- Research Article
- Open Access

# Existence of Periodic and Almost Periodic Solutions of Abstract Retarded Functional Difference Equations in Phase Spaces

- Claudio Vidal
^{1}Email author

**2009**:380568

https://doi.org/10.1155/2009/380568

© Claudio Vidal 2009

**Received: **20 November 2008

**Accepted: **10 June 2009

**Published: **16 June 2009

## Abstract

The existence of periodic, almost periodic, and asymptotically almost periodic of periodic and almost periodic of abstract retarded functional difference equations in phase spaces is obtained by using stability properties of a bounded solution.

## Keywords

- Positive Integer
- Phase Space
- Periodic Solution
- Functional Differential Equation
- Bounded Solution

## 1. Introduction

assuming that this system possesses a bounded solution with some property of stability. In (1.1) , and denotes an abstract phase space which we will define later.

The abstract space was introduced by Hale and Kato [1] to study qualitative theory of functional differential equations with unbounded delay. There exists a lot of literature devoted to this subject; we refer the reader to Corduneanu and Lakshmikantham [2], Hino et al. [3]. The theory of abstract retarded functional difference equations in phase space has attracted the attention of several authors in recent years. We only mention here Murakami [4, 5], Elaydi et al. [6], Cuevas and Pinto [7, 8], Cuevas and Vidal [9], and Cuevas and Del Campo [10].

As usual, we denote by , , and the set of all integers, the set of all nonnegative integers, and the set of all nonpositive integers, respectively. Let be the -dimensional complex Euclidean space with norm . the set .

If
*is a function, we define for*
, the function
by
,
. Furthermore
is the function given for
, with
.

Axiom (C) says that any element of the Banach space of the bounded functions equipped with the supremum norm is on .

Remark 1.1.

Remark 1.2.

We will denote by ( , and ) or simply by , the solution of (1.1) passing through , that is, , and the functional equation (1.1) is satisfied.

During this paper we will assume that the sequences and are bounded. The paper is organized as follows. In Section 2 we see some important implications of the fading memory spaces. Section 3 is devoted to recall definitions and some important basic results about almost periodic sequences, asymptotically almost periodic sequences, and uniformly asymptotically almost periodic functions. In Section 4 we analyze separately the cases where is periodic and when it is almost periodic. Thus, in Section 4.1 assuming that the system (1.1) is periodic and the existence of a bounded solution (particular solution) which is uniformly stable and the phase space satisfies only the axioms (A)–(C), we prove the existence of an almost periodic solution and an asymptotically almost periodic solution. If additionally the particular solution is uniformly asymptotically stable, we prove the existence of a periodic solution. Similarly, in Section 4.2 considering that system (1.1) is almost periodic and the existence of a bounded solution and whenever the phase space satisfies the axioms (A)–(C), but here it is also necessary that verifies the fading memory property. If the particular solution is asymptotically almost periodic, then system (1.1) has an almost periodic solution. While, if the particular solution is uniformly asymptotically stable, we prove the existence of an asymptotically almost periodic solution.

In [11, 12] the problem of existence of almost periodic solutions for functional difference equations is considered in the first case for the discrete Volterra equation and in the second reference for the functional difference equations with finite delay; in both cases the authors assume the existence of a bounded solution with a property of stability that gives information about the existence of an almost periodic solution. In an analogous way in [13] the problem of the existence of almost periodic solutions for functional difference equations with infinite delay is considered. These results can be applied to several kinds of discrete equations. However, our approach differs from Hamaya's because, firstly, in our work we consider both cases, namely, when is periodic and when it is almost periodic in the first variable. And secondly, we analyze very carefully the implications of the existence of a bounded solution of (1.1) with each property: uniformly stable, uniformly asymptotically stable, and globally uniformly stable.

Furthermore, we cite the articles [14–16] which are devoted to study almost periodic solutions of difference equations, but a little is known about almost periodic solutions, and in particular, for periodic solutions of nonlinear functional difference equations in phase space via uniform stability, uniformly asymptotically stability, and globally uniformly stability properties of a bounded solution.

## 2. Fading Memory Spaces and Implications

Definition 2.1.

A phase space that satisfies axioms (A)-(B) and ( ) or ( ) and such that the semigroup is strongly stable is called a fading memory space.

Remark 2.2.

Remember that a strongly continuous semigroup is strongly stable if for all , as .

Thus, we have the following result.

Lemma 2.3.

Let , with , where is a fading memory space. If as , then as .

Proof.

Then, by definition as because . On the other hand, by hypothesis, as , so it follows from Axiom (C') that . Therefore, we conclude that as .

## 3. Notations and Preliminary Results

In this section, we review the definitions of (uniformly) almost periodic, asymptotically almost periodic sequence, which have been discussed by several authors and present some related properties.

For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in [3, 17, 18] for the continuous case. For the discrete case we mention [11, 12].

Definition 3.1.

is called the -translation number of . We will denote by the set of all such sequences. We will write that is a.p. if .

Definition 3.2.

where is an almost periodic sequence, and as . We will denote by the set of all such sequences. We will write that is a.a.p. if .

In general, we will consider a Banach space.

Definition 3.3.

Denote by all such sequences, and is said to be an almost periodic (a.p.) in .

Definition 3.4.

A sequence , (or ), , equivalently, a function (or, ) is called asymptotically almost periodic if , where and (or, ) satisfying as (or, ). Denote by (or all such sequences, and is said to be an asymptotically almost periodic on (or on ) (a.a.p.) in .

Remark 3.5.

Almost periodic sequences can be also defined for any sequence ( ) or by requiring that consecutive integers are in .

Definition 3.6.

is called the -translation number of . We will denote by the set of all such sequences. In brief we will write that is u.a.p. if .

Definition 3.7.

for some sequence , where is any compact set in .

For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in [3, 17, 18] for the continuous case. For the discrete case we mention [11, 12]. With the objective to make this manuscript self contained we decided to include the majority of the proofs.

- (a)
- (b)

Lemma 3.9. (a) If is an a.p. sequence, then is bounded.

(b) is an a.p. sequence if and only if for any sequence there exists a subsequence such that converges uniformly on as . Furthermore, the limits sequence is also an almost periodic sequence.

(d) , (or, ) is an a.a.p. sequence if and only if for any sequence (or, ) such that and as (or, as ), there exists a subsequence such that converges uniformly on (or ) as .

Lemma 3.10.

where is an a.p. sequence while as , is unique.

Lemma 3.11.

Let be almost periodic in uniformly for and continuous in . Then is bounded and uniformly continuous on for any compact set in .

Lemma 3.12.

Let be the same as in the previous lemma. Then, for any sequence , there exist a subsequence of and a function continuous in such that uniformly on as , where is any compact set in . Moreover, is also almost periodic in uniformly for .

Lemma 3.13.

Let be the same as in the previous lemma. Then, there exists a sequence , as such that uniformly on as , where is any compact set in .

Lemma 3.14.

Let be almost periodic in uniformly for and continuous in , and let be an almost periodic sequence in such that for all , where is a compact set in . Then is almost periodic in .

Lemma 3.15.

Let be almost periodic in uniformly for and continuous in , and let be an almost periodic sequence in such that for all , where is a compact set in and for . Then is almost periodic in .

Remark 3.16.

If is a.a.p., then the decomposition , in the definition of an a.a.p. function, is unique (see [18]).

## 4. Existence of Almost Periodic Solutions

From now on we will assume that the system (1.1) has a unique solution for a given initial condition on and without loss of generality , thus .

- (H1)
- (H2)

For this bounded solution , there is an such that for all . So, we will have to assume that for all , and . Next, we will point out the definitions of stability for functional difference equations adapting it from the continuous case according to Hino et al. in [3].

Definition 4.1.

- (i)
*stable*, if for any and any integer , there is such that implies that for all , where is any solution of (1.1); - (ii)
*uniformly stable*, abbreviated as " '', if for any and any integer , there is ( does not depend on ) such that implies that for all , where is any solution of (1.1); - (iii)
*uniformly asymptotically stable*, abbreviated as " '', if it is uniformly stable and there is such that for any , there is a positive integer such that if and , then for all , where is any solution of (1.1); - (iv)
*globally uniformly asymptotically stable*, abbreviated as " '', if it is uniformly stable and as , whenever is any solution of (1.1).

Remark 4.2.

### 4.1. The Periodic Case

- (H3)

Lemma 4.3.

Suppose that condition ( ) holds. If is a bounded solution of (1.1) such that , then is also bounded in .

Proof.

Lemma 4.4.

Suppose that condition ( ) holds. Let be a sequence in such that for all . Assume that as for every and , then in as for each . In particular, if as uniformly in , then in as uniformly in .

Proof.

for each . Therefore, we have concluded the proof.

Theorem 4.5.

Suppose that condition ( ) and (H1)–(H3) hold. If the bounded solution of (1.1) is , then is an a.a.p. sequence in , equivalently, (1.1) has an a.a.p. solution.

Proof.

through . It is clear that if is , then is also with the same pair as the one for .

This implies that for any positive integer sequence , as , there is a subsequence of for which converges uniformly on as . Thus, the conclusion of the theorem follows from Lemma 3.9(d).

Before proving our following result we remark that if is a.a.p. then there are unique sequences such that , with a.p. and as as . By Lemma 3.9(a) it follows that is bounded and thus . Hence, by Axiom (C) we must have that for all . In particular, for all .

Theorem 4.6.

Suppose that and (H1)–(H3) hold and the bounded solution of (1.1) is , then system (1.1) has an a.p. solution, which is also .

Proof.

as , we have for , that is, the system (1.1) has an almost periodic solution, and so we have proved the first statement of the theorem.

for all , which implies that for all if because is arbitrary. This proves that is .

In the case when we have an asymptotically stable solution of (1.1) we obtain the following result.

Theorem 4.7.

Suppose that and (H1)–(H3) hold and the bounded solution of (1.1) is , then the system (1.1) has a periodic solution of period for some positive integer , which is also .

Proof.

which implies that for all because is a.p.

For the integer sequence , , we have . Then uniformly for all as , and again by Lemma 4.4, uniformly in as . Since , we have for , which implies that (1.1) has a periodic solution of period .

as if , because , , and satisfy (1.1). This completes the proof.

Finally, if the particular solution is , we will prove that system (1.1) has a periodic solution.

Theorem 4.8.

Suppose that and (H1)–(H3) hold and that the bounded solution of (1.1) is , then the system (1.1) has a periodic solution of period .

Proof.

By Theorem 4.5, is a.a.p. Then ), where ( ) is an a.p. sequence and as . Notice that is also a solution of (1.1) satisfying . Since is , we have that as , which implies that for all . Using same technique as in the proof of Theorem 4.7, we can show that is a -periodic solution of (1.1).

### 4.2. The Almost Periodic Case

- (H4)

By we denote the uniform closure of , that is, . Note that by Lemma 3.12 and by Lemma 3.13.

Lemma 4.9.

Suppose that Axiom (C) is true, and that is an a.p. sequence with , then is a.p.

Proof.

Lemma 4.10.

Suppose that is a fading memory space and is a.a.p. with , then is a.a.p.

Proof.

Since is a.a.p. there are unique sequences and such that is a.p. and as . Then by Lemma 4.9 it follows that is a.p., and by Lemma 2.3 it follows that as . Therefore, is a.a.p.

Theorem 4.11.

Suppose that conditions , (H1)-(H2), and (H4) hold and that is a fading memory space. If the bounded solution of (1.1) is an a.a.p. sequence, then the system (1.1) has an a.p. solution.

Proof.

and from the previous considerations the first term of the right-hand side of (4.23) tends to zero as and since as , we have that for all , which implies that (1.1) has an a.p. solution passing through , where for .

We are now in a position to prove the following result.

Theorem 4.12.

Suppose that the assumptions , (H1), (H2), and (H4) hold, and that is a fading memory space. If the bounded solution of (1.1) is , then is a.a.p. Consequently, (1.1) has an a.p. solution which is .

Proof.

and hence by Axiom A(ii) for all if . This implies that the bounded solution of (1.1) is a.a.p. by Lemma 3.9(d). Furthermore, (1.1) has an a.p. solution, which is by Theorem 4.11. This ends the proof.

## Appendix

## Authors’ Affiliations

## References

- Hale JK, Kato J:
**Phase space for retarded equations with infinite delay.***Funkcialaj Ekvacioj*1978,**21**(1):11-41.MATHMathSciNetGoogle Scholar - Corduneanu C, Lakshmikantham V:
**Equations with unbounded delay: a survey.***Nonlinear Analysis: Theory, Methods & Applications*1980,**4**(5):831-877. 10.1016/0362-546X(80)90001-2MATHMathSciNetView ArticleGoogle Scholar - Hino Y, Murakami S, Naito T:
*Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics*.*Volume 1473*. Springer, Berlin, Germany; 1991:x+317.Google Scholar - Murakami S:
**Representation of solutions of linear functional difference equations in phase space.***Nonlinear Analysis: Theory, Methods & Applications*1997,**30**(2):1153-1164. 10.1016/S0362-546X(97)00296-4MATHMathSciNetView ArticleGoogle Scholar - Murakami S:
**Some spectral properties of the solution operator for linear Volterra difference systems.**In*New Developments in Difference Equations and Applications (Taipei, 1997)*. Gordon and Breach, Amsterdam, The Netherlands; 1999:301-311.Google Scholar - Elaydi S, Murakami S, Kamiyama E:
**Asymptotic equivalence for difference equations with infinite delay.***Journal of Difference Equations and Applications*1999,**5**(1):1-23. 10.1080/10236199908808167MATHMathSciNetView ArticleGoogle Scholar - Cuevas C, Pinto M:
**Asymptotic behavior in Volterra difference systems with unbounded delay.***Journal of Computational and Applied Mathematics*2000,**113**(1-2):217-225. 10.1016/S0377-0427(99)00257-5MATHMathSciNetView ArticleGoogle Scholar - Cuevas C, Pinto M:
**Convergent solutions of linear functional difference equations in phase space.***Journal of Mathematical Analysis and Applications*2003,**277**(1):324-341. 10.1016/S0022-247X(02)00570-XMATHMathSciNetView ArticleGoogle Scholar - Cuevas C, Vidal C:
**Discrete dichotomies and asymptotic behavior for abstract retarded functional difference equations in phase space.***Journal of Difference Equations and Applications*2002,**8**(7):603-640. 10.1080/10236190290032499MATHMathSciNetView ArticleGoogle Scholar - Cuevas C, Del Campo L:
**An asymptotic theory for retarded functional difference equations.***Computers & Mathematics with Applications*2005,**49**(5-6):841-855. 10.1016/j.camwa.2004.06.032MATHMathSciNetView ArticleGoogle Scholar - Song Y, Tian H:
**Periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay.***Journal of Computational and Applied Mathematics*2007,**205**(2):859-870. 10.1016/j.cam.2005.12.042MATHMathSciNetView ArticleGoogle Scholar - Song Y:
**Periodic and almost periodic solutions of functional difference equations with finite delay.***Advances in Difference Equations*2007,**2007:**-15.Google Scholar - Hamaya Y:
**Existence of an almost periodic solution in a difference equation with infinite delay.***Journal of Difference Equations and Applications*2003,**9**(2):227-237. 10.1080/1023619021000035836MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, O'Regan D, Wong PJY:
**Constant-sign periodic and almost periodic solutions of a system of difference equations.***Computers & Mathematics with Applications*2005,**50**(10-12):1725-1754. 10.1016/j.camwa.2005.03.020MATHMathSciNetView ArticleGoogle Scholar - Ignatyev AO, Ignatyev OA:
**On the stability in periodic and almost periodic difference systems.***Journal of Mathematical Analysis and Applications*2006,**313**(2):678-688. 10.1016/j.jmaa.2005.04.001MATHMathSciNetView ArticleGoogle Scholar - Zhang S, Liu P, Gopalsamy K:
**Almost periodic solutions of nonautonomous linear difference equations.***Applicable Analysis*2002,**81**(2):281-301. 10.1080/0003681021000021961MATHMathSciNetView ArticleGoogle Scholar - Fink AM:
*Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377*. Springer, Berlin, Germany; 1974:viii+336.Google Scholar - Zaidman S:
*Almost-Periodic Functions in Abstract Spaces, Research Notes in Mathematics*.*Volume 126*. Pitman, Boston, Mass, USA; 1985:iii+133.Google Scholar

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