Almost Periodic Solutions of Nonlinear Discrete Volterra Equations with Unbounded Delay
© S. K. Choi and N. Koo. 2008
Received: 30 June 2008
Accepted: 14 October 2008
Published: 28 October 2008
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).
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 .
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  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 .
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  for the integro-differential equation into results for the discrete Volterra equation.
Definition 2.1 (see ).
Definition 2.2 (see ).
Definition 2.3 (see ).
Definition 2.4 (see ).
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 .
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 .
Note that and for any , we can assume the almost periodicity of and , respectively .
Definition 2.6 (see ).
is called the limiting equation of (2.7).
respectively. This definition is a discrete analogue of Hamaya's definition in .
3. Main Results
Definition 3.1 (see ).
It is known  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 .
Throughout this paper, we impose the following assumptions.
Recently Song  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  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 . Many results have been obtained concerning total stability [3, 7, 9, 12–15].
Definition 3.4 (see ).
Hamaya  defined the following stability notion.
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 . 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 ).
where is any solution of the limiting equation (2.12) of (2.7), which passes through such that for all .
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 .
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.
Finally, in view of Theorems 3.10 and 3.2, we obtain the following.
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.
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.
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