Almost Periodic Solutions of Nonlinear Discrete Volterra Equations with Unbounded Delay

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

Hamaya [1] discussed the relationship between stability under disturbances from hull and total stability for the integro-differential equation

(1.1)

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].

Song and Tian [3] studied periodic and almost periodic solutions of discrete Volterra equations with unbounded delay of the form

(1.2)

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]).

A continuous function is called almost periodic in uniformly for if for any there corresponds a number such that any interval of length contains a for which

(2.1)

for all and .

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

Definition 2.2 (see [9]).

is said to bealmost 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

(2.2)

for all and all .

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

Definition 2.3 (see [3]).

A continuous function is said to bealmost 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

(2.3)

for all , uniformly for .

Definition 2.4 (see [3]).

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

Definition 2.5.

Let be continuous for , for any . is said to bealmost 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

(2.4)

for all and all .

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

Let denote the space of all -valued bounded functions on with

(2.5)

for any .

Let for any integer . For any , we define the notation by the relation

(2.6)

for .

Consider the discrete Volterra equation with unbounded delay

(2.7)

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 .

For any , we set

(2.8)

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].

In view of almost periodicity, for any sequence with as , there exists a subsequence such that

(2.9)

uniformly on for any compact set ,

(2.10)

uniformly on for any compact set , and are also almost periodic in uniformly for , and almost periodic in uniformly for , respectively. We define the hull

(2.11)

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

Definition 2.6 (see [3]).

If , then the equation

(2.12)

is called the limiting equation of (2.7).

For the compact set in , , we define and by

(2.13)

where

(2.14)

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

3. Main Results

Definition 3.1 (see [3]).

A function is calledasymptotically 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:

(H) for any and any compact set , there exists such that

(3.1)

whenever is continuous and for all .

Also, Islam [10] showed that asymptotic almost periodicity implies almost periodicity for the bounded solution of the almost periodic integral equation

(3.2)

Throughout this paper, we impose the following assumptions.

(H1) For any and any , there exists an integer such that

(3.3)

whenever for all .

(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.

Since is asymptotically almost periodic, it has the decomposition

(3.4)

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 .

Note that, by almost periodicity,

(3.5)

uniformly on , where is a compact set in , and

(3.6)

uniformly on , where is a compact subset of .

Let . Then, we obtain

(3.7)

This implies that is a solution of

(3.8)

For since

(3.9)

Moreover, for any , there exists a such that for all . Thus

(3.10)

as whenever . Hence,

(3.11)

Now, we show that

(3.12)

as . Note that, for some , and for all and . From (H1), there exists an integer such that

(3.13)

for any . Then, we have

(3.14)

by (3.13).

Since is continuous for and on as , we obtain

(3.15)

It follows from the continuity of that

(3.16)

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]).

The bounded solution of (1.1) is said to betotally stable if for any , there exists a such that if and is any continuous function which satisfies on , then

(3.17)

where is a solution of

(3.18)

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

Hamaya [1] defined the following stability notion.

Definition 3.5.

The bounded solution of (1.1) is said to be stable under disturbances from with respect to if for any , there exists an such that

(3.19)

whenever , and for some , where is a solution through of the limiting equation

(3.20)

of (1.1) such that for all .

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]).

The bounded solution of (2.7) is said to betotally stable if for any there exists a such that if and is a sequence such that for all , then

(3.21)

where is any solution of

(3.22)

such that for all .

Definition 3.7.

The bounded solution of (2.7) is said to bestable under disturbances from with respect to if for any , there exists an such that if and for some , then

(3.23)

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 given and let be the number for total stability of . In view of , there exists an such that

(3.24)

whenever for all . Also, since satisfies , we have

(3.25)

whenever for all . We choose such that and set

(3.26)

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 .

For every , we set

(3.27)

Then, is a solution of the perturbation

(3.28)

such that for all . We claim that for all . From

(3.29)

we have

(3.30)

Thus

(3.31)

when for . Since

(3.32)

we obtain

(3.33)

and thus

(3.34)

This implies that

(3.35)

where , as long as . Therefore, we have

(3.36)

as long as . Consequently, we obtain that for all . Since is totally stable, we have

(3.37)

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 .

Consider the limiting equation

(3.38)

where . Assume that

(3.39)

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

Putting , is a solution of

(3.40)

passing through such that for all . If we set , then is a solution of

(3.41)

Since

(3.42)

we have

(3.43)

Since is stable under disturbances from , we obtain

(3.44)

that is,

(3.45)

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

Now, from the almost periodicity, there exists a subsequence of , which we denote by again, such that converges uniformly on and converges uniformly on , where is a compact subset of , as . It follows that for any , there exists a such that implies

(3.46)

for all , where is a positive integer such that

(3.47)

Since

(3.48)

we have

(3.49)

whenever . We can assume that converges uniformly on any compact interval in . Thus, there exists a such that whenever . To show that is asymptotically almost periodic, we will show that

(3.50)

if , where is a solution of

(3.51)

such that for all and . Since

(3.52)

whenever , we have

(3.53)

from the fact that is stable under disturbances from with respect to . Consequently, we obtain

(3.54)

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.