Total Stability in Nonlinear Discrete Volterra Equations with Unbounded Delay
We study the total stability in nonlinear discrete Volterra equations with unbounded delay, as a discrete analogue of the results for integrodifferential equations by Y. Hamaya (1990).
The concepts of stability and asymptotic stability introduced by Lyapunov could be called stabilities under sudden perturbations. The perturbation suddenly moves the systems from its equilibrium state but then immediately disappears. Stability says that the effect of this will not be great if the sudden perturbation is not too great. Asymptotic stability states, in addition, that if the sudden perturbation is not great, the effect of the perturbation will tend to disappear. In practice, however, the perturbations are not simply impulses, and this led Duboshin (1940) and Malkin (1944) to consider what they called stability under constantly acting perturbations, today known as total stability. This says that if the perturbation is not too large and if the system is not too far from the origin initially it will remain near the origin. Total stability can be described roughly as the property that a bounded perturbation has a bounded effect on the solution . Many results have been obtained concerning total stability [1–9].
In , Hamaya discussed the relationship between total stability and stability under disturbances from hull for the integrodifferential equation 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 integrodifferential 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 .
Song and Tian  studied periodic and almost periodic solutions of discrete Volterra equations with unbounded delay of the form 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.
Choi and Koo  investigated the existence of an almost periodic solution of (1.2) as a discretization of the results in . The purpose of this paper is to study the total stability for the discrete Volterra equation of the form To do this, we will employ to change Hamaya's results in  for the integrodifferential equation into results for the discrete Volterra equation (1.3).
We denote by , respectively, the set of integers, the set of nonnegative integers, and the set of nonpositive integers. Let denote Euclidean space.
Definition 2.1 (see ). A continuous function is said to be 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 by , such that for all , uniformly for .
Definition 2.2 (see ). Let be continuous for , for any , where . is said to be 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 all and all .
For the basic results of almost periodic functions, see [8, 14, 15].
Let denote the space of all -valued bounded functions on with
for any .
Consider the discrete Volterra equation with unbounded delay of the form under certain conditions for and (see below). We assume that, given , there is a solution of (1.3) such that for , passing through .
Let be any compact subset of such that for all and for all .
For any , we set 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 .
In view of almost periodicity, for any sequence with as , there exists a subsequence such that uniformly on for any compact set , uniformly on for any compact set . We define the hull Note that and for any , we can assume the almost periodicity of and , respectively .
3. Main Results
We deal with the discrete Volterra equation with unbounded delay of the form Throughout this paper we assume the following.(H1) is continuous in for every and is almost periodic in uniformly for . (H2) is continuous in for any , and is almost periodic in uniformly for . Moreover, for any and any there exists a number such that for all whenever for all . (H3) is continuous in for every . is defined as for . Furthermore, for any , there exists a function with the property that as and whenever for all .(H4)Equation (3.1) has a bounded solution defined on , through , such that for some ,
Total stability 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.
Definition 3.2. The bounded solution of (3.1) is said to be totally stable if for any there exists a such that if and is a function such that for all , then for all , where is any solution of such that for all .
Definition 3.3. 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, for all .
It is known  that the decomposition in Definition 3.3 is unique. Moreover, is asymptotically almost periodic if and only if for any integer sequence with as , there exists a subsequence for which converges uniformly for as .
Theorem 3.4. Under the assumptions (H1)–(H4), if is a bounded and totally stable solution of (3.1), then it is asymptotically almost periodic.
Proof. Let be an integer sequence with as . Set Then is a solution of
and for Clearly, is totally stable with the same number for the total stability of . We can assume that converges uniformly on any finite set in as by taking a subsequence if necessary. Then there exists a number such that whenever . Put
Then is a solution of
and for all . We will show that there exists a number such that for all whenever .
Note that for all , there exists a number such that . It is clear that and for all .
In view of (H2), there exists a number such that
From the almost periodicity of and , respectively, there exists a number for which whenever . Since as , we obtain that if , then Then, by (3.10), and (3.11), we have Therefore we have by (3.12), (3.13), and (3.14). Since is totally stable, we obtain that for all if . This implies that for all , for all and all . It follows that for any with as there exists a subsequence such that converges uniformly on as , that is, is asymptotically almost periodic. This completes the proof.
Remark 3.5. Hino et al.  showed that for the functional differential equation the solution of the limiting equation of (3.17) is asymptotically almost periodic if is totally stable. Here means that there exists a sequence as , such that uniformly on any compact set in and uniformly on any compact interval in the set of nonnegative real numbers, where the space is the fading memory space by Hale and Kato .
Proof. From , there exists a sequence as , such that uniformly on , uniformly on for any compact set , and uniformly on any compact set in as . Set Then it is clear that is a solution of
such that for all , where . Note that for all for some . Let be a function such that for all . By (H2), there exists a number such that
where is the number for the total stability of . Also, we have
for the same since . Hence, by the same argument as in the proof of Theorem 3.4, there exists a positive integer such that if , then
Then is a solution of
such that for . Note that for by (3.22). From (3.22) and the fact that is totally stable, we have
Let . To show that is totally stable we will show that if , and for , then for all , where is a solution of
such that for all . Suppose that this is not the case. Then there exists an integer such that We set . Then is a solution of defined on such that for all . Also, is a solution of where Note that for all , and for . Thus for . Also, we have from (3.26). Thus we obtain Since is totally stable, we have On the other hand, (3.26) implies that Hence, if and for , then we obtain This contradicts (3.28). Therefore for all when , and for all , where . Consequently, is totally stable.
The following definitions are the discrete analogues of Hamaya's definitions in .
Definition 3.8. The bounded solution of (3.1) is said to be totally asymptotically stable if it is totally stable and there exists a and for any there exists an and a such that if and is any function which satisfies for , then for all , where is a solution of such that for all .
Note that the total asymptotic stability is equivalent to the uniform asymptotic stability whenever .
Theorem 3.9. Under the assumptions (H1)–(H4), if the bounded solution of (3.1) is attracting in and totally stable, then it is totally asymptotically stable.
Proof. Let be the number for the attracting of in and let , where is the number for the total stability of . Suppose that is not totally asymptotically stable. Then there exists a number with and exist sequences , and such that and for all , where is a solution of
such that for all and with for . Note that and for . Then we have
for all and since is totally stable. Also, there exists an integer number such that if , then for all .
We claim that on if . If we assume that on , then for since is totally stable. This contradicts , because .
Now, for the sequence , taking a subsequence if necessary, there exists a .
If we set , then is the defined on . There exists a subsequence of , which we denote by again, and a function such that uniformly on any compact set in as such that for all . Moreover, we can show that is a solution of
such that for all , by the same method as in [13, Theorem 3.1] . We have Then, by letting , we obtain Since is attracting in , we have as . This contradicts . Hence is totally asymptotically stable. This completes the proof of the theorem.
The authors are thankful to the anonymous referees for their valuable comments and corrections to improve this paper.
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