research article almost automorphic functions of order and...

Research ArticleAlmost Automorphic Functions of Order 119899 and Applications toDynamic Equations on Time Scales

Gisegravele Mophou12 Gaston Mandata NrsquoGueacutereacutekata3 and Aril Milce14

1Laboratoire CEREGMIA Universite des Antilles et de la Guyane Campus Fouillole 97159 Pointe-a-Pitre Guadeloupe2Laboratoire MAINEGE Universite Ouaga 3S 06 BP 10347 Ouagadougou 06 Burkina Faso3Department of Mathematics Morgan State University Baltimore MD 21251 USA4Departement de Mathematiques Ecole Normale Superieure Universite drsquoEtat drsquoHaıti Rue Monseigneur GuillouxPort-au-Prince Haiti

Correspondence should be addressed to Gisele Mophou giselemophouuniv-agfr

Received 30 August 2014 Revised 24 November 2014 Accepted 4 December 2014 Published 31 December 2014

Academic Editor Qi-Ru Wang

Copyright copy 2014 Gisele Mophou et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We revisit the notion on almost automorphic functions on time scales given by Lizama and Mesquita (2013) Then we present thenotion of almost automorphic functions of order 119899 Finally we apply this notion to study the existence and uniqueness and theglobal stability of almost automorphic solution of first order to a dynamical equation with finite time varying delay

1 Introduction

The concept of time scales was initiated in 1988 by Hilger inhis outstanding PhD thesis [1] The purpose of such theorywas to unify both continuous and discrete analysis Conse-quently using time scales in studying dynamic systems prev-ents from proving results separately for differential equationsand difference equations Since then several papers weredevoted to dynamical systems on time scales [2ndash8] We referalso readers to the excellent book by Bohner and Peterson[9] and their edited book [10] which contains high qualitycontributions to the theory

It was natural to study almost periodic time scales as wellas almost periodic differential equations on almost periodictime scales [3] Our initial motivation for the current studycomes from [4] where the authors studied the existence andexponential stability of almost periodic solutions of a neutral-type BAM neural network with delays on time scales usingexponential dichotomy of linear dynamical systems

Recently Lizama and Mesquita introduced the notion ofalmost automorphic functions on time scales in their work[11] The purpose of this paper is twofold First we would liketo revisit Lizama andMesquitarsquos paper in light of the followingremarks

Let T be a time scale It is said to be invariant undertranslations if

Π = 120591 isin R 119905 ∓ 120591 isin T forall119905 isin T = 0 (1)

We prove in Lemma 25 that Π sub T hArr 0 isin T However we observe that the inclusion may be strict

Indeed let us consider the time scale 119875119886119887

= ⋃infin

119896=minusinfin[119896(119886 +

119887) 119896(119886 + 119887) + 119886] where 0 lt 119886 lt 119887 it is obviously invariantunder translations and it contains 0 and 119886 but not minus119886 Then119886 notin Π This also proves that the invariant under translationstime scales 119875

119886119887is not symmetric

For this reason several results in [11] hold only if the timescale is symmetric

Secondlywewould like to study the existence and stabilityof almost automorphic solutions of the following linear dyna-mic system with finite delay

119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119909

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

119905 isin T 119894 = 1 2 119898

119909119894(119904) = 120593 (119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(2)

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 410210 13 pageshttpdxdoiorg1011552014410210

2 Discrete Dynamics in Nature and Society

where T is an appropriate time scale 119898 ge 2 is the numberof neurons in the network 119909

119894(119905) denote the activation of the

119894th neuron at time 119905 and 119886119894represents the rate with which

the 119894th neurons will rest their potential to the resting state inisolation when they are disconnected from the network andthe external inputs at time 119905 The119898times119898matrix 119861(119905) = (119887

119894119895(119905))

represents the connection strengths between neurons at time119905 119887

119894119895is an element of feedback templates at time 119905 119891

119895is the

activation function 120591119894119895is the transmission delay at time 119905 and

satisfies 119905 minus 120591119894119895isin T 119868

119894(119905) denote the bias of the 119894th neuron at

time 119905 120579 = max(119894119895)

sup119905isinT120591119894119895(119905) and [119886 119887]T = 119905 | 119905 isin [119886 119887]capT

We organized our paper as follows In Section 2 we recallsome definitions and recent results on time scales In Sec-tion 3 we present properties of almost automorphic functionson symmetric time scales along with a composition theoremIn Section 4 we introduce and present elementary propertiesof almost automorphic functions on time scales of order 119899and study differentiation and integration of such functionsin Section 5 and superposition of operators on the space ofsuch functions in Section 6 Finally in Section 7 we study theexistence uniqueness and global stability of system (2)

2 Preliminaries

In this section we recall some definitions and recent resultson time scales

Definition 1 A time scale is an arbitrary nonempty closedsubset of real numbers

Definition 2 Let T be a time scaleThe forward and backwardjump operators 120590 120588 T rarr T and the graininess 120583 T rarr

[0 +infin) are defined respectively by

120590 (119905) = inf 119904 isin T 119904 gt 119905 120588 (119905) = sup 119904 isin T 119904 lt 119905

120583 (119905) = 120590 (119905) minus 119905

(3)

In Definition 5 we put inf 0 = sup T and sup 0 = inf T

Definition 3 Let T be time scales

(i) A point 119905 isin T is called right-dense if 119905 lt sup T and120590(119905) = 119905

(ii) A point 119905 isin T is called left-dense if 119905 gt inf T and120588(119905) = 119905

Definition 4 (see [9]) A function 119891 T rarr R is called reg-ulated provided its right-sided limits exist (finite) at all right-dense points in T and its left-sided limits exist (finite) at allleft-dense points in T

Definition 5 (see [9]) A function 119891 T rarr R is called rd-continuous provided it is continuous at right-dense points inT and its left-sided limits exist (finite) at left-dense points inT

We will denote the set of rd-continuous function 119891 T rarr

R by 119862rd = 119862rd(T R) From now on we define the set T119896 by

T119896

=

T [120588 (sup T) sup T] if sup T lt infin

T if sup T = infin

(4)

Definition 6 (see [9]) Let119891 T rarr R be a function and 119905 isin T119896We define119891Δ(119905) to be the number (provided it exists) with theproperty that given any 120598 gt 0 there exists a neighborhood119880of 119905 such that

10038161003816100381610038161003816119891 (120590 (119905)) minus 119891 (119904) minus 119891

Δ

(119905) [120590 (119905) minus 119904]

10038161003816100381610038161003816le 120598 |120590 (119905) minus 119904|

forall119904 isin 119880

(5)

We call 119891Δ(119905) the delta (or Hilger) derivative of 119891 at 119905Moreover we say that 119891 is delta (or Hilger) differentiable

(or in short differentiable) on T119896 provided 119891Δ(119905) exists for all119905 isin T119896The function119891Δ T119896 rarr R is called the (delta) deriva-tive of 119891 on T119896

Next we recall some easy and useful relationships con-cerning the delta derivative

Theorem 7 (see [9]) Assume 119891 T rarr R is a function and let119905 isin T119896 Then one has the following

(i) If 119891 is differentiable at 119905 then 119891 is continuous at 119905(ii) If 119891 is continuous at 119905 and 119905 is right-scattered then 119891 is

differentiable at 119905 with

119891Δ

(119905) =

119891 (120590 (119905)) minus 119891 (119905)

120583 (119905)

(6)

(iii) If 119905 is right-dense then119891 is differentiable at 119905 if and onlyif the limit

lim119904rarr119905

119891 (119905) minus 119891 (119904)

119905 minus 119904

(7)

exists as a finite number In this case

119891Δ

(119905) = lim119904rarr119905

119891 (119905) minus 119891 (119904)

119905 minus 119904

(8)

(iv) If 119891 is differentiable at 119905 then

119891 (120590 (119905)) = 119891 (119905) + 120583 (119905) 119891Δ

(119905) (9)

Theorem 8 (see [9]) Assume 119891 119892 T rarr R are differentiableat 119905 isin T119896 Then

(i) the sum 119891 + 119892 T rarr R is differentiable at 119905 with

(119891 + 119892)Δ

(119905) = 119891Δ

(119905) + 119892Δ

(119905) (10)

(ii) for any constant 120572 120572119891 T rarr R is differentiable at 119905with

(120572119891)Δ

(119905) = 120572119891Δ

(119905) (11)

Discrete Dynamics in Nature and Society 3

(iii) the product 119891119892 T rarr R is differentiable at 119905 with

(119891119892)Δ

(119905) = 119891Δ

(119905) 119892 (119905) + 119891 (120590 (119905)) 119892Δ

(119905)

= 119891 (119905) 119892Δ

(119905) + 119891Δ

(119905) 119892 (120590 (119905))

(12)

We define higher order derivatives of a function on timescale in the usual way

Definition 9 (see [9]) For a function 119891 T rarr R one will talkabout the second derivative119891ΔΔ provided 119891Δ is differentiableon T119896

2

= (T119896)119896 with derivative 119891ΔΔ = (119891

Δ

)Δ

T1198962

rarr RSimilarly one defines higher order derivatives 119891Δ

119899

T119896119899

rarr

R Finally for 119905 isin T one denotes 1205902(119905) = 120590(120590(119905)) and1205882

(119905) = 120588(120588(119905)) and 120590119899(119905) and 120588119899(119905) are defined accordinglyFor convenience one also puts 1205880(119905) = 120590

0

(119905) = 119905 and T1198960

= T

Theorem 10 (see [9] (Leibniz formula)) Let 119878(119899)119896

be the setconsisting of all possible strings of length 119899 containing exactly119896 times 120590 and 119899 minus 119896 times Δ If 119891Λ exists for all Λ isin 119878

(119899)

119896 then

(119891119892)Δ119899

=

119899

sum

119896=0

( sum

Λisin119878(119899)

119896

119891Λ

)119892Δ119896

(13)

holds for all 119899 isin N

The following results on chain rule can be found in [9]

Theorem 11 (chain rule) Let 119891 R rarr R be continuously dif-ferentiable and suppose 119892 T rarr R is delta differentiable onT119896 Then 119891 ∘ 119892 T rarr R is delta differentiable and the formula

(119891 ∘ 119892)Δ

(119905) = [int

1

0

1198911015840

(119892 (119905) + ℎ120583 (119905) 119892Δ

(119905)) 119889ℎ] times 119892Δ

(119905)

(14)

holds for all 119905 isin T119896

Theorem 12 (chain rule) Assume that ] T rarr R is strictlyincreasing and T = ](T) is a time scale Let 120596 T rarr X whereX is a Banach space If ]Δ(119905) and 120596

Δ

(119905) exist for 119905 isin T119896 then

(120596 ∘ ])Δ = ]Δ sdot 120596Δ

∘ ] (15)

where Δ denote the Δ in T119896

Definition 13 If 119886 isin T sup T = +infin and 119891 is a 119862rd functionon [119886 +infin) then the improper integral of 119891 is defined by

int

infin

119886

119891 (119905) Δ119905 = lim119887rarr+infin

int

119887

119886

119891 (119905) Δ119905 (16)

provided this limit exists In this case the improper integralis said to converge

Lemma 14 (see [4]) Let 119886 isin T119896 119887 isin T and assume that119891 T times T119896 rarr R is continuous at (119905 119905) where 119905 isin T119896 with

119905 gt 119886 Assume also that 119891Δ(119905 sdot) is rd-continuous on [119886 120590(119905)]Suppose that for each 120598 gt 0 there exists a neighborhood 119880 of120591 isin [119886 120590(119905)] such that10038161003816100381610038161003816119891 (120590 (119905) 120591) minus 119891 (119904 120591) minus 119891

Δ

(119905 120591) [120590 (119905) minus 119904]

10038161003816100381610038161003816le 120598 |120590 (119905) minus 119904|


(17)

where 119891Δ denotes the derivative of 119891 with respect to the firstvariable Then

119892 (119905) = int

119905

119886

119891 (119905 120591) Δ120591 119894119898119901119897119894119890119904

119892Δ

(119905) = int

119905

119886

119891Δ

(119905 120591) Δ120591 + 119891 (120590 (119905) 119905)

(18)

We now present some definitions and results useful forthe study of some dynamical systems

Definition 15 (see [9]) One says that a function 119901 T rarr R isregressive provided

1 + 120583 (119905) 119901 (119905) = 0 forall119905 isin T119896

(19)

The set of all regressive functions will be denoted byR =

R(T R)

Definition 16 One defines the setR+ of all positively regres-sive elements ofR by

R+

= R+

(T R)

= 119901 isin R 1 + 120583 (119905) 119901 (119905) gt 0 forall119905 isin T (20)

Definition 17 (see [9]) If 119901 isin R then one defines thegeneralized exponential function by

119890119901(119905 119904) = exp(int

119905

119904

120585120583(120591)

(119901 (120591)) Δ120591) for 119905 119904 isin T (21)

where the cylinder transformation 120585ℎ C

ℎrarr Z

ℎis given by

120585ℎ((119911)) =

1

ℎ

log (1 + 119911ℎ) (22)

where log is the principal logarithm function For ℎ = 0 wedefine 120585

0(119911) = 119911 for all 119911 isin C

The generalized exponential functions have the followingproperties

Lemma 18 (see [9]) Assume that 119901 119902 T rarr R are two regre-ssive functions Then

(i) 1198900(119905 119904) equiv 1 and 119890

119901(119905 119905) equiv 1

(ii) 119890119901(119905 119904) = 1119890

119901(119904 119905) = 119890

⊖119901(119904 119905)

(iii) 119890119901(119905 119904)119890

119901(119904 119903) = 119890

119901(119905 119903)

(iv) [119890119901(119905 119904)]

Δ

= 119901(119905)119890119901(119905 119904)


Lemma 19 (see [9]) Assume that 119901 isin R+ Then

(i) 119890119901(119905 119904) gt 0 for all 119905 119904 isin T

(ii) if 119901(119905) le 119902(119905) for all 119905 ge 119904 119905 119904 isin T then 119890119901(119905 119904) le

119890119902(119905 119904) for all 119905 ge 119904

Lemma 20 (see [9]) If 119901 isin R and 119886 119887 119888 isin T then

(i) [119890119901(119888 sdot)]

Δ

= minus119901[119890119901(119888 sdot)]

120590

(ii) int119887119886

119901(119905)119890119901(119888 120590(119905))Δ119905 = 119890

119901(119888 119886) minus 119890

119901(119888 119887) for all 119905 ge 119904

Proposition 21 (see [9]) Let 119901 T rarr R be rd-continuousand regressive 119905

0isin T and 119910

0isin R Then the unique solution of

the initial value problem

119910Δ

(119905) = 119901 (119905) 119910 (119905) + ℎ (119905) 119910 (1199050) = 119910

0(23)

is given by

119910 (119905) = 119890119901(119905 119905

0) + int

119905

1199050

119890119901(119905 120590 (119904)) ℎ (119904) Δ119904 (24)

We now present some definitions about matrix-valuedfunctions on T

Definition 22 Let 119860 be an119898 times 119898matrix-valued function onT One says that 119860 is rd-continuous on T if each entry of 119860 isrd-continuous on T One denotes by Crd(T R

119898times119898

) the classof all rd-continuous119898 times 119898matrix-valued functions on T

We say that 119860 is delta differentiable on T if each entry of119860 is delta differentiable on T And in this case we have

119860 (120590 (119905)) = 119860 (119905) + 120583 (119905) 119860Δ

(119905) (25)

Definition 23 An 119898 times 119898 matrix-valued function 119860 is calledregressive if

119868 + 120583 (119905) 119860 (119905) is invertible forall119905 isin T119896

(26)

The class of all such regressive rd-continuous functions isdenoted by

R (T R119898times119898

) (27)

3 Almost Automorphic Functions of Order 119899on Time Scales

From now on (X sdot ) is a (real or complex) Banach space

Definition 24 (see [11]) A time scale T is called invariantunder translations if


Lemma 25 Let T be an invariant under translations timescale Then one has

(i) Π sub T hArr 0 isin T (ii) Π cap T = 0 hArr 0 notin T

Proof

(i) In view of the definition ofΠ if 0 isin T then for all 120591 isinΠ we have 120591 isin T Thus Π sub T Conversely assumethatΠ sub T Then for any 120591 isin Π we have 0 = 120591minus120591 isin T Thus 0 isin T

(ii) It is clear that if Π cap T = 0 then 0 notin T Now assumethat 0 notin T If 120591 isin ΠcapT then we have 119905minus120591 isin T for any119905 isin T particularly for 119905 = 120591 we have 120591 minus 120591 = 0 isin T This contradicts the fact that 0 notin T Thus Π cap T = 0

We have the following properties of the points in T for-ward jump operator and the graininess function when thetime scales are invariant under translations

Lemma 26 (see [2]) Let T be an invariant under translationtime scale If 119905 is right-dense (resp right-scattered) then forevery ℎ isin Π 119905 + ℎ is right-dense (resp right-scattered)

Lemma 27 (see [2]) Let T be an invariant under translationstime scale and ℎ isin Π Then

(i) 120590(119905 + ℎ) = 120590(119905) + ℎ and 120590(119905 minus ℎ) = 120590(119905) minus ℎ for every119905 isin T

(ii) 120583(119905 + ℎ) = 120583(119905) = 120583(119905 minus ℎ) for every 119905 isin T

Remark 28 As we pointed out in Section 1 time scales invari-ant under translations are not automatically symmetric Sincealmost automorphic functions are defined on symmetricdomains some definitions and results in [11] on these func-tions will be given with additional assumption on the timescale More precisely we will assume that the time scale issymmetric and invariant under translations

Definition 29 Let X be Banach space and let T be a sym-metric time scale which is invariant under translations Thenthe rd-continuous function 119891 T rarr X is called almostautomorphic on T if for every sequence (119904

119899) onΠ there exists

a subsequence (120591119899) sub (119904

119899) such that

119891 (119905) = lim119899rarrinfin

119891 (119905 + 120591119899) (29)

is well defined for each 119905 isin T and

lim119899rarrinfin

119891 (119905 minus 120591119899) = 119891 (119905) (30)

for each 119905 isin T

We denote by 119860119860T (X) the space of all almost automor-phic functions on time scales 119891 T rarr X

Remark 30 In view of Lemma 27 if T is a symmetric timescale which is invariant under translations then the graininessfunction 120583 T rarr R

+is an almost automorphic function

We have the following properties

Theorem 31 Let T be a symmetric time scale which is invari-ant under translation Assume that the rd-continuous functions


119891 119892 T rarr X are almost automorphic on T Then the followingassertions hold

(i) 119891 + 119892 is almost automorphic on time scales(ii) 120582119891 is almost automorphic on T for every scalar 120582(iii) for each 119897 isin Π the function 119891

119897 T rarr X defined by

119891119897(119905) = 119891(119905 + 119897) is almost automorphic on time scales

(iv) 119891 T rarr X defined by

119891(119905) = 119891(minus119905) is almost automo-rphic on time scales

(v) sup119905isinT119891(119905) lt infin that is 119891 is a bounded function

(vi) sup119905isinT119891(119905) = sup

119905isinT119891(119905) where

119891 (119905) = lim119899rarrinfin

119891 (119905 + 120591119899) lim

119899rarrinfin

119891 (119905 minus 120591119899) = 119891 (119905) (31)

Proof See [11]

We have the following remark on the property given in[11]

Remark 32 Notice that

(i) in order to give a sense to (iii) we consider 119897 as anelement of Π instead of 119897 isin T as in [11]

(ii) we need the symmetry of the time scale T to obtainthat minus119905 isin T if 119905 isin T that is to give a sense to the defin-ition of 119891 in (iv)

Remark 33 The space 119860119860T (X) equipped with the normsup

119905isinT119891(119905) is a Banach space (see [11] pp 2280)

Lemma 34 If 119891 isin 119860119860T (X) the range 119877119891 = 119891(119905) 119905 isin T isrelatively compact inX

Proof Let 119886 isin T be fixed and let (1199091015840119899)119899be a sequence in 119877

119891

Then for any 119899 isin N there is 1199051015840119899isin T such that 1199091015840

119899= 119891(119905

1015840

119899)

By invariance under translations of T for each 119899 isin N we canfind 1205721015840

119899isin Π such that 1199051015840

119899= 119886 + 120572

1015840

119899 Hence for all 119899 isin N we

have 1199091015840119899= 119891(119886 + 120572

1015840

119899) Since 119891 is almost automorphic on time

scale there exists a subsequence (120572119899)119899of (1205721015840

119899)119899such that

lim119899rarrinfin

119891 (119886 + 120572119899) = 119891 (119886) (32)

Thus the subsequence (119909119899= 119891(119886 + 120572

119899)) converges to 119891(119886)

Therefore 119877119891is relatively compact inX

Theorem 35 (see [11]) Let T be a symmetric time scale whichis invariant under translations Let also 119906 T rarr C and 119891

T rarr X be two almost automorphic functions on time scalesThen the function 119906119891 T rarr X defined by (119906119891)(119905) = 119906(119905)119891(119905)

is almost automorphic on time scales

Theorem 36 (see [11]) Let T be a symmetric time scale whichis invariant under translations and let (119891

119899) be a sequence of

almost automorphic functions such that lim119899rarr+infin

119891119899(119905) = 119891(119905)

converges uniformly for each 119905 isin T Then 119891 is an almostautomorphic function

Theorem 37 (see [11]) Let T be a symmetric time scale whichis invariant under translations and let X and Y be Banachspaces Suppose 119891 T rarr X is an almost automorphic functionon time scales and 120601 X rarr Y is a continuous function thenthe composite function 120601 ∘ 119891 T rarr Y is almost automorphicon time scales

Definition 38 (see [11]) Let X be a (real or complex) Banachspace and let T be a symmetric time scale which is invariantunder translations Then a rd-continuous function 119891 T times

X rarr X is called almost automorphic on 119905 isin T for each 119909 isin X

if for every sequence (119904119899) on Π there exists a subsequence

(120591119899) sub (119904

119899) such that

119891 (119905 119909) = lim119899rarrinfin

119891 (119905 + 120591119899 119909) (33)

is well defined for each 119905 isin T 119909 isin X and

lim119899rarrinfin

119891 (119905 minus 120591119899 119909) = 119891 (119905 119909) (34)

for each 119905 isin T and 119909 isin X

Theorem 39 (see [11]) Let T be a symmetric time scale whichis invariant under translations and let 119891 119892 T times X rarr X bealmost automorphic functions on time scale in 119905 for each 119909 inX Then the following assertions hold

(i) 119891 + 119892 is almost automorphic on time scale in 119905 for each119909 inX

(ii) 120582119891 is almost automorphic function on time scale in 119905 foreach 119909 inX where 120582 is an arbitrary scalar

(iii) sup119905isinT119891(119905 119909) = 119872

119909lt infin for each 119909 isin X

(iv) sup119905isinT119891(119905 119909) = 119873

119909lt infin for each 119909 isin X where 119891 is

the function in Definition 38

Theorem 40 Let T be a symmetric time scale which is invari-ant under translations and let 119891 T times X rarr X be almostautomorphic functions on time scale in 119905 for each 119909 in X andsatisfy Lipschitz condition in 119909 uniformly in 119905 that is

1003817100381710038171003817119891 (119905 119909) minus 119891 (119905 119910)

1003817100381710038171003817le 119871

1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (35)

for all 119909 119910 isin X Assume that 120601 T rarr X almost automorphicon time scale Then the function 119880 T rarr X defined by 119880(119905) =119891(119905 120601(119905)) is almost automorphic on time scale

We can now introduce the notion of almost automorphicfunctions of order 119899 on time scales


We denote by 119862119899

rd(T X) the linear space of all functions119891 T rarr X that are 119899th differentiable on T and 119891

Δ119899

is rd-continuous We denote by 119861119899

rd(T X) the subspace of119862119899

rd(T X) consisting of such function 119891 T rarr X for which

sup119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) lt +infin (36)


where119891Δ119894

denotes the 119894th derivative of119891 119894 = 1 2 119899119891Δ0

=

119891 and 119891Δ1

= 119891Δ In the space 119861119899rd(T X) we introduce the

norm

10038171003817100381710038171198911003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) for 119891 isin 119861119899

rd (T X) (37)

Then we have the following

Proposition 41 119861119899rd(T X) equipped with the norm definedabove is a Banach space

Proof Itrsquos clear that 119861119899rd(T X) is a linear space and that (37)is a norm on 119861119899rd(X) Now let (119891119895)119895 be a Cauchy sequence in119861119899

rd(T X) Then for any 120598 gt 0 there exists 119873 isin N such thatfor all 119895 isin N 119895 gt 119873 we have

10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598

forall119901 isin N

(38)

In other words given 120598 gt 0 there is119873 isin N such that for all119895 isin N 119895 gt 119873

sup119901isinN

10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup

119901isinN119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598 (39)

In particular for all 119905 isin T (119891Δ119894

119895(119905)) 119894 = 0 1 119899 are Cauchy

sequences inXwhich is a Banach spaceThus if we denote by119891 the limit of 119891Δ

0

119895(119905) we have that

119891Δ119894

119895(119905) 997888rarr 119891

Δ119894

(119905) 119894 = 0 1 119899 (40)

Since for 119894 = 0 1 119899100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

le 120598 forall119895 119901 isin N forall119905 isin T (41)

passing to the limit in these above 119899+1 relations as 119901 rarr +infinwe obtain for 119894 = 0 1 119899

100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817

le 120598 forall119895 isin N forall119905 isin T (42)

This means that for all 119895 isin N

10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899le sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817

) le 120598 (43)

which on the one hand proves that 119891 isin 119861119899

rd(T X) since forany 119895 isin N

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

le

10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899+

10038171003817100381710038171003817119891119895

10038171003817100381710038171003817119899 forall119905 isin T (44)

and on the other hand shows that

119891119895997888rarr 119891 as 119895 997888rarr +infin (45)

Definition 42 LetX be a (real or complex) Banach space andlet T be a symmetric time scale which is invariant under tran-slations Then a rd-continuous function 119891 T times X rarr X iscalled almost automorphic if 119891(119905 119909) is almost automorphicin 119905 isin T uniformly for each 119909 isin 119861 where 119861 is any boundedsubset ofX

We denote by 119860119860(T times XX) the space of all almostautomorphic functions on time scales 119891 T timesX rarr X

Definition 43 A function 119891 isin 119862119899

rd(T X) is said to be 119862119899

rd-almost automorphic (briefly 119862

119899

rd-aa) if 119891 119891Δ119894

belong to119860119860T (X) for all 119894 = 1 119899

Denote by 119860119860(119899)

T (X) the set ofC119899-aa functionsDirectly from the above definitions it follows that

119860119860(119899+1)

T (X) sub 119860119860(119899)

T (X) Moreover putting 119899 = 0 we have119860119860

(0)

T (X) = 119860119860T (X)

Lemma 44 We have 119860119860(119899)

T (X) sub 119861119899

rd(T X)

Proof It is straightforward from the definition of an almostautomorphic function on time scales (see Theorem 31)

Proposition 45 A linear combination of 119862119899

rd-aa functions isa119862119899

rd-aa function Moreover letX be a Banach space over thefield K (K = R or C) Let 119891 119892 isin 119860119860

(119899)

T (X) ] isin 119860119860(119899)

T (K)and 120582 isin K Assume that ]Λ exists for all Λ isin 119878

(119899)

119896and is almost

automorphic on time scale Then the following functions arealso in 119860119860(119899)

T (X)

(i) 119891 + 119892

(ii) 120582119891

(iii) ]119891

(iv) 119891119886(119905) = 119891(119905 + 119886) where 119886 isin Π is fixed

Proof For the proof of (i) and (ii) one proceeds as in [11]To prove (iii) we use the Leibnitz formula on time scalesthe definition and the properties of an almost automorphicfunction we get the result easily

Now let us prove (iv) For any 119886 isin Π if we considerthe function V T rarr R defined by V(119905) = 119886 + 119905 then wehave 119891

119886(119905) = (119891 ∘ V)(119905) for all 119905 in T It is clear that V is

strictly increasing V(T) = T and VΔ(119905) = 1 for all 119905 isin T Using Theorem 12 we obtain (119891

119886)Δ

(119905) = VΔ(119905) sdot (119891Δ ∘ V)(119905) =119891Δ

(V(119905)) = 119891Δ

(119886 + 119905) = 119891Δ

119886(119905) for each 119905 isin T Hence for 119891Δ

being an almost automorphic function on time scale we ded-uce that (119891

119886)Δ

= 119891Δ

119886is almost also automorphic on time scale

Similarly we prove that (119891119886)Δ119894

is almost automorphic for 119894 =1 2 119899 Thus 119891

119886isin 119860119860

119899

T (X)

Theorem 46 If a sequence (119891119901)119901isinN of 119862119899

rd-aa functions issuch that 119891

119901minus 119891

119899rarr 0 as 119896 rarr +infin then 119891 is 119862119899

rd-aafunction


Proof From the assumption it is clear that 119891 isin 119861119899

rd(T X)Moreover 119891Δ

119894

119901rarr 119891

Δ119894

uniformly on T for each 119894 = 0 119899ThusTheorem 36 allows us to say that119891 is a119862119899

rd-aa function

Corollary 47 119860119860(119899)

T (X) considered with norm (37) turns outto be a Banach space

Proof In viewof Proposition 45 andLemma44119860119860(119899)

T (X) is alinear subspace of 119861119899rd(T X) Let (119891119901) 119891119901 isin 119860119860

(119899)

T (X) 119901 isin Nbe a Cauchy sequence Then there exists 119891 isin 119861

119899

rd(T X) suchthat lim

119896rarrinfin119891

119901minus119891

119899= 0 ByTheorem 46 119891 isin 119860119860

(119899)

T (X) soit is a Banach space

5 Differentiation and Integration

The first result in this section gives a sufficient conditionwhich guarantees that the derivative of a function 119891 isin

119860119860(119899)

T (X) is also a 119862119899

rd-aa function

Theorem 48 Let T be a symmetric time scale which isinvariant under translations Let alsoX be a Banach space and119891 T rarr X an almost automorphic function on T Assume that119891 isΔ-differentiable onT and119891Δ is uniformly continuousThen119891Δ is also almost automorphic on time scales

Proof Assume that the points of T are right-dense Then forT being invariant under translations we obtain T = R Hence119891Δ

= 1198911015840 and since119891Δ is uniformly continuous it follows from

Theorem 241 in [12] that 119891Δ is almost automorphic on timescale

Now let us suppose that T has at least a right-scatteredpoint 119905

0 then we have

119891Δ

(1199050) =

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

(46)

Given a sequence (1205721015840119899) isin Π since 119891 is almost automorphic

there is a subsequence (120572119899)119899such that

lim119899rarrinfin

119891Δ

(1199050+ 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)

= lim119899rarrinfin

119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119892 (1199050)

(47)

since Lemma 27 holds On the other hand we have

lim119899rarrinfin

119892 (1199050minus 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119891Δ

(1199050)

(48)

The proof is completed

Theorem 49 If 119891 isin 119860119860(119899)

T (X) and 119891Δ119899+1

is uniformly contin-uous then 119891Δ isin 119860119860(119899)

T (X)

Proof In view of Theorem 48 we have 119891Δ119899+1

isin 119860119860T (X)This means that 119891 is in 119860119860(119899+1)

T (X) Then it follows that 119891Δ isin119860119860

(119899)

T (X)

Similarly as in [12] we introduce some useful notationsin order to facilitate the proof

Notation 1 Let T be a symmetric time scale which is invariantunder translations If 119891 T rarr X is a function and a sequence120572 = (120572

119899) sub Π is such that

lim119899rarrinfin

119891 (119905 + 120572119899) = 119892 (119905) pointwise on T (49)

we will write 119879119904119891 = 119892

Remark 50 (see [12]) Consider the following

(i) 119879119904is a linear operator

Given a fixed sequence 120572 = (120572119899) sub Π the domain

of 119879119904is 119863(119879

119904) = 119891 T rarr X such that 119879

119904119891 exists

119863(119879119904) is a linear set

(ii) Let us write minus119904 = (minus120572119899) and suppose that 119891 isin 119863(119879

119904)

and119879119904119891 isin 119863(119879

minus119904)The product operator119860

119904= 119879

minus119904119879119904119891

is well defined It is also a linear operator(iii) 119860

119904maps bounded functions into bounded functions

and for an almost authorphic function on time scale119891 we get 119860

119904119891 = 119891

Now we are ready to enunciate and prove Bohl-Bohrrsquostype theorem known from the literature for almost automor-phic functions on time scale The proof is inspired by theproof of Theorem 244 in [12]

Theorem 51 Let T be a symmetric time scale containing zeroand invariant under translations Let also 119891 isin 119860119860T (X) Oneconsiders the function 119865 T rarr X defined by 119865(119905) = int

119905

0

119891(119904)Δ119904Then 119865 isin 119860119860T (X) if and only if 119877119865 is relatively compact inX

Proof In view of Lemma 34 it suffices to prove that 119865 isin

119860119860T (X) if 119877119865 is relatively compact inXAssume that 119877

119865is relatively compact in X and let (11990410158401015840

119899) sub

Π Then there exists a subsequence (1199041015840119899) of (11990410158401015840

119899) such that

lim119899rarrinfin

119891 (119905 + 1199041015840

119899) = 119892 (119905)

lim119899rarrinfin

119892 (119905 minus 1199041015840

119899) = 119891 (119905)

(50)


pointwise on T and

lim119899rarrinfin

119865 (1199041015840

119899) = 120572

1 (51)

for some 1205721isin X

We get for every 119905 isin T

119865 (119905 + 1199041015840

119899) = int

119905+1199041015840

119899

0

119891 (119903) Δ119903

= int

1199041015840

119899

0

119891 (119903) Δ119903 + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

= 119865 (1199041015840

119899) + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

(52)

Making the change of variable 120575 = 119903 minus 1199041015840

119899 we obtain

119865 (119905 + 1199041015840

119899) = 119865 (119904

1015840

119899) + int

119905

0

119891 (120575 + 1199041015840

119899) Δ120575 (53)

If we apply the Lebesgue dominated theorem to this latteridentity we get

lim119899rarrinfin

119865 (119905 + 1199041015840

119899) = 120572

1+ int

119905

0

119892 (120590) Δ120590 (54)

for each 119905 isin T Let us observe that the range of the function119866(119905) = 120572

1+ int

119905

0

119892(119903)Δ119903 is also relatively compact and

sup119905isinT

(119866 (119905)) le sup119905isinT

119865 (119905) (55)

so that we can extract a subsequence (119904119899) of (1199041015840

119899) such that

lim119899rarrinfin

119866 (minus119904119899) = 120572

2 (56)

for some 1205722isin X Now we can write

119866 (119905 minus 119904119899) = 119866 (minus119904

119899) + int

119905

0

119892 (119903 minus 119904119899) Δ119903 (57)

so that

lim119899rarrinfin

119866 (119905 minus 119904119899) = 120572

2+ int

119905

0

119891 (119903) Δ119903 = 1205722+ 119865 (119905) (58)

Let us prove now that 1205722= 0 Using Notation 1 we get

119860119904119865 = 120572

2+ 119865 (59)

where 119904 = (119904119899) Now it is easy to observe that 119865 and 120572

2belong

to the domain of 119860119904 therefore 119860

119904119865 is also in the domain of

119860119904and we deduce the equation

1198602

119904119865 = 119860

1199041205722+ 119860

119904119865 = 120572

2+ 120572

2+ 119865 = 2120572

2+ 119865 (60)

We can continue indefinitely the process to get

119860119899

119904119865 = 119899120572

2+ 119865 forall119899 = 1 2 (61)

But we have

sup119905isinT

1003817100381710038171003817119860119899

119904119865 (119905)

1003817100381710038171003817le sup

119905isinT

119865 (119905) (62)

and 119865(119905) is a bounded functionThis leads to contradiction if 120572

2= 0 Hence 120572

2= 0

and using Remark 50 we have 119860119904119865 = 119865 so 119865 is almost

automorphic The proof is complete

Theorem 52 If 119891 isin 119860119860T (X) and the range 119877119865is relatively

compact then 119865 isin 119860119860(1)

T (X)

Proof If 119891 isin 119860119860T (X) and the range 119877119865 is relatively compactthen in view of Theorem 51 119865 isin 119860119860T (X) Since 119865(119905) =

int

119905

0

119891(119904)Δ119904 119865Δ(119905) = 119891(119905) for 119905 isin T Thus 119865Δ isin 119860119860T (X) andconsequently 119865 isin 119860119860

(1)

T (X)

Theorem 52 is a special case of the following

Theorem 53 If 119891 isin 119860119860(119899)

T (X) and the range 119877119865is relatively

compact then 119865 isin 119860119860(119899+1)

T (X)

Proof If119891 isin 119860119860(119899)

T (X) and the range119877119865is relatively compact

then in view of Theorem 51 119865 isin 119860119860T (X) Therefore 119865Δ =

119891 isin 119860119860(119899)

T (X) This means that 119865 isin 119860119860(119899+1)

T (X)

Corollary 54 If 119891Δ isin 119860119860(119899)



T (X)

Proof We know that119891(119905) = 119891(0)+int

119905

0

119891Δ

(119904)Δ119904 for each 119905 isin T In view of Theorem 53 we have 119891 isin 119860119860

(119899+1)

T (X)

6 Superposition Operators

In this sectionX1andX

2are two Banach spaces

Proposition 55 Let 119860 X1rarr X

2be a bounded linear ope-

rator and 119891 isin 119860119860(119899)

T (X1) Then we have 119860119891 isin 119860119860

(119899)

T (X2)

Proof Since119860 is a bounded linear operator we have (119860119891)Δ119899

=

119860119891Δ119899

Therefore observing the fact that 119891Δ119894

isin 119860119860T (X1) for

each 119894 = 0 1 119899 because 119891 isin 119860119860(119899)

T (X1) it follows from

Theorem 37 that 119860119891Δ119894

isin 119860119860T (X2) for all 119894 = 0 1 119899

Hence 119860119891 isin 119860119860(119899)

T (X2)

For every 119891 isin 119860119860(119899)

T (X1) we define the function119866 T rarr

X2as follows

119866 (119905) = int

119905

0

119860119891 (119904) Δ119904 (63)

where 119860 X1rarr X

2is a bounded linear operator We have

the following

Corollary 56 Let119860 X1rarr X

2be a bounded linear operator

with a relatively compact range Then X2-valued function 119866

defined above is in 119860119860(119899+1)

T (X2)


Proof According to Proposition 55 119860119891 isin 119860119860(119899)

T (X2) for

every 119891 isin 119860119860(119899)

T (X1) Since 119860 is a bounded linear operator

the range 119877119860

of the operator 119860 contains the range 119877119866

of 119866 Hence 119877119866is relatively compact and it follows from

Theorem 53 that 119866 isin 119860119860(119899+1)

T (X2)

Remark 57 In Corollary 56 if operator 119860 is compact (or ofthe finite rank) the stated result holds

Now we will consider the superposition of operator (theautonomous case) acting on the space 119860119860(119899)

T (X) Using thisfact we will prove the following result with the Frechetderivative

Theorem 58 If 120601 isin C1

(RR) and119891 isin 119860119860(1)

T (R) then 120601∘119891 isin

119860119860(1)

T (R)

Proof First we observe that the result holds if for 119899 = 0 inview of Theorem 37 we have that 120601 ∘ 119891 isin 119860119860T (R) if 120601 isin

C1

(RR) and 119891 isin 119860119860T (R) So it suffices to show that (120601 ∘119891)

Δ

isin 119860119860T (R) to complete the proof of the theoremByTheorem 11 for each 119905 isin T we have

(120601 ∘ 119891)Δ

(119905) = [int

1

0

1206011015840

(119891 (119905) + ℎ120583 (119905) 119891Δ

(119905)) 119889ℎ]119891Δ

(119905)

(64)

Since 119891 isin 119860119860(1)

T (R) and 120583 isin 119860119860T (R+) we have that for

any ℎ isin [0 1] the function 119905 997891rarr 119891(119905) + ℎ120583(119905)119891Δ

(119905) belongsto 119860119860T (R) Therefore for 1206011015840 being continuous Theorem 37allows us to say that the function 119905 997891rarr 120601

1015840

(119891(119905) + ℎ120583(119905)119891Δ

(119905))

is also in 119860119860T (R) for any ℎ isin [0 1] and consequently thefunction 119905 997891rarr int

1

0

1206011015840

(119891(119905) + ℎ120583(119905)119891Δ

(119905))119889ℎ is 119860119860T (R) It thenfollows from Theorem 35 that (120601 ∘ 119891)

Δ

isin 119860119860T (R) since119891Δ

isin 119860119860T (R)

7 Applications to First-Order DynamicEquations on Time Scales

Definition 59 (see [5]) Let 119860(119905) be 119898 times 119898 rd-continuousmatrix-valued function on T The linear system

119909Δ

(119905) = 119860 (119905) 119909 (119905) 119905 isin T (65)

is said to admit an exponential dichotomy on T if there existpositive constants 119870 120572 projection 119875 and the fundamentalsolution matrix119883(119905) of (65) satisfying

10038161003816100381610038161003816119883(119905)119875119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119905 119904) 119904 119905 isin T 119905 ge 119904

10038161003816100381610038161003816119883(119905)(119868 minus 119875)119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119904 119905) 119904 119905 isin T 119904 ge 119905

(66)

where | sdot |T is the matrix norm on T This means that if 119860 =

(119886119894119895)119898times119898

then we can take |119860|T = (sum119898

119894=1sum119898

119895=1|119886119894119895|2

)12

Consider the following system

119909Δ

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin T (67)

where 119860 is an 119898 times 119898 matrix-valued function which isregressive on T and 119891 T rarr R119898 is rd-continuous

Theorem 60 (see [11]) Let T be a symmetric time scale whichis invariant under translation and let 119860 isin R(T R119898times119898

) bealmost automorphic and nonsingular on T and (119860minus1

(119905))119905isinT and

(119868 + 120583(119905)119860(119905))minus1

119905isinT are bounded Assume that linear system

(65) admits an exponential dichotomy and 119891 isin 119862rd(T R119898

) isan almost automorphic function on time scales Then system(67) has an almost automorphic solution as follows

119909 (119905) = int

119905

minusinfin

119883 (119905) 119875119883minus1

(120590 (119904)) 119891 (119904) Δ119904

minus int

infin

119905

119883 (119905) (119868 minus 119875)119883minus1

(120590 (119904)) 119891 (119904) Δ119904

(68)

where119883(119905) is the fundamental solution matrix of (65)

Lemma 61 (see [3]) Let 119888119894 T rarr (0 +infin) be an almost

automorphic function minus119888119894isin R+ andmin

1le119894le119898inf

119905isinT 119888119894(119905) gt

0 Then the linear system

119909Δ

(119905) = diag (minus1198881(119905) minus119888

2(119905) minus119888

119898(119905)) 119909 (119905) (69)

admits an exponential dichotomy on T

In view of Lemma 61 andTheorem 60 we have the follow-ing result

Lemma 62 Let T be a symmetric time scale which is invariantunder translation Let 119860(119905) = diag(minus119888

1(119905) minus119888

2(119905) minus119888

119898(119905))

be such that the functions 119888119894 T rarr (0 +infin) 119894 = 1 2 119898 are

almost automorphic minus119888119894isin R+ andmin

1le119894le119898inf

119905isinT 119888119894(119905) gt 0119894 = 1 2 119898 Assume also that 119891 isin 119862rd(T R

119898

) is an almostautomorphic function on time scales Then system (67) has analmost automorphic solution as follows

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (70)


1(119905) minus119888

2(119905) minus119888

119898(119905))

be such that the functions 119888119894 T rarr (0 +infin) 119894 = 1 2 119898

are 1198621

rd-almost automorphic minus119888119894isin R+ and min

1le119894le119898inf

119905isinT

119888119894(119905) gt 0 119894 = 1 2 119898 Assume also that 119891 is a 1198621

rd-almostautomorphic function on time scales Then system (67) has a1198621

rd-almost automorphic solution as follows

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (71)

Proof By Lemma 62

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (72)

is a 119862rd-almost automorphic solution to system (67) Nowsince

119909Δ

119894(119905) = minus119888

119894(119905) int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904

+ 119891119894(119905) 119894 = 1 2 119898

(73)


using on the one hand the fact that functions 119905 997891rarr int

119905

minusinfin

119890minus119888119894

(119905 120590(119904))119891119894(119904)Δ119904 119905 997891rarr 119888

119894(119905) and 119905 997891rarr 119891

119894are 119862rd-almost

automorphic and on the other hand the fact thatTheorem 35holds we deduce that 119909Δ

119894(119905) 119894 = 1 2 119898 are also 119862rd-

almost automorphic The proof of Lemma 63 is then com-plete

In the following we will consider119860119860(1)

T (R)with the norm sdot

1obtained by taking 119899 = 1 in (37)

Set 119864 = 120595 = (1205951 120595

2 120595

119898)119879

| 120595119894isin 119860119860

(1)

T (R) 119894 =

1 2 119898 Then with the norm 120595119864= max

1le119894le119898120595

1 119864 is

a Banach space

Definition 64 Let 119911lowast(119905) = (119909lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 be a1198621

rd-almost automorphic solution of (2) with initial value120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879 Assume that there existsa positive constant 120582 with minus120582 isin R+ such that for 119905

0isin

[minus120579 0]T there exists119872 gt 1 such that for an arbitrary solution119911(119905) = (119909

1(119905) 119909

2(119905) 119909

119898(119905))

119879 of (2) with initial value120593(119905) =(120593

1(119905) 120593

2(119905) 120593

119898(119905))

119879 119911 satisfies1003816100381610038161003816

1003817100381710038171003817119911(119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161le 119872

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864119890minus120582(119905 119905

0)

119905 isin [minus120579 +infin)T 119905 ge 1199050

(74)

where1003816100381610038161003816

1003817100381710038171003817119911 (119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161

= max1le119894le119898

[1003816100381610038161003816119909119894(119905) minus 119909

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(119909119894minus 119909

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

]

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864

= max1le119894le119898

sup119905isin[minus1205790]T

(1003816100381610038161003816120593119894(119905) minus 120593

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(120593119894minus 120593

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

)

(75)

Then the solution 119911lowast is said to be exponentially stable

In what follows we will give sufficient condition for theexistence of 119862119899

rd-almost automorphic solutions of (2)Let

119864 = 120601 (119905) = (1206011(119905) 120601

2(119905) 120601

119898(119905))

119879

| 120601119894isin 119860119860

(1)

T (R)

119894 = 1 2 119898

(76)

Then it is clear that 119864 endowed with the norm 120601119864

=

max1le119894le119898

1206011198941 where sdot

1is obtained by taking 119899 = 1 in

(37) is a Banach spaceWe make the following assumption

(H1) Assume 119886119894 119887119894119895 119868119894isin 119860119860

(1)

T (R+

) 119905minus120591119894119895isin 119860119860

(1)

T (T)minus119886119894isin

R+ 119894 119895 = 1 2 119898 and min1le119894le119898

inf119905isinT 119886119894(119905) gt 0

(H2) There exists a positive constant 119872119895 119895 = 1 2 119898

such that10038161003816100381610038161003816119891119895(119909)

10038161003816100381610038161003816le 119872

119895 119895 = 1 2 119898 (77)

(H3) 119891119895 isin 119860119860(1)

T (R) 119895 = 1 2 119898 and there exists apositive constant119867119891

119895such that

10038161003816100381610038161003816119891119895(119906) minus 119891

119895(V)

10038161003816100381610038161003816le 119867

119891

119895|119906 minus V|

forall119906 V isin R 119895 = 1 2 119898

(78)

For convenience for a 1198621

rd-almost automorphic function119891 T rarr R we set 119891 = sup

119905isinT |119891(119905)| and by 119891 = inf119905isinT |119891(119905)|

Theorem 65 Assume that (1198671) and (119867

3) hold Assume also

that

max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

lt 1 (79)

Then (2) has a unique 1198621

rd-almost automorphic solution in1198640= 120601(119905) isin 119864 | 120601

119864le 119871 where 119871 is a positive constant

satisfying

119871 = max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) (80)

Proof For any 120593 isin 119864 we consider the following 1198621

rd-almostautomorphic system

119909Δ

119894= minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119905 120593

119895(119905 minus 120591

119894119895(119905)))

+ 119868119894(119905) 119905 isin T 119894 = 1 2 119898

(81)

Since (1198671) holds it follows from Lemma 61 that the system

119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) 119894 = 1 2 119898 (82)

admits an exponential dichotomy on T Thus by Lemma 63system (2) has a 1198621

rd-almost automorphic solution

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(83)

Nowwe prove that the followingmapping is a contractionon 119864

0

Γ 119864 997888rarr 119864

120593 = (1205931 120593

2 120593

119898)119879

997891997888rarr (Γ120593) (119905) = [(Γ120593)1(119905) (Γ120593)

2(119905) (Γ120593)

119898(119905) ]

119879

(84)


where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)

To this end we proceed in two steps

Step 1We prove that if 120593 isin 1198640then (Γ120593) isin 119864

0

Let 120593 isin 1198640 then using (119867

2) and the fact that 119887

119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)

Step 2We prove that Γ is a contraction on 1198640

Let 120593 and 120601 be in 1198640 Then using (119867

3) and the fact that

119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)

Because max1le119894le119898

[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891

119895] lt 1 Γ is a contraction on 119864

0 We then deduce by the

fixed point theorem of Banach that Γ has a unique solution

in 1198640 Consequently system (2) has a unique 119862

1

rd-almostautomorphic solution in 119864

0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621

rd-almost automorphic solution of (2) is globallyexponentially stable

Proof According toTheorem 65 system (2) has a1198621

rd-almostautomorphic solution 119909lowast(119905) = (119909

lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 withinitial value

120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))

119879 is an arbi-trary solution of (2) with initial value 120593(119905) = (120593

1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898

Then it follows from (2) that

119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)

Multiplying both sides of the first equation in (93) by 119890minus119886119894

(119905 120590(119904)) and integrating on [1199050 119905]T where 1199050 isin [minus120579 0]T we

obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)


This means that for 119894 = 1 2 119898

119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)

Then choose 120582 lt min1le119894le119898

119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)

One proves by contradiction as in [8] that1003816100381610038161003816

1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864 forall119905 isin (119905

0 +infin)

T

(97)

This means that the 1198621

rd-almost automorphic solution 119909lowast of

(2) is globally exponentially stable

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referee for hisher carefulreading and valuable suggestions Aril Milce received a PhDscholarship from the French Embassy in Haiti He is alsosupported by ldquoEcole Normale Superieurerdquo an entity of theldquoUniversite drsquoEtat drsquoHaıtirdquo

References

[1] S Hilger Ein Makettenkalkul mit Anwendung auf Zentrums-mannigfaltigkeiten [PhD thesis] Universitat Wurzburg 1988

[2] C Lizama J GMesquita and R Ponce ldquoA connection betweenalmost periodic functions defined on time scales and RrdquoApplicable Analysis vol 93 no 12 pp 2547ndash2558 2014

[3] Y Li and C Wang ldquoAlmost periodic functions on time scalesand applicationsrdquo Discrete Dynamics in Nature and Society vol2011 Article ID 727068 20 pages 2011

[4] Y Li and L Yang ldquoAlmost periodic solutions for neutral-typeBAM neural networks with delays on time scalesrdquo Journal ofApplied Mathematics vol 2013 Article ID 942309 13 pages2013

[5] Y Li and C Wang ldquoUniformly almost periodic functions andalmost periodic solutions to dynamic equations on time scalesrdquoAbstract and Applied Analysis vol 2011 Article ID 341520 22pages 2011

[6] J Zhang M Fan and H Zhu ldquoExistence and roughness ofexponential dichotomies of linear dynamic equations on timescalesrdquo Computers and Mathematics with Applications vol 59no 8 pp 2658ndash2675 2010

[7] V Lakshmikantham and A S Vatsala ldquoHybrid systems on timescalesrdquo Journal of Computational and Applied Mathematics vol141 no 1-2 pp 227ndash235 2002

[8] L Yang Y Li andWWu ldquo119862119899-almost periodic functions and anapplication to a Lasota-Wazewkamodel on time scalesrdquo Journalof Applied Mathematics vol 2014 Article ID 321328 10 pages2014

[9] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001

[10] M Bohner and A PetersonAdvances in Dynamic Equations onTime Scales Birkhauser Boston Mass USA 2003

[11] C Lizama and J G Mesquita ldquoAlmost automorphic solutionsof dynamic equations on time scalesrdquo Journal of FunctionalAnalysis vol 265 no 10 pp 2267ndash2311 2013

[12] GM NrsquoGuerekataAlmost Automorphic and Almost PeriodicityFunctions in Abstract Spaces Kluwer Academic New York NYUSA 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


where T is an appropriate time scale 119898 ge 2 is the numberof neurons in the network 119909

119894(119905) denote the activation of the

119894th neuron at time 119905 and 119886119894represents the rate with which

the 119894th neurons will rest their potential to the resting state inisolation when they are disconnected from the network andthe external inputs at time 119905 The119898times119898matrix 119861(119905) = (119887

119894119895(119905))

represents the connection strengths between neurons at time119905 119887

119894119895is an element of feedback templates at time 119905 119891

119895is the

activation function 120591119894119895is the transmission delay at time 119905 and

satisfies 119905 minus 120591119894119895isin T 119868

119894(119905) denote the bias of the 119894th neuron at

time 119905 120579 = max(119894119895)

sup119905isinT120591119894119895(119905) and [119886 119887]T = 119905 | 119905 isin [119886 119887]capT

We organized our paper as follows In Section 2 we recallsome definitions and recent results on time scales In Sec-tion 3 we present properties of almost automorphic functionson symmetric time scales along with a composition theoremIn Section 4 we introduce and present elementary propertiesof almost automorphic functions on time scales of order 119899and study differentiation and integration of such functionsin Section 5 and superposition of operators on the space ofsuch functions in Section 6 Finally in Section 7 we study theexistence uniqueness and global stability of system (2)

2 Preliminaries

In this section we recall some definitions and recent resultson time scales

Definition 1 A time scale is an arbitrary nonempty closedsubset of real numbers

Definition 2 Let T be a time scaleThe forward and backwardjump operators 120590 120588 T rarr T and the graininess 120583 T rarr

[0 +infin) are defined respectively by

120590 (119905) = inf 119904 isin T 119904 gt 119905 120588 (119905) = sup 119904 isin T 119904 lt 119905

120583 (119905) = 120590 (119905) minus 119905

(3)

In Definition 5 we put inf 0 = sup T and sup 0 = inf T

Definition 3 Let T be time scales

(i) A point 119905 isin T is called right-dense if 119905 lt sup T and120590(119905) = 119905

(ii) A point 119905 isin T is called left-dense if 119905 gt inf T and120588(119905) = 119905

Definition 4 (see [9]) A function 119891 T rarr R is called reg-ulated provided its right-sided limits exist (finite) at all right-dense points in T and its left-sided limits exist (finite) at allleft-dense points in T

Definition 5 (see [9]) A function 119891 T rarr R is called rd-continuous provided it is continuous at right-dense points inT and its left-sided limits exist (finite) at left-dense points inT

We will denote the set of rd-continuous function 119891 T rarr

R by 119862rd = 119862rd(T R) From now on we define the set T119896 by

T119896

=

T [120588 (sup T) sup T] if sup T lt infin

T if sup T = infin

(4)

Definition 6 (see [9]) Let119891 T rarr R be a function and 119905 isin T119896We define119891Δ(119905) to be the number (provided it exists) with theproperty that given any 120598 gt 0 there exists a neighborhood119880of 119905 such that

10038161003816100381610038161003816119891 (120590 (119905)) minus 119891 (119904) minus 119891

Δ

(119905) [120590 (119905) minus 119904]

10038161003816100381610038161003816le 120598 |120590 (119905) minus 119904|


(5)

We call 119891Δ(119905) the delta (or Hilger) derivative of 119891 at 119905Moreover we say that 119891 is delta (or Hilger) differentiable

(or in short differentiable) on T119896 provided 119891Δ(119905) exists for all119905 isin T119896The function119891Δ T119896 rarr R is called the (delta) deriva-tive of 119891 on T119896

Next we recall some easy and useful relationships con-cerning the delta derivative

Theorem 7 (see [9]) Assume 119891 T rarr R is a function and let119905 isin T119896 Then one has the following

(i) If 119891 is differentiable at 119905 then 119891 is continuous at 119905(ii) If 119891 is continuous at 119905 and 119905 is right-scattered then 119891 is

differentiable at 119905 with

119891Δ

(119905) =

119891 (120590 (119905)) minus 119891 (119905)

120583 (119905)

(6)

(iii) If 119905 is right-dense then119891 is differentiable at 119905 if and onlyif the limit

lim119904rarr119905

119891 (119905) minus 119891 (119904)

119905 minus 119904

(7)

exists as a finite number In this case

119891Δ

(119905) = lim119904rarr119905

119891 (119905) minus 119891 (119904)

119905 minus 119904

(8)

(iv) If 119891 is differentiable at 119905 then

119891 (120590 (119905)) = 119891 (119905) + 120583 (119905) 119891Δ

(119905) (9)

Theorem 8 (see [9]) Assume 119891 119892 T rarr R are differentiableat 119905 isin T119896 Then

(i) the sum 119891 + 119892 T rarr R is differentiable at 119905 with

(119891 + 119892)Δ

(119905) = 119891Δ

(119905) + 119892Δ

(119905) (10)

(ii) for any constant 120572 120572119891 T rarr R is differentiable at 119905with

(120572119891)Δ

(119905) = 120572119891Δ

(119905) (11)



(119891119892)Δ

(119905) = 119891Δ

(119905) 119892 (119905) + 119891 (120590 (119905)) 119892Δ

(119905)

= 119891 (119905) 119892Δ

(119905) + 119891Δ

(119905) 119892 (120590 (119905))

(12)



2


Δ

)Δ

T1198962


119899

T119896119899

rarr



0

(119905) = 119905 and T1198960

= T



(119899)

119896 then

(119891119892)Δ119899

=

119899

sum

119896=0

( sum

Λisin119878(119899)

119896

119891Λ

)119892Δ119896

(13)




(119891 ∘ 119892)Δ

(119905) = [int

1

0

1198911015840

(119892 (119905) + ℎ120583 (119905) 119892Δ

(119905)) 119889ℎ] times 119892Δ

(119905)

(14)



Δ


(120596 ∘ ])Δ = ]Δ sdot 120596Δ

∘ ] (15)



int

infin

119886

119891 (119905) Δ119905 = lim119887rarr+infin

int

119887

119886

119891 (119905) Δ119905 (16)




Δ

(119905 120591) [120590 (119905) minus 119904]

10038161003816100381610038161003816le 120598 |120590 (119905) minus 119904|


(17)


119892 (119905) = int

119905

119886

119891 (119905 120591) Δ120591 119894119898119901119897119894119890119904

119892Δ

(119905) = int

119905

119886

119891Δ

(119905 120591) Δ120591 + 119891 (120590 (119905) 119905)

(18)



1 + 120583 (119905) 119901 (119905) = 0 forall119905 isin T119896

(19)


R(T R)


R+

= R+

(T R)



119890119901(119905 119904) = exp(int

119905

119904

120585120583(120591)

(119901 (120591)) Δ120591) for 119905 119904 isin T (21)


ℎrarr Z

ℎis given by

120585ℎ((119911)) =

1

ℎ

log (1 + 119911ℎ) (22)


0(119911) = 119911 for all 119911 isin C



(i) 1198900(119905 119904) equiv 1 and 119890

119901(119905 119905) equiv 1

(ii) 119890119901(119905 119904) = 1119890

119901(119904 119905) = 119890

⊖119901(119904 119905)

(iii) 119890119901(119905 119904)119890

119901(119904 119903) = 119890

119901(119905 119903)

(iv) [119890119901(119905 119904)]

Δ

= 119901(119905)119890119901(119905 119904)



(i) 119890119901(119905 119904) gt 0 for all 119905 119904 isin T


119890119902(119905 119904) for all 119905 ge 119904


(i) [119890119901(119888 sdot)]

Δ

= minus119901[119890119901(119888 sdot)]

120590

(ii) int119887119886

119901(119905)119890119901(119888 120590(119905))Δ119905 = 119890

119901(119888 119886) minus 119890

119901(119888 119887) for all 119905 ge 119904


0isin T and 119910



119910Δ

(119905) = 119901 (119905) 119910 (119905) + ℎ (119905) 119910 (1199050) = 119910

0(23)

is given by

119910 (119905) = 119890119901(119905 119905

0) + int

119905

1199050

119890119901(119905 120590 (119904)) ℎ (119904) Δ119904 (24)



119898times119898



119860 (120590 (119905)) = 119860 (119905) + 120583 (119905) 119860Δ

(119905) (25)



(26)


R (T R119898times119898

) (27)







Proof












119899) such that

119891 (119905) = lim119899rarrinfin

119891 (119905 + 120591119899) (29)


lim119899rarrinfin

119891 (119905 minus 120591119899) = 119891 (119905) (30)















(vi) sup119905isinT119891(119905) = sup

119905isinT119891(119905) where

119891 (119905) = lim119899rarrinfin

119891 (119905 + 120591119899) lim

119899rarrinfin

119891 (119905 minus 120591119899) = 119891 (119905) (31)

Proof See [11]









119891


119899= 119891(119905

1015840

119899)


119899isin Π such that 1199051015840

119899= 119886 + 120572

1015840


have 1199091015840119899= 119891(119886 + 120572

1015840



119899)119899such that

lim119899rarrinfin

119891 (119886 + 120572119899) = 119891 (119886) (32)


119899)) converges to 119891(119886)








119891119899(119905) = 119891(119905)






(120591119899) sub (119904

119899) such that

119891 (119905 119909) = lim119899rarrinfin

119891 (119905 + 120591119899 119909) (33)


lim119899rarrinfin

119891 (119905 minus 120591119899 119909) = 119891 (119905 119909) (34)





(iii) sup119905isinT119891(119905 119909) = 119872


(iv) sup119905isinT119891(119905 119909) = 119873




1003817100381710038171003817119891 (119905 119909) minus 119891 (119905 119910)

1003817100381710038171003817le 119871

1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (35)






Δ119899




sup119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) lt +infin (36)


where119891Δ119894


=

119891 and 119891Δ1


norm

10038171003817100381710038171198911003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) for 119891 isin 119861119899

rd (T X) (37)





10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598

forall119901 isin N

(38)


sup119901isinN

10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup


(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598 (39)


119895(119905)) 119894 = 0 1 119899 are Cauchy


0

119895(119905) we have that

119891Δ119894

119895(119905) 997888rarr 119891

Δ119894

(119905) 119894 = 0 1 119899 (40)

Since for 119894 = 0 1 119899100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817



100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817



10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899le sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817

) le 120598 (43)



119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

le

10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899+

10038171003817100381710038171003817119891119895

10038171003817100381710038171003817119899 forall119905 isin T (44)








119899

rd-aa) if 119891 119891Δ119894


Denote by 119860119860(119899)


119860119860(119899+1)

T (X) sub 119860119860(119899)


(0)

T (X) = 119860119860T (X)

Lemma 44 We have 119860119860(119899)

T (X) sub 119861119899

rd(T X)





(119899)

T (X) ] isin 119860119860(119899)


(119899)

119896and is almost


T (X)

(i) 119891 + 119892

(ii) 120582119891

(iii) ]119891






119886)Δ

(119905) = VΔ(119905) sdot (119891Δ ∘ V)(119905) =119891Δ

(V(119905)) = 119891Δ

(119886 + 119905) = 119891Δ



119886)Δ

= 119891Δ




119886isin 119860119860

119899

T (X)



119901minus 119891


rd-aafunction




119894

119901rarr 119891

Δ119894


rd-aa function

Corollary 47 119860119860(119899)




(119899)


119899



119901minus119891

119899= 0 ByTheorem 46 119891 isin 119860119860

(119899)




119860119860(119899)

T (X) is also a 119862119899

rd-aa function






0 then we have

119891Δ

(1199050) =

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

(46)



lim119899rarrinfin

119891Δ

(1199050+ 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119892 (1199050)

(47)


lim119899rarrinfin

119892 (1199050minus 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119891Δ

(1199050)

(48)


Theorem 49 If 119891 isin 119860119860(119899)

T (X) and 119891Δ119899+1


T (X)




(119899)

T (X)




lim119899rarrinfin

119891 (119905 + 120572119899) = 119892 (119905) pointwise on T (49)

we will write 119879119904119891 = 119892




of 119879119904is 119863(119879

119904) = 119891 T rarr X such that 119879

119904119891 exists

119863(119879119904) is a linear set


119904)

and119879119904119891 isin 119863(119879


119904= 119879

minus119904119879119904119891




119904119891 = 119891



119905

0





119899) sub


119899) such that

lim119899rarrinfin

119891 (119905 + 1199041015840

119899) = 119892 (119905)

lim119899rarrinfin

119892 (119905 minus 1199041015840

119899) = 119891 (119905)

(50)


pointwise on T and

lim119899rarrinfin

119865 (1199041015840

119899) = 120572

1 (51)



119865 (119905 + 1199041015840

119899) = int

119905+1199041015840

119899

0

119891 (119903) Δ119903

= int

1199041015840

119899

0

119891 (119903) Δ119903 + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

= 119865 (1199041015840

119899) + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

(52)


119899 we obtain

119865 (119905 + 1199041015840

119899) = 119865 (119904

1015840

119899) + int

119905

0

119891 (120575 + 1199041015840

119899) Δ120575 (53)


lim119899rarrinfin

119865 (119905 + 1199041015840

119899) = 120572

1+ int

119905

0

119892 (120590) Δ120590 (54)


1+ int

119905

0


sup119905isinT

(119866 (119905)) le sup119905isinT

119865 (119905) (55)


119899) such that

lim119899rarrinfin

119866 (minus119904119899) = 120572

2 (56)


119866 (119905 minus 119904119899) = 119866 (minus119904

119899) + int

119905

0

119892 (119903 minus 119904119899) Δ119903 (57)

so that

lim119899rarrinfin

119866 (119905 minus 119904119899) = 120572

2+ int

119905

0

119891 (119903) Δ119903 = 1205722+ 119865 (119905) (58)


119860119904119865 = 120572

2+ 119865 (59)


2belong




1198602

119904119865 = 119860

1199041205722+ 119860

119904119865 = 120572

2+ 120572

2+ 119865 = 2120572

2+ 119865 (60)


119860119899

119904119865 = 119899120572

2+ 119865 forall119899 = 1 2 (61)

But we have

sup119905isinT

1003817100381710038171003817119860119899

119904119865 (119905)

1003817100381710038171003817le sup

119905isinT

119865 (119905) (62)


2= 0 Hence 120572

2= 0





T (X)


int

119905

0


(1)

T (X)


Theorem 53 If 119891 isin 119860119860(119899)



T (X)

Proof If119891 isin 119860119860(119899)



119891 isin 119860119860(119899)


T (X)




T (X)


119905

0

119891Δ


(119899+1)

T (X)






rator and 119891 isin 119860119860(119899)


(119899)

T (X2)


=

119860119891Δ119899


isin 119860119860T (X1) for



Theorem 37 that 119860119891Δ119894

isin 119860119860T (X2) for all 119894 = 0 1 119899

Hence 119860119891 isin 119860119860(119899)

T (X2)

For every 119891 isin 119860119860(119899)


X2as follows

119866 (119905) = int

119905

0

119860119891 (119904) Δ119904 (63)



the following





T (X2)



T (X2) for

every 119891 isin 119860119860(119899)


the range 119877119860




T (X2)





(RR) and119891 isin 119860119860(1)

T (R) then 120601∘119891 isin

119860119860(1)

T (R)


C1


Δ


(120601 ∘ 119891)Δ

(119905) = [int

1

0

1206011015840

(119891 (119905) + ℎ120583 (119905) 119891Δ

(119905)) 119889ℎ]119891Δ

(119905)

(64)

Since 119891 isin 119860119860(1)




1015840

(119891(119905) + ℎ120583(119905)119891Δ

(119905))


1

0

1206011015840

(119891(119905) + ℎ120583(119905)119891Δ


Δ

isin 119860119860T (R) since119891Δ

isin 119860119860T (R)



119909Δ

(119905) = 119860 (119905) 119909 (119905) 119905 isin T (65)


10038161003816100381610038161003816119883(119905)119875119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119905 119904) 119904 119905 isin T 119905 ge 119904

10038161003816100381610038161003816119883(119905)(119868 minus 119875)119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119904 119905) 119904 119905 isin T 119904 ge 119905

(66)


(119886119894119895)119898times119898


119894=1sum119898

119895=1|119886119894119895|2

)12


119909Δ

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin T (67)




(119905))119905isinT and

(119868 + 120583(119905)119860(119905))minus1




119909 (119905) = int

119905

minusinfin

119883 (119905) 119875119883minus1

(120590 (119904)) 119891 (119904) Δ119904

minus int

infin

119905

119883 (119905) (119868 minus 119875)119883minus1

(120590 (119904)) 119891 (119904) Δ119904

(68)




1le119894le119898inf

119905isinT 119888119894(119905) gt


119909Δ

(119905) = diag (minus1198881(119905) minus119888

2(119905) minus119888

119898(119905)) 119909 (119905) (69)




1(119905) minus119888

2(119905) minus119888

119898(119905))



1le119894le119898inf


119898


119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (70)


1(119905) minus119888

2(119905) minus119888

119898(119905))


are 1198621


1le119894le119898inf

119905isinT




119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (71)

Proof By Lemma 62

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (72)


119909Δ

119894(119905) = minus119888

119894(119905) int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904

+ 119891119894(119905) 119894 = 1 2 119898

(73)



119905

minusinfin

119890minus119888119894

(119905 120590(119904))119891119894(119904)Δ119904 119905 997891rarr 119888

119894(119905) and 119905 997891rarr 119891



119894(119905) 119894 = 1 2 119898 are also 119862rd-





Set 119864 = 120595 = (1205951 120595

2 120595

119898)119879

| 120595119894isin 119860119860

(1)

T (R) 119894 =


1le119894le119898120595

1 119864 is

a Banach space


1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 be a1198621


(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))


0isin


1(119905) 119909

2(119905) 119909

119898(119905))


1(119905) 120593

2(119905) 120593

119898(119905))

119879 119911 satisfies1003816100381610038161003816

1003817100381710038171003817119911(119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161le 119872

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864119890minus120582(119905 119905

0)


(74)

where1003816100381610038161003816

1003817100381710038171003817119911 (119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161

= max1le119894le119898

[1003816100381610038161003816119909119894(119905) minus 119909

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(119909119894minus 119909

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

]

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864

= max1le119894le119898


(1003816100381610038161003816120593119894(119905) minus 120593

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(120593119894minus 120593

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

)

(75)




119864 = 120601 (119905) = (1206011(119905) 120601

2(119905) 120601

119898(119905))

119879

| 120601119894isin 119860119860

(1)

T (R)

119894 = 1 2 119898

(76)


=

max1le119894le119898

1206011198941 where sdot



(H1) Assume 119886119894 119887119894119895 119868119894isin 119860119860

(1)

T (R+

) 119905minus120591119894119895isin 119860119860

(1)


R+ 119894 119895 = 1 2 119898 and min1le119894le119898

inf119905isinT 119886119894(119905) gt 0


such that10038161003816100381610038161003816119891119895(119909)

10038161003816100381610038161003816le 119872

119895 119895 = 1 2 119898 (77)

(H3) 119891119895 isin 119860119860(1)


119895such that

10038161003816100381610038161003816119891119895(119906) minus 119891

119895(V)

10038161003816100381610038161003816le 119867

119891

119895|119906 minus V|

forall119906 V isin R 119895 = 1 2 119898

(78)





3) hold Assume also

that

max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

lt 1 (79)




satisfying

119871 = max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) (80)



119909Δ

119894= minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119905 120593

119895(119905 minus 120591

119894119895(119905)))

+ 119868119894(119905) 119905 isin T 119894 = 1 2 119898

(81)


119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) 119894 = 1 2 119898 (82)



119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(83)


0

Γ 119864 997888rarr 119864

120593 = (1205931 120593

2 120593

119898)119879

997891997888rarr (Γ120593) (119905) = [(Γ120593)1(119905) (Γ120593)

2(119905) (Γ120593)

119898(119905) ]

119879

(84)


where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)



0



119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)




119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119891119892)Δ

(119905) = 119891Δ

(119905) 119892 (119905) + 119891 (120590 (119905)) 119892Δ

(119905)

= 119891 (119905) 119892Δ

(119905) + 119891Δ

(119905) 119892 (120590 (119905))

(12)



2


Δ

)Δ

T1198962


119899

T119896119899

rarr



0

(119905) = 119905 and T1198960

= T



(119899)

119896 then

(119891119892)Δ119899

=

119899

sum

119896=0

( sum

Λisin119878(119899)

119896

119891Λ

)119892Δ119896

(13)




(119891 ∘ 119892)Δ

(119905) = [int

1

0

1198911015840

(119892 (119905) + ℎ120583 (119905) 119892Δ

(119905)) 119889ℎ] times 119892Δ

(119905)

(14)



Δ


(120596 ∘ ])Δ = ]Δ sdot 120596Δ

∘ ] (15)



int

infin

119886

119891 (119905) Δ119905 = lim119887rarr+infin

int

119887

119886

119891 (119905) Δ119905 (16)




Δ

(119905 120591) [120590 (119905) minus 119904]

10038161003816100381610038161003816le 120598 |120590 (119905) minus 119904|


(17)


119892 (119905) = int

119905

119886

119891 (119905 120591) Δ120591 119894119898119901119897119894119890119904

119892Δ

(119905) = int

119905

119886

119891Δ

(119905 120591) Δ120591 + 119891 (120590 (119905) 119905)

(18)



1 + 120583 (119905) 119901 (119905) = 0 forall119905 isin T119896

(19)


R(T R)


R+

= R+

(T R)



119890119901(119905 119904) = exp(int

119905

119904

120585120583(120591)

(119901 (120591)) Δ120591) for 119905 119904 isin T (21)


ℎrarr Z

ℎis given by

120585ℎ((119911)) =

1

ℎ

log (1 + 119911ℎ) (22)


0(119911) = 119911 for all 119911 isin C



(i) 1198900(119905 119904) equiv 1 and 119890

119901(119905 119905) equiv 1

(ii) 119890119901(119905 119904) = 1119890

119901(119904 119905) = 119890

⊖119901(119904 119905)

(iii) 119890119901(119905 119904)119890

119901(119904 119903) = 119890

119901(119905 119903)

(iv) [119890119901(119905 119904)]

Δ

= 119901(119905)119890119901(119905 119904)



(i) 119890119901(119905 119904) gt 0 for all 119905 119904 isin T


119890119902(119905 119904) for all 119905 ge 119904


(i) [119890119901(119888 sdot)]

Δ

= minus119901[119890119901(119888 sdot)]

120590

(ii) int119887119886

119901(119905)119890119901(119888 120590(119905))Δ119905 = 119890

119901(119888 119886) minus 119890

119901(119888 119887) for all 119905 ge 119904


0isin T and 119910



119910Δ

(119905) = 119901 (119905) 119910 (119905) + ℎ (119905) 119910 (1199050) = 119910

0(23)

is given by

119910 (119905) = 119890119901(119905 119905

0) + int

119905

1199050

119890119901(119905 120590 (119904)) ℎ (119904) Δ119904 (24)



119898times119898



119860 (120590 (119905)) = 119860 (119905) + 120583 (119905) 119860Δ

(119905) (25)



(26)


R (T R119898times119898

) (27)







Proof












119899) such that

119891 (119905) = lim119899rarrinfin

119891 (119905 + 120591119899) (29)


lim119899rarrinfin

119891 (119905 minus 120591119899) = 119891 (119905) (30)















(vi) sup119905isinT119891(119905) = sup

119905isinT119891(119905) where

119891 (119905) = lim119899rarrinfin

119891 (119905 + 120591119899) lim

119899rarrinfin

119891 (119905 minus 120591119899) = 119891 (119905) (31)

Proof See [11]









119891


119899= 119891(119905

1015840

119899)


119899isin Π such that 1199051015840

119899= 119886 + 120572

1015840


have 1199091015840119899= 119891(119886 + 120572

1015840



119899)119899such that

lim119899rarrinfin

119891 (119886 + 120572119899) = 119891 (119886) (32)


119899)) converges to 119891(119886)








119891119899(119905) = 119891(119905)






(120591119899) sub (119904

119899) such that

119891 (119905 119909) = lim119899rarrinfin

119891 (119905 + 120591119899 119909) (33)


lim119899rarrinfin

119891 (119905 minus 120591119899 119909) = 119891 (119905 119909) (34)





(iii) sup119905isinT119891(119905 119909) = 119872


(iv) sup119905isinT119891(119905 119909) = 119873




1003817100381710038171003817119891 (119905 119909) minus 119891 (119905 119910)

1003817100381710038171003817le 119871

1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (35)






Δ119899




sup119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) lt +infin (36)


where119891Δ119894


=

119891 and 119891Δ1


norm

10038171003817100381710038171198911003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) for 119891 isin 119861119899

rd (T X) (37)





10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598

forall119901 isin N

(38)


sup119901isinN

10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup


(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598 (39)


119895(119905)) 119894 = 0 1 119899 are Cauchy


0

119895(119905) we have that

119891Δ119894

119895(119905) 997888rarr 119891

Δ119894

(119905) 119894 = 0 1 119899 (40)

Since for 119894 = 0 1 119899100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817



100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817



10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899le sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817

) le 120598 (43)



119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

le

10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899+

10038171003817100381710038171003817119891119895

10038171003817100381710038171003817119899 forall119905 isin T (44)








119899

rd-aa) if 119891 119891Δ119894


Denote by 119860119860(119899)


119860119860(119899+1)

T (X) sub 119860119860(119899)


(0)

T (X) = 119860119860T (X)

Lemma 44 We have 119860119860(119899)

T (X) sub 119861119899

rd(T X)





(119899)

T (X) ] isin 119860119860(119899)


(119899)

119896and is almost


T (X)

(i) 119891 + 119892

(ii) 120582119891

(iii) ]119891






119886)Δ

(119905) = VΔ(119905) sdot (119891Δ ∘ V)(119905) =119891Δ

(V(119905)) = 119891Δ

(119886 + 119905) = 119891Δ



119886)Δ

= 119891Δ




119886isin 119860119860

119899

T (X)



119901minus 119891


rd-aafunction




119894

119901rarr 119891

Δ119894


rd-aa function

Corollary 47 119860119860(119899)




(119899)


119899



119901minus119891

119899= 0 ByTheorem 46 119891 isin 119860119860

(119899)




119860119860(119899)

T (X) is also a 119862119899

rd-aa function






0 then we have

119891Δ

(1199050) =

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

(46)



lim119899rarrinfin

119891Δ

(1199050+ 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119892 (1199050)

(47)


lim119899rarrinfin

119892 (1199050minus 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119891Δ

(1199050)

(48)


Theorem 49 If 119891 isin 119860119860(119899)

T (X) and 119891Δ119899+1


T (X)




(119899)

T (X)




lim119899rarrinfin

119891 (119905 + 120572119899) = 119892 (119905) pointwise on T (49)

we will write 119879119904119891 = 119892




of 119879119904is 119863(119879

119904) = 119891 T rarr X such that 119879

119904119891 exists

119863(119879119904) is a linear set


119904)

and119879119904119891 isin 119863(119879


119904= 119879

minus119904119879119904119891




119904119891 = 119891



119905

0





119899) sub


119899) such that

lim119899rarrinfin

119891 (119905 + 1199041015840

119899) = 119892 (119905)

lim119899rarrinfin

119892 (119905 minus 1199041015840

119899) = 119891 (119905)

(50)


pointwise on T and

lim119899rarrinfin

119865 (1199041015840

119899) = 120572

1 (51)



119865 (119905 + 1199041015840

119899) = int

119905+1199041015840

119899

0

119891 (119903) Δ119903

= int

1199041015840

119899

0

119891 (119903) Δ119903 + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

= 119865 (1199041015840

119899) + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

(52)


119899 we obtain

119865 (119905 + 1199041015840

119899) = 119865 (119904

1015840

119899) + int

119905

0

119891 (120575 + 1199041015840

119899) Δ120575 (53)


lim119899rarrinfin

119865 (119905 + 1199041015840

119899) = 120572

1+ int

119905

0

119892 (120590) Δ120590 (54)


1+ int

119905

0


sup119905isinT

(119866 (119905)) le sup119905isinT

119865 (119905) (55)


119899) such that

lim119899rarrinfin

119866 (minus119904119899) = 120572

2 (56)


119866 (119905 minus 119904119899) = 119866 (minus119904

119899) + int

119905

0

119892 (119903 minus 119904119899) Δ119903 (57)

so that

lim119899rarrinfin

119866 (119905 minus 119904119899) = 120572

2+ int

119905

0

119891 (119903) Δ119903 = 1205722+ 119865 (119905) (58)


119860119904119865 = 120572

2+ 119865 (59)


2belong




1198602

119904119865 = 119860

1199041205722+ 119860

119904119865 = 120572

2+ 120572

2+ 119865 = 2120572

2+ 119865 (60)


119860119899

119904119865 = 119899120572

2+ 119865 forall119899 = 1 2 (61)

But we have

sup119905isinT

1003817100381710038171003817119860119899

119904119865 (119905)

1003817100381710038171003817le sup

119905isinT

119865 (119905) (62)


2= 0 Hence 120572

2= 0





T (X)


int

119905

0


(1)

T (X)


Theorem 53 If 119891 isin 119860119860(119899)



T (X)

Proof If119891 isin 119860119860(119899)



119891 isin 119860119860(119899)


T (X)




T (X)


119905

0

119891Δ


(119899+1)

T (X)






rator and 119891 isin 119860119860(119899)


(119899)

T (X2)


=

119860119891Δ119899


isin 119860119860T (X1) for



Theorem 37 that 119860119891Δ119894

isin 119860119860T (X2) for all 119894 = 0 1 119899

Hence 119860119891 isin 119860119860(119899)

T (X2)

For every 119891 isin 119860119860(119899)


X2as follows

119866 (119905) = int

119905

0

119860119891 (119904) Δ119904 (63)



the following





T (X2)



T (X2) for

every 119891 isin 119860119860(119899)


the range 119877119860




T (X2)





(RR) and119891 isin 119860119860(1)

T (R) then 120601∘119891 isin

119860119860(1)

T (R)


C1


Δ


(120601 ∘ 119891)Δ

(119905) = [int

1

0

1206011015840

(119891 (119905) + ℎ120583 (119905) 119891Δ

(119905)) 119889ℎ]119891Δ

(119905)

(64)

Since 119891 isin 119860119860(1)




1015840

(119891(119905) + ℎ120583(119905)119891Δ

(119905))


1

0

1206011015840

(119891(119905) + ℎ120583(119905)119891Δ


Δ

isin 119860119860T (R) since119891Δ

isin 119860119860T (R)



119909Δ

(119905) = 119860 (119905) 119909 (119905) 119905 isin T (65)


10038161003816100381610038161003816119883(119905)119875119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119905 119904) 119904 119905 isin T 119905 ge 119904

10038161003816100381610038161003816119883(119905)(119868 minus 119875)119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119904 119905) 119904 119905 isin T 119904 ge 119905

(66)


(119886119894119895)119898times119898


119894=1sum119898

119895=1|119886119894119895|2

)12


119909Δ

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin T (67)




(119905))119905isinT and

(119868 + 120583(119905)119860(119905))minus1




119909 (119905) = int

119905

minusinfin

119883 (119905) 119875119883minus1

(120590 (119904)) 119891 (119904) Δ119904

minus int

infin

119905

119883 (119905) (119868 minus 119875)119883minus1

(120590 (119904)) 119891 (119904) Δ119904

(68)




1le119894le119898inf

119905isinT 119888119894(119905) gt


119909Δ

(119905) = diag (minus1198881(119905) minus119888

2(119905) minus119888

119898(119905)) 119909 (119905) (69)




1(119905) minus119888

2(119905) minus119888

119898(119905))



1le119894le119898inf


119898


119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (70)


1(119905) minus119888

2(119905) minus119888

119898(119905))


are 1198621


1le119894le119898inf

119905isinT




119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (71)

Proof By Lemma 62

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (72)


119909Δ

119894(119905) = minus119888

119894(119905) int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904

+ 119891119894(119905) 119894 = 1 2 119898

(73)



119905

minusinfin

119890minus119888119894

(119905 120590(119904))119891119894(119904)Δ119904 119905 997891rarr 119888

119894(119905) and 119905 997891rarr 119891



119894(119905) 119894 = 1 2 119898 are also 119862rd-





Set 119864 = 120595 = (1205951 120595

2 120595

119898)119879

| 120595119894isin 119860119860

(1)

T (R) 119894 =


1le119894le119898120595

1 119864 is

a Banach space


1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 be a1198621


(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))


0isin


1(119905) 119909

2(119905) 119909

119898(119905))


1(119905) 120593

2(119905) 120593

119898(119905))

119879 119911 satisfies1003816100381610038161003816

1003817100381710038171003817119911(119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161le 119872

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864119890minus120582(119905 119905

0)


(74)

where1003816100381610038161003816

1003817100381710038171003817119911 (119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161

= max1le119894le119898

[1003816100381610038161003816119909119894(119905) minus 119909

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(119909119894minus 119909

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

]

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864

= max1le119894le119898


(1003816100381610038161003816120593119894(119905) minus 120593

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(120593119894minus 120593

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

)

(75)




119864 = 120601 (119905) = (1206011(119905) 120601

2(119905) 120601

119898(119905))

119879

| 120601119894isin 119860119860

(1)

T (R)

119894 = 1 2 119898

(76)


=

max1le119894le119898

1206011198941 where sdot



(H1) Assume 119886119894 119887119894119895 119868119894isin 119860119860

(1)

T (R+

) 119905minus120591119894119895isin 119860119860

(1)


R+ 119894 119895 = 1 2 119898 and min1le119894le119898

inf119905isinT 119886119894(119905) gt 0


such that10038161003816100381610038161003816119891119895(119909)

10038161003816100381610038161003816le 119872

119895 119895 = 1 2 119898 (77)

(H3) 119891119895 isin 119860119860(1)


119895such that

10038161003816100381610038161003816119891119895(119906) minus 119891

119895(V)

10038161003816100381610038161003816le 119867

119891

119895|119906 minus V|

forall119906 V isin R 119895 = 1 2 119898

(78)





3) hold Assume also

that

max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

lt 1 (79)




satisfying

119871 = max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) (80)



119909Δ

119894= minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119905 120593

119895(119905 minus 120591

119894119895(119905)))

+ 119868119894(119905) 119905 isin T 119894 = 1 2 119898

(81)


119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) 119894 = 1 2 119898 (82)



119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(83)


0

Γ 119864 997888rarr 119864

120593 = (1205931 120593

2 120593

119898)119879

997891997888rarr (Γ120593) (119905) = [(Γ120593)1(119905) (Γ120593)

2(119905) (Γ120593)

119898(119905) ]

119879

(84)


where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)



0



119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)




119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(i) 119890119901(119905 119904) gt 0 for all 119905 119904 isin T


119890119902(119905 119904) for all 119905 ge 119904


(i) [119890119901(119888 sdot)]

Δ

= minus119901[119890119901(119888 sdot)]

120590

(ii) int119887119886

119901(119905)119890119901(119888 120590(119905))Δ119905 = 119890

119901(119888 119886) minus 119890

119901(119888 119887) for all 119905 ge 119904


0isin T and 119910



119910Δ

(119905) = 119901 (119905) 119910 (119905) + ℎ (119905) 119910 (1199050) = 119910

0(23)

is given by

119910 (119905) = 119890119901(119905 119905

0) + int

119905

1199050

119890119901(119905 120590 (119904)) ℎ (119904) Δ119904 (24)



119898times119898



119860 (120590 (119905)) = 119860 (119905) + 120583 (119905) 119860Δ

(119905) (25)



(26)


R (T R119898times119898

) (27)







Proof












119899) such that

119891 (119905) = lim119899rarrinfin

119891 (119905 + 120591119899) (29)


lim119899rarrinfin

119891 (119905 minus 120591119899) = 119891 (119905) (30)















(vi) sup119905isinT119891(119905) = sup

119905isinT119891(119905) where

119891 (119905) = lim119899rarrinfin

119891 (119905 + 120591119899) lim

119899rarrinfin

119891 (119905 minus 120591119899) = 119891 (119905) (31)

Proof See [11]









119891


119899= 119891(119905

1015840

119899)


119899isin Π such that 1199051015840

119899= 119886 + 120572

1015840


have 1199091015840119899= 119891(119886 + 120572

1015840



119899)119899such that

lim119899rarrinfin

119891 (119886 + 120572119899) = 119891 (119886) (32)


119899)) converges to 119891(119886)








119891119899(119905) = 119891(119905)






(120591119899) sub (119904

119899) such that

119891 (119905 119909) = lim119899rarrinfin

119891 (119905 + 120591119899 119909) (33)


lim119899rarrinfin

119891 (119905 minus 120591119899 119909) = 119891 (119905 119909) (34)





(iii) sup119905isinT119891(119905 119909) = 119872


(iv) sup119905isinT119891(119905 119909) = 119873




1003817100381710038171003817119891 (119905 119909) minus 119891 (119905 119910)

1003817100381710038171003817le 119871

1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (35)






Δ119899




sup119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) lt +infin (36)


where119891Δ119894


=

119891 and 119891Δ1


norm

10038171003817100381710038171198911003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) for 119891 isin 119861119899

rd (T X) (37)





10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598

forall119901 isin N

(38)


sup119901isinN

10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup


(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598 (39)


119895(119905)) 119894 = 0 1 119899 are Cauchy


0

119895(119905) we have that

119891Δ119894

119895(119905) 997888rarr 119891

Δ119894

(119905) 119894 = 0 1 119899 (40)

Since for 119894 = 0 1 119899100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817



100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817



10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899le sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817

) le 120598 (43)



119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

le

10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899+

10038171003817100381710038171003817119891119895

10038171003817100381710038171003817119899 forall119905 isin T (44)








119899

rd-aa) if 119891 119891Δ119894


Denote by 119860119860(119899)


119860119860(119899+1)

T (X) sub 119860119860(119899)


(0)

T (X) = 119860119860T (X)

Lemma 44 We have 119860119860(119899)

T (X) sub 119861119899

rd(T X)





(119899)

T (X) ] isin 119860119860(119899)


(119899)

119896and is almost


T (X)

(i) 119891 + 119892

(ii) 120582119891

(iii) ]119891






119886)Δ

(119905) = VΔ(119905) sdot (119891Δ ∘ V)(119905) =119891Δ

(V(119905)) = 119891Δ

(119886 + 119905) = 119891Δ



119886)Δ

= 119891Δ




119886isin 119860119860

119899

T (X)



119901minus 119891


rd-aafunction




119894

119901rarr 119891

Δ119894


rd-aa function

Corollary 47 119860119860(119899)




(119899)


119899



119901minus119891

119899= 0 ByTheorem 46 119891 isin 119860119860

(119899)




119860119860(119899)

T (X) is also a 119862119899

rd-aa function






0 then we have

119891Δ

(1199050) =

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

(46)



lim119899rarrinfin

119891Δ

(1199050+ 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119892 (1199050)

(47)


lim119899rarrinfin

119892 (1199050minus 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119891Δ

(1199050)

(48)


Theorem 49 If 119891 isin 119860119860(119899)

T (X) and 119891Δ119899+1


T (X)




(119899)

T (X)




lim119899rarrinfin

119891 (119905 + 120572119899) = 119892 (119905) pointwise on T (49)

we will write 119879119904119891 = 119892




of 119879119904is 119863(119879

119904) = 119891 T rarr X such that 119879

119904119891 exists

119863(119879119904) is a linear set


119904)

and119879119904119891 isin 119863(119879


119904= 119879

minus119904119879119904119891




119904119891 = 119891



119905

0





119899) sub


119899) such that

lim119899rarrinfin

119891 (119905 + 1199041015840

119899) = 119892 (119905)

lim119899rarrinfin

119892 (119905 minus 1199041015840

119899) = 119891 (119905)

(50)


pointwise on T and

lim119899rarrinfin

119865 (1199041015840

119899) = 120572

1 (51)



119865 (119905 + 1199041015840

119899) = int

119905+1199041015840

119899

0

119891 (119903) Δ119903

= int

1199041015840

119899

0

119891 (119903) Δ119903 + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

= 119865 (1199041015840

119899) + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

(52)


119899 we obtain

119865 (119905 + 1199041015840

119899) = 119865 (119904

1015840

119899) + int

119905

0

119891 (120575 + 1199041015840

119899) Δ120575 (53)


lim119899rarrinfin

119865 (119905 + 1199041015840

119899) = 120572

1+ int

119905

0

119892 (120590) Δ120590 (54)


1+ int

119905

0


sup119905isinT

(119866 (119905)) le sup119905isinT

119865 (119905) (55)


119899) such that

lim119899rarrinfin

119866 (minus119904119899) = 120572

2 (56)


119866 (119905 minus 119904119899) = 119866 (minus119904

119899) + int

119905

0

119892 (119903 minus 119904119899) Δ119903 (57)

so that

lim119899rarrinfin

119866 (119905 minus 119904119899) = 120572

2+ int

119905

0

119891 (119903) Δ119903 = 1205722+ 119865 (119905) (58)


119860119904119865 = 120572

2+ 119865 (59)


2belong




1198602

119904119865 = 119860

1199041205722+ 119860

119904119865 = 120572

2+ 120572

2+ 119865 = 2120572

2+ 119865 (60)


119860119899

119904119865 = 119899120572

2+ 119865 forall119899 = 1 2 (61)

But we have

sup119905isinT

1003817100381710038171003817119860119899

119904119865 (119905)

1003817100381710038171003817le sup

119905isinT

119865 (119905) (62)


2= 0 Hence 120572

2= 0





T (X)


int

119905

0


(1)

T (X)


Theorem 53 If 119891 isin 119860119860(119899)



T (X)

Proof If119891 isin 119860119860(119899)



119891 isin 119860119860(119899)


T (X)




T (X)


119905

0

119891Δ


(119899+1)

T (X)






rator and 119891 isin 119860119860(119899)


(119899)

T (X2)


=

119860119891Δ119899


isin 119860119860T (X1) for



Theorem 37 that 119860119891Δ119894

isin 119860119860T (X2) for all 119894 = 0 1 119899

Hence 119860119891 isin 119860119860(119899)

T (X2)

For every 119891 isin 119860119860(119899)


X2as follows

119866 (119905) = int

119905

0

119860119891 (119904) Δ119904 (63)



the following





T (X2)



T (X2) for

every 119891 isin 119860119860(119899)


the range 119877119860




T (X2)





(RR) and119891 isin 119860119860(1)

T (R) then 120601∘119891 isin

119860119860(1)

T (R)


C1


Δ


(120601 ∘ 119891)Δ

(119905) = [int

1

0

1206011015840

(119891 (119905) + ℎ120583 (119905) 119891Δ

(119905)) 119889ℎ]119891Δ

(119905)

(64)

Since 119891 isin 119860119860(1)




1015840

(119891(119905) + ℎ120583(119905)119891Δ

(119905))


1

0

1206011015840

(119891(119905) + ℎ120583(119905)119891Δ


Δ

isin 119860119860T (R) since119891Δ

isin 119860119860T (R)



119909Δ

(119905) = 119860 (119905) 119909 (119905) 119905 isin T (65)


10038161003816100381610038161003816119883(119905)119875119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119905 119904) 119904 119905 isin T 119905 ge 119904

10038161003816100381610038161003816119883(119905)(119868 minus 119875)119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119904 119905) 119904 119905 isin T 119904 ge 119905

(66)


(119886119894119895)119898times119898


119894=1sum119898

119895=1|119886119894119895|2

)12


119909Δ

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin T (67)




(119905))119905isinT and

(119868 + 120583(119905)119860(119905))minus1




119909 (119905) = int

119905

minusinfin

119883 (119905) 119875119883minus1

(120590 (119904)) 119891 (119904) Δ119904

minus int

infin

119905

119883 (119905) (119868 minus 119875)119883minus1

(120590 (119904)) 119891 (119904) Δ119904

(68)




1le119894le119898inf

119905isinT 119888119894(119905) gt


119909Δ

(119905) = diag (minus1198881(119905) minus119888

2(119905) minus119888

119898(119905)) 119909 (119905) (69)




1(119905) minus119888

2(119905) minus119888

119898(119905))



1le119894le119898inf


119898


119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (70)


1(119905) minus119888

2(119905) minus119888

119898(119905))


are 1198621


1le119894le119898inf

119905isinT




119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (71)

Proof By Lemma 62

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (72)


119909Δ

119894(119905) = minus119888

119894(119905) int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904

+ 119891119894(119905) 119894 = 1 2 119898

(73)



119905

minusinfin

119890minus119888119894

(119905 120590(119904))119891119894(119904)Δ119904 119905 997891rarr 119888

119894(119905) and 119905 997891rarr 119891



119894(119905) 119894 = 1 2 119898 are also 119862rd-





Set 119864 = 120595 = (1205951 120595

2 120595

119898)119879

| 120595119894isin 119860119860

(1)

T (R) 119894 =


1le119894le119898120595

1 119864 is

a Banach space


1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 be a1198621


(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))


0isin


1(119905) 119909

2(119905) 119909

119898(119905))


1(119905) 120593

2(119905) 120593

119898(119905))

119879 119911 satisfies1003816100381610038161003816

1003817100381710038171003817119911(119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161le 119872

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864119890minus120582(119905 119905

0)


(74)

where1003816100381610038161003816

1003817100381710038171003817119911 (119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161

= max1le119894le119898

[1003816100381610038161003816119909119894(119905) minus 119909

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(119909119894minus 119909

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

]

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864

= max1le119894le119898


(1003816100381610038161003816120593119894(119905) minus 120593

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(120593119894minus 120593

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

)

(75)




119864 = 120601 (119905) = (1206011(119905) 120601

2(119905) 120601

119898(119905))

119879

| 120601119894isin 119860119860

(1)

T (R)

119894 = 1 2 119898

(76)


=

max1le119894le119898

1206011198941 where sdot



(H1) Assume 119886119894 119887119894119895 119868119894isin 119860119860

(1)

T (R+

) 119905minus120591119894119895isin 119860119860

(1)


R+ 119894 119895 = 1 2 119898 and min1le119894le119898

inf119905isinT 119886119894(119905) gt 0


such that10038161003816100381610038161003816119891119895(119909)

10038161003816100381610038161003816le 119872

119895 119895 = 1 2 119898 (77)

(H3) 119891119895 isin 119860119860(1)


119895such that

10038161003816100381610038161003816119891119895(119906) minus 119891

119895(V)

10038161003816100381610038161003816le 119867

119891

119895|119906 minus V|

forall119906 V isin R 119895 = 1 2 119898

(78)





3) hold Assume also

that

max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

lt 1 (79)




satisfying

119871 = max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) (80)



119909Δ

119894= minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119905 120593

119895(119905 minus 120591

119894119895(119905)))

+ 119868119894(119905) 119905 isin T 119894 = 1 2 119898

(81)


119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) 119894 = 1 2 119898 (82)



119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(83)


0

Γ 119864 997888rarr 119864

120593 = (1205931 120593

2 120593

119898)119879

997891997888rarr (Γ120593) (119905) = [(Γ120593)1(119905) (Γ120593)

2(119905) (Γ120593)

119898(119905) ]

119879

(84)


where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)



0



119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)




119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















(vi) sup119905isinT119891(119905) = sup

119905isinT119891(119905) where

119891 (119905) = lim119899rarrinfin

119891 (119905 + 120591119899) lim

119899rarrinfin

119891 (119905 minus 120591119899) = 119891 (119905) (31)

Proof See [11]









119891


119899= 119891(119905

1015840

119899)


119899isin Π such that 1199051015840

119899= 119886 + 120572

1015840


have 1199091015840119899= 119891(119886 + 120572

1015840



119899)119899such that

lim119899rarrinfin

119891 (119886 + 120572119899) = 119891 (119886) (32)


119899)) converges to 119891(119886)








119891119899(119905) = 119891(119905)






(120591119899) sub (119904

119899) such that

119891 (119905 119909) = lim119899rarrinfin

119891 (119905 + 120591119899 119909) (33)


lim119899rarrinfin

119891 (119905 minus 120591119899 119909) = 119891 (119905 119909) (34)





(iii) sup119905isinT119891(119905 119909) = 119872


(iv) sup119905isinT119891(119905 119909) = 119873




1003817100381710038171003817119891 (119905 119909) minus 119891 (119905 119910)

1003817100381710038171003817le 119871

1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (35)






Δ119899




sup119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) lt +infin (36)


where119891Δ119894


=

119891 and 119891Δ1


norm

10038171003817100381710038171198911003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) for 119891 isin 119861119899

rd (T X) (37)





10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598

forall119901 isin N

(38)


sup119901isinN

10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup


(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598 (39)


119895(119905)) 119894 = 0 1 119899 are Cauchy


0

119895(119905) we have that

119891Δ119894

119895(119905) 997888rarr 119891

Δ119894

(119905) 119894 = 0 1 119899 (40)

Since for 119894 = 0 1 119899100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817



100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817



10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899le sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817

) le 120598 (43)



119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

le

10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899+

10038171003817100381710038171003817119891119895

10038171003817100381710038171003817119899 forall119905 isin T (44)








119899

rd-aa) if 119891 119891Δ119894


Denote by 119860119860(119899)


119860119860(119899+1)

T (X) sub 119860119860(119899)


(0)

T (X) = 119860119860T (X)

Lemma 44 We have 119860119860(119899)

T (X) sub 119861119899

rd(T X)





(119899)

T (X) ] isin 119860119860(119899)


(119899)

119896and is almost


T (X)

(i) 119891 + 119892

(ii) 120582119891

(iii) ]119891






119886)Δ

(119905) = VΔ(119905) sdot (119891Δ ∘ V)(119905) =119891Δ

(V(119905)) = 119891Δ

(119886 + 119905) = 119891Δ



119886)Δ

= 119891Δ




119886isin 119860119860

119899

T (X)



119901minus 119891


rd-aafunction




119894

119901rarr 119891

Δ119894


rd-aa function

Corollary 47 119860119860(119899)




(119899)


119899



119901minus119891

119899= 0 ByTheorem 46 119891 isin 119860119860

(119899)




119860119860(119899)

T (X) is also a 119862119899

rd-aa function






0 then we have

119891Δ

(1199050) =

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

(46)



lim119899rarrinfin

119891Δ

(1199050+ 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119892 (1199050)

(47)


lim119899rarrinfin

119892 (1199050minus 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119891Δ

(1199050)

(48)


Theorem 49 If 119891 isin 119860119860(119899)

T (X) and 119891Δ119899+1


T (X)




(119899)

T (X)




lim119899rarrinfin

119891 (119905 + 120572119899) = 119892 (119905) pointwise on T (49)

we will write 119879119904119891 = 119892




of 119879119904is 119863(119879

119904) = 119891 T rarr X such that 119879

119904119891 exists

119863(119879119904) is a linear set


119904)

and119879119904119891 isin 119863(119879


119904= 119879

minus119904119879119904119891




119904119891 = 119891



119905

0





119899) sub


119899) such that

lim119899rarrinfin

119891 (119905 + 1199041015840

119899) = 119892 (119905)

lim119899rarrinfin

119892 (119905 minus 1199041015840

119899) = 119891 (119905)

(50)


pointwise on T and

lim119899rarrinfin

119865 (1199041015840

119899) = 120572

1 (51)



119865 (119905 + 1199041015840

119899) = int

119905+1199041015840

119899

0

119891 (119903) Δ119903

= int

1199041015840

119899

0

119891 (119903) Δ119903 + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

= 119865 (1199041015840

119899) + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

(52)


119899 we obtain

119865 (119905 + 1199041015840

119899) = 119865 (119904

1015840

119899) + int

119905

0

119891 (120575 + 1199041015840

119899) Δ120575 (53)


lim119899rarrinfin

119865 (119905 + 1199041015840

119899) = 120572

1+ int

119905

0

119892 (120590) Δ120590 (54)


1+ int

119905

0


sup119905isinT

(119866 (119905)) le sup119905isinT

119865 (119905) (55)


119899) such that

lim119899rarrinfin

119866 (minus119904119899) = 120572

2 (56)


119866 (119905 minus 119904119899) = 119866 (minus119904

119899) + int

119905

0

119892 (119903 minus 119904119899) Δ119903 (57)

so that

lim119899rarrinfin

119866 (119905 minus 119904119899) = 120572

2+ int

119905

0

119891 (119903) Δ119903 = 1205722+ 119865 (119905) (58)


119860119904119865 = 120572

2+ 119865 (59)


2belong




1198602

119904119865 = 119860

1199041205722+ 119860

119904119865 = 120572

2+ 120572

2+ 119865 = 2120572

2+ 119865 (60)


119860119899

119904119865 = 119899120572

2+ 119865 forall119899 = 1 2 (61)

But we have

sup119905isinT

1003817100381710038171003817119860119899

119904119865 (119905)

1003817100381710038171003817le sup

119905isinT

119865 (119905) (62)


2= 0 Hence 120572

2= 0





T (X)


int

119905

0


(1)

T (X)


Theorem 53 If 119891 isin 119860119860(119899)



T (X)

Proof If119891 isin 119860119860(119899)



119891 isin 119860119860(119899)


T (X)




T (X)


119905

0

119891Δ


(119899+1)

T (X)






rator and 119891 isin 119860119860(119899)


(119899)

T (X2)


=

119860119891Δ119899


isin 119860119860T (X1) for



Theorem 37 that 119860119891Δ119894

isin 119860119860T (X2) for all 119894 = 0 1 119899

Hence 119860119891 isin 119860119860(119899)

T (X2)

For every 119891 isin 119860119860(119899)


X2as follows

119866 (119905) = int

119905

0

119860119891 (119904) Δ119904 (63)



the following





T (X2)



T (X2) for

every 119891 isin 119860119860(119899)


the range 119877119860




T (X2)





(RR) and119891 isin 119860119860(1)

T (R) then 120601∘119891 isin

119860119860(1)

T (R)


C1


Δ


(120601 ∘ 119891)Δ

(119905) = [int

1

0

1206011015840

(119891 (119905) + ℎ120583 (119905) 119891Δ

(119905)) 119889ℎ]119891Δ

(119905)

(64)

Since 119891 isin 119860119860(1)




1015840

(119891(119905) + ℎ120583(119905)119891Δ

(119905))


1

0

1206011015840

(119891(119905) + ℎ120583(119905)119891Δ


Δ

isin 119860119860T (R) since119891Δ

isin 119860119860T (R)



119909Δ

(119905) = 119860 (119905) 119909 (119905) 119905 isin T (65)


10038161003816100381610038161003816119883(119905)119875119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119905 119904) 119904 119905 isin T 119905 ge 119904

10038161003816100381610038161003816119883(119905)(119868 minus 119875)119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119904 119905) 119904 119905 isin T 119904 ge 119905

(66)


(119886119894119895)119898times119898


119894=1sum119898

119895=1|119886119894119895|2

)12


119909Δ

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin T (67)




(119905))119905isinT and

(119868 + 120583(119905)119860(119905))minus1




119909 (119905) = int

119905

minusinfin

119883 (119905) 119875119883minus1

(120590 (119904)) 119891 (119904) Δ119904

minus int

infin

119905

119883 (119905) (119868 minus 119875)119883minus1

(120590 (119904)) 119891 (119904) Δ119904

(68)




1le119894le119898inf

119905isinT 119888119894(119905) gt


119909Δ

(119905) = diag (minus1198881(119905) minus119888

2(119905) minus119888

119898(119905)) 119909 (119905) (69)




1(119905) minus119888

2(119905) minus119888

119898(119905))



1le119894le119898inf


119898


119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (70)


1(119905) minus119888

2(119905) minus119888

119898(119905))


are 1198621


1le119894le119898inf

119905isinT




119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (71)

Proof By Lemma 62

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (72)


119909Δ

119894(119905) = minus119888

119894(119905) int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904

+ 119891119894(119905) 119894 = 1 2 119898

(73)



119905

minusinfin

119890minus119888119894

(119905 120590(119904))119891119894(119904)Δ119904 119905 997891rarr 119888

119894(119905) and 119905 997891rarr 119891



119894(119905) 119894 = 1 2 119898 are also 119862rd-





Set 119864 = 120595 = (1205951 120595

2 120595

119898)119879

| 120595119894isin 119860119860

(1)

T (R) 119894 =


1le119894le119898120595

1 119864 is

a Banach space


1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 be a1198621


(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))


0isin


1(119905) 119909

2(119905) 119909

119898(119905))


1(119905) 120593

2(119905) 120593

119898(119905))

119879 119911 satisfies1003816100381610038161003816

1003817100381710038171003817119911(119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161le 119872

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864119890minus120582(119905 119905

0)


(74)

where1003816100381610038161003816

1003817100381710038171003817119911 (119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161

= max1le119894le119898

[1003816100381610038161003816119909119894(119905) minus 119909

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(119909119894minus 119909

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

]

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864

= max1le119894le119898


(1003816100381610038161003816120593119894(119905) minus 120593

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(120593119894minus 120593

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

)

(75)




119864 = 120601 (119905) = (1206011(119905) 120601

2(119905) 120601

119898(119905))

119879

| 120601119894isin 119860119860

(1)

T (R)

119894 = 1 2 119898

(76)


=

max1le119894le119898

1206011198941 where sdot



(H1) Assume 119886119894 119887119894119895 119868119894isin 119860119860

(1)

T (R+

) 119905minus120591119894119895isin 119860119860

(1)


R+ 119894 119895 = 1 2 119898 and min1le119894le119898

inf119905isinT 119886119894(119905) gt 0


such that10038161003816100381610038161003816119891119895(119909)

10038161003816100381610038161003816le 119872

119895 119895 = 1 2 119898 (77)

(H3) 119891119895 isin 119860119860(1)


119895such that

10038161003816100381610038161003816119891119895(119906) minus 119891

119895(V)

10038161003816100381610038161003816le 119867

119891

119895|119906 minus V|

forall119906 V isin R 119895 = 1 2 119898

(78)





3) hold Assume also

that

max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

lt 1 (79)




satisfying

119871 = max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) (80)



119909Δ

119894= minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119905 120593

119895(119905 minus 120591

119894119895(119905)))

+ 119868119894(119905) 119905 isin T 119894 = 1 2 119898

(81)


119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) 119894 = 1 2 119898 (82)



119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(83)


0

Γ 119864 997888rarr 119864

120593 = (1205931 120593

2 120593

119898)119879

997891997888rarr (Γ120593) (119905) = [(Γ120593)1(119905) (Γ120593)

2(119905) (Γ120593)

119898(119905) ]

119879

(84)


where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)



0



119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)




119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where119891Δ119894


=

119891 and 119891Δ1


norm

10038171003817100381710038171198911003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

) for 119891 isin 119861119899

rd (T X) (37)





10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598

forall119901 isin N

(38)


sup119901isinN

10038171003817100381710038171003817119891119895+119901

minus 119891119895

10038171003817100381710038171003817119899= sup


(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817

) le 120598 (39)


119895(119905)) 119894 = 0 1 119899 are Cauchy


0

119895(119905) we have that

119891Δ119894

119895(119905) 997888rarr 119891

Δ119894

(119905) 119894 = 0 1 119899 (40)

Since for 119894 = 0 1 119899100381710038171003817100381710038171003817

119891Δ119894

119901+119895(119905) minus 119891

Δ119894

119895(119905)

100381710038171003817100381710038171003817



100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817



10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899le sup

119905isinT

(

119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

119895(119905) minus 119891

Δ119894

(119905)

100381710038171003817100381710038171003817

) le 120598 (43)



119899

sum

119894=0

100381710038171003817100381710038171003817

119891Δ119894

(119905)

100381710038171003817100381710038171003817

le

10038171003817100381710038171003817119891119895minus 119891

10038171003817100381710038171003817119899+

10038171003817100381710038171003817119891119895

10038171003817100381710038171003817119899 forall119905 isin T (44)








119899

rd-aa) if 119891 119891Δ119894


Denote by 119860119860(119899)


119860119860(119899+1)

T (X) sub 119860119860(119899)


(0)

T (X) = 119860119860T (X)

Lemma 44 We have 119860119860(119899)

T (X) sub 119861119899

rd(T X)





(119899)

T (X) ] isin 119860119860(119899)


(119899)

119896and is almost


T (X)

(i) 119891 + 119892

(ii) 120582119891

(iii) ]119891






119886)Δ

(119905) = VΔ(119905) sdot (119891Δ ∘ V)(119905) =119891Δ

(V(119905)) = 119891Δ

(119886 + 119905) = 119891Δ



119886)Δ

= 119891Δ




119886isin 119860119860

119899

T (X)



119901minus 119891


rd-aafunction




119894

119901rarr 119891

Δ119894


rd-aa function

Corollary 47 119860119860(119899)




(119899)


119899



119901minus119891

119899= 0 ByTheorem 46 119891 isin 119860119860

(119899)




119860119860(119899)

T (X) is also a 119862119899

rd-aa function






0 then we have

119891Δ

(1199050) =

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

(46)



lim119899rarrinfin

119891Δ

(1199050+ 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119892 (1199050)

(47)


lim119899rarrinfin

119892 (1199050minus 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119891Δ

(1199050)

(48)


Theorem 49 If 119891 isin 119860119860(119899)

T (X) and 119891Δ119899+1


T (X)




(119899)

T (X)




lim119899rarrinfin

119891 (119905 + 120572119899) = 119892 (119905) pointwise on T (49)

we will write 119879119904119891 = 119892




of 119879119904is 119863(119879

119904) = 119891 T rarr X such that 119879

119904119891 exists

119863(119879119904) is a linear set


119904)

and119879119904119891 isin 119863(119879


119904= 119879

minus119904119879119904119891




119904119891 = 119891



119905

0





119899) sub


119899) such that

lim119899rarrinfin

119891 (119905 + 1199041015840

119899) = 119892 (119905)

lim119899rarrinfin

119892 (119905 minus 1199041015840

119899) = 119891 (119905)

(50)


pointwise on T and

lim119899rarrinfin

119865 (1199041015840

119899) = 120572

1 (51)



119865 (119905 + 1199041015840

119899) = int

119905+1199041015840

119899

0

119891 (119903) Δ119903

= int

1199041015840

119899

0

119891 (119903) Δ119903 + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

= 119865 (1199041015840

119899) + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

(52)


119899 we obtain

119865 (119905 + 1199041015840

119899) = 119865 (119904

1015840

119899) + int

119905

0

119891 (120575 + 1199041015840

119899) Δ120575 (53)


lim119899rarrinfin

119865 (119905 + 1199041015840

119899) = 120572

1+ int

119905

0

119892 (120590) Δ120590 (54)


1+ int

119905

0


sup119905isinT

(119866 (119905)) le sup119905isinT

119865 (119905) (55)


119899) such that

lim119899rarrinfin

119866 (minus119904119899) = 120572

2 (56)


119866 (119905 minus 119904119899) = 119866 (minus119904

119899) + int

119905

0

119892 (119903 minus 119904119899) Δ119903 (57)

so that

lim119899rarrinfin

119866 (119905 minus 119904119899) = 120572

2+ int

119905

0

119891 (119903) Δ119903 = 1205722+ 119865 (119905) (58)


119860119904119865 = 120572

2+ 119865 (59)


2belong




1198602

119904119865 = 119860

1199041205722+ 119860

119904119865 = 120572

2+ 120572

2+ 119865 = 2120572

2+ 119865 (60)


119860119899

119904119865 = 119899120572

2+ 119865 forall119899 = 1 2 (61)

But we have

sup119905isinT

1003817100381710038171003817119860119899

119904119865 (119905)

1003817100381710038171003817le sup

119905isinT

119865 (119905) (62)


2= 0 Hence 120572

2= 0





T (X)


int

119905

0


(1)

T (X)


Theorem 53 If 119891 isin 119860119860(119899)



T (X)

Proof If119891 isin 119860119860(119899)



119891 isin 119860119860(119899)


T (X)




T (X)


119905

0

119891Δ


(119899+1)

T (X)






rator and 119891 isin 119860119860(119899)


(119899)

T (X2)


=

119860119891Δ119899


isin 119860119860T (X1) for



Theorem 37 that 119860119891Δ119894

isin 119860119860T (X2) for all 119894 = 0 1 119899

Hence 119860119891 isin 119860119860(119899)

T (X2)

For every 119891 isin 119860119860(119899)


X2as follows

119866 (119905) = int

119905

0

119860119891 (119904) Δ119904 (63)



the following





T (X2)



T (X2) for

every 119891 isin 119860119860(119899)


the range 119877119860




T (X2)





(RR) and119891 isin 119860119860(1)

T (R) then 120601∘119891 isin

119860119860(1)

T (R)


C1


Δ


(120601 ∘ 119891)Δ

(119905) = [int

1

0

1206011015840

(119891 (119905) + ℎ120583 (119905) 119891Δ

(119905)) 119889ℎ]119891Δ

(119905)

(64)

Since 119891 isin 119860119860(1)




1015840

(119891(119905) + ℎ120583(119905)119891Δ

(119905))


1

0

1206011015840

(119891(119905) + ℎ120583(119905)119891Δ


Δ

isin 119860119860T (R) since119891Δ

isin 119860119860T (R)



119909Δ

(119905) = 119860 (119905) 119909 (119905) 119905 isin T (65)


10038161003816100381610038161003816119883(119905)119875119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119905 119904) 119904 119905 isin T 119905 ge 119904

10038161003816100381610038161003816119883(119905)(119868 minus 119875)119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119904 119905) 119904 119905 isin T 119904 ge 119905

(66)


(119886119894119895)119898times119898


119894=1sum119898

119895=1|119886119894119895|2

)12


119909Δ

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin T (67)




(119905))119905isinT and

(119868 + 120583(119905)119860(119905))minus1




119909 (119905) = int

119905

minusinfin

119883 (119905) 119875119883minus1

(120590 (119904)) 119891 (119904) Δ119904

minus int

infin

119905

119883 (119905) (119868 minus 119875)119883minus1

(120590 (119904)) 119891 (119904) Δ119904

(68)




1le119894le119898inf

119905isinT 119888119894(119905) gt


119909Δ

(119905) = diag (minus1198881(119905) minus119888

2(119905) minus119888

119898(119905)) 119909 (119905) (69)




1(119905) minus119888

2(119905) minus119888

119898(119905))



1le119894le119898inf


119898


119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (70)


1(119905) minus119888

2(119905) minus119888

119898(119905))


are 1198621


1le119894le119898inf

119905isinT




119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (71)

Proof By Lemma 62

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (72)


119909Δ

119894(119905) = minus119888

119894(119905) int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904

+ 119891119894(119905) 119894 = 1 2 119898

(73)



119905

minusinfin

119890minus119888119894

(119905 120590(119904))119891119894(119904)Δ119904 119905 997891rarr 119888

119894(119905) and 119905 997891rarr 119891



119894(119905) 119894 = 1 2 119898 are also 119862rd-





Set 119864 = 120595 = (1205951 120595

2 120595

119898)119879

| 120595119894isin 119860119860

(1)

T (R) 119894 =


1le119894le119898120595

1 119864 is

a Banach space


1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 be a1198621


(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))


0isin


1(119905) 119909

2(119905) 119909

119898(119905))


1(119905) 120593

2(119905) 120593

119898(119905))

119879 119911 satisfies1003816100381610038161003816

1003817100381710038171003817119911(119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161le 119872

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864119890minus120582(119905 119905

0)


(74)

where1003816100381610038161003816

1003817100381710038171003817119911 (119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161

= max1le119894le119898

[1003816100381610038161003816119909119894(119905) minus 119909

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(119909119894minus 119909

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

]

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864

= max1le119894le119898


(1003816100381610038161003816120593119894(119905) minus 120593

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(120593119894minus 120593

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

)

(75)




119864 = 120601 (119905) = (1206011(119905) 120601

2(119905) 120601

119898(119905))

119879

| 120601119894isin 119860119860

(1)

T (R)

119894 = 1 2 119898

(76)


=

max1le119894le119898

1206011198941 where sdot



(H1) Assume 119886119894 119887119894119895 119868119894isin 119860119860

(1)

T (R+

) 119905minus120591119894119895isin 119860119860

(1)


R+ 119894 119895 = 1 2 119898 and min1le119894le119898

inf119905isinT 119886119894(119905) gt 0


such that10038161003816100381610038161003816119891119895(119909)

10038161003816100381610038161003816le 119872

119895 119895 = 1 2 119898 (77)

(H3) 119891119895 isin 119860119860(1)


119895such that

10038161003816100381610038161003816119891119895(119906) minus 119891

119895(V)

10038161003816100381610038161003816le 119867

119891

119895|119906 minus V|

forall119906 V isin R 119895 = 1 2 119898

(78)





3) hold Assume also

that

max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

lt 1 (79)




satisfying

119871 = max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) (80)



119909Δ

119894= minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119905 120593

119895(119905 minus 120591

119894119895(119905)))

+ 119868119894(119905) 119905 isin T 119894 = 1 2 119898

(81)


119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) 119894 = 1 2 119898 (82)



119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(83)


0

Γ 119864 997888rarr 119864

120593 = (1205931 120593

2 120593

119898)119879

997891997888rarr (Γ120593) (119905) = [(Γ120593)1(119905) (Γ120593)

2(119905) (Γ120593)

119898(119905) ]

119879

(84)


where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)



0



119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)




119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894

119901rarr 119891

Δ119894


rd-aa function

Corollary 47 119860119860(119899)




(119899)


119899



119901minus119891

119899= 0 ByTheorem 46 119891 isin 119860119860

(119899)




119860119860(119899)

T (X) is also a 119862119899

rd-aa function






0 then we have

119891Δ

(1199050) =

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

(46)



lim119899rarrinfin

119891Δ

(1199050+ 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119892 (1199050)

(47)


lim119899rarrinfin

119892 (1199050minus 120572

119899) = lim

119899rarrinfin

119891 (120590 (1199050+ 120572

119899)) minus 119891 (119905

0+ 120572

119899)

120583 (1199050+ 120572

119899)


119891 (120590 (1199050) + 120572

119899) minus 119891 (119905

0+ 120572

119899)

120583 (1199050)

=

119891 (120590 (1199050)) minus 119891 (119905

0)

120583 (1199050)

= 119891Δ

(1199050)

(48)


Theorem 49 If 119891 isin 119860119860(119899)

T (X) and 119891Δ119899+1


T (X)




(119899)

T (X)




lim119899rarrinfin

119891 (119905 + 120572119899) = 119892 (119905) pointwise on T (49)

we will write 119879119904119891 = 119892




of 119879119904is 119863(119879

119904) = 119891 T rarr X such that 119879

119904119891 exists

119863(119879119904) is a linear set


119904)

and119879119904119891 isin 119863(119879


119904= 119879

minus119904119879119904119891




119904119891 = 119891



119905

0





119899) sub


119899) such that

lim119899rarrinfin

119891 (119905 + 1199041015840

119899) = 119892 (119905)

lim119899rarrinfin

119892 (119905 minus 1199041015840

119899) = 119891 (119905)

(50)


pointwise on T and

lim119899rarrinfin

119865 (1199041015840

119899) = 120572

1 (51)



119865 (119905 + 1199041015840

119899) = int

119905+1199041015840

119899

0

119891 (119903) Δ119903

= int

1199041015840

119899

0

119891 (119903) Δ119903 + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

= 119865 (1199041015840

119899) + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

(52)


119899 we obtain

119865 (119905 + 1199041015840

119899) = 119865 (119904

1015840

119899) + int

119905

0

119891 (120575 + 1199041015840

119899) Δ120575 (53)


lim119899rarrinfin

119865 (119905 + 1199041015840

119899) = 120572

1+ int

119905

0

119892 (120590) Δ120590 (54)


1+ int

119905

0


sup119905isinT

(119866 (119905)) le sup119905isinT

119865 (119905) (55)


119899) such that

lim119899rarrinfin

119866 (minus119904119899) = 120572

2 (56)


119866 (119905 minus 119904119899) = 119866 (minus119904

119899) + int

119905

0

119892 (119903 minus 119904119899) Δ119903 (57)

so that

lim119899rarrinfin

119866 (119905 minus 119904119899) = 120572

2+ int

119905

0

119891 (119903) Δ119903 = 1205722+ 119865 (119905) (58)


119860119904119865 = 120572

2+ 119865 (59)


2belong




1198602

119904119865 = 119860

1199041205722+ 119860

119904119865 = 120572

2+ 120572

2+ 119865 = 2120572

2+ 119865 (60)


119860119899

119904119865 = 119899120572

2+ 119865 forall119899 = 1 2 (61)

But we have

sup119905isinT

1003817100381710038171003817119860119899

119904119865 (119905)

1003817100381710038171003817le sup

119905isinT

119865 (119905) (62)


2= 0 Hence 120572

2= 0





T (X)


int

119905

0


(1)

T (X)


Theorem 53 If 119891 isin 119860119860(119899)



T (X)

Proof If119891 isin 119860119860(119899)



119891 isin 119860119860(119899)


T (X)




T (X)


119905

0

119891Δ


(119899+1)

T (X)






rator and 119891 isin 119860119860(119899)


(119899)

T (X2)


=

119860119891Δ119899


isin 119860119860T (X1) for



Theorem 37 that 119860119891Δ119894

isin 119860119860T (X2) for all 119894 = 0 1 119899

Hence 119860119891 isin 119860119860(119899)

T (X2)

For every 119891 isin 119860119860(119899)


X2as follows

119866 (119905) = int

119905

0

119860119891 (119904) Δ119904 (63)



the following





T (X2)



T (X2) for

every 119891 isin 119860119860(119899)


the range 119877119860




T (X2)





(RR) and119891 isin 119860119860(1)

T (R) then 120601∘119891 isin

119860119860(1)

T (R)


C1


Δ


(120601 ∘ 119891)Δ

(119905) = [int

1

0

1206011015840

(119891 (119905) + ℎ120583 (119905) 119891Δ

(119905)) 119889ℎ]119891Δ

(119905)

(64)

Since 119891 isin 119860119860(1)




1015840

(119891(119905) + ℎ120583(119905)119891Δ

(119905))


1

0

1206011015840

(119891(119905) + ℎ120583(119905)119891Δ


Δ

isin 119860119860T (R) since119891Δ

isin 119860119860T (R)



119909Δ

(119905) = 119860 (119905) 119909 (119905) 119905 isin T (65)


10038161003816100381610038161003816119883(119905)119875119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119905 119904) 119904 119905 isin T 119905 ge 119904

10038161003816100381610038161003816119883(119905)(119868 minus 119875)119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119904 119905) 119904 119905 isin T 119904 ge 119905

(66)


(119886119894119895)119898times119898


119894=1sum119898

119895=1|119886119894119895|2

)12


119909Δ

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin T (67)




(119905))119905isinT and

(119868 + 120583(119905)119860(119905))minus1




119909 (119905) = int

119905

minusinfin

119883 (119905) 119875119883minus1

(120590 (119904)) 119891 (119904) Δ119904

minus int

infin

119905

119883 (119905) (119868 minus 119875)119883minus1

(120590 (119904)) 119891 (119904) Δ119904

(68)




1le119894le119898inf

119905isinT 119888119894(119905) gt


119909Δ

(119905) = diag (minus1198881(119905) minus119888

2(119905) minus119888

119898(119905)) 119909 (119905) (69)




1(119905) minus119888

2(119905) minus119888

119898(119905))



1le119894le119898inf


119898


119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (70)


1(119905) minus119888

2(119905) minus119888

119898(119905))


are 1198621


1le119894le119898inf

119905isinT




119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (71)

Proof By Lemma 62

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (72)


119909Δ

119894(119905) = minus119888

119894(119905) int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904

+ 119891119894(119905) 119894 = 1 2 119898

(73)



119905

minusinfin

119890minus119888119894

(119905 120590(119904))119891119894(119904)Δ119904 119905 997891rarr 119888

119894(119905) and 119905 997891rarr 119891



119894(119905) 119894 = 1 2 119898 are also 119862rd-





Set 119864 = 120595 = (1205951 120595

2 120595

119898)119879

| 120595119894isin 119860119860

(1)

T (R) 119894 =


1le119894le119898120595

1 119864 is

a Banach space


1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 be a1198621


(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))


0isin


1(119905) 119909

2(119905) 119909

119898(119905))


1(119905) 120593

2(119905) 120593

119898(119905))

119879 119911 satisfies1003816100381610038161003816

1003817100381710038171003817119911(119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161le 119872

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864119890minus120582(119905 119905

0)


(74)

where1003816100381610038161003816

1003817100381710038171003817119911 (119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161

= max1le119894le119898

[1003816100381610038161003816119909119894(119905) minus 119909

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(119909119894minus 119909

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

]

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864

= max1le119894le119898


(1003816100381610038161003816120593119894(119905) minus 120593

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(120593119894minus 120593

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

)

(75)




119864 = 120601 (119905) = (1206011(119905) 120601

2(119905) 120601

119898(119905))

119879

| 120601119894isin 119860119860

(1)

T (R)

119894 = 1 2 119898

(76)


=

max1le119894le119898

1206011198941 where sdot



(H1) Assume 119886119894 119887119894119895 119868119894isin 119860119860

(1)

T (R+

) 119905minus120591119894119895isin 119860119860

(1)


R+ 119894 119895 = 1 2 119898 and min1le119894le119898

inf119905isinT 119886119894(119905) gt 0


such that10038161003816100381610038161003816119891119895(119909)

10038161003816100381610038161003816le 119872

119895 119895 = 1 2 119898 (77)

(H3) 119891119895 isin 119860119860(1)


119895such that

10038161003816100381610038161003816119891119895(119906) minus 119891

119895(V)

10038161003816100381610038161003816le 119867

119891

119895|119906 minus V|

forall119906 V isin R 119895 = 1 2 119898

(78)





3) hold Assume also

that

max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

lt 1 (79)




satisfying

119871 = max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) (80)



119909Δ

119894= minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119905 120593

119895(119905 minus 120591

119894119895(119905)))

+ 119868119894(119905) 119905 isin T 119894 = 1 2 119898

(81)


119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) 119894 = 1 2 119898 (82)



119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(83)


0

Γ 119864 997888rarr 119864

120593 = (1205931 120593

2 120593

119898)119879

997891997888rarr (Γ120593) (119905) = [(Γ120593)1(119905) (Γ120593)

2(119905) (Γ120593)

119898(119905) ]

119879

(84)


where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)



0



119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)




119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










pointwise on T and

lim119899rarrinfin

119865 (1199041015840

119899) = 120572

1 (51)



119865 (119905 + 1199041015840

119899) = int

119905+1199041015840

119899

0

119891 (119903) Δ119903

= int

1199041015840

119899

0

119891 (119903) Δ119903 + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

= 119865 (1199041015840

119899) + int

119905+1199041015840

119899

1199041015840

119899

119891 (119903) Δ119903

(52)


119899 we obtain

119865 (119905 + 1199041015840

119899) = 119865 (119904

1015840

119899) + int

119905

0

119891 (120575 + 1199041015840

119899) Δ120575 (53)


lim119899rarrinfin

119865 (119905 + 1199041015840

119899) = 120572

1+ int

119905

0

119892 (120590) Δ120590 (54)


1+ int

119905

0


sup119905isinT

(119866 (119905)) le sup119905isinT

119865 (119905) (55)


119899) such that

lim119899rarrinfin

119866 (minus119904119899) = 120572

2 (56)


119866 (119905 minus 119904119899) = 119866 (minus119904

119899) + int

119905

0

119892 (119903 minus 119904119899) Δ119903 (57)

so that

lim119899rarrinfin

119866 (119905 minus 119904119899) = 120572

2+ int

119905

0

119891 (119903) Δ119903 = 1205722+ 119865 (119905) (58)


119860119904119865 = 120572

2+ 119865 (59)


2belong




1198602

119904119865 = 119860

1199041205722+ 119860

119904119865 = 120572

2+ 120572

2+ 119865 = 2120572

2+ 119865 (60)


119860119899

119904119865 = 119899120572

2+ 119865 forall119899 = 1 2 (61)

But we have

sup119905isinT

1003817100381710038171003817119860119899

119904119865 (119905)

1003817100381710038171003817le sup

119905isinT

119865 (119905) (62)


2= 0 Hence 120572

2= 0





T (X)


int

119905

0


(1)

T (X)


Theorem 53 If 119891 isin 119860119860(119899)



T (X)

Proof If119891 isin 119860119860(119899)



119891 isin 119860119860(119899)


T (X)




T (X)


119905

0

119891Δ


(119899+1)

T (X)






rator and 119891 isin 119860119860(119899)


(119899)

T (X2)


=

119860119891Δ119899


isin 119860119860T (X1) for



Theorem 37 that 119860119891Δ119894

isin 119860119860T (X2) for all 119894 = 0 1 119899

Hence 119860119891 isin 119860119860(119899)

T (X2)

For every 119891 isin 119860119860(119899)


X2as follows

119866 (119905) = int

119905

0

119860119891 (119904) Δ119904 (63)



the following





T (X2)



T (X2) for

every 119891 isin 119860119860(119899)


the range 119877119860




T (X2)





(RR) and119891 isin 119860119860(1)

T (R) then 120601∘119891 isin

119860119860(1)

T (R)


C1


Δ


(120601 ∘ 119891)Δ

(119905) = [int

1

0

1206011015840

(119891 (119905) + ℎ120583 (119905) 119891Δ

(119905)) 119889ℎ]119891Δ

(119905)

(64)

Since 119891 isin 119860119860(1)




1015840

(119891(119905) + ℎ120583(119905)119891Δ

(119905))


1

0

1206011015840

(119891(119905) + ℎ120583(119905)119891Δ


Δ

isin 119860119860T (R) since119891Δ

isin 119860119860T (R)



119909Δ

(119905) = 119860 (119905) 119909 (119905) 119905 isin T (65)


10038161003816100381610038161003816119883(119905)119875119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119905 119904) 119904 119905 isin T 119905 ge 119904

10038161003816100381610038161003816119883(119905)(119868 minus 119875)119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119904 119905) 119904 119905 isin T 119904 ge 119905

(66)


(119886119894119895)119898times119898


119894=1sum119898

119895=1|119886119894119895|2

)12


119909Δ

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin T (67)




(119905))119905isinT and

(119868 + 120583(119905)119860(119905))minus1




119909 (119905) = int

119905

minusinfin

119883 (119905) 119875119883minus1

(120590 (119904)) 119891 (119904) Δ119904

minus int

infin

119905

119883 (119905) (119868 minus 119875)119883minus1

(120590 (119904)) 119891 (119904) Δ119904

(68)




1le119894le119898inf

119905isinT 119888119894(119905) gt


119909Δ

(119905) = diag (minus1198881(119905) minus119888

2(119905) minus119888

119898(119905)) 119909 (119905) (69)




1(119905) minus119888

2(119905) minus119888

119898(119905))



1le119894le119898inf


119898


119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (70)


1(119905) minus119888

2(119905) minus119888

119898(119905))


are 1198621


1le119894le119898inf

119905isinT




119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (71)

Proof By Lemma 62

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (72)


119909Δ

119894(119905) = minus119888

119894(119905) int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904

+ 119891119894(119905) 119894 = 1 2 119898

(73)



119905

minusinfin

119890minus119888119894

(119905 120590(119904))119891119894(119904)Δ119904 119905 997891rarr 119888

119894(119905) and 119905 997891rarr 119891



119894(119905) 119894 = 1 2 119898 are also 119862rd-





Set 119864 = 120595 = (1205951 120595

2 120595

119898)119879

| 120595119894isin 119860119860

(1)

T (R) 119894 =


1le119894le119898120595

1 119864 is

a Banach space


1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 be a1198621


(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))


0isin


1(119905) 119909

2(119905) 119909

119898(119905))


1(119905) 120593

2(119905) 120593

119898(119905))

119879 119911 satisfies1003816100381610038161003816

1003817100381710038171003817119911(119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161le 119872

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864119890minus120582(119905 119905

0)


(74)

where1003816100381610038161003816

1003817100381710038171003817119911 (119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161

= max1le119894le119898

[1003816100381610038161003816119909119894(119905) minus 119909

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(119909119894minus 119909

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

]

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864

= max1le119894le119898


(1003816100381610038161003816120593119894(119905) minus 120593

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(120593119894minus 120593

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

)

(75)




119864 = 120601 (119905) = (1206011(119905) 120601

2(119905) 120601

119898(119905))

119879

| 120601119894isin 119860119860

(1)

T (R)

119894 = 1 2 119898

(76)


=

max1le119894le119898

1206011198941 where sdot



(H1) Assume 119886119894 119887119894119895 119868119894isin 119860119860

(1)

T (R+

) 119905minus120591119894119895isin 119860119860

(1)


R+ 119894 119895 = 1 2 119898 and min1le119894le119898

inf119905isinT 119886119894(119905) gt 0


such that10038161003816100381610038161003816119891119895(119909)

10038161003816100381610038161003816le 119872

119895 119895 = 1 2 119898 (77)

(H3) 119891119895 isin 119860119860(1)


119895such that

10038161003816100381610038161003816119891119895(119906) minus 119891

119895(V)

10038161003816100381610038161003816le 119867

119891

119895|119906 minus V|

forall119906 V isin R 119895 = 1 2 119898

(78)





3) hold Assume also

that

max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

lt 1 (79)




satisfying

119871 = max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) (80)



119909Δ

119894= minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119905 120593

119895(119905 minus 120591

119894119895(119905)))

+ 119868119894(119905) 119905 isin T 119894 = 1 2 119898

(81)


119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) 119894 = 1 2 119898 (82)



119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(83)


0

Γ 119864 997888rarr 119864

120593 = (1205931 120593

2 120593

119898)119879

997891997888rarr (Γ120593) (119905) = [(Γ120593)1(119905) (Γ120593)

2(119905) (Γ120593)

119898(119905) ]

119879

(84)


where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)



0



119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)




119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











T (X2) for

every 119891 isin 119860119860(119899)


the range 119877119860




T (X2)





(RR) and119891 isin 119860119860(1)

T (R) then 120601∘119891 isin

119860119860(1)

T (R)


C1


Δ


(120601 ∘ 119891)Δ

(119905) = [int

1

0

1206011015840

(119891 (119905) + ℎ120583 (119905) 119891Δ

(119905)) 119889ℎ]119891Δ

(119905)

(64)

Since 119891 isin 119860119860(1)




1015840

(119891(119905) + ℎ120583(119905)119891Δ

(119905))


1

0

1206011015840

(119891(119905) + ℎ120583(119905)119891Δ


Δ

isin 119860119860T (R) since119891Δ

isin 119860119860T (R)



119909Δ

(119905) = 119860 (119905) 119909 (119905) 119905 isin T (65)


10038161003816100381610038161003816119883(119905)119875119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119905 119904) 119904 119905 isin T 119905 ge 119904

10038161003816100381610038161003816119883(119905)(119868 minus 119875)119883

minus1

(119904)

10038161003816100381610038161003816Tle 119870119890

⊖120572(119904 119905) 119904 119905 isin T 119904 ge 119905

(66)


(119886119894119895)119898times119898


119894=1sum119898

119895=1|119886119894119895|2

)12


119909Δ

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin T (67)




(119905))119905isinT and

(119868 + 120583(119905)119860(119905))minus1




119909 (119905) = int

119905

minusinfin

119883 (119905) 119875119883minus1

(120590 (119904)) 119891 (119904) Δ119904

minus int

infin

119905

119883 (119905) (119868 minus 119875)119883minus1

(120590 (119904)) 119891 (119904) Δ119904

(68)




1le119894le119898inf

119905isinT 119888119894(119905) gt


119909Δ

(119905) = diag (minus1198881(119905) minus119888

2(119905) minus119888

119898(119905)) 119909 (119905) (69)




1(119905) minus119888

2(119905) minus119888

119898(119905))



1le119894le119898inf


119898


119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (70)


1(119905) minus119888

2(119905) minus119888

119898(119905))


are 1198621


1le119894le119898inf

119905isinT




119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (71)

Proof By Lemma 62

119909119894(119905) = int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904 119894 = 1 2 119898 (72)


119909Δ

119894(119905) = minus119888

119894(119905) int

119905

minusinfin

119890minus119888119894(119905 120590 (119904)) 119891

119894(119904) Δ119904

+ 119891119894(119905) 119894 = 1 2 119898

(73)



119905

minusinfin

119890minus119888119894

(119905 120590(119904))119891119894(119904)Δ119904 119905 997891rarr 119888

119894(119905) and 119905 997891rarr 119891



119894(119905) 119894 = 1 2 119898 are also 119862rd-





Set 119864 = 120595 = (1205951 120595

2 120595

119898)119879

| 120595119894isin 119860119860

(1)

T (R) 119894 =


1le119894le119898120595

1 119864 is

a Banach space


1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 be a1198621


(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))


0isin


1(119905) 119909

2(119905) 119909

119898(119905))


1(119905) 120593

2(119905) 120593

119898(119905))

119879 119911 satisfies1003816100381610038161003816

1003817100381710038171003817119911(119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161le 119872

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864119890minus120582(119905 119905

0)


(74)

where1003816100381610038161003816

1003817100381710038171003817119911 (119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161

= max1le119894le119898

[1003816100381610038161003816119909119894(119905) minus 119909

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(119909119894minus 119909

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

]

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864

= max1le119894le119898


(1003816100381610038161003816120593119894(119905) minus 120593

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(120593119894minus 120593

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

)

(75)




119864 = 120601 (119905) = (1206011(119905) 120601

2(119905) 120601

119898(119905))

119879

| 120601119894isin 119860119860

(1)

T (R)

119894 = 1 2 119898

(76)


=

max1le119894le119898

1206011198941 where sdot



(H1) Assume 119886119894 119887119894119895 119868119894isin 119860119860

(1)

T (R+

) 119905minus120591119894119895isin 119860119860

(1)


R+ 119894 119895 = 1 2 119898 and min1le119894le119898

inf119905isinT 119886119894(119905) gt 0


such that10038161003816100381610038161003816119891119895(119909)

10038161003816100381610038161003816le 119872

119895 119895 = 1 2 119898 (77)

(H3) 119891119895 isin 119860119860(1)


119895such that

10038161003816100381610038161003816119891119895(119906) minus 119891

119895(V)

10038161003816100381610038161003816le 119867

119891

119895|119906 minus V|

forall119906 V isin R 119895 = 1 2 119898

(78)





3) hold Assume also

that

max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

lt 1 (79)




satisfying

119871 = max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) (80)



119909Δ

119894= minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119905 120593

119895(119905 minus 120591

119894119895(119905)))

+ 119868119894(119905) 119905 isin T 119894 = 1 2 119898

(81)


119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) 119894 = 1 2 119898 (82)



119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(83)


0

Γ 119864 997888rarr 119864

120593 = (1205931 120593

2 120593

119898)119879

997891997888rarr (Γ120593) (119905) = [(Γ120593)1(119905) (Γ120593)

2(119905) (Γ120593)

119898(119905) ]

119879

(84)


where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)



0



119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)




119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905

minusinfin

119890minus119888119894

(119905 120590(119904))119891119894(119904)Δ119904 119905 997891rarr 119888

119894(119905) and 119905 997891rarr 119891



119894(119905) 119894 = 1 2 119898 are also 119862rd-





Set 119864 = 120595 = (1205951 120595

2 120595

119898)119879

| 120595119894isin 119860119860

(1)

T (R) 119894 =


1le119894le119898120595

1 119864 is

a Banach space


1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))

119879 be a1198621


(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))


0isin


1(119905) 119909

2(119905) 119909

119898(119905))


1(119905) 120593

2(119905) 120593

119898(119905))

119879 119911 satisfies1003816100381610038161003816

1003817100381710038171003817119911(119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161le 119872

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864119890minus120582(119905 119905

0)


(74)

where1003816100381610038161003816

1003817100381710038171003817119911 (119905) minus 119911

lowast

(119905)1003817100381710038171003817

10038161003816100381610038161

= max1le119894le119898

[1003816100381610038161003816119909119894(119905) minus 119909

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(119909119894minus 119909

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

]

1003817100381710038171003817120593 minus 120593

lowast1003817100381710038171003817119864

= max1le119894le119898


(1003816100381610038161003816120593119894(119905) minus 120593

lowast

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(120593119894minus 120593

lowast

119894)Δ

(119905)

100381610038161003816100381610038161003816

)

(75)




119864 = 120601 (119905) = (1206011(119905) 120601

2(119905) 120601

119898(119905))

119879

| 120601119894isin 119860119860

(1)

T (R)

119894 = 1 2 119898

(76)


=

max1le119894le119898

1206011198941 where sdot



(H1) Assume 119886119894 119887119894119895 119868119894isin 119860119860

(1)

T (R+

) 119905minus120591119894119895isin 119860119860

(1)


R+ 119894 119895 = 1 2 119898 and min1le119894le119898

inf119905isinT 119886119894(119905) gt 0


such that10038161003816100381610038161003816119891119895(119909)

10038161003816100381610038161003816le 119872

119895 119895 = 1 2 119898 (77)

(H3) 119891119895 isin 119860119860(1)


119895such that

10038161003816100381610038161003816119891119895(119906) minus 119891

119895(V)

10038161003816100381610038161003816le 119867

119891

119895|119906 minus V|

forall119906 V isin R 119895 = 1 2 119898

(78)





3) hold Assume also

that

max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

lt 1 (79)




satisfying

119871 = max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) (80)



119909Δ

119894= minus119886

119894(119905) 119909

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) 119891

119895(119905 120593

119895(119905 minus 120591

119894119895(119905)))

+ 119868119894(119905) 119905 isin T 119894 = 1 2 119898

(81)


119909Δ

119894(119905) = minus119886

119894(119905) 119909

119894(119905) 119894 = 1 2 119898 (82)



119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(83)


0

Γ 119864 997888rarr 119864

120593 = (1205931 120593

2 120593

119898)119879

997891997888rarr (Γ120593) (119905) = [(Γ120593)1(119905) (Γ120593)

2(119905) (Γ120593)

119898(119905) ]

119879

(84)


where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)



0



119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)




119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

(Γ120593)119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(85)



0



119894119895 119868119894isin

119860119860(1)

T (R) we have

1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904)

times

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816+1003816100381610038161003816119868119894(119904)1003816100381610038161003816)Δ119904

le (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times(

119898

sum

119895=1

119887119894119895(119904)

times119891119895(120593

119895(119904 minus 120591

119894119895(119904))))Δ119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

minus119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) 119868

119894(119904) Δ119904

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) 119891

119895(120593

119895(119905 minus 120591

119894119895(119905))) + 119868

119894(119905)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894) + (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)

= (

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894

119886119894

)

(86)

Consequently

1003817100381710038171003817(Γ120593)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905)1003816100381610038161003816+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

(

119898

sum

119895=1

119887119894119895119872

119895+ 119868

119894)(1 +

119886119894+ 1

119886119894

) = 119871

(87)




119887119894119895isin 119860119860

(1)

T (R) we obtain

1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus 119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904)

10038161003816100381610038161003816119891119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904)))

10038161003816100381610038161003816

]

]

Δ119904

le int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

10038161003816100381610038161003816120593119895(119904 minus 120591

119894119895(119904))

minus120601119895(119904 minus 120591

119894119895(119904))

10038161003816100381610038161003816

]

]

Δ119904


le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

int

119905

minusinfin

119890minus119886119894(119905 120590 (119904)) Δ119904

le

1

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus 119886119894(119905) int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times[

[

119898

sum

119895=1

119887119894119895(119904) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

]

]

Δ119904

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

sum

119895=1

119887119894119895(119905) (119891

119895(120593

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(120601

119895(119904 minus 120591

119894119895(119904))))

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119886119894

119886119894

[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

+[

[

119898

sum

119895=1

119887119894119895119867

119891

119895

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

]

]

=[

[

119898

sum

119895=1

119887119894119895(119904)119867

119891

119895

]

]

[1 +

119886119894

119886119894

]1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(88)

Hence

1003817100381710038171003817(Γ120593) minus (Γ120601)

1003817100381710038171003817119864

= max1le119894le119898

sup119905isinT

[1003816100381610038161003816(Γ120593)

119894(119905) minus (Γ120601)

119894(119905)1003816100381610038161003816

+

100381610038161003816100381610038161003816

(Γ120593)Δ

119894(119905) minus (Γ120601)

Δ

119894(119905)

100381610038161003816100381610038161003816

]

le max1le119894le119898

[

[

1

119886119894

119898

sum

119895=1

119887119894119895119867

119891

119895 (1 +

119886119894

119886119894

)

119898

sum

119895=1

119887119894119895119867

119891

119895

]

]

1003817100381710038171003817120593 minus 120601

10038171003817100381710038171198640

(89)


[(1119886119894) sum

119898

119895=1119887119894119895119867

119891

119895 (1 + (119886

119894119886

119894)) sum

119898

119895=1119887119894119895

119867119891





1


0

119909120593

119894(119905) = int

119905

minusinfin

119890minus119886119894(119905 120590 (119904))

times (

119898

sum

119895=1

119887119894119895(119904) 119891

119895(120593

119895(119904 minus 120591

119894119895(119904))) + 119868

119894(119904))Δ119904

119894 = 1 2 119898

(90)


3) hold Assume also

that

(1198674) 120579

119894=

119898

sum

119895=1

119887119894119895119867

119891

119895ge

119886119894minus 119886

119894119886119894

119886119894+ 119886

119894

119894 = 1 2 119898 (91)

Then the 1198621




lowast

1(119905) 119909

lowast

2(119905) 119909

lowast

119898(119905))


120593lowast

(119904) = (120593lowast

1(119905) 120593

lowast

2(119905) 120593

lowast

119898(119905))

119879

(92)

Assume that 119911(119905) = (1199091(119905) 119909

2(119905) 119909

119898(119905))


1(119905)

1205932(119905) 120593

119898(119905))

119879 Let 119910119894(119905) = 119909

119894(119905) minus 119909

lowast

119894(119905) 119894 = 1 2 119898


119910Δ

119894(119905) = minus119886

119894(119905) 119910

119894(119905) +

119898

sum

119895=1

119887119894119895(119905) [119891

119895(119909

119895(119905 minus 120591

119894119895(119905)))

minus119891119895(119909

lowast

119895(119905 minus 120591

119894119895(119905)))]

119905 isin T 119894 = 1 2 119898

119910119894(119904) = 120593 (119904) minus 120593

lowast

(119904) 119904 isin [minus120579 0]T 119894 = 1 2 119898

(93)



obtain for 119894 = 1 2 119898

int

119905

1199050

119910Δ

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

= int

119905

1199050

minus119886119894(119904) 119910

119894(119904) 119890

minus119886119894(119905 120590 (119904)) Δ119904

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(94)



119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119910119894(119905) = 119910

119894(1199050) 119890

minus119886119894

(119905 1199050)

+

119898

sum

119895=1

int

119905

1199050

119887119894119895(119904) [119891

119895(119909

119895(119904 minus 120591

119894119895(119904)))

minus119891119895(119909

lowast

119895(119904 minus 120591

119894119895(119904)))] 119890

minus119886119894(119905 120590 (119904)) Δ119904

(95)


119886119894and

119872 gt max1le119894le119898

119886119894119886119894

119886119894minus (119886

119894+ 119886

119894) 120579

119894

119886119894

119886119894minus 120579

119894

(96)


1003817100381710038171003817119910 (119905)

1003817100381710038171003817

10038161003816100381610038161le 119872119890

⊖120582(119905 119905

0)1003817100381710038171003817120593 minus 120593


0 +infin)

T

(97)






Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article almost automorphic functions of order and...

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