pseudo almost periodic solutions of an iterative...

Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 1, pp. 515–524 DOI: 10.18514/MMN.2017.2047

PSEUDO ALMOST PERIODIC SOLUTIONS OF AN ITERATIVEEQUATION WITH VARIABLE COEFFICIENTS

HOU YU ZHAO AND MICHAL FECKAN

Received 09 June, 2016

Abstract. In this paper, the exponential dichotomy, and Tikhonov and Banach fixed point the-orems are used to study the existence and uniqueness of pseudo almost periodic solutions ofnonhomogeneous iterative functional differential equations of the form x0.t/ D �1.t/x.t/C

�2.t/xŒ2�.t/C : : :C�n.t/x

Œn�.t/Cf .t/.

2010 Mathematics Subject Classification: 39B12; 39B82

Keywords: iterative functional differential equation, pseudo almost periodic solutions, fixed pointtheorem

1. INTRODUCTION

Recently, iterative functional differential equations of the form

x0.t/DH.xŒ0�.t/;xŒ1�.t/;xŒ2�.t/; : : : ;xŒn�.t//

have appeared in several papers, where xŒ0�.t/D t , xŒi�.t/D x.xŒi�1�.t//, i � 1. In[4], Cooke points out that it is highly desirable to establish the existence and stabilityproperties of periodic solutions for equations of the form

x0.t/Cax.t �h.t;x.t///D F.t/;

in which the lag h.t;x.t// implicitly involves x.t/. Stephan [14] studies the existenceof periodic solutions of equation

x0.t/Cax.t � rC�h.t;x.t///D F.t/:

Eder [6] considers the iterative functional differential equation

x0.t/D xŒ2�.t/

This work was partially supported by the National Natural Science Foundation of China (GrantNo. 11326120, 11501069), Foundation of Chongqing Municipal Education Commission (Grant No.KJ1400528), the Grants VEGA-MS 1/0071/14, VEGA-SAV 2/0153/16 and by the Slovak Researchand Development Agency under the contract No. APVV-14-0378.

c 2017 Miskolc University Press

516 H. Y. ZHAO AND M. FECKAN

and obtains that every solution either vanishes identically or is strictly monotonic.FeLckan [7] studies the equation

x0.t/D f .xŒ2�.t//

by obtaining an existence theorem for solutions satisfying x.0/D 0: Later, Wang andSi [17] study

x0.xŒr�.t//D c0´C c1x.t/C c2xŒ2�.t/C : : :CxŒn�.t/;

and show the existence theorem of analytic solutions. In particularly, Si and Cheng[12] discusses the smooth solutions of equation of

x0.t/D �1x.t/C�2xŒ2�.t/C : : :C�nx

Œn�.t/Cf .t/:

For some various properties of solutions for several iterative functional differentialequations, we refer the interested reader to [10, 11, 13, 15].

On the other hand, the existence of pseudo almost periodic solutions is amongthe most attractive topics in qualitative theory of differential equations due to theirapplications, especially in biology, economics and physics [1,3,5,18,20]. As pointedout by Ait Das and Ezzinbi [1], it is an interesting thing to study the pseudo almostperiodic systems with delays. It is obviously that iterative functional differentialequations are special type state-dependent delay differential equations. In [9], Liupointed that the properties of the almost periodic functions do not always hold in theset of pseudo almost periodic functions and given an example: when x.t/ is a pseudoalmost periodic function, x.x.t// may not be a pseudo almost periodic function.To the best of our knowledge, there are few results about pseudo almost periodicsolutions for iterative functional differential equations except [9] and [16].

In the present work, we propose an existence result for pseudo almost periodicsolutions of iterative functional differential equations

x0.t/D �1.t/x.t/C�2.t/xŒ2�.t/C : : :C�n.t/x

Œn�.t/Cf .t/: (1.1)

by using Tikhonov fixed theorem. Uniqueness of the solution is achieved by Banachfixed point theorem.

This paper is organized as follows. In Section 2 we give two theorems which areused in Section 3 to establish the main existence result. In Section 4, we show that(1.1) has a unique pseudo almost periodic solution under some suitable conditionsand present an example to illustrate the theory. Related problems are also studied in[2].

PSEUDO ALMOST PERIODIC SOLUTIONS 517

2. PRELIMINARIES

Let C.R;R/ denote the set of all continuous functions from R to R endowed withthe usual metric

d.f;g/D

1XmD1

2�mkf �gkm

1Ckf �gkmfor kf �gkm D max

t2Œ�m;m�jf .t/�g.t/j;

so the topology on C.R;R/ is the uniform convergence on each compact intervalsof R. We also consider the set BC.R;R/ of all bounded and continuous functionsfrom R to R with the norm kf k D supt2R jf .t/j, so the topology on BC.R;R/ is theuniform convergence on R. We denote by AP.R;R/ the set of all almost periodicfunctions from R to R. Define the set

PAP0.R;R/Dng 2 BC.R;R/

ˇlim

T!C1

1

2T

Z T

�T

jg.t/jdt D 0o:

A function ' 2 BC.R;R/ is called pseudo almost periodic if it can be expressedas ' D hC g, where h 2 AP.R;R/ and g 2 PAP0.R;R/. The collection of suchfunctions will be denoted by PAP.R;R/. In particular, .PAP.R;R/;k�k/ is a Banachspace [19]. For M;L > 0, define

BC .M;L/Dn' 2 C.R;R/

ˇj'.t/j �M; j'.t2/�'.t1/j � Ljt2� t1j;

for all t; t1; t2 2 Ro;

BS .M;L/D S.R;R/\BC .M;L/; S 2 fPAP;AP;PAP0g:

From [9], it is easy to see that BS .M;L/, S 2 fPAP;AP;PAP0;C g are closed con-vex and bounded subsets of BC.R;R/ and C.R;R/, respectively. Furthermore, bythe Arzela-Ascoli theorem, the subset BC .M;L/ is compact in C.R;R/.

On the other hand, the subset BC .M;L/ is not precompact in BC.R;R/, since

a sequencenM sech L.tCn/

M

o1nD12 BC .M;L/ has no a convergent subsequence in

BC.R;R/.Furthermore, the subset BAP .M;L/ is not compact in C.R;R/, since we consider

the function from [8]

g.t/D

1XkD1

1

ksin2

t

2k:

From 1k

sin2 t2k �

1kt2

4k and that the seriesP1kD1

1kt2

4k convergents, we see that

gn.t/D

nXkD1

1

ksin2

t

2k! g.t/


in C.R;R/. Similarly we derive

g0n.t/D

nXkD1

1

k2k�1sin

t

2kcos

t

2k! g0.t/

in BC.R;R/ and kg0nk � 2, kg0k � 2. Take

'n.t/Dminf1;gn.t/g D1Cgn.t/�j1�gn.t/j

2:

Clearly 'n 2BAP .1;2/ and 'n! ' Dminf1;gg in C.R;R/. Next we know from [8]that limsupt!1g.t/ D1 and liminft!1g.t/ D 0. So applying [8, Lemma 2.1],we know that ' …AP.R;R/. Hence the subset BAP .1;2/ is not complete in C.R;R/.

By taking the sequencen'n

�L2M

t�o1nD1

, we see that BAP .M;L/ is not complete inC.R;R/ for any M > 0 and L > 0. Since BAP .M;L/ � BC .M;L/, BAP .M;L/ isjust precompact in C.R;R/. We have also shown thatBAP .M;L/ are not precompactin BC.R;R/, but they are complete.

Furthermore, for any k 2N, we have 12k�

R 2k�0 '.t/dt � 1

2k�

R 2k�0 sin2 t

2D12

, so' … PAP0.R;R/. But we do not know ' … PAP.R;R/.

Finally, taking

n.t/D

8<:

0 for jt j � nC1;1 for jt j � n;

tCnC1 for t j 2 Œ�n�1;�n�;nC1� t for t � Œn;nC1�;

we see that n 2 BPAP0.1;1/ and n ! 1 in C.R;R/, so BPAP0

.M;L/ are notcomplete in C.R;R/. We have also shown that BPAP0

.M;L/ are not precompact inBC.R;R/, but they are complete.

Summarizing, we arrive at the following result.

Theorem 1. The subsets BS .M;L/, S 2 fPAP;AP;PAP0;C g are not precom-pact in BC.R;R/. They are precompact in C.R;R/. Furthermore, BAP .M;L/and BPAP0

.M;L/ are just precompact in C.R;R/, while BC .M;L/ is compact inC.R;R/.

So (non-)compactness of BPAP .M;L/ in C.R;R/ is still open.Finally, we recall Tikhonov fixed point theorem.

Theorem 2. (Tikhonov) Let˝ be a non-empty compact convex subset of a locallyconvex topological vector space X . Then any continuous function A W˝!˝ has afixed point.


3. EXISTENCE RESULT

In this section, the existence of pseudo almost periodic solutions of equation (1.1)will be studied. Throughout this paper, it will be assumed that �1 W R! .�1;0/ isan almost periodic function, �i W R! R are pseudo almost periodic functions, and

�C1 D supt2R

�1.t/ < 0;�Ci D sup

t2Rj�i .t/j;

where i D 2; : : : ;n.Now we apply Theorem 2. Let X be either PAP.R;R/ or BC.R;R/ but fixed. If

' 2X , from Corollary 5.4 in [19], it follows 'Œ2� 2X . Consider an auxiliary equation

x0.t/D �1.t/x.t/C�2.t/'Œ2�.t/C : : :C�n.t/'

Œn�.t/Cf .t/; (3.1)

where f 2 PAP.R;R/ and �1.t/ < 0. It is easy to see that the linear equation

x0.t/D �1.t/x.t/

admits an exponential dichotomy on R, by Theorem 2.3 in [18], we know that (3.1)has exactly one solution

x'.t/D

Z t

�1

eR t

s �1.u/du� nXiD2

�i .s/'Œi�.s/Cf .s/

�ds

in X .Now take f 2 BPAP .M 0;L0/ for constants M 0 and L0. Let A W BC .M;L/!

BC.R;R/ be defined by

.A'/.t/D

Z t

�1

eR t

s �1.u/du� nXiD2

�i .s/'Œi�.s/Cf .s/

�ds; t 2 R: (3.2)

Note A W BPAP .M;L/! PAP.R;R/.

Lemma 1. For any '; 2 BC .M;L/; t1; t2 2 R, the following inequality hold,

jj'Œn�� Œn�jj �

n�1XjD0

Lj k'� k; nD 1;2; : : : : (3.3)

Proof. It can be obtained by direct calculation by the definition of BC .M;L/. �

Lemma 2. Operator A W BC .M;L/! C.R;R/ is continuous.

Proof. Let 'j ! '0 as j !1 for 'j 2 BC .M;L/, j 2N0 DN[f0g uniformlyon any compact interval Œ�m;m�, m 2N of R. Set

hj .s/D

nXiD2

�i .s/'Œi�j .s/Cf .s/; j 2N0:


Then hj ! h0 uniformly on Œ�m;m�. Next we haveˇeR�m

s �1.u/duhj .s/ˇ�

�M

nXiD2

�Ci Ckf k�e��

C

1 .sCm/; s 2 Œ�1;�m�:

Since Z �m�1

e��C

1 .sCm/ds D�1

�C1<1;

we can apply the Lebesgue dominated convergence theorem to obtain .A'j /.�m/!.A'0/.�m/. From (3.1),

x0j .t/�x00.t/D �1.t/.xj .t/�x0.t//Chj .t/�h0.t/ (3.4)

for xj .t/D .A'j /.t/, k 2N0. Integrating the both sides of (3.4) from �m to t , wehave

xj .t/�x0.t/D xj .�m/�x0.�m/C

Z t

�m

�1.s/.xj .s/�x0.s//ds

C

Z t

�m

.hj .s/�h0.s//ds

and

jxj .t/�x0.t/j � jxj .�m/�x0.�m/jC2mkhj �h0km��C1

Z t

�m

jxj .s/�x0.s/jds

for any t 2 Œ�m;m�. Then Gronwall’s inequality implies

kxj �x0km ��jxj .�m/�x0.�m/jC2mkhj �h0km

�e�2�

C

1 m;

which means

kA'j �A'0km ��jA'j .�m/�A'0.�m/jC2mkhj �h0km

�e�2�

C

1 m:

Hence A'j .t/! A'0.t/ uniformly on t 2 Œ�m;m�. Since m 2N is arbitrarily, wegetA'j !A'0 in C.R;R/, i.e.,A WBC .M;L/!C.R;R/ is continuous. This provesthe continuity of A. �

Of course, then A W BPAP .M;L/! C.R;R/ is continuous as well.

Theorem 3. Suppose

��C1 �

nXiD2

�Ci > 0: (3.5)

Then for any f 2 BPAP .M 0;L0/, Eq. (1.1) has a solution

' 2 BPAP .M;L/� C.R;R/


for the constants M and L given by

M D�M 0

�C1 CPniD2�

Ci

; LD2M 0�C1

�C1 CPniD2�

Ci

: (3.6)

Proof. For any '; 2 BC .M;L/; t; t1; t2 2 R, from (3.6) we haveˇ.A'/.t/

ˇ�

nXiD2

�Ci

ˇˇZ t

�1

eR t

s �1.u/du'Œi�.s/ds

ˇˇC

ˇˇZ t

�1

eR t

s �1.u/duf .s/ds

ˇˇ

� �M

�C1

nXiD2

�Ci �M 0

�C1DM:

Without loss of generality, assume t2 � t1, thenˇ.A'/.t2/� .A /.t1/

ˇ�

nXiD2

�Ci

ˇˇZ t2

�1

eR t2

s �1.u/du'Œi�.s/ds�

Z t1

�1

eR t1

s �1.u/du'Œi�.s/ds

ˇˇ

C

ˇˇZ t2

�1

eR t2

s �1.u/duf .s/ds�

Z t1

�1

eR t1

s �1.u/duf .s/ds

ˇˇ

�

nXiD2

�Ci

"ˇZ t1

�1

eR t1

s �1.u/du�eR t2

t1�1.u/du

�1�'Œi�.s/ds

ˇC

ˇ Z t2

t1

eR t2

s �1.u/du'Œi�.s/dsˇ#

C

"ˇZ t1

�1

eR t1

s �1.u/du�eR t2

t1�1.u/du

�1�f .s/ds

ˇC

ˇ Z t2

t1

eR t2

s �1.u/duf .s/dsˇ#

�M

nXiD2

�Ci

"j�C1 jjt2� t1j

ˇ Z t1

�1

e�C

1 .t1�s/dsˇC

ˇ Z t2

t1

e�C

1 .t2�s/dsˇ#

CM 0

"j�C1 jjt2� t1j

ˇ Z t1

�1

e�C

1 .t1�s/dsˇC

ˇ Z t2

t1

e�C

1 .t2�s/dsˇ#

� 2.M

nXiD2

�Ci CM0/jt2� t1j D Ljt2� t1j:

This shows that A' 2 BC .M;L/. Hence A W BC .M;L/! BC .M;L/ and also A WBPAP .M;L/!BPAP .M;L/. By Lemma 2 we getA WBPAP .M;L/!BPAP .M;L/.So all conditions of Tikhonov fixed theorem are satisfied forAwith˝DBPAP .M;L/and X D C.R;R/. Thus there exists a fixed point ' in BPAP .M;L/ of A. This is


equivalent to say that ' is a solution of (1.1) in BPAP .M;L/. This completes theproof. �

The solution ' of Theorem 3 is a limit in C.R;R/ of a sequence of pseudo al-most periodic functions in BPAP .M;L/, i.e., ' is a uniform approximation on eachcompact intervals of R of a sequence of pseudo almost periodic functions. If f 2BAP .M

0;L0/ then we know by Theorem 1 only that in general, the solution ' ofTheorem 3 belongs just to f 2 BAP .M;L/.

4. UNIQUENESS RESULT

In this section, uniqueness of (1.1) will be proved, moreover, an example will beshowed.

Theorem 4. Suppose (3.5). Then for any f 2 BPAP .M 0;L0/, Eq. (1.1) has theunique solution ' 2 BPAP .M;L/ provided that it holds

� D�1

�C1

nXiD2

i�1XjD0

�Ci Lj < 1 (4.1)

for the constants M and L given by (3.6).

Proof. We already known that A W BPAP .M;L/! BPAP .M;L/ where we con-sider BPAP .M;L/ in BC.R;R/. For '; 2 BPAP .M;L/, by (3.2) and (3.3)

j'.t/� .t/j Dˇ.A'/.t/� .A /.t/

ˇ�

nXiD2

�Ci

Z t

�1

eR t

s �1.u/duˇ'Œi�.s/� Œi�.s/

ˇds

� �1

�C1

nXiD2

i�1XjD0

�Ci Ljk'� k D � k'� k;

thusk'� k � � k'� k:

From (4.1), we know � <1 andBPAP .M;L/ is closed in the Banach spaceBC.R;R/.The Banach fixed point theorem gives a unique fixed point of A in BPAP .M;L/. Theproof is finished. �

Example 1. Now, we apply Theorem 4. Consider the following equation:

x0.t/ D �.jsin tC sinp3t jC6/x.t/C

1

5

�jcos tC cos

p5t jC8

�x.x.t//

C1

7

�sin tC sin

p2tC sech t

�; (4.2)

where �1.t/D�.jsin tCsinp3t jC6/, �2.t/D 1

5

�jcos tCcos

p5t jC8

�and f .t/D

17

�sin tC sin

p2tC sech t

�. Take M 0 D 3

7and L0 D 4

7. A simple calculation yields


�C1 D �6 and �C2 D 2, so (3.5) holds: ��C1 ��C2 D 4 > 0. Then by (3.6) we derive

M D 328

and LD 97

. Thus (4.1) gives � D 1621< 1. Hence by Theorem 4, Eq. (4.2)

has a unique solution in BPAP�328; 97

�.

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Authors’ addresses

Hou Yu ZhaoSchool of Mathematics, Chongqing Normal University, Chongqing 401331, P.R.China

Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, CanadaE-mail address: [email protected]

Michal FeckanDepartment of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratis-

lava, Mlynska dolina, 842 48 Bratislava, Slovakia

Mathematical Institute of Slovak Academy of Sciences, Stefanikova 49, 814 73 Bratislava, Slov-akia

E-mail address: [email protected]

pseudo almost periodic solutions of an iterative...

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