ¼ f0g implies I ¼ f0gor J ¼ f0g,R is1-prime if, for every two right idealsK and LofR,KL ¼ f0g impliesK ¼ f0gor L ¼ f0g.R is2-prime if, for every two right R-subgroups A and B of R, AB ¼ f0g implies A ¼ f0g orB ¼ f0g. R is 3-prime if, for all x; y∈R, xRy ¼ f0g implies x ¼ 0 or y ¼ 0 and R is equiprime(e-prime) if, for any 0≠ a; x; y∈R, xca ¼ yca for all c∈R implies x ¼ y. These five kinds ofprimeness are equivalent in the class of rings. But in the class of near-rings, we have: (1)Ris equiprime implies that R is zero-symmetric 3-prime, (2) R is 3-prime implies that R is2-prime, (3) R is zero-symmetric 2-prime implies that R is 1-prime and (4) R is 1-prime impliesthat R is 0-prime. For details about these kinds and their examples and relationships see[1–3,5–7] and [10]. A near-ring (a ring) R is called 3-semiprime (semiprime) if, for all x∈R,xRx ¼ f0g implies x ¼ 0. An idealP ofR is: (i) a 0-prime ideal ofR if for every two idealsAandB of R,AB⊆P implies thatA⊆P or B⊆P, (ii) a 1-prime ideal of R if for every two right idealsAandBofR,AB⊆P implies thatA⊆P orB⊆P, (iii) a 2-prime ideal ofR if for every two rightR-subgroups A and B of R, AB⊆P implies that A⊆P or B⊆P, (iv) a 3-prime ideal of R if fora; b∈R, aRb⊆P implies that a∈P or b∈P, (v) an e-prime (equiprime) ideal of R if for everya∈R−P and x; y∈R, xca− yca∈P for all c∈R implies that x− y∈P. Clearly that any near-ring is a υ-prime ideal of itself, where υ∈ f0; 1; 2; 3; eg. It is well-known that (ii) implies (i) and(iv) implies (iii). Also, for zero-symmetric near-rings we have (iii) implies (ii). An ideal I of R iscalled completely prime if, for a; b∈R, ab∈ I implies that a∈ I or b∈ I. If the zero ideal iscompletely prime, then we say that R is completely prime. Then R is completely prime if andonly if R is without zero divisors. For more details about prime ideals, see [2,4,5] and [10].

In [1], the authors gave us a short historical view about the primeness of near-rings. Wewill use it and add some information to it.

Several different generalizations of primeness for rings have been introduced for near-rings. In [6], Holcombe studied three different concepts of primeness, which he called 0-prime,1-prime and 2-prime. In [5], Groenewald obtained further results for these and introducedfurther notion which he called 3-primeness. In [2], Booth, Groenewald and Veldsman gaveanother definition, called equiprimeness, or e-primeness. In [10], Veldsmanmademore studieson equiprime near-rings. In [1], Booth and Groenewald gave an element-wise characterizationof the radical associated with ν -primeness for ν ¼ 1; 2; 3; e.

In this paper we extend the idea of primeness that they did and give some new results forthe primeness of near-rings. Firstly, we introduce new concepts called set-divisors, idealdivisors, etc. These concepts are generalizations of the concept of zero divisors and giveanother characterization of different kinds of the primeness in near-rings and hence in rings.Also, we study the 3-primeness and give new characterizations of 3-prime (3-semiprime) near-rings and hence for prime (semiprime) rings. These characterizationsmake 3-primeness lookslike the forms of the other kinds of primeness. In fact, we show that a near-ring (a ring) is3-prime (prime) if and only if UV ¼ f0g implies U ¼ f0g or V ¼ f0g, where U and V aresemigroup left ideals of R. Hence, a ring is prime if and only if it is without zero-semigroupright (left) ideal divisors. A similar result is made for 3-semiprime near-rings (semiprimerings) and we conclude that: for a near-ring R, if r2 ≠ 0 for all r∈R− f0g, then R is3-semiprime. We show that some kinds of near-rings are 3-prime if and only if they are

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2-prime. Also, we introduce a new kind of primeness in near-rings (the sixth one) calledK-primeness and we show that it is totally different from the other kinds of primeness and itlies between 3-primeness and e-primeness. Depending on that, we give two chains ofprimeness in the class of zero-symmetric near-rings for comparison. In the last part of thepaper, we study different kinds of prime ideals. We introduce a new kind of prime idealscalled K-prime ideals andwe show that they are different from the other kinds of prime ideals.they lie between 3-prime ideal and e-prime ideals. Also, we give a new characterization of3-prime ideals and show that P is a 3-prime ideal of R if and only if UV ⊆P implies U ⊆P orV ⊆P, where U and V are semigroup left ideals of R. We introduce new concepts calledset-attractors, ideal-attractors, etc. which are generalizations of the new concepts above(set-divisors, etc.).Wemake a connection between these concepts and different kinds of primeideals in near-rings to give a new characterization of these prime ideals. Finally, we use theseconcepts to show that: P is a completely prime ideal of R if and only if R is without external Pset-attractors.

2. On prime near-ringsLet R be a near-ring. It is clear that R is without zero divisors if and only ifAB ¼ f0g impliesA ¼ f0g or B ¼ f0g, whereA and B are non-empty subsets of R. This observation gives us ahint of a new definition.

Definition 2.1. Let R be a non-zero near-ring.

(1) LetAbe a non-empty subset of R. We say thatA is a left zero-set divisor (a right zero-set divisor) of R if there exists a non-empty non-zero subset B of R such that AB ¼ f0g(BA ¼ f0g). We say thatA is a zero-set divisor of R ifA is a left or a right zero-set divisor ofR.

(2) Let A be an ideal of R. We say that A is a left zero-ideal divisor (a right zero-idealdivisor) of R if there exists a non-zero ideal B of R such that AB ¼ f0g (BA ¼ f0g). We saythat A is a zero-ideal divisor of R if A is a left or a right zero-ideal divisor of R.

We can do same definitions if A is a left (right) ideal, a left (right) R -subgroup, a two-sidedR-subgroup, a semigroup left (right) ideal or a semigroup ideal.

Definition 2.1 generalizes the concept of zero divisors in rings and near-rings. So, we havethe following remark.

Remark 2.1. From Definition 2.1, we can rewrite the definitions of different kinds of theprimeness as follows:

Let R be a near-ring. Then

(1) R is completely prime if and only if R is without zero divisors if and only if R iswithout zero-set divisors.

(2) R is 0-prime if and only if R is without zero-ideal divisors.

(3) R is 1-prime if and only if R is without zero-right ideal divisors.

(4) R is 2-prime if and only if R is without zero-right R-subgroup divisors.

Remark 2.1 enhances a question: Canwe get a definition of 3-primeness like that mentioned inRemark 2.1? The following result answers this question.

Theorem 2.1. Let R be a near-ring. Then the following statements are equivalent:

(i) R is 3-prime.

(ii) aU ¼ f0g implies a ¼ 0 or U ¼ f0g, where a∈R and U is a semigroup left ideal of R.


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(iii) AU ¼ f0g implies A ¼ f0gor U ¼ f0g, where A is a non-empty subset of R and U isa semigroup left ideal of R.

(iv) UV ¼ f0g implies U ¼ f0g or V ¼ f0g, where U and V are semigroup left idealsof R.

Proof. (i) implies (ii), (ii) implies (iii) and (iii) implies (iv) are clear.To prove that (iv) implies (i), we will use the contradiction. For that purpose, suppose R is

not 3-prime. So there exist non-zero elements x; y∈R such that xRy ¼ f0g. Thus,RxRy ¼ f0g. But Rx and Ry are semigroup left ideals of R, so Rx ¼ f0g or Ry ¼ f0g by(iv). Hence,Rf0; xg ¼ f0gorRf0; yg ¼ f0gand either f0; xgor f0; yg is a semigroup left idealof R. But R is also a semigroup left ideal of R. Thus, f0; xg ¼ 0, f0; yg ¼ f0g or R ¼ f0g by(iv), a contradiction with that x; y;R are all non-zero. So R is 3-prime and (iv) implies (i). -

For zero-symmetric near-rings, we have the following extra result.

Theorem 2.2. Let R be a zero-symmetric near-ring. Then the following statements areequivalent:

(i) R is 3-prime.

(ii) Ua ¼ f0g implies a ¼ 0 orU ¼ f0g, where a∈R andU is a semigroup right ideal ofR.

(iii) UA ¼ f0g impliesU ¼ f0g or,A ¼ f0gwhereU is a semigroup right ideal of R andA is a non-empty subset of R.

(iv) UV ¼ f0g implies U ¼ f0g orV ¼ f0g, where U and V are semigroup right idealsof R.

(v) UV ¼ f0g implies U ¼ f0g orV ¼ f0g, where U is a semigroup right ideal of R andV is a semigroup left ideal of R.

Now, we can add (5) to Remark 2.1:

(5) R is 3-prime if and only if UV ¼ f0g implies U ¼ f0g or V ¼ f0g, where U and Vare semigroup left ideals of R if and only if R is without zero-semigroup left ideal divisors.

Since any ring is a zero-symmetric near-ring, we have the following result:

Corollary 2.3. A ring is prime if and only if it is without zero-semigroup right (left) idealdivisors.

Using the same idea, the following result gives us a result for 3-semiprime zero-symmetricnear-rings.

Theorem 2.4. Let R be a zero-symmetric near-ring. Then the following statements areequivalent:

(i) R is 3-semiprime.

(ii) aU ¼ f0g implies a ¼ 0, where a∈U and U is a semigroup left ideal of R.

(iii) Ua ¼ f0g implies a ¼ 0, where a∈U and U is a semigroup right ideal of R.

(iv) U 2 ¼ f0g implies U ¼ f0g, where U is a semigroup left ideal of R.

(v) U 2 ¼ f0g implies U ¼ f0g, where U is a semigroup right ideal of R.

Proof. (i) implies (ii). Suppose (i) holds. Let U be a semigroup left ideal of R such thataU ¼ f0g, where a∈U. Then for all v∈U, we have aRv ¼ f0g. Thus, aRa ¼ f0g and a ¼ 0by (i).

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(i) implies (iii) can be proved by the same way.(ii) implies (iv) and (iii) implies (v) are clear.(iv) implies (v). Suppose that (iv) holds and U 2 ¼ f0g, where U is a semigroup right

ideal of R. So uRu ¼ f0g for all u∈U and hence RuRu ¼ f0g. But Ru is a semigroup leftideal of R. So Ru ¼ f0g for all u∈U by (iv). So f0; ug is a semigroup left ideal of R andf0; ugf0; ug ¼ f0g for all u∈U. So u ¼ 0 by (iv) and hence U ¼ f0g.

(v) implies (i). Suppose that (v) holds and that xRx ¼ f0g for some x∈R. Thus,xRxR ¼ f0g. But xR is a semigroup right ideal of R, so xR ¼ f0g by (v). Hence,f0; xgf0; xg ¼ f0g. But f0; xg is a semigroup right ideal of R. Thus, f0; xg ¼ f0g by (v)and hence x ¼ 0. So R is 3-semiprime and (v) implies (i). -

Corollary 2.5. A ring R is semiprime if and only if U 2 ¼ f0g implies U ¼ f0g, where U is asemigroup right (left) ideal of R.

But in the general case of 3-semiprime near-rings, we have only the following result.

Theorem 2.6. Let R be a near-ring. Then the following statements are equivalent:

(i) R is 3-semiprime.

(ii) aU ¼ f0g implies a ¼ 0, where a∈U and U is a semigroup left ideal of R.

(iii) U 2 ¼ f0g implies U ¼ f0g, where U is a semigroup left ideal of R.

Unfortunately, we cannot remove theword “zero-symmetric” in Theorems 2.2 and 2.4. Thefollowing example is the near-ring in [9, Appendix, E, 22] and it shows that the condition“zero-symmetric” in Theorems 2.2 and 2.4 is not redundant.

Example 1. Let ðR;þÞbe the Klein’s four group f0; a; b; cg. Then it is an abelian group suchthat xþ x ¼ 0 for all x∈R and xþ y ¼ z for all different non-zero elements x; y; z∈R. Definethe multiplication on R as follows:

$ 0 a b c

0 0 a 0 a

a 0 a 0 a

b 0 a 0 a

c 0 a b c

ClearlyR is an abelian non-zero-symmetric near-ring. The only semigroup right ideals ofRareR, f0; ag and f0; a; bg. So R satisfies the conditions “UV ¼ f0g impliesU ¼ f0g orV ¼ f0g,whereU andV are semigroup right ideals ofR” and “U 2 ¼ f0g impliesU ¼ f0g, whereU is asemigroup right ideal of R”. But R is not 3-semiprime as bRb ¼ f0g. From Theorem 2.6, wecan deduce that there is a non-zero semigroup left ideal V of R such that V 2 ¼ f0g andυV ¼ f0g, where υ∈V − f0g. It is easy to find out that V ¼ f0; bg and v ¼ b.

From the above example, observe thatf0; a; bgb ¼ f0; a; bgfbg ¼ f0g:

So, we cannot use this example for (ii) or (iii) in Theorem 2.2 and for (iii) in Theorem 2.4. In fact,removing “zero-symmetric” from those parts is an open problem.

Corollary 2.7. Let R be a near-ring. If r2 ≠ 0 for all r∈R− f0g, then R is 3-semiprime.

Proof. Suppose there exists a non-zero semigroup left ideal U of R such that aU ¼ f0g,where a∈U. That means a2 ¼ 0. By hypothesis, a ¼ 0 and hence R is 3-semiprime. -

Example 2. Let R ¼ ℤ6. Then R is semiprime since r2 ≠ 0 for all r∈R− f0g.


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Example 3. Let R ¼ f0; 2; 4; 6; 8; 10; 12g the subring of ℤ14. Then R is semiprime sincer2 ≠ 0 for all r∈R− f0g.

The converse of Corollary 2.7 is not true as the following example shows.

Example 4. Let R ¼ M2ðℤ2Þ. Then R is a prime ring and hence semiprime, but�0 10 0

��0 10 0

�¼

�0 00 0

�:

For commutative near-rings, we have the converse and we get the following result.

Corollary 2.8. Let R be a commutative near-ring. Then r2 ≠ 0 for all r∈R− f0g if and only ifR is 3-semiprime.

We conclude this section by the following results about the relation between 2-primenessand 3-primeness. The fact that R is 3-prime implies R is 2-prime is well-known. The followingresults have the converse.

Theorem 2.9. Let R be a zero-symmetric near-ring such that 2R ¼ f0g. Then R is 3-prime ifand only if R is 2-prime.

Proof. Suppose that xRy ¼ f0g. Thus, xRyR ¼ f0g. But xR and yR are right R-subgroups ofR. So xR ¼ f0gor yR ¼ f0g as R is 2-prime. Hence, f0; xgR ¼ f0g or f0; ygR ¼ f0g and theneither f0; xg or f0; yg is a right R-subgroup of R. But R is also a right R-subgroup of R. Thus,f0; xg ¼ 0, f0; yg ¼ f0g or R ¼ f0g. Hence, x ¼ 0 or y ¼ 0 and R is 3-prime. -

Theorem 2.10 Any distributive near-ring R is 3-prime if and only if it is 2-prime.

Proof. Suppose that R is 2-prime and xRy ¼ f0g for some x; y∈R. So xRyR ¼ f0g andhence xR ¼ f0g or yR ¼ f0g. So AR ¼ f0g or BR ¼ f0g, where A ¼ fnxjn∈ℤg andB ¼ fnyjn∈ℤg. So A and B are right R-subgroups of R and hence A ¼ f0g or B ¼ f0g.Therefore, x ¼ 0 or y ¼ 0 and R is 3-prime. -

3. K-prime near-ringsIn this section, we will introduce a new kind of primeness of near-rings called K-primeness.Firstly, we will begin with the following result.

Theorem 3.1. Let R be a ring. Then the following statements are equivalent:

(i) R is prime.

(ii) for any 0≠ a; x; y∈R, xsa ¼ yra for all s; r∈R− f0g implies x ¼ y.

Proof. A ring R is prime if and only if it is equiprime, so we will use the definition ofequiprimeness, i.e. for any 0≠ a; x; y∈R, xca ¼ yca for all c∈R implies x ¼ y.

(i) implies (ii) is clear.(ii) implies (i). Suppose (ii) holds. If for all c∈R, xca ¼ yca for 0≠ a; x; y∈R, then

ðx− yÞca ¼ 0 ¼ 0ra for all c; r∈R. So x ¼ y by (ii). -Part (ii) enhances the following definition for near-rings.

Definition 3.1. Let R be a near-ring. We say that R is K-prime if, for any 0≠ a; x; y∈R,xsa ¼ yra for all s; r∈R− f0g implies x ¼ y.

As we mentioned before for rings, a ring is prime if and only if it is equiprime. So we havethe following result.

Corollary 3.2. A ring R is prime if and only if it is K-prime.The following result shows that every K-prime near-ring is zero-symmetric 3-prime.

Theorem 3.3. Let R be a K-prime near-ring. Then R is zero-symmetric 3-prime.

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Proof. Firstly, we will show thatR is zero-symmetric. IfR is not zero-symmetric, then it hasat least one non-zero constant element c (see [8, Theorem 1.15). For different elements x; yofR,we have that xsc ¼ yrc ¼ c for all s; r∈R− f0g, a contradiction with the hypothesis. So R iszero-symmetric. Now, suppose xRy ¼ f0g for some x; y∈R. So xcy ¼ 0 for all c∈R. If y≠ 0,then xcy ¼ 0ry for all c; r∈R. So x ¼ 0 from the hypothesis and hence R is 3-prime. -

In the case of near-rings, we have only that e-primeness implies K-primeness as shownin the proof of Theorem 3.1 (since an e-prime near-ring is zero-symmetric [10]). But theconverse is not true as we will show in the next example. We will use the near-ringmentioned in [9, Appendix, F, 7] in the next example.

Example 5. Let ðR;þÞ be the cyclic group ℤ5 and define the multiplication on R as follows:

$ 0 1 2 3 4

0 0 0 0 0 01 0 1 2 3 42 0 4 3 2 13 0 1 2 3 44 0 4 3 2 1

So R is an abelian near-ring which is not a ring (as ð1þ 1Þ2 ¼ 3≠ 4 ¼ 2þ 2 ¼ ð1Þ2þ ð1Þ2).Clearly that R is without zero divisors. Hence, R is 3-prime. R is not equiprime. Indeed,1c1 ¼ 3c1 ¼ c1 for all c∈R. But if 0≠ a; x; y∈R such that xsa ¼ yra for all s; r∈R− f0g, thenx ¼ y. Clearly that is true if x or y is equal to zero, since R is without zero divisors. That is the onlypossible case. In fact, if xsa ¼ yra for all s; r∈R− f0g and x; y; a are all non-zero, then from thetable we can choose so; ro ∈R− f0g to satisfy that xso ¼ 1 and yro ¼ 2. Hence, a ¼ 2a whichimplies that a ¼ 0 (from the table), a contradiction with 0≠ a. Therefore, K-primeness does notimply e-primeness.

Also, we can find zero-symmetric 3-prime near-rings which are not K-prime, as thefollowing example shows.

Example 6. LetRbe a trivial zero-symmetric near-ring of order greater than 2. ClearlyR is 3-prime. Taking two non-zero elements x and y such that x≠ y, we have xsx ¼ yrx ¼ x for alls; r∈R− f0g. So R is not K-prime.

Theorem 3.1, Theorem 3.3 and the examples after them show that K-primeness is a newkind of primeness.

Observe that K-primeness lies between 3-primeness and e-primeness (equiprimeness). Sowe have the following chain of primeness in the class of zero-symmetric near-rings:

The class of e-prime near-rings⊆ The class of K-prime near-rings⊆ The class of 3-prime near-rings⊆ The class of 2-prime near-rings⊆ The class of 1-prime near-rings⊆ The class of 0-prime near-rings

Remark 3.1. Observe that:(i) It is well-known thatMoðGÞ is e-prime (see [10]) and hence K-prime. Observe that it has

zero divisors.(ii) SinceMðGÞ is not zero-symmetric, so it is not K-prime (and hence not e-prime), but it

has zero divisors.


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(iii) Let N be any near-field. Then N is e-prime and hence K-prime. Indeed, for any0≠ a; x; y∈R such that xca ¼ yca for all c∈R, we have that x ¼ y by choosing c ¼ a−1.Observe that N is without zero divisors.

(iv) Example 6 shows a 3-prime near-ring without zero divisors which is not K-prime(and hence not e-prime).

From the above parts in Remark 3.1, there is no relation between e-primeness (K-primeness) and the existence of zero divisors in near-rings. So, we have another chain of theprimeness in the class of zero-symmetric near-rings:

The class of completely prime near-rings⊆ The class of 3-prime near-rings⊆ The class of 2-prime near-rings⊆ The class of 1-prime near-rings⊆ The class of 0-prime near-rings

4. On prime idealsThe next definition introduces K-prime ideals.

Definition 4.1. LetRbe a near-ring and P an ideal ofR. Then P is a K-prime ideal ofR if forevery a∈R−P and x; y∈R, xra− ysa∈P for all r; s∈R−P implies x− y∈P.

Clearly R is K-prime if and only if f0g is a K-prime ideal of R.The relationship between K-prime ideals and other kinds of prime ideals is stated in the

following result.

Theorem 4.1. Let R be a near-ring with an ideal P.

(i) If P is a K-prime ideal of R, then P is a 3-prime ideal of R.

(ii) If P is an e-prime ideal of R, then P is a K-prime ideal of R.

Proof. (i) Firstly, we will show that P contains all the constant elements of R. Let c be aconstant element in R. If c∈R−P, then

xrc� ysc ¼ c� c ¼ 0∈P

for all x; y∈R and r; s∈R−P. So x− y∈P and hence x− 0 ¼ x∈P for all x∈R. Thus,P ¼ R, a contradiction with c∉P. So c∈P.

Now, suppose aRb⊆P for some a; b∈R and b∉P. From above, any element s∈R−P is azero-symmetric element. So 0sb ¼ 0∈P for all s∈R−P. So arb− 0sb∈P for all r; s∈R−P.Thus, a∈P by the hypothesis and P is 3-prime.

(ii) Firstly, observe that if r∈P and s∈R is a zero-symmetric element, then

rs ¼ ðr þ 0Þs� 0s∈P:

Suppose xra− ysa∈P for all r; s∈R−P, where a∈R−P and x; y∈R. So xca− yca∈P forall c∈R−P. Now, suppose c∈P. As a∉P, we have that a is a zero-symmetric element (see [10]).So ca∈P and hence xca− yca∈P. ButP is e-prime. So x− y∈P andP is a K-prime ideal ofR.-

The next result generalizes Theorem 2.1 for 3-prime ideals.

Theorem 4.2. Let R be a near-ring and P an ideal of R. Then the following statements areequivalent:

(i) P is a 3-prime ideal of R.(ii) BU ⊆P implies B⊆P or U ⊆P, where B is a non-empty subset of R and U is a

semigroup left ideal of R.(iii) UV ⊆P implies U ⊆P or V ⊆P, where U and V are semigroup left ideals of R.

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Proof. (i) implies (ii). Suppose (i) holds. LetU be a semigroup left ideal of R and B be a non-empty subset of R such that BU ⊆P. If B?P, then there exists b∈B−P such that bRu⊆Pfor all u∈U. Thus, U ⊆P by (i).

(ii) implies (iii) is clear.(iii) implies (i). To prove it, we will use the contradiction. Suppose that (iii) holds and P is

not a 3-prime ideal. So there exist x; y R−P such that xRy⊆P. Thus, RxRy⊆P. So Rx⊆P orRy⊆P by (iii). Hence, RðP ∪ fxgÞ⊆P or RðP ∪ fygÞ⊆P and then P ∪ fxg or P ∪ fyg is asemigroup left ideal of R. But R itself is also a semigroup left ideal of R. Thus, P ∪ fxg⊆P,P ∪ fyg⊆P or R⊆P by (iii), a contradiction with that x; y∈R−P. So P is 3-prime and (iii)implies (i). -

Remark4.1. FromTheorem 4.2, a new characterization of 3-prime ideals can be written asfollows:

(*) P is a 3-prime ideal of R if for every two semigroup left ideals A and B of R, AB⊆Pimplies A⊆P or B⊆P.

Using Theorem 4.2 and its proof, we can prove the following result which generalizesTheorem 2.2 for 3-prime ideals.

Theorem 4.3. Let R be a zero-symmetric near-ring and P an ideal of R. Then the followingstatements are equivalent:

(i) P is a 3-prime ideal of R.

(ii) UB⊆P implies U ⊆P or B⊆P, where U is a semigroup right ideal of R and B is anon-empty subset of R.

(iii) UV ⊆P implies U ⊆P or V ⊆P, where U and V are semigroup right ideals of R.

We cannot eliminate the condition “zero-symmetric” in Theorem 4.3 as the followingexample shows:

Example 7. Observe that f0g is not a 3-prime ideal in Example 1 although it satisfies thecondition “If UV ⊆ f0g, then U ⊆ f0g or V ⊆ f0g, where U and V are semigroup right idealsof R”. This shows that “zero-symmetric” in Theorem 4.3 is not redundant.

Now, we would like to generalize Definition 2.1.

Definition 4.2. Let R be a near-ring with an ideal I.

(i) Let A be a non-empty subset of R. We say that A is a left I set-attractor (a right Iset-attractor) ofR if there exists a non-empty subsetB ofR andB? I such thatAB⊆ I (BA⊆ I).We say that A is an I set-attractor of R if A is a left or a right I set-attractor of R.

(ii) LetAbe an ideal of R. We say thatA is a left I ideal-attractor (a right I ideal-attractor)of R if there exists an ideal B of R and B? I such that AB⊆ I (BA⊆ I). We say that A is an Iideal-attractor of R if A is a left or a right I ideal-attractor of R.

We can do the same definitions if A is a left (right) ideal of R, a left (right, two-sided)R-subgroup of R, a semigroup ideal of R or a semigroup left (right) ideal of R.

Example 8. Let R be a near-ring with an ideal I ≠R. Any non-empty subset of I is a right Iset-attractor of R and hence an I set-attractor of R. In particular, I is an I set-attractor of R.Also, if there exist an ideal (a left (right) ideal, a left R-subgroup, a semigroup left ideal) B of Rsuch that B? I, then I is an I ideal-attractor (I left (right) ideal-attractor, I left R-subgroup-attractor, I semigroup left ideal-attractor) of R.

Definition 4.3. Let R be a near-ring with an ideal P. If A is a P set-attractor (P ideal-attractor, etc.) ofR, then we say thatA is an internal P set-attractor (P ideal-attractor, etc.) ofR


241

ifA⊆P. IfA?P, then we say thatA is an external P set-attractor (P ideal-attractor, etc.) ofR.If R does not have any external P set-attractors (P ideal-attractors, etc.), then we say that R iswithout external P set-attractors (P ideal-attractors, etc.), i.e. for a P set-attractor (P ideal-attractor, etc.) A of R, we have that A⊆P

Example 9. (i) Any near-ring R is without external (or internal) R-set attractors.(ii) Any near-ring without zero divisors is without external f0g-set attractors.(iii) Let R be the ring ℤ4. Take P to be the ideal f0; 2g. Then R is without external P

set-attractors.(iv) Let R be the ring ℤ6. Take P to be the ideal f0g. Then f2g, f3g and f4g are external P

set-attractors and f0g is an internal P set-attractor.

Theorem 4.4. Let R be a near-ring with an ideal P. Then the following statements areequivalent:

(i) R is without external P set-attractors.

(ii) P is a completely prime ideal of R.

Proof. (i) implies (ii), Suppose (i) holds and ab∈P for some a; b∈R. So fagfbg⊆P. If a∉P,then b∈P by (i) and P is completely prime.

(ii) implies (i). Suppose (ii) holds andA is a P set-attractor of R. So there exists a non-emptysubset B of R and B?P such that AB⊆P or BA⊆P. Suppose the case is AB⊆P. Takey∈B−P. So xy∈P for all x∈A and then A⊆P by (ii). By the same way we can do for theother case. So R is without external P set-attractors. -

Remark 4.2. (i) If I ¼ f0g in Definition 4.2, then we have Definition 2.1.(ii) From the above two definitions, Theorem 4.2 and 4.4, we can rewrite the statements of

different kinds of prime ideals as follows:Let R be a near-ring with an ideal P. Then

(1) P is completely prime if and only ifR is without external P set-attractors if and only iffor every two non-empty subsets A and B of R, AB⊆P implies A⊆P or B⊆P.

(2) P is 0-prime if and only if R is without external P ideal-attractors.

(3) R is 1-prime if and only if R is without external P right ideal-attractors.

(4) R is 2-prime if and only if R is without external P right R-subgroup-attractors.

(5) R is 3-prime if and only if R is without external P semigroup left ideal-attractors.

References

[1] G.L. Booth, N.J. Groenewald, Different Prime Ideals Innear-Rings. II. Rings and Radicals(Shijiazhuang, 1994), 131–140. in: Pitman Res. Notes Math. Ser., vol. 346, Longman, Harlow, 1996.

[2] G.L. Booth, N.J. Groenewald, S. Veldsman, A Kurosh-Amitsurprime radical for near-rings, Comm.Algebra 18 (9) (1990) 3111–3122.

[3] G. Ferrero, C. Cotti Ferrero, Nearrings, in: Some Developments linked to Semigroups and Groups,Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.

[4] N.J. Groenewald, Note on the completely prime radical innear-rings, in: Near-Rings and Near-Fields (T€ubingen, 1985), 97–100, in: North-Holland Math. Stud., vol. 137, North-Holland,Amsterdam, 1987.

[5] N.J. Groenewald, Different prime ideals in near-rings, Comm. Algebra 19 (10) (1991) 2667–2675.

[6] W.L.M. Holcombe, Primitive Near-Rings (Doctoral dissertation), Uneversity of Leeds, 1970.

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[7] M. Holcombe, A hereditary radical for near-rings, Studia Sci. Math. Hungar. 17 (1–4) (1982)453–456.

[8] J.D.P. Meldrum, Near-rings and their linkswith groups, in: Research Notes in Mathematics, vol.134, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[9] G. Pilz, Near-Rings. The Theory and its applications, second ed., in: North-Holland MathematicsStudies, vol. 23, North-Holland Publishing Co, Amsterdam, 1983.

[10] S. Veldsman, On equiprime near-rings, Comm. Algebra 20 (9) (1992) 2569–2587.

Corresponding authorKhalid H. Al-Shaalan can be contacted at: [email protected]



243


Multivariate Hardy and Littlewoodinequalities on time scales

Ammara Nosheen and Aneela NawazDepartment of Mathematics, University of Lahore (Sargodha Campus),

Sargodha, Pakistan, and

Khuram Ali Khan and Khalid Mahmood AwanDepartment of Mathematics, University of Sargodha, Sargodha, Pakistan

AbstractIn the paper we extend some Hardy and Littlewood type inequalities on time scales for the function of nvariables. Special cases of obtained results include generalized Wirtinger, Hardy and Littlewood typeinequalities.

Keywords Hardy and Littlewood inequalities, Wirtinger type inequality, Time scales calculus


1. IntroductionThe discrete Hardy inequality [8] was proved and published byHardy himself. It states that ifðcnÞ is a sequence of non-negative real numbers which are not identically zero, then for everyreal number p > 1, one has that

X∞k¼1

�c1 þ c2 þ c3 þ � � � þ ck

k

�p<

�p

p� 1

�pX∞k¼1

c pk :

The classical Hardy inequality [9] states that if f ≥ 0 and integrable over any finite intervalð0; rÞ and f d is integrable and convergent over ð0;∞Þ then for d > 1,Z ∞

0

�1

r

Z r

0

f ðτÞdτ�d

dr ≤

�d

d � 1

�d Z ∞

0

f dðrÞdr; (1)

Inequalities ofmultivariateHardy andLittlewood

245

JEL Classification — primary 26D15; secondary 39A13; 34N05© Ammara Nosheen, Aneela Nawaz, Khuram Ali Khan and Khalid Mahmood Awan. Published in

the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article ispublished under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce,distribute, translate and create derivative works of this article (for both commercial and non-commercialpurposes), subject to full attribution to the original publication and authors. The full terms of this licensemay be seen at http://creativecommons.org/licences/by/4.0/legalcode


Funding: This research did not receive any specific grant from funding agencies in the public,commercial, or not-for-profit sectors.

The publisher wishes to inform readers that the article “Multivariate Hardy and Littlewoodinequalities on time scales” was originally published by the previous publisher of the Arab Journal ofMathematical Sciences and the pagination of this article has been subsequently changed. There has beenno change to the content of the article. This change was necessary for the journal to transition from theprevious publisher to the new one. The publisher sincerely apologises for any inconvenience caused. Toaccess and cite this article, please use Nosheen, A., Nawaz, A., Khan, K. A., Awan, K. M. (2019),“Multivariate Hardy and Littlewood inequalities on time scales”,Arab Journal ofMathematical Sciences,Vol. 26 No. 1/2, pp. 245-263. The original publication date for this paper was 27/12/2019.



Received 19 September 2019Revised 9 December 2019



Vol. 26 No. 1/2, 2020pp. 245-263


DOI 10.1016/j.ajmsc.2019.12.003

equality holds if and only if f ðrÞ ¼ 0 almost everywhere. Hardy inequality (1) has beengeneralized by Hardy himself in [11], where he exposed that, for any integrable functionf ðyÞ > 0 on ð0;∞Þ and d > 1, the following holdZ ∞

0

1

yn

�Z ∞

y

f ðhÞdh�d

dy ≤

�d

1� n

�d Z ∞

0

1

yn−df dðyÞdy; n < 1; (2)

Z ∞

0

1

yn

�Z y

0

f ðhÞdh�d

dy ≤

�d

n� 1

�d Z ∞

0

1

yn−df dðyÞdy; n > 1: (3)

Hardy and Littlewood [10] demonstrate the discrete versions of (2) and (3). In particular theyproved that if d > 1 and ðpmÞ is a sequence of non-negative terms then

X∞m¼1

1

mj

X∞i¼m

pi

!d

≤ NX∞m¼1

1

mj−dpdm; j < 1;

X∞m¼1

1

mj

Xmi¼1

pi

!d

≤ NX∞m¼1

1

mj−dpdm; j > 1;

where N is a non-negative constant. Time scales calculus [12] was introduced in 1988 by theGerman mathematician Stefan Hilger, which unifies sums and integrals. Some extension ofHardy type inequalities on time scales can be found in [2–4].

S. H. Saker et al. [13] proved some Hardy and Littlewood type inequalities on time scales inthe following form:

Theorem 1.1. Let T be a time scale with a∈ ð0;∞ÞT and p; q > 0 such that p=q≥ 2 and

γ > 1. Furthermore assume that g is a nonnegative and the delta integralR∞a

tpq−γgp=qðtÞΔt

exists. Let

ΛðtÞ ¼Z t

a

gðsÞΔs; for any t ∈ ½a;∞�T: (4)

Then one gets

Z ∞

a

1

tγðΛσðtÞÞp=qΔt ≤ 2

pq−2pkγ

qðγ � 1Þ�Z ∞

a

1

tγ�pq

gp=qðtÞΔt�p

q

3

�Z ∞

a

ΛσðtÞΛp=q

tγΔt

�p−qp

þ 2pq−2pkγ

qðγ � 1ÞZ ∞

a

μpq � 1

tγ−1gp=qðtÞΔt:

Theorem 1.2. Let T be a time scale with a∈ ð0;∞ÞT and p; q > 0 such that p=q≥ 2and γ > 1. Furthermore assume that g is a nonnegative function and the delta integralR∞a

tpq−γgp=qðtÞΔt exist. Let ΛðtÞ be as defined in (4). Then

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246

Z ∞

a

1

tγðΛσðtÞÞp=qΔt ≤

�2pq�1pkγ

qðγ � 1Þ�p=q Z ∞

a

1

tγ−pq

gp=qðtÞΔt:

Theorem 1.3. Let T be a time scale with a∈ ð0;∞ÞT and p; q > 0 such that p=q > 1 andγ > 1. Furthermore assume that g is a nonnegative function and the delta integralR∞a

tpq−γgp=qðtÞΔt exists. Let ΛðtÞ be as defined in (4). ThenZ ∞

a

1

tγðΛσðtÞÞp=qΔt ≤

�pkγ

qðγ � 1Þ�p=q Z ∞

a

1

tγ−pq

gp=qðtÞΔt:

Theorem 1.4. Let T be a time scale with a∈ ð0;∞ÞT and p; q > 0 such that p=q > 1 and

γ < 1. Furthermore assume that g is a nonnegative and delta integralR∞a

ðσðtÞÞpq−γ gp=qðtÞΔtexists. Let

ΩðtÞ ¼Z ∞

t

gðsÞΔs; for any t ∈ ½a;∞�T:

Then one gets Z ∞

a

ðΩðtÞÞp=qσγðtÞ ≤

�p

qð1� γÞ�p=q Z ∞

a

gp=qðtÞðσðtÞÞγ−p

q

Δt:

In this paper we extend results of Theorem 1.1 to Theorem 1.4 for the function of nvariables.

2. PreliminariesIn this section, we recall the following concepts from theory of time scales [5,7]. A time scale isan arbitrary, non empty closed subset of real numbers. Set of integers and Cantor set areexamples of time scales, while rational numbers, complex numbers and open intervalbetween 0 and 1 not time scales. LetT be a time scale, for t ∈T, forward and backward jumpoperators are defined by

σðtÞ :¼ inffa∈T; a > tg; ρðtÞ :¼ supfa∈T; a < tg;respectively. The conventions for these operators are inf f ¼ supT and supf ¼ infT.If σðtÞ > t, then t is right-scattered and if ρðtÞ < t, then t is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated points.

If σðtÞ ¼ t, then t is right-dense and if ρðtÞ ¼ t, then t is left-dense. Points that are right-dense and left-dense at the same time are called dense points. The functionsμ : T→ℝ; ν : T→ℝ defined by μðtÞ ¼ σðtÞ− t and νðtÞ ¼ t − ρðtÞ are called forward andbackward graininess functions, respectively.

A function g : T→ℝ is said to be right-dense continuous (rd-continuous) provided g iscontinuous at right-dense points and at left-dense points in T, left-hand limits exist and arefinite. The set of all such rd-continuous functions is denoted by CrdðTÞ. For any functiong : T→ℝ, the notation gσðtÞ denotes gðσðtÞÞ. The delta derivative (also Hilger derivative)gΔðtÞ exists if and only if for every e > 0 there exists a neighborhood U of t such that

j gðσðtÞÞ � gðsÞ � gΔðtÞðσðtÞ � sÞ j ≤ j σðtÞ � s j; for all s; t in U:


247

Assume that h : T→ℝ, if HΔðtÞ ¼ hðtÞ, then the Cauchy (delta) integral of h. defined byZ t

a

hðsÞΔs :¼ HðtÞ � HðaÞ:

Integration by parts formula [7, Theorem1.77]:If a; b∈T and u; v∈CrdðTÞ, thenZ b

a

uðtÞvΔðtÞΔt ¼ ½uðtÞvðtÞ�ba �Z b

a

uΔðtÞvσðtÞΔt: (5)

Chain rule 1 [7, Theorem 1.90]:Assume that f : ℝ→ℝ is continuously differentiable and suppose g : T→ℝ is delta

differentiable. Then f og : T→ℝ is delta differentiable and

ðf ogÞΔðtÞ ¼�Z 1

0

f0 ðgðtÞ þ hμðtÞgΔðtÞÞdh

�gΔðtÞ (6)

holds.Chain rule 2 [7, Theorem 1.87]:If f and g satisfy the conditions of Chain rule 1, Then f og : T→ℝ is delta differentiable

and there exists c in the real interval ½t; σðtÞ� such that

ð f ogÞΔðtÞ ¼ f0 ðgðcÞÞgΔðtÞ: (7)

H€older’s inequality [7, Theorem 6.13]:For continuous real-valued functions g : T→ℝ, h : T→ℝ, let a; b∈T, p > 1 and

1pþ 1

q¼ 1, then Z b

a

gðtÞhðtÞdt ¼�Z b

a

gpðtÞdt�1=p�Z b

a

hqðtÞdt�1=q

: (8)

Fubini’s Theorem on time scales [6]:Let ðψ ; M ; μΔÞ and ðΓ; N ; λΔÞ be two finite dimensional time scales measure spaces. If

Λ : ψ 3Γ→ℝ is a μΔ 3 λΔ-integrable function. The function ςðt2Þ ¼RψΛðt1; t2ÞΔt1 exists for

any t1 ∈Γ and ξðt1Þ ¼RΓ Λðt1; t2ÞΔt2 exists for t2 ∈ψ, thenZψΔt1

ZΓ

Λðt1; t2ÞΔt2 ¼ZΓ

Δt2

ZψΛðt1; t2ÞΔt1: (9)

We assume throughout that all the functions are non-negative and the integralsconsidered exist.

In this paper, we use the following notations. We assume that there exists constant ki > 0with

si

σiðsiÞ ≥1

kifor si ≥ ai; i∈ f1; . . . ; ng: (10)

Λσ1��σjk ðt1; . . . ; tnÞ¼: Λσ1 ��σj

k ¼: Λkðσ1ðt1Þ; . . . ; σjðtjÞ; tjþ1; . . . ; tnÞ; k; j∈ f1; . . . ; ngZ ∞

a1

. . .

Z ∞

an

f ðt1; . . . ; tnÞΔt1; . . . ;Δtn ¼:Z ∞Yn

i¼1ai

f ðt1; . . . ; tnÞYni¼1

Δti:

AJMS26,1/2

248

3. Hardy and Littlewood-type inequalities for p/q ≥ 2 and γ > 1The following inequalities are used to prove next results.

aλ þ bλ ≤ ðaþ bÞλ ≤ 2λ−1ðaλ þ bλÞ for a; b ≥ 0; λ ≥ 1: (11)

2λ−1ðaλ þ bλÞ ≤ ðaþ bÞλ ≤ aλ þ bλ for a; b ≥ 0; 0 ≤ λ ≤ 1: (12)

Theorem3.1. Assume i∈ f1; . . . ; ng,Ti is a time scale with ai ∈ ð0;∞ÞTiand γi > 1, further

assume g : ½a1;∞ÞT13 � � �3 ½an;∞ÞTn

→ℝþ is such that the delta integralsR∞Qn

i¼1ai

Qni¼1ðtiÞ

pq−γi gp=qðt1; . . . ; tnÞΔti for any ðt1; . . . ; tnÞ∈ ½a1;∞ÞT1

3 � � �3 ½an;∞ÞTnexist,

define

Λkðt1; . . . ; tnÞ ¼Z ∞Yk

j¼1aj

gðs1; . . . ; snÞYkj¼1

Δsj; k∈ f1; . . . ; ng; (13)

then for p; q > 0 and p=q≥ 2Z ∞Yn

i¼1ai

ðΛσ1 ...σnn Þp=qYn

i¼1tγii

Yni¼1

Δti

≤Xnr¼1

Ynj¼rþ1

cj~cr

Z ∞Yn

j¼rþ1aj

Ynj¼rþ1

ðμjðtjÞÞðp=q−1Þ

tγj−1

j

Z ∞Yr�1

i¼1ai

Yr−1i¼1

1

tγii

3

(Z ∞

ar

ðΛσ1 ��σr−1r−1 Þp=qtγr− p=qr

Δtr

)q=p�Λσ1...σrr ÞðΛσ1 ...σr−1

r Þp=qΔtrp−q

pYr−1i¼1

ΔtiYnj¼rþ1

Δtj

þYni¼1

~ci

Z ∞Yni¼1

ai

Yni¼1

ðμiðtiÞÞðp=q−1Þtγi−1i

gp=qðt1; . . . ; tnÞYni¼1

Δti

(14)

holds, where ~cr ¼ cr p=q; cr ¼ 2p=q−2kγrr

γr−1:

Proof. To prove the result, we use the principle of mathematical induction. For n ¼ 1 thestatement is true by Theorem 1.1. Let the statement be true for 1≤ n≤ k.

To prove the result for n ¼ kþ 1. The left-hand side of (14) can be written as,Z ∞Ykþ1

i¼1ai

1Ykþ1

i¼1tγii

ðΛσ1 ��σkþ1

kþ1 Þp=qYkþ1

i¼1

Δti: (15)

DenoteR∞

akþ1

ðΛσ1 ��σkþ1kþ1

Þp=q

tγkþ1kþ1

Δtkþ1 ¼ Ikþ1. Apply (5) with vΔtkþ1

uðtkþ1Þ ¼ 1

tγkþ1kþ1

and

vσkþ1ðtkþ1Þ ¼ ðΛσ1��σkþ1

kþ1 Þp=q by keeping fix ðt1; . . . ; tkÞ∈ ½a1;∞ÞT13 � � �3 ½ak;∞ÞTk

.

Ikþ1 ¼�uðtkþ1ÞððΛσ1 ��σk

kþ1 Þp=qÞ� ∞akþ1

Z ∞

akþ1

−uðtkþ1Þ v

Δtkþ1

ðΛσ1��σkkþ1 Þp=qΔtkþ1; (16)


249

where,uðtkþ1Þ ¼

Z ∞

tkþ1

−1

sγkþ1

kþ1

Δskþ1: (17)

Use chain rule (6) and the fact that σkþ1ðskþ1Þ ≥ skþ1 to get

v

Δskþ1

� 1

sγkþ1−1

kþ1

!¼ ðγkþ1 � 1Þ

Z 1

0

½hkþ1σkþ1ðskþ1Þ þ ð1� hkþ1Þskþ1�−γkþ1dhkþ1

≥ðγkþ1 � 1Þσγkþ1

kþ1 ðskþ1Þ:(18)

(10) together with (18) gives

v

Δskþ1

−

1

sγkþ1−1

kþ1

!≥ðγkþ1 � 1Þkγkþ1

kþ1 sγkþ1

kþ1

:

Therefore Z ∞

tkþ1

� 1

sγkþ1

kþ1

Δskþ1

≥

Z ∞

tkþ1

� kγkþ1

kþ1

γkþ1 � 1

v

Δskþ1

� 1

sγkþ1−1

kþ1

!Δskþ1 ¼ −

kγkþ1

kþ1

γkþ1 � 1

1

tγkþ1−1

kþ1

!:

(19)

(17) together with (19) gives

−uðtkþ1Þ ¼ −

Z ∞

tkþ1

−1

sγkþ1

kþ1

Δskþ1 ≤kγkþ1

kþ1

γkþ1 � 1

1

tγkþ1−1

kþ1

!: (20)

From (13), (16), (17), (20), we have (note that ukþ1ð∞Þ ¼ 0 and Λkþ1ðt1; . . . ; tk; akþ1Þ ¼ 0)

Ikþ1 ¼kγkþ1

kþ1

γkþ1 � 1

Z ∞

akþ1

1

tγkþ1−1

kþ1

v

Δtkþ1

ðΛσ1 ��σkkþ1 Þp=qΔtkþ1: (21)

Apply chain rule 1 (6) on the right-hand side of (21)

v

Δtkþ1

ðΛσ1��σkkþ1 Þp=q

¼ p

q

v

Δtkþ1

Λσ1 ��σkkþ1

Z 1

0

�Λkþ1 þ hkþ1μkþ1ðtkþ1Þ v

Δtkþ1

Λσ1 ��σkkþ1

�pq−1

dhkþ1:

(22)

Use right part of (11) on the right-hand side of (22),

v

Δtkþ1

ðΛσ1 ��σkkþ1 Þp=q

≤p

q2p=q−2ðΛσ1 ��σk

kþ1 Þp=q−1 v

Δtkþ1

ðΛσ1 ��σkkþ1 Þ

þ p

q2p=q−2ðμkþ1ðtkþ1ÞÞp=q−1ð v

Δtkþ1

Λσ1 ��σkkþ1 Þ

p=q

:

(23)

AJMS26,1/2

250

Substitute (23) into (21)

Ikþ1 ≤p2p=q−2k

γkþ1

kþ1

qðγkþ1 � 1ÞZ ∞

akþ1

1

tγkþ1−1

kþ1

ðΛσ1 ��σkkþ1 Þp=q−1 v

Δtkþ1

Λσ1 ��σkkþ1 Δtkþ1

þ p2p=q−2kγkþ1

kþ1


akþ1

1

tγkþ1−1

kþ1

ðμkþ1ðtkþ1ÞÞp=q−1ð v

Δtkþ1

Λσ1 ��σkkþ1 Þ

p=q

Δtkþ1:

(24)

Since

v

Δtkþ1

Λσ1��σkkþ1 ¼ Λσ1 ��σk

k ≥ 0: (25)

Use (25) in (24)

Ikþ1 ≤p2p=q−2k

γkþ1

kþ1


akþ1

1

tγkþ1−1

kþ1

ðΛσ1 ��σkkþ1 Þp=q−1Λσ1 ��σk

k Δtkþ1

þ p2p=q−2kγkþ1

kþ1


akþ1

1

tγkþ1−1

kþ1

ðμkþ1ðtkþ1ÞÞp=q−1ðΛσ1 ��σkk Þp=qΔtkþ1:

(26)

Substitute (26) in (15)Z ∞Ykþ1

i¼1ai

1Ykþ1

i¼1tγii


kþ1 Þp=qYkþ1

i¼1

Δti

≤

Z ∞Yk

i¼1ai

1Yk

i¼1tγii

p2p=q−2kγkþ1

kþ1


akþ1

1

tγkþ1−1

kþ1

ðΛσ1 ��σkkþ1 Þp=q−1Λσ1��σk

k

Yki¼1

Δtkþ1Δti

þZ ∞Yk

i¼1ai

1Yk

i¼1tγii

p2p=q−2kγkþ1

kþ1


akþ1

ðμkþ1ðtkþ1ÞÞp=q−1tγkþ1−1

kþ1

ðΛσ1 ��σkk Þp=q

Yki¼1

Δtkþ1Δti:

(27)

Exchange integrals on right-hand side of (27) k -times by using (9)

¼ p2p=q−2kγkþ1

kþ1


akþ1

1

tγkþ1−1

kþ1

Z ∞Yk

i¼1ai

1Yk

i¼1tγii


k

Yki¼1

ΔtiΔtkþ1

þ p2p=q−2kγkþ1

kþ1

qðγkþ1 � 1Þðμkþ1ðtkþ1ÞÞp=q−1Z ∞

akþ1

1

tγkþ1−1

kþ1

3

Z ∞Yk

i¼1ai

1Yk

i¼1tγii


Yki¼1

ΔtiΔtkþ1:

(28)

Use the induction hypothesis with Λσ1��σkk in (28) for fixed tkþ1 ∈Tkþ1 and again apply (9)

k-times to get


251

¼ p2p=q−2kγkþ1

kþ1


akþ1

1

tγkþ1−1

kþ1

Z ∞Yk

i¼1ai

1Yk

i¼1tγii


k

Yki¼1

ΔtiΔtkþ1

þ p2p=q−2kγkþ1

kþ1

qðγkþ1 � 1Þðμkþ1ðtkþ1ÞÞp=q−1Z ∞

akþ1

1

tγkþ1−1

kþ1

3Xkr¼1

Ykj¼rþ1

cj~cr

Z ∞Yk

j¼rþ1aj

Ykj¼rþ1


tγj−1

j

3

Z ∞Yr�1

i¼1aj

Yr−1i¼1

1

tγii

(Z ∞

ar

ðΛσ1 ��σr−1r−1 Þp=qtγr−p=qr

Δtr

)q=p

3�

Λσ1 ��σrr ÞðΛσ1��σr−1

r Þp=qΔtrp−q

pYr−1i¼1

ΔtiYkj¼rþ1

Δtj

þYki¼1

~ci

Z ∞Yk

i¼1aj

Yki¼1


gp=qðt1; . . . ; tkÞYki¼1

Δti:

Hence Z ∞Ykþ1

i¼1ai

ðΛσ1 ...σnn Þp=qYkþ1

i¼1tγii

Ykþ1

i¼1

Δti

≤Xkþ1

r¼1

Ykþ1

j¼rþ1

cj~cr

Z ∞Ykþ1

j¼rþ1aj

Ykþ1

j¼rþ1


tγj−1

j

Z ∞Yr�1

i¼1ai

Yr−1i¼1

1

tγii

3

(Z ∞

ar

ðΛσ1 ��σr−1r−1 Þp=qtγr−p=qr

Δtr

)q=p�Λσ1 ...σrr ðΛσ1 ...σr−1

r Þp=qΔtrp−q

pYr−1i¼1

ΔtiYkþ1

j¼rþ1

Δtj

þYkþ1

i¼1

~ci

Z ∞Ykþ1

i¼1ai

Ykþ1

i¼1


gp=qðt1; . . . ; tkþ1ÞYkþ1

i¼1

Δti:

Hence by induction principle, the statement is true ∀ n∈ℕ. ,

Theorem 3.2. Assume i∈ f1; . . . ; ng, Ti is a time scale with ai ∈ ð0;∞ÞTiand γi > 1,

further assume g : ½a1;∞ÞT13 � � �3 ½an;∞ÞTn


i¼1aiQn

i¼1ðtiÞpq−γi gp=qðt1; . . . ; tnÞ

Qni¼1 Δti exist. Let Λkðt1; . . . ; tnÞ be defined in (13), then for p; q > 0

and p=q≥ 2Z ∞Yn

i¼1ai

1Yn

i¼1tγii

ðΛσ1 ��σnn Þp=q

Yni¼1

Δti

≤

�p

q

�npq Yn

i¼1

�2pq�1k

γii

ðγi � 1Þ�p=q Z ∞Yn

i¼1ai

Yni¼1

1

tγi−p=qi


Δti;

(29)

holds.

AJMS26,1/2

252


To prove the result for n ¼ kþ 1. Proceed it as in the proof of Theorem 3.1 up to (21). Applychain rule 1 (6) on the right-hand side of (21) yields

v

Δtkþ1

ðΛσ1 ��σkkþ1 Þp=q

¼�p

q

�v

Δtkþ1

Λσ1 ��σkkþ1

Z 1

0

�hkþ1Λ

σ1 ��σkþ1

kþ1 þ ð1� hkþ1ÞΛσ1 ��σkkþ1

�pq−1dhkþ1:

(30)

Use (11) on the right-hand side of (30),

≤

�p

q

�2pq−2ðΛσ1 ��σkþ1

kþ1 Þpq−1

v

Δtkþ1

Λσ1 ��σkkþ1 þ

�p

q

�2pq−2ðΛσ1 ��σk

kþ1 Þpq−1 v

Δtkþ1

Λσ1 ��σkkþ1 ;

use the fact σkþ1ðtkþ1Þ≥ tkþ1

¼�p

q


kþ1 Þpq−1

v

Δtkþ1

Λσ1 ��σkkþ1 þ

�p

q


kþ1 Þpq−1

v

Δtkþ1

Λσ1 ��σkkþ1

¼�p

q


kþ1 Þpq−1

v

Δtkþ1

Λσ1 ��σkkþ1 :

(31)

Since

v

Δtkþ1

Λσ1 ��σkkþ1 ¼ Λσ1��σk

k ≥ 0: (32)

Use (32) in (31) and substitute in (21) to get

Ikþ1 ≤p2

pq−1k

γkþ1

kþ1


akþ1

1

tγkþ1−1

kþ1


kþ1 Þpq−1Λσ1 ��σk

k Δtkþ1: (33)

Apply H€older’s inequality on the right-hand side of (33) with indices p=q and p=ðp− qÞ

Ikþ1 ≤p2

pq−1k

γkþ1

kþ1

qðγkþ1 � 1Þ

8>>>><>>>>:

Z ∞

akþ1

8<:tγkþ1ðp−qq Þkþ1

tγkþ1−1

kþ1

Λσ1 ��σkk

9=;

p=q

Δtkþ1

9>>>>=>>>>;

q=p

3 fIkþ1gp−qp :

After simplification, we get

Ikþ1 ≤

p2

pq�1k

γkþ1

kþ1

qðγkþ1 � 1Þ

!p=q Z ∞

akþ1


t−pqþγkþ1

kþ1

Δtkþ1: (34)


253

Substitute (34) into (15)Z ∞Ykþ1

i¼1ai

1Ykþ1

i¼1tγii


kþ1 Þp=qYkþ1

i¼1

Δti

≤

Z ∞Yk

i¼1ai

1Yk

i¼1tγii

p2

pq�1k

γkþ1

kþ1

qðγkþ1 � 1Þ

!p=q Z ∞

akþ1


t−pqþγkþ1

kþ1

Δtkþ1:

(35)


p2

pq�1k

γkþ1

kþ1

qðγkþ1 � 1Þ

!p=q Z ∞

akþ1

1

t−pqþγkþ1

kþ1

(Z ∞Yk

i¼1ai

1Yk

i¼1tγii


Yki¼1

Δti

)Δtkþ1: (36)

Use the induction hypothesis forΛσ1��σkk in (36) for fixed tkþ1 ∈Tkþ1 and again apply (9) k times

to get Z ∞Ykþ1

i¼1ai

1Ykþ1

i¼1tγii


kþ1 Þp=qYkþ1

i¼1

Δti

≤

�p

q

�ðkþ1Þpq Ykþ1

i¼1

2pq�1k

γii

γi � 1

!p=q Z ∞Ykþ1

i¼1ai

Ykþ1

i¼1

1

tγi−p=qi

gp=qðt1; . . . ; tkþ1ÞYki¼1

Δti:


Corollary 3.3. As a special case of Theorem 3.2, when T1 ¼ � � � ¼ Tn ¼ ℝ, p=q ¼ λ > 1and γi < 1, (29) becomes the following Wirtinger type inequalityZ ∞Yn

i¼1ai

1Yn

i¼1tγii

ðGðt1; . . . ; tnÞÞλYni¼1

dti

≤Yni¼1

�λ2λ�1

1� γi

�λ Z ∞Yn

i¼1ai

1Yn

i¼1tγi−λ1

�vn

vt1 � � � vtnGλðt1; . . . ; tnÞ

�Yni¼1

dti;

where Gðt1; . . . ; tnÞ¼:R tiQn

i¼1aigðs1; . . . ; snÞ

Qn

i¼1dsi.

When γ1 ¼ � � � ¼ γn ¼ λ > 1, we have another Hardy type inequality for function ofn-variables Z ∞Yn

i¼1ai

1Yn

i¼1ti

Z tiYn

i¼1ai

gðs1; . . . ; snÞYni¼1

dsi

!λYni¼1

dti

≤

�λ2λ�1

λ� 1

�λ

gλðt1; . . . ; tnÞYni¼1

dti:

AJMS26,1/2

254

Remark 3.4. Assume that T1 ¼ � � � ¼ Tn ¼ ℕ in Theorem 3.2, p=q ¼ λ > 1, ai > 1,γi > 1 for i∈ f1; . . . ; ng, further assume that

P∞

m1¼1 . . .P∞

mn¼1 gλðm1; . . . ;mnÞ is convergent.

(29) becomes the following discrete Hardy and Littlewood inequality

X∞m1¼1

� � �X∞mn¼1

1

mγ11 . . .m

γnn

Xm1

k1¼1

� � �Xmn

kn¼1

gðk1; . . . ; knÞ!λ

≤Yni¼1

�2λ�1λγi � 1

�λ X∞m1¼1

� � �X∞mn¼1

1Yn

i¼1m

γi−λi

gλðm1; . . . ;mnÞ:

4. Hardy and Littlewood-type inequalities for p/q ≥ 1 and γ > 1

Theorem 4.1. Assume i∈ f1; . . . ; ng, Ti is a time scale with ai ∈ ð0;∞ÞTiand γi < 1,


→ ℝþ is such that the delta integralsR∞Qn

i¼1aiQn

i¼1 tipq−γi gp=qðt1; . . . ; tnÞ

Qni¼1 Δti exist. Let Λkðt1; . . . ; tnÞ be defined in (13), then for p; q > 0

and p=q > 1Z ∞Yn

i¼1ai

1Yn

i¼1tiγiðΛσ1 ��σn

n Þp=qYni¼1

Δti

≤

�p

q

�npq Yn

i¼1

�kγii

γi � 1

�p=q Z ∞Yn

i¼1ai

Yni¼1

1

tγi−p=qi


Δti;

(37)

holds, where n is a positive integer.

Proof. To prove the result, we use the principle of mathematical induction. For n ¼ 1the statement is true by Theorem 1.3. Let the statement be true for 1≤ n≤ k.

To prove the result for n ¼ kþ 1. Proceed it as in the proof of Theorem 3.1 up to (21).Apply the chain rule 2 (7) to get

v

Δtkþ1

ðΛσ1 ��σkkþ1 Þp=q ¼ p

qðΛσ1 ��σk

kþ1 ðt1; . . . ; tk; ckþ1ÞÞpq−1

v

Δtkþ1

Λσ1��σkkþ1 ;

where ckþ1 ∈ ½tkþ1; σkþ1ðtkþ1Þ�. Sincev

Δtkþ1

Λσ1 ��σkkþ1 ¼: Λσ1 ��σk

k ≥ 0;

and σkþ1ðtkþ1Þ≥ ckþ1, one has that

v

Δtkþ1

Λσ1 ��σkkþ1 ≤

p

q

Λσ1 ��σkþ1

kþ1

�pq−1Λσ1 ��σk

k : (38)


Ikþ1 ≤pk

γkþ1

kþ1


akþ1

Λσ1 ��σkþ1

kþ1

�pq−1

tγkþ1−1

kþ1

Λσ1 ��σkk Δtkþ1: (39)


255

Apply H€older’s inequality on the right-hand side of (39) with indices p=q and p=ðp− qÞ

Ikþ1 ≤pk

γkþ1

kþ1

qðγkþ1 � 1Þ

8><>:Z ∞

akþ1

8<:tγkþ1

p−qp

�kþ1

tγkþ1−1

kþ1

Λσ1;...;σkk

9=;

p=q

Δtkþ1

9>=>;

q=p

3 fIkþ1gp−qp :


Ikþ1 ≤

�pk

γkþ1

kþ1

qðγkþ1 � 1Þ�p=q Z ∞

akþ1

Λσ1 ;...;σkk

�p=qt−pqþγkþ1

kþ1

Δtkþ1: (40)


i¼1ai

1Ykþ1

i¼1tγii

Λσ1 ��σkþ1

kþ1

�p=qYkþ1

i¼1

Δti

≤

Z ∞Yk

i¼1ai

1Yk

i¼1tγii

�pk

γkþ1

kþ1


akþ1

Λσ1 ;...;σkk

�p=qtγkþ1−

pq

kþ1

Δtkþ1:

(41)


¼�

pkγkþ1

kþ1


akþ1

1

tγkþ1−

pq

kþ1

(Z ∞Yk

i¼1ai

1Yk

i¼1tγii


Yki¼1

Δti

)Δtkþ1: (42)

Use the induction hypothesis with ðΛσ1��σkk Þp=q in (42) for fixed tkþ1 ∈Tkþ1 and again apply (9)

k-times to getZ ∞Ykþ1

i¼1ai

1Ykþ1

i¼1tγii


kþ1 Þp=qYkþ1

i¼1

Δti

≤

�p

q

�ðkþ1Þpq Ykþ1

i¼1

�kγii

γi � 1

�p=q Z ∞Ykþ1

i¼1ai

Ykþ1

i¼1

1

tγi−p=qi

gp=qðt1; . . . ; tkþ1ÞYkþ1

i¼1

Δti:


Corollary 4.2. As a special case of Theorem 4.1, when T1 ¼ � � � ¼ Tn ¼ ℝ, p=q ¼ λ > 1and γ1; . . . ; γn < 1, (37) becomes the following Wirtinger type inequality,Z ∞Yn

i¼1ai

1Yn

i¼1tγii

Gλðt1; . . . ; tnÞYni¼1

dti

≤Yni¼1

�λ

1� γi

�λ Z ∞Yn

i¼1ai

1Yn

i¼1tγi−λ1

�vn

vt1 � � � vtn Gλðt1; . . . ; tnÞ

�Yni¼1

dti;

where Gðt1; . . . ; tnÞ¼:R tiQn

i¼1aigðs1; . . . ; snÞ

Qn

i¼1 Δsi .

AJMS26,1/2

256

When γ1 ¼ � � � ¼ γn ¼ λ > 1, we have the classical Hardy type inequality for function ofn -variables Z ∞Yn

i¼1ai

1Yn

i¼1ti

Z tiYn

i¼1ai


dsi

!λYni¼1

dti

≤

� λλ� 1

�λgλðt1; . . . ; tnÞ

Yni¼1

dti:

Corollary 4.3. Assume that T1 ¼ � � � ¼ Tn ¼ ℕ in Theorem 4.1, p=q ¼ λ > 1, ai > 1,γi > 1 for i∈ f1; . . . ; ng, further assume that

P∞

m1¼1 � � �P∞

mn¼1 gλðm1; . . . ;mnÞ is

convergent. Note that in this case mi

σiðmiÞ ¼mi

miþ1 therefore12 ≤

mi

miþ1 ≤ 1, and we get following

discrete Hardy and Littlewood inequality

X∞m1¼1

� � �X∞mn¼1

1

mγ11 . . .m

γnn

Xm1

k1¼1

� � �Xmn

kn¼1

gðk1; . . . ; knÞ!λ

≤Yni¼1

�2λλ

γi � 1

�λ X∞m1¼1

� � �X∞mn¼1

1Yn

i¼1m

γi−λi

gλðm1; . . . ;mnÞ:

Remark 4.4. Assume i∈ f1; . . . ; ng, Ti is a time scale with ai ∈ ð0;∞ÞTiand γi < 1,


→ℝþ is such that the delta integrals

R∞Qn

i¼1ai

Qni¼1 σiðtiÞ

pq−γi�σiðtiÞti

�pqðγi−1Þ

gp=qn ðt1; . . . ; tnÞ

Qni¼1 Δti exist. Let Λkðt1; . . . ; tnÞ be defined

in Theorem 3.1, then for p; q > 0 and p=q > 1Z ∞Yn

i¼1ai

ðΛσ1��σnn Þp=qYn

i¼1ðσiðtiÞÞγi

Yni¼1

Δti

≤

�p

q

�npq Yn

i¼1

�1

γi � 1

�p=q Z ∞Yn

i¼1ai

gp=qðt1; . . . ; tnÞYn

i¼1σγi−

pqðtiÞ

Yni¼1

�σiðtiÞti

�pqðγi−1ÞYn

i¼1

Δti;

holds.

Proof. Replace left-hand side of (37) in Theorem 4.1 byZ ∞Yn

i¼1ai

ðΛσ1 ��σnn Þp=qYn

i¼1ðσiðtiÞÞγi

Yni¼1

Δti;

and proceed as in the proof of Theorem 4.1. ,

5. Hardy and Littlewood-type inequalities for p/q ≤ 2 and γ > 1

Theorem 5.1. Assume i∈ f1; . . . ; ng, Ti is a time scale with ai ∈ ð0;∞ÞTiand γi < 1,


→ℝþ is such that the delta integrals


257

R∞Qn

i¼1ai

Qni¼1ðtiÞ

pq−γi gp=qðt1; . . . ; tnÞ

Qni¼1 Δti exist. Let Λkðt1; . . . ; tnÞ be defined in (13), then for

p; q > 0 and p=q≤ 2Z ∞Yn

i¼1ai

1Yn

i¼1tγii

ðΛσ1 ��σnn Þp=q

Yni¼1

Δti

≤

�p

q

�npq Yn

i¼1

�2k

γii

ðγi � 1Þ�p=q Z ∞Yn

i¼1ai

1Yn

i¼1tγi−

pq

i


Δti:

(43)

Proof. Proceed as in the proof of Theorem 3.2 and apply inequality (12) in (21) to get (43).,

Remark 5.2. As a special case of Theorem 5.1, when T1 ¼ � � � ¼ Tn ¼ ℝ, p=q ¼ λ > 1and γ1; . . . ; γn < 1, we have the following Hardy type inequality

Z ∞Yn

i¼1ai

1Yn

i¼1tγii

Z tiYn

i¼1ai


dsi

!λYni¼1

dti

≤Yni¼1

�2λ

1� γi

�λ Z ∞Yn

i¼1ai

1Yn

i¼1tγi−λi

gλðt1; . . . ; tnÞYni¼1

dti:

Remark 5.3. Assume that T1 ¼ � � � ¼ Tn ¼ ℕ in Theorem 5.1, p=q ¼ λ > 1, ai > 1,γi > 1 for i∈ f1; . . . ; ng, further assume that

P∞

m1¼1 � � �P∞

mn¼1 gλðm1; . . . ;mnÞ is

convergent. In this case, (43) becomes the following discrete Hardy and Littlewood inequality

X∞m1¼1

� � �X∞mn¼1

1

mγ11 . . .m

γnn

Xm1

k1¼1

� � �Xmn

kn¼1

gðk1; . . . ; knÞ!λ

≤Yni¼1

�2λ

γi � 1

�λ X∞m1¼1

� � �X∞mn¼1

1Yn

i¼1m

γi−λi

gλðm1; . . . ;mnÞ:

6. Hardy and Littlewood-type inequalities for p/q > 1 and γ < 1

Theorem 6.1. Assume i∈ f1; . . . ; ng,Ti is a time scale with ai ∈ ð0;∞ÞTiand γi < 1,



i¼1ai

Qni¼1 ðσiðtiÞÞ


Qni¼1 Δti exist, for any ðt1; . . . ; tnÞ∈ ½a1;∞ÞT1

3 � � �3½an;∞ÞTn

, define

Ωkðt1; . . . ; tnÞ ¼Z tjYk

j¼1aj

gðs1; . . . ; snÞYkj¼1

Δsj; k∈ f1; . . . ; ng (44)

then for p; q > 0 and p=q > 1

AJMS26,1/2

258

Z ∞Yn

i¼1ai

Ωp=qn ðt1; . . . ; tnÞYn

i¼1σγii ðtiÞ

Yni¼1

Δti

≤

�p

q

�npq Yn

i¼1

�1

1� γi

�p=q Z ∞Yn

i¼1ai

1Yn

i¼1ðσiðtiÞÞγi−p=q


Δti

(45)

holds, where n is any positive integer.


To prove the result for n ¼ kþ 1. The left-hand side of (45) can be written asZ ∞Yn

i¼1ai

Ωp=qkþ1ðt1; . . . ; tkþ1ÞYkþ1

i¼1σγii ðtiÞ

Ykþ1

i¼1

Δti (46)

DenoteR∞akþ1

Ωp=q

kþ1ðt1;...;tkþ1Þ

σγkþ1kþ1

ðtkþ1ÞΔtkþ1 ¼ Ikþ1. Apply (5) with v

Δtkþ1vðtkþ1Þ ¼ 1

σγkþ1kþ1

ðtkþ1Þand uðtkþ1Þ ¼

Ωp=qkþ1ðt1; . . . ; tkþ1Þ. Thus

Ikþ1 ¼ vðtkþ1ÞΩp=qkþ1ðt1; . . . ; tkþ1Þj∞akþ1

þZ ∞

akþ1

vσkþ1ðtkþ1Þð− v

Δtkþ1

Ωp=qkþ1ðt1; . . . ; tkþ1ÞÞΔtkþ1;

(47)

where vðtkþ1Þ ¼R tkþ1

akþ11=σγkþ1

kþ1ðskþ1ÞΔskþ1. Use chain rule (6) and the fact that σkþ1ðskþ1Þ≥ skþ1

to get

v

Δskþ1

ðs1−γkþ1

kþ1 Þ ¼ ð1� γkþ1ÞZ 1

0

½hkþ1σkþ1ðskþ1Þ þ ð1� hkþ1Þskþ1�−γkþ1dhkþ1

≥ ð1� γkþ1Þ1

σγkþ1

kþ1 ðskþ1Þ;

which gives

vσkþ1ðtkþ1Þ ¼Z σkþ1ðtkþ1Þ

akþ1

1

σγkþ1

kþ1 ðskþ1ÞΔskþ1 ≤1

ð1� γkþ1Þðσkþ1ðtkþ1ÞÞ1−γkþ1 : (48)

Combine (47), (48) and use the facts Ωkþ1ðt1; . . . ; tk;∞Þ ¼ 0, vðakþ1Þ ¼ 0 to get

Ikþ1 ≤1

ð1� γkþ1ÞZ ∞

akþ1

� vΔtkþ1

Ωp=qkþ1ðt1; . . . ; tkþ1Þ

ðσkþ1ðtkþ1ÞÞγkþ1−1Δtkþ1: (49)

Apply chain rule 2 (7) to find

−v

Δtkþ1

Ωp=qkþ1ðt1; . . . ; tkþ1Þ ¼ −

�p

q

�Ω

pq−1

kþ1ðt1; . . . ; tk; ckþ1Þ v

Δtkþ1

Ωkþ1ðt1; . . . ; tkþ1Þ;

where, ckþ1 ∈ ½tkþ1; σkþ1ðtkþ1Þ�. Since


259

v

Δtkþ1

Ωkþ1ðt1; . . . ; tkþ1Þ ¼ −

Z ∞Yk

i¼1ai

gðs1; . . . ; sk; tkþ1ÞYki¼1

Δsi

¼: Ωkðt1; . . . ; tkþ1Þ ≤ 0;

and ckþ1 ≥ tkþ1, one has that

−v

Δtkþ1

Ωp=qkþ1ðt1; . . . ; tkþ1Þ≤ p

qΩ

pq−1

kþ1ðt1; . . . ; tkþ1ÞΩk ðt1; . . . ; tkþ1Þ: (50)


Ikþ1 ≤p

qð1� γkþ1ÞZ ∞

akþ1

Ωpq−1

kþ1ðt1; . . . ; tkþ1Þðσkþ1ðtkþ1ÞÞγkþ1−1

Ωkðt1; . . . ; tkþ1ÞΔtkþ1: (51)

Apply H€older’s inequality on the right-hand side of (51) with indices p=q and p=ðp− qÞ toobtain

Ikþ1 ≤p

qð1� γkþ1Þ

"Z ∞

akþ1

"ðσγkþ1

kþ1 ðtkþ1ÞÞp�qp

ðσkþ1ðtkþ1ÞÞγkþ1�1Ωkðt1; . . . ; tkþ1Þ

#p=qΔtkþ1

#q=p

3 ½Ikþ1�p−qp :


Ikþ1 ≤

�p

qð1� γkþ1Þ�p=q Z ∞

akþ1

Ωp=qk ðt1; . . . ; tkþ1Þ

ðσkþ1ðtkþ1ÞÞ−pqþγkþ1

Δtkþ1: (52)


i¼1ai


i¼1σγii ðtiÞ

Ykþ1

i¼1

Δti

≤

Z ∞Yk

i¼1ai

1Yk

i¼1σγii ðtiÞ

�p


akþ1

Ωp=qk ðt1; . . . ; tkþ1Þ


Ykþ1

i¼1

Δti:

(53)

Exchange integrals on right-hand side of (53) k-times by using (9)

¼�

p


akþ1

1


3

(Z ∞Yk

i¼1ai

Ωp=qk ðt1; . . . ; tkþ1ÞYk

i¼1σγii ðtiÞ

Yki¼1

Δti

)Δtkþ1:

(54)

AJMS26,1/2

260

Use the induction hypothesis for Ωkðt1; . . . ; tkþ1Þ in (54) instead for Ωkðt1; . . . ; tkÞ for fixedtkþ1 ∈Tkþ1 and again apply (9) k times to get

Z ∞Ykþ1

i¼1ai


i¼1σγii ðtiÞ

Ykþ1

i¼1

Δti

≤

�p

q

�ðkþ1Þpq Ykþ1

i¼1

�1

1� γi

�p=q Z ∞Ykþ1

i¼1ai

Ykþ1

i¼1

1

ðσiðtiÞÞγi−p=qgp=qðt1; . . . ; tkþ1Þ

Ykþ1

i¼1

Δti:


Corollary 6.2. Under the conditions of Theorem 6.1, we get the following inequality

Z ∞Yn

i¼1ai

1Yn

i¼1σγii ðtiÞ

ðΩnðσ1ðt1Þ; . . . ; σnðtnÞÞÞp=qYni¼1

Δti

≤

�p

q

�npq Yn

i¼1

�1

1� γi

�p=q Z ∞Yn

i¼1ai

Yni¼1

1

ðσiðtiÞÞγi−p=qgp=qðt1; . . . ; tnÞ

Yni¼1

Δti:

(55)

Proof. The fact vn Ωn

Δt1 ��Δtn ≤ 0 implies

Z ∞Yn

i¼1ai

1Yn

i¼1σγii ðtiÞ

ðΩnðσ1ðt1Þ; . . . ; σnðtnÞÞÞp=qYni¼1

Δti

≤

Z ∞Yn

i¼1ai

1Yn

i¼1σγii ðtiÞ

ðΩnðt1; . . . ; tnÞÞp=qYni¼1

Δti:

(56)

Now use (45) in (56) to get (55). ,

Remark 6.3. Consider T1 ¼ � � � ¼ Tn ¼ ℝ, p=q ¼ λ > 1 and γ1; . . . ; γn < 1, in Theorem6.1. Denote Gðt1; . . . ; tnÞ ¼

R∞Qn

i¼1tigðs1; . . . ; snÞ

Qni¼1 dsi. Thus, (45) takes the form

Z ∞Yn

i¼1ai

1Yn

i¼1tγii

ðGλðt1; . . . ; tnÞÞYni¼1

dti

≤Yni¼1

�λ

1� γi

�λ Z ∞Yn

i¼1ai

1Yn

i¼1ðtiÞγi−λ

vn

vt1 . . . vtnGλðt1; . . . ; tnÞ

Yni¼1

dti;

which can be considered as a generalization of Wirtinger’s inequality [1].

Remark 6.4. As a special case of Theorem 6.1, assume that T1 ¼ � � � ¼ Tn ¼ ℕ,p=q ¼ λ > 1, a1 ¼ � � � ¼ an ¼ 1 and γ1; . . . ; γn < 1. In this case (55) becomes the followingdiscrete Hardy and Littlewood inequality


261

X∞m1¼1

� � �X∞mn¼1

1Yn

i¼1ðmi þ 1Þγi

X∞k1¼m1þ1

� � �X∞

kn¼mnþ1

gðk1; . . . ; knÞ!λ

≤Yni¼1

�λ

1� γi

�λ X∞m1¼1

� � �X∞mn¼1

1Yn

i¼1ðmi þ 1Þγi−λ

gλðm1; . . . ;mnÞ:

7. Hardy and Littlewood-type inequalities for p/q ≤ 2 and γ < 1

Theorem 7.1. Assume i∈ f1; . . . ; ng,Ti is a time scale with ai ∈ ð0;∞ÞTiand γi < 1,



i¼1ai

Qni¼1 ðσiðtiÞÞ


Qni¼1 Δti exist, then for p; q > 0 and p=q≤ 2. Then

Z ∞Yn

i¼1ai

1Yn

i¼1σγii ðtiÞ

Z ∞Yn

i¼1ti


Δsi

!p=qYni¼1

Δti

≤

�p

q

�npq Yn

i¼1

�2

1� γi

�p=q Z ∞Yn

i¼1ai

Yni¼1

1

ðσiðtiÞÞγi−p=qgp=qðt1; . . . ; tnÞ

Yni¼1

Δti:

(57)

Proof: Use (12) and proceed as in the proof of Theorem 6.1 to get (57). ,

Remark 7.2. In Theorem 7.1, when T1 ¼ � � � ¼ Tn ¼ ℝ, p=q ¼ λ > 1 and γi < 1, (57)becomes the following Wirtinger type inequality,Z ∞Yn

i¼1ai

1Yn

i¼1tγii

ðGðt1; . . . ; tnÞÞλYni¼1

dti

≤Yni¼1

�2λ

1� γi

�λ Z ∞Yn

i¼1ai

1Yn

i¼1ðtiÞγi−λ

ð vn

vt1 . . . vtnGðt1; . . . ; tnÞÞλ

Yni¼1

dti;

where Gðt1; . . . ; tnÞ¼:R∞Qn

i¼1tigðs1; . . . ; snÞ

Qn

i¼1 dsi.

Remark 7.3. In Theorem 7.1, assume that T1 ¼ � � � ¼ Tn ¼ ℕ, p=q ¼ λ > 1,a1 ¼ � � � ¼ an ¼ 1 and γi < 1. (57) becomes the following discrete Hardy and Littlewoodinequality

X∞m1¼1

� � �X∞mn¼1

1Yn

i¼1ðmi þ 1Þγi

X∞k1¼m1þ1

� � �X∞

kn¼mnþ1

gðk1; . . . ; knÞ!λ

≤Yni¼1

�2λ

1� γi

�λ X∞m1¼1

� � �X∞mn¼1

1Yn

i¼1ðmi þ 1Þγi−λ

gλðm1; . . . ;mnÞ:

References

[1] R.P. Agarwal, M. Bohner, D. O’Regan, S.H. Saker, Some Wirtinger-type inequalities on time scalesand their applications, Pacific J. Math. 252 (2011) 1–26.

AJMS26,1/2

262

[2] W. Ahmad, K.A. Khan, A. Nosheen, M.A. Sultan, Copson, Leindler type inequalities of function ofseveral variables on time scales, Punjab Univ. J. Math. 51 (8) (2019) 157–168.

[3] M.S. Ashraf, K.A. Khan, A. Nosheen, Hardy-Copson type inequalities on time scales for thefunctions of n independent variables, Int. J. Anal. Appl. 17 (2) (2019) 244–259, http://dx.doi.org/10.28924/2291-8639-17-2019-244.

[4] J. Baric, R. Bibi, M. Bohner, A. Nosheen, J. Pecaric, Jensen inequalities and their applications ontime scales, in: Element, Zagreb, Croatia, (2015).

[5] M. Bohner, S.G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer Int. Publ.Switzerland, (2016) http://dx.doi.org/10.1007/978-3-319-47620-9.

[6] M. Bohner, A. Nosheen, J. Pecaric, A. Younas, Some dynamic Hardy type inequalities on timescales, J. Math. Inequal. 8 (1) (2014) 185–199.

[7] M. Bohner, A. Petereson, Dynamic Equations on Time Scales; An Introduction with Applications,Birkhauser, Boston, (2001).

[8] G.H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (3–4) (1920) 314–317, http://dx.doi.org/10.1007/BF01199965.

[9] G.H. Hardy, Notes on some points in the integral caluclus, LX. An inequality between integrals,Messenger Math. 54 (1925) 150–156.

[10] G.H. Hardy, Notes on some points in the integral calculus, Messenger Math. 57 (1928) 12–16.

[11] G.H. Hardy, J.E. Littlewood, Elementary theorems concerning power series with positivecoefficients and moment constants of positive functions, J. Reine Angew. Math. 157 (1927)141–158.

[12] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus,Results Maths (1990) 18–56.

[13] S.H. Saker, O’Regan Donal, Hardy and Littlewood Inequalities on time scales, Bull. Malays. Math.Sci. Soc. 39 (2) (2016) 527–543.

Corresponding authorAmmara Nosheen can be contacted at: [email protected]



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Volume 26 Issue 1/2 2020

Number 1/2

1 Editorial advisory board

3 Existence of mild solutions for fractional non-instantaneous impulsive integro-differential equations with nonlocal conditionsArshi Meraj and Dwijendra N. Pandey

15 Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionalsKhuram Ali Khan, Tasadduq Niaz, Ðilda Pečarić and Josip Pečarić

41 Some new fractional integral inequalities for generalized relative semi-m-(r; h

1, h

2)-

preinvex mappings via generalized Mittag-Leffler functionArtion Kashuri and Rozana Liko

57 Unbalanced multi-drawing urn with random addition matrixAguech Rafik and Selmi Olfa

75 Approximation of fixed point of multivalued ρ-quasi-contractive mappings in modular function spacesGodwin Amechi Okeke and Safeer Hussain Khan

95 The implicit midpoint rule for nonexpansive mappings in 2-uniformly convex hyperbolic spacesH. Fukhar-ud-din and A.R. Khan

107 On abstract Hilfer fractional integrodifferential equations with boundary conditionsSabri T.M. Thabet, Bashir Ahmad and Ravi P. Agarwal

127 Remarks on the critical nonlinear high-order heat equationTarek Saanouni

153 Approximative K-atomic decompositions and frames in Banach spacesShah Jahan

167 Existence of self-similar solutions of the two-dimensional Navier–Stokes equation for non-Newtonian fluidsDongming Wei and Samer Al-Ashhab

179 Coupled fixed points and coupled best proximity points for cyclic Ćirić type operatorsAdrian Magdaş

197 Nonlinear Jordan centralizer of strictly upper triangular matricesDriss Aiat Hadj Ahmed

203 Subcommuting and comparable iterative roots of order preserving homeomorphismsVeerapazham Murugan and Murugan Suresh Kumar

211 Strong consistency of a kernel-based rule for spatially dependent dataAhmad Younso, Ziad Kanaya and Nour Azhari

227 Generators and number fields for torsion points of a special elliptic curveHasan Sankari and Mustafa Bojakli

233 On the primeness of near-ringsKhalid H. Al-Shaalan

245 Multivariate Hardy and Littlewood inequalities on time scalesAmmara Nosheen, Aneela Nawaz, Khuram Ali Khan and Khalid Mahmood Awan

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