# Almost Periodic Solutions of Nonlinear Discrete Volterra Equations with Unbounded Delay

- Sung Kyu Choi
^{1}and - Namjip Koo
^{1}Email author

**2008**:692713

**DOI: **10.1155/2008/692713

© S. K. Choi and N. Koo. 2008

**Received: **30 June 2008

**Accepted: **14 October 2008

**Published: **28 October 2008

## Abstract

We study the existence of almost periodic solutions for nonlinear discrete Volterra equations with unbounded delay, as a discrete analogue of the results for integro-differential equations by Y. Hamaya (1993).

## 1. Introduction

where is continuous and is almost periodic in uniformly for , and is continuous and is almost periodic in uniformly for . He showed that for a periodic integro-differential equation, uniform stability and stability under disturbances from hull are equivalent. Also, he showed the existence of an almost periodic solution under the assumption of total stability in [2].

where is continuous in for every , and for any , is continuous for . They showed that under some suitable conditions, if the bounded solution of (1.2) is totally stable, then it is an asymptotically almost periodic solution of (1.2), and (1.2) has an almost periodic solution. Also, Song [4] proved that if the bounded solution of (1.2) is uniformly asymptotically stable, then (1.2) has an almost periodic solution.

Equation (1.2) is a discrete analogue of the integro-differential equation (1.1), and (1.2) is a summation equation that is a natural analogue of this integro-differential equation. For the asymptotic properties of discrete Volterra equations, see [5].

In this paper, in order to obtain an existence theorem for an almost periodic solution of a discrete Volterra equations with unbounded delay, we will employ to change Hamaya's results in [1] for the integro-differential equation into results for the discrete Volterra equation.

## 2. Preliminaries

We denote by , respectively, the set of real numbers, the set of nonnegative real numbers, and the set of nonpositive real numbers. Let denote -dimensional Euclidean space.

Definition 2.1 (see [6]).

*almost periodic in*

*uniformly for*if for any there corresponds a number such that any interval of length contains a for which

Let and let be a function which is defined and continuous for and .

Definition 2.2 (see [9]).

*almost periodic in*

*uniformly for*if for any and any compact set in , there exists an such that any interval of length contains a for which

We denote by , respectively, the set of integers, the set of nonnegative integers, and the set of nonpositive integers.

Definition 2.3 (see [3]).

*almost periodic in*

*uniformly for*if for every and every compact set , there corresponds an integer such that among consecutive integers there is one, here denoted , such that

Definition 2.4 (see [3]).

Let
. A set
is said to be*compact* if there is a finite integer set
and compact set
such that
.

Definition 2.5.

*almost periodic in*

*uniformly for*if for any and any compact set , there exists a number such that any discrete interval of length contains a for which

For the basic results of almost periodic functions, see [6–8].

where is continuous in for every and is almost periodic in uniformly for , is continuous in for any and is almost periodic in uniformly for . We assume that, given , there is a solution of (2.7) such that for , passing through . Denote by this solution .

Let be any compact subset of such that for all and for all .

where . Then, defines a metric on the space . Note that the induced topology by is the same as the topology of convergence on any finite subset of [3].

*hull*

Note that and for any , we can assume the almost periodicity of and , respectively [3].

Definition 2.6 (see [3]).

is called the *limiting equation* of (2.7).

respectively. This definition is a discrete analogue of Hamaya's definition in [1].

## 3. Main Results

Definition 3.1 (see [3]).

A function
is called*asymptotically almost periodic* if it is a sum of an almost periodic function
and a function
defined on
which tends to zero as
, that is
.

It is known [8] that the decomposition in Definition 3.1 is unique, and is asymptotically almost periodic if and only if for any integer sequence with as , there exists a subsequence for which converges uniformly for as .

Hamaya [9] proved that if the bounded solution of the integro-differential equation (1.1) is asymptotically almost periodic, then is almost periodic under the following assumption:

whenever is continuous and for all .

Throughout this paper, we impose the following assumptions.

(H2) Equation (2.7) has a bounded solution , that is, for some , passing through , where .

Note that assumption (H1) holds for any . Also, we assume that the compact set in satisfies for all and for all , where is any solution of the limiting equation of (2.12) and (2.7) .

Theorem 3.2.

Under assumptions and , if the bounded solution is asymptotically almost periodic, then (2.7) has an almost periodic solution.

Proof.

where is almost periodic in and as . Let be a sequence such that as , as , and is also almost periodic. We will prove that is a solution of (2.7) for .

uniformly on , where is a compact subset of .

by (3.13).

as . Therefore, is an almost periodic solution of (2.7) for .

Remark 3.3.

Recently Song [4] obtained a more general result than that of Theorem 3.2, that is, under the assumption of asymptotic almost periodicity of a bounded solution of (2.7), he showed the existence of an almost periodic solution of the limiting equation (2.12) of (2.7).

Total stability introduced by Malkin [11] in 1944 requires that the solution of is "stable" not only with respect to the small perturbations of the initial conditions, but also with respect to the perturbations, small in a suitable sense, of the right-hand side of the equation [11]. Many results have been obtained concerning total stability [3, 7, 9, 12–15].

Definition 3.4 (see [1]).

*totally stable*if for any , there exists a such that if and is any continuous function which satisfies on , then

such that for all . Here, is defined by for any .

Hamaya [1] defined the following stability notion.

Definition 3.5.

*stable under disturbances*from with respect to if for any , there exists an such that

The concept of stability under disturbances from hull was introduced by Sell [16, 17] for the ordinary differential equation. Hamaya proved that Sell's definition is equivalent to Hamaya's definition in [1]. Also, he showed that total stability implies stability under disturbances from hull in [1, Theorem 1]. To prove the discrete analogue for this result, we list definitions.

Definition 3.6 (see [3]).

*totally stable*if for any there exists a such that if and is a sequence such that for all , then

Definition 3.7.

*stable under disturbances*from with respect to if for any , there exists an such that if and for some , then

where is any solution of the limiting equation (2.12) of (2.7), which passes through such that for all .

Theorem 3.8.

Under assumptions and , if the bounded solution of (2.7) is totally stable, then it is stable under disturbances from with respect to .

Proof.

Let be any solution of the limiting equation (2.12), passing through , such that for all . Note that for all by the assumption on . We suppose that and . We will show that for all .

This shows that is stable under disturbances from with respect to .

Remark 3.9.

Yoshizawa [15, Lemma 5] proved that the total stability of a bounded solution of the functional differential equation implies the stability under disturbances from hull. For a similar result for the integro-differential equation (1.1), see [1,Theorem 1].

Yoshizawa showed the existence of asymptotically almost periodic solution by stability under disturbances from hull for the nonlinear differential equation and the functional differential equation in [7, Theorem 12.4] and [15, Theorem 5], respectively.

Also, as the discrete case, Zhang and Zheng [18, Theorem 3.2] obtained the similar result for the functional difference equation . For the discrete Volterra equation (2.7), we get the following result.

Theorem 3.

Under assumptions and , if the bounded solution of (2.7) is stable under disturbances from with respect to , then is asymptotically almost periodic.

Proof.

For any sequence with as , let . Then, is a solution of (3.8) passing through where for all , as in the proof of Theorem 3.2. We claim that is stable under disturbances from with respect to for .

for some , where is any solution of (3.38). We will show that for all .

This shows that is stable under disturbances from with respect to for .

whenever . Therefore, is asymptotically almost periodic.

Finally, in view of Theorems 3.10 and 3.2, we obtain the following.

Corollary 3.11.

Under assumptions and if the bounded solution of (2.7) is stable under disturbances from with respect to , then (2.7) has an almost periodic solution.

Remark 3.12.

Song and Tian obtained the result for the existence of almost periodic solution to (2.7) by showing that if the bounded solution of (2.7) is totally stable, then it is an asymptotically almost periodic solution in [3, Theorem 4.4]. Note that total stability implies stability under disturbances from hull for (2.7) in view of Theorem 3.8.

## Declarations

### Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions which led to an important improvement of original manuscript. This work was supported by the Second Stage of Brain Korea 21 Project in 2008.

## Authors’ Affiliations

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