Abstract and Applied Analysis
Volume 2009 (2009), Article ID 976369, 13 pages
doi:10.1155/2009/976369
Research Article

Total Stability in Nonlinear Discrete Volterra Equations with Unbounded Delay

Sung Kyu Choi,1 Yoon Hoe Goo,2 Dong Man Im,3 and Namjip Koo1

1Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea
2Department of Mathematics, Hanseo University, Seosan 352-820, South Korea
3Department of Mathematics Education, Cheongju University, Cheongju 360-764, South Korea

Received 15 December 2008; Revised 20 February 2009; Accepted 24 March 2009

Academic Editor: Elena Litsyn

Copyright © 2009 Sung Kyu Choi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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).

1. Introduction

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 [1]. Many results have been obtained concerning total stability [1–9].

In [10], Hamaya discussed the relationship between total stability and stability under disturbances from hull for the integrodifferential equation where 𝑓 ∶ ℝ × ℝ 𝑑 → ℝ 𝑑 is continuous and is almost periodic in 𝑡 uniformly for 𝑥 ∈ ℝ 𝑑 , and 𝐹 ∶ ℝ × ( − ∞ , 0 ] × ℝ 𝑑 × ℝ 𝑑 → ℝ 𝑑 is continuous and is almost periodic in 𝑡 uniformly for ( 𝑠 , 𝑥 , 𝑦 ) ∈ 𝑅 ∗ = ( − ∞ , 0 ] × ℝ 𝑑 × ℝ 𝑑 . 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 [11].

Song and Tian [12] studied periodic and almost periodic solutions of discrete Volterra equations with unbounded delay of the form where 𝑓 ∶ ℤ × ℝ 𝑑 → ℝ 𝑑 is continuous in 𝑥 ∈ ℝ 𝑑 for every 𝑛 ∈ ℤ , and for any 𝑗 , 𝑛 ∈ ℤ ( 𝑗 ≤ 𝑛 ) , 𝐵 ∶ ℤ × ℤ × ℝ 𝑑 × ℝ 𝑑 → ℝ 𝑑 is continuous for 𝑥 , 𝑦 ∈ ℝ 𝑑 . 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 [13] proved that if the bounded solution of (1.2) is uniformly asymptotically stable, then (1.2) has an almost periodic solution.

Choi and Koo [9] investigated the existence of an almost periodic solution of (1.2) as a discretization of the results in [10]. 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 [2] for the integrodifferential equation into results for the discrete Volterra equation (1.3).

2. Preliminaries

We denote by ℤ , ℤ + , ℤ − , respectively, the set of integers, the set of nonnegative integers, and the set of nonpositive integers. Let ℝ 𝑑 denote 𝑑 - d i m e n s i o n a l Euclidean space.

Definition 2.1 (see [12]). A continuous function 𝑓 ∶ ℤ × ℝ 𝑑 → ℝ 𝑑 is said to be almost periodic in 𝑛 ∈ ℤ uniformly for 𝑥 ∈ ℝ 𝑑 if for every 𝜀 > 0 and every compact set 𝐾 ⊂ ℝ 𝑑 , there corresponds an integer 𝑁 = 𝑁 ( 𝜀 , 𝐾 ) > 0 such that among 𝑁 consecutive integers there is one, here denoted by 𝑝 , such that for all 𝑛 ∈ ℤ , uniformly for 𝑥 ∈ ℝ 𝑑 .

Definition 2.2 (see [12]). Let 𝐵 ∶ ℤ × ℤ ∗ → ℝ 𝑑 be continuous for 𝑥 , 𝑦 ∈ ℝ 𝑑 , for any 𝑛 ∈ ℤ , 𝑗 ∈ ℤ − , where ℤ ∗ = ℤ − × ℝ 𝑑 × ℝ 𝑑 . 𝐵 ( 𝑛 , 𝑗 , 𝑥 , 𝑦 ) is said to be almost periodic in 𝑛 uniformly for ( 𝑗 , 𝑥 , 𝑦 ) ∈ 𝑍 ∗ if for any 𝜀 > 0 and any compact set 𝐾 ∗ ⊂ 𝑍 ∗ , there exists a number 𝑙 = 𝑙 ( 𝜀 , 𝐾 ∗ ) > 0 such that any discrete interval of length 𝑙 contains a 𝜏 for which for all 𝑛 ∈ ℤ and all ( 𝑗 , 𝑥 , 𝑦 ) ∈ 𝐾 ∗ .
For the basic results of almost periodic functions, see [8, 14, 15].
Let 𝑙 − ( ℝ 𝑑 ) denote the space of all ℝ 𝑑 -valued bounded functions on ℤ − with
for any 𝜙 ∈ 𝑙 − ( ℝ 𝑑 ) .

Consider the discrete Volterra equation with unbounded delay of the form under certain conditions for 𝑓 , 𝐵 , and ℎ (see below). We assume that, given 𝜙 ∈ 𝑙 − ( ℝ 𝑑 ) , there is a solution 𝑥 of (1.3) such that 𝑥 ( 𝑛 ) = 𝜙 ( 𝑛 ) for 𝑛 ∈ ℤ − , passing through ( 0 , 𝜙 0 ) , 𝜙 0 ∈ 𝑙 − ( ℝ 𝑑 ) .

Let 𝐾 be any compact subset of ℝ 𝑑 such that 𝜙 ( 𝑗 ) ∈ 𝐾 for all 𝑗 ≤ 0 and 𝑥 ( 𝑛 ) ∈ 𝐾 for all 𝑛 ≥ 1 .

For any 𝜙 , 𝜓 ∈ 𝑙 − ( ℝ 𝑑 ) , we set where 𝜌 𝑞 ( 𝜙 , 𝜑 ) = m a x − 𝑞 ≤ 𝑗 ≤ 0 | 𝜙 ( 𝑗 ) − 𝜓 ( 𝑗 ) | , 𝑞 ≥ 0 . Then 𝜌 defines a metric on the space 𝑙 − ( ℝ 𝑑 ) . Note that the induced topology by 𝜌 is the same as the topology of convergence on any finite subset of ℤ − [12].

In view of almost periodicity, for any sequence ( 𝑛 𝑘 ) ⊂ ℤ + with 𝑛 𝑘 → ∞ as 𝑘 → ∞ , there exists a subsequence ( 𝑛 𝑘 ) ⊂ ( 𝑛 𝑘 ) such that uniformly on ℤ × 𝑆 for any compact set 𝑆 ⊂ ℝ 𝑑 , uniformly on ℤ × 𝑆 ∗ for any compact set 𝑆 ∗ ⊂ 𝑍 ∗ . We define the hull Note that ( 𝑓 , 𝐵 ) ∈ 𝐻 ( 𝑓 , 𝐵 ) and for any ( 𝑔 , 𝐷 ) ∈ 𝐻 ( 𝑓 , 𝐵 ) , we can assume the almost periodicity of 𝑔 and 𝐷 , respectively [12].

3. Main Results

We deal with the discrete Volterra equation with unbounded delay of the form Throughout this paper we assume the following.

(H1) 𝑓 ∶ ℤ × ℝ 𝑑 → ℝ 𝑑 is continuous in 𝑥 ∈ ℝ 𝑑 for every 𝑛 ∈ ℤ and is almost periodic in 𝑛 ∈ ℤ uniformly for 𝑥 ∈ ℝ 𝑑 . (H2) 𝐵 ∶ ℤ × ℤ ∗ → ℝ 𝑑 is continuous in 𝑥 , 𝑦 ∈ ℝ 𝑑 for any 𝑛 ∈ ℤ , 𝑗 ∈ ℤ − and is almost periodic in 𝑛 ∈ ℤ uniformly for ( 𝑗 , 𝑥 , 𝑦 ) ∈ ℤ ∗ = ℤ − × ℝ 𝑑 × ℝ 𝑑 . Moreover, for any 𝜀 > 0 and any 𝜏 > 0 there exists a number 𝑀 = 𝑀 ( 𝜀 , 𝜏 ) > 0 such that for all 𝑛 ∈ ℤ whenever | 𝑥 ( 𝑗 ) | ≤ 𝜏 for all 𝑗 ∈ ℤ − . (H3) ℎ ∶ ℤ × 𝑙 − ( ℝ 𝑑 ) → ℝ 𝑑 is continuous in 𝜙 ∈ 𝑙 − ( ℝ 𝑑 ) for every 𝑛 ∈ ℤ . 𝑥 𝑛 ∈ 𝑙 − ( ℝ 𝑑 ) is defined as 𝑥 𝑛 ( 𝑗 ) = 𝑥 ( 𝑛 + 𝑗 ) for 𝑗 ∈ ℤ − . Furthermore, for any 𝑟 > 0 , there exists a function 𝛼 𝑟 ∶ ℤ → ℝ 𝑑 with the property that 𝛼 𝑟 ( 𝑛 ) → 0 as 𝑛 → ∞ and whenever | 𝜙 ( 𝑗 ) | ≤ 𝑟 for all 𝑗 ∈ ℤ − .(H4)Equation (3.1) has a bounded solution 𝑢 ( 𝑛 ) = 𝑢 ( 𝑛 , 𝜙 0 ) defined on ℤ + , through ( 0 , 𝜙 0 ) , 𝜙 0 ∈ 𝑙 − ( ℝ 𝑑 ) such that for some 0 ≤ 𝑀 < ∞ ,

Note that for any ( 𝑔 , 𝐷 ) ∈ 𝐻 ( 𝑓 , 𝐵 ) , 𝐷 ( 𝑛 , 𝑗 , 𝑥 , 𝑦 ) satisfies (H2) with 𝐵 = 𝐷 [12]. The limiting equation of (3.1) is defined as where ( 𝑔 , 𝐷 ) ∈ 𝐻 ( 𝑓 , 𝐵 ) . We assume that for any solution 𝑣 ( 𝑛 ) of (3.5), 𝑣 ( 𝑛 ) ∈ 𝐾 for all 𝑛 ≥ 1 , where 𝐾 is the above-mentioned compact set in ℝ 𝑑 .

Theorem 3.1 (see [12]). Under the assumptions (H1)–(H4), if 𝑢 ( 𝑛 ) = 𝑢 ( 𝑛 , 𝜙 0 ) , 𝜙 0 ∈ 𝑙 − ( ℝ 𝑑 ) is a bounded solution of (3.1), passing through ( 0 , 𝜙 0 ) and ( 𝑔 , 𝐷 ) ∈ 𝐻 ( 𝑓 , 𝐵 ) , then the limiting equation (3.5) of (3.1) has a bounded solution on ℤ .

Total stability requires that the solution of 𝑥 ( 𝑡 ) = 𝑓 ( 𝑡 , 𝑥 ) 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 𝑢 ( 𝑛 ) of (3.1) is said to be totally stable if for any 𝜀 > 0 there exists a 𝛿 = 𝛿 ( 𝜀 ) > 0 such that if 𝑛 0 ≥ 0 , 𝜌 ( 𝑢 𝑛 0 , 𝑥 𝑛 0 ) < 𝛿 , and 𝑝 ( 𝑛 ) is a function such that | 𝑝 ( 𝑛 ) | < 𝛿 for all 𝑛 ≥ 𝑛 0 , then 𝜌 ( 𝑢 𝑛 , 𝑥 𝑛 ) < 𝜀 for all 𝑛 ≥ 𝑛 0 , where 𝑥 ( 𝑛 ) is any solution of such that 𝑥 𝑛 0 ( 𝑗 ) ∈ 𝐾 for all 𝑗 ∈ ℤ − .

Definition 3.3. A function 𝜙 ∶ ℤ → ℝ 𝑑 is called asymptotically almost periodic if it is a sum of an almost periodic function 𝜙 1 and a function 𝜙 2 defined on ℤ which tends to zero as 𝑛 → ∞ , that is, 𝜙 ( 𝑛 ) = 𝜙 1 ( 𝑛 ) + 𝜙 2 ( 𝑛 ) for all 𝑛 ∈ ℤ .

It is known [15] that the decomposition 𝜙 = 𝜙 1 + 𝜙 2 in Definition 3.3 is unique. Moreover, 𝜙 is asymptotically almost periodic if and only if for any integer sequence ( 𝜏 𝑘 ) with 𝜏 𝑘 → ∞ as 𝑘 → ∞ , there exists a subsequence ( 𝜏 𝑘 ) ⊂ ( 𝜏 𝑘 ) for which 𝜙 ( 𝑛 + 𝜏 𝑘 ) converges uniformly for 𝑛 ∈ ℤ as 𝑘 → ∞ [15].

Theorem 3.4. Under the assumptions (H1)–(H4), if 𝑢 ( 𝑛 ) is a bounded and totally stable solution of (3.1), then it is asymptotically almost periodic.

Proof. Let ( 𝑛 𝑘 ) be an integer sequence with 𝑛 𝑘 → ∞ as 𝑘 → ∞ . Set 𝑢 𝑘 ( 𝑛 ) = 𝑢 ( 𝑛 + 𝑛 𝑘 ) , 𝑘 = 1 , 2 , … . Then 𝑢 𝑘 ( 𝑛 ) is a solution of and 𝑢 𝑘 ( 𝑛 ) ∈ 𝐾 for 𝑘 = 1 , 2 , … . Clearly, 𝑢 𝑘 ( 𝑛 ) is totally stable with the same number ( 𝜀 , 𝛿 ) for the total stability of 𝑢 ( 𝑛 ) . We can assume that ( 𝑢 𝑘 ( 𝑛 ) ) converges uniformly on any finite set in ℤ − as 𝑘 → ∞ by taking a subsequence if necessary. Then there exists a number 𝑘 1 ( 𝜀 ) > 0 such that 𝜌 ( 𝑢 𝑘 0 , 𝑢 𝑚 0 ) < 𝛿 ( 𝜀 ) whenever 𝑘 , 𝑚 ≥ 𝑘 1 ( 𝜀 ) . Put Then 𝑢 𝑚 ( 𝑛 ) = 𝑢 ( 𝑛 + 𝑛 𝑚 ) is a solution of and 𝑢 𝑚 ( 𝑛 ) ∈ 𝐾 for all 𝑛 ∈ ℤ . We will show that there exists a number 𝑘 2 ( 𝜀 ) > 0 such that | 𝑝 ( 𝑛 ) | < 𝛿 ( 𝜀 ) for all 𝑛 ≥ 0 whenever 𝑘 , 𝑚 ≥ 𝑘 2 ( 𝜀 ) .
Note that for all 𝑥 ∈ 𝐾 , there exists a number 𝑐 > 0 such that | 𝑥 | ≤ 𝑐 . It is clear that | 𝑢 𝑘 ( 𝑛 ) | ≤ 𝑐 and | 𝑢 𝑚 ( 𝑛 ) | ≤ 𝑐 for all 𝑛 ∈ ℤ .
In view of (H2), there exists a number 𝑀 = 𝑀 ( 𝑐 , 𝜀 ) > 0 such that
From the almost periodicity of 𝑓 ( 𝑛 , 𝑥 ) and 𝐵 ( 𝑛 , 𝑗 , 𝑥 , 𝑦 ) , respectively, there exists a number 𝑘 2 ( 𝜀 ) ≥ 𝑘 1 ( 𝜀 ) for which whenever 𝑘 , 𝑚 ≥ 𝑘 2 ( 𝜀 ) . Since ℎ ( 𝑛 , 𝜙 ) → 0 as 𝑛 → ∞ , we obtain that if 𝑘 , 𝑚 ≥ 𝑘 2 ( 𝜀 ) , then Then, by (3.10), and (3.11), we have Therefore we have 𝑘 , 𝑚 ≥ 𝑘 2 ( 𝜀 ) , by (3.12), (3.13), and (3.14). Since 𝑢 𝑘 ( 𝑛 ) is totally stable, we obtain that 𝜌 ( 𝑢 𝑘 𝑛 , 𝑢 𝑚 𝑛 ) < 𝜀 for all 𝑛 ≥ 0 if 𝑘 , 𝑚 ≥ 𝑘 2 ( 𝜀 ) . This implies that for all 𝑘 , 𝑚 ≥ 𝑘 2 ( 𝜀 ) , for all 𝜀 ≤ ( 1 / 4 ) and all 𝑛 ≥ 0 . It follows that for any ( 𝑛 𝑘 ) ⊂ ℤ with 𝑛 𝑘 → ∞ as 𝑘 → ∞ there exists a subsequence ( 𝑛 𝑘 𝑗 ) ⊂ ( 𝑛 𝑘 ) such that ( 𝑢 ( 𝑛 + 𝑛 𝑘 𝑗 ) ) converges uniformly on ℤ + as 𝑗 → ∞ , that is, 𝑢 ( 𝑛 ) is asymptotically almost periodic. This completes the proof.

Remark 3.5. Hino et al. [5] showed that for the functional differential equation the solution 𝑣 ( 𝑡 ) of the limiting equation of (3.17) is asymptotically almost periodic if 𝑣 ( 𝑡 ) is totally stable. Here ( 𝑣 , ̂ 𝑔 ) ∈ 𝐻 ( 𝑢 , 𝑓 ) means that there exists a sequence ( 𝑡 𝑘 ) , 𝑡 𝑘 → ∞ as 𝑘 → ∞ , such that 𝑓 ( 𝑡 + 𝑡 𝑘 , 𝜙 ) → ̂ 𝑔 ( 𝑡 , 𝜙 ) ∈ 𝐻 ( 𝑓 ) uniformly on any compact set in 𝐵 and 𝑢 ( 𝑡 + 𝑡 𝑘 ) → 𝑣 ( 𝑡 ) uniformly on any compact interval in the set of nonnegative real numbers, where the space 𝐵 is the fading memory space by Hale and Kato [16].

Theorem 3.6. Assume that (H1)–(H4). If the solution 𝑣 ( 𝑛 ) of (3.5) satisfying ( 𝑣 , 𝑔 , 𝐷 ) ∈ 𝐻 ( 𝑢 , 𝑓 , 𝐵 ) is totally stable, then the bounded solution 𝑢 ( 𝑛 ) of (3.1) is also totally stable.

Proof. From ( 𝑣 , 𝑔 , 𝐷 ) ∈ 𝐻 ( 𝑢 , 𝑓 , 𝐵 ) , there exists a sequence ( 𝑛 𝑘 ) ⊂ ℤ , 𝑛 𝑘 → ∞ as 𝑘 → ∞ , such that 𝑓 ( 𝑛 + 𝑛 𝑘 , 𝑥 ) → 𝑔 ( 𝑛 , 𝑥 ) uniformly on ℤ × 𝐾 , 𝐵 ( 𝑛 + 𝑛 𝑘 , 𝑗 , 𝑥 , 𝑦 ) → 𝐷 ( 𝑛 , 𝑗 , 𝑥 , 𝑦 ) uniformly on ℤ × 𝑆 ∗ × 𝐾 × 𝐾 for any compact set 𝑆 ∗ ⊂ ℤ − , and 𝑢 ( 𝑛 + 𝑛 𝑘 ) → 𝑣 ( 𝑛 ) uniformly on any compact set in ℝ as 𝑘 → ∞ . Set 𝑢 𝑘 ( 𝑛 ) = 𝑢 ( 𝑛 + 𝑛 𝑘 ) , 𝑘 = 1 , 2 , … . Then it is clear that 𝑢 𝑘 ( 𝑛 ) is a solution of such that 𝑢 𝑘 0 ( 𝑗 ) ∈ 𝐾 for all 𝑗 ≤ 0 , where 𝑢 𝑘 0 ( 𝑗 ) = 𝑢 𝑘 ( 0 + 𝑗 ) = 𝑢 ( 𝑗 + 𝑛 𝑘 ) . Note that for all 𝑥 ∈ 𝐾 , | 𝑥 | ≤ 𝑐 for some 𝑐 > 0 . Let 𝑥 ( 𝜏 ) be a function such that 𝑥 ( 𝜏 ) ∈ 𝐾 for all 𝜏 ≤ 𝑛 . By (H2), there exists a number 𝑀 = 𝑀 ( 𝑐 , 𝜀 ) > 0 such that where 𝛿 ( ⋅ ) is the number for the total stability of 𝑣 ( 𝑛 ) . Also, we have for the same 𝑀 since 𝐵 ( 𝑛 + 𝑛 𝑘 , 𝑗 , 𝑥 , 𝑦 ) → 𝐷 ( 𝑛 , 𝑗 , 𝑥 , 𝑦 ) . Hence, by the same argument as in the proof of Theorem 3.4, there exists a positive integer 𝑘 0 ( 𝜀 ) such that if 𝑘 ≥ 𝑘 0 ( 𝜀 ) , then Put Then 𝑢 𝑘 ( 𝑛 ) is a solution of such that 𝑢 𝑘 0 ( 𝑗 ) ∈ 𝐾 for 𝑗 ≤ 0 . Note that | 𝑟 ( 𝑛 ) | < 𝛿 ( ( 1 / 2 ) 𝛿 ( 𝜀 / 2 ) ) for 𝑛 ≥ 0 by (3.22). From (3.22) and the fact that 𝑣 ( 𝑛 ) is totally stable, we have for all 𝑛 ≥ 0 .
Let 𝑚 = 𝑘 0 ( 𝜀 ) . To show that 𝑢 ( 𝑛 ) is totally stable we will show that if 𝑛 0 ≥ 0 , 𝜌 ( 𝑢 𝑛 0 , 𝑦 𝑛 0 ) < ( 1 / 2 ) 𝛿 ( 𝜀 / 2 ) , and | 𝑝 ( 𝑛 ) | < ( 1 / 2 ) 𝛿 ( 𝜀 / 2 ) for 𝑛 ≥ 𝑛 0 , then 𝜌 ( 𝑢 𝑛 , 𝑦 𝑛 ) < 𝜀 for all 𝑛 ≥ 𝑛 0 , where 𝑦 ( 𝑛 ) is a solution of
such that 𝑦 𝑛 0 ( 𝑗 ) ∈ 𝐾 for all 𝑗 ≤ 0 . Suppose that this is not the case. Then there exists an integer 𝜎 > 𝑛 0 such that We set 𝑧 ( 𝑛 ) = 𝑦 ( 𝑛 + 𝑛 𝑚 ) . Then 𝑧 ( 𝑛 ) is a solution of defined on [ 𝑛 0 − 𝑛 𝑚 , 𝜎 − 𝑛 𝑚 ] such that 𝑧 𝑛 0 − 𝑛 𝑚 ( 𝑗 ) = 𝑦 𝑛 0 ( 𝑗 ) ∈ 𝐾 for all 𝑗 ≤ 0 . Also, 𝑧 ( 𝑛 ) is a solution of where Note that | 𝑧 ( 𝑛 ) | ≤ 𝑐 for all 𝑛 ≤ 𝜎 − 𝑛 𝑚 , and | 𝑝 ( 𝑛 + 𝑛 𝑚 ) | < ( 1 / 2 ) 𝛿 ( 𝜀 / 2 ) for 𝑛 ≥ 𝑛 0 − 𝑛 𝑚 . Thus | 𝑞 ( 𝑛 ) | < 𝛿 ( 𝜀 / 2 ) for 𝑛 0 − 𝑛 𝑚 ≤ 𝑛 ≤ 𝜎 − 𝑛 𝑚 . Also, we have from (3.26). Thus we obtain Since 𝑣 ( 𝑛 ) is totally stable, we have On the other hand, (3.26) implies that Hence, if 𝑛 0 ≥ 0 , 𝜌 ( 𝑢 𝑛 0 , 𝑦 𝑛 0 1 ) < ( 2 𝜀 ) 𝛿 ( 2 ) , and | 𝑝 ( 𝑛 ) | < ( 1 / 2 ) 𝛿 ( 𝜀 / 2 ) for 𝑛 ≥ 𝑛 0 , then we obtain This contradicts (3.28). Therefore 𝜌 ( 𝑢 𝑛 , 𝑦 𝑛 ) < 𝜀 for all 𝑛 ≥ 𝑛 0 when 𝑛 0 ≥ 0 , 𝜌 ( 𝑢 𝑛 0 , 𝑦 𝑛 0 ) < 𝛿 ∗ ( 𝜀 ) and | 𝑝 ( 𝑛 ) | < 𝛿 ∗ ( 𝜀 ) for all 𝑛 ≥ 𝑛 0 , where 𝛿 ∗ ( 𝜀 ) = ( 1 / 2 ) 𝛿 ( 𝜀 / 2 ) . Consequently, 𝑢 ( 𝑛 ) is totally stable.

The following definitions are the discrete analogues of Hamaya's definitions in [2].

Definition 3.7. The bounded solution 𝑢 ( 𝑛 ) of (3.1) is said to be attracting in 𝐻 ( 𝑓 , 𝐵 ) if there exists a 𝛿 0 > 0 such that for any 𝑛 0 ≥ 0 and any ( 𝑣 , 𝑔 , 𝐷 ) ∈ 𝐻 ( 𝑢 , 𝑓 , 𝐵 ) , 𝜌 ( 𝑣 𝑛 0 , 𝑥 𝑛 0 ) < 𝛿 0 implies 𝜌 ( 𝑣 𝑛 , 𝑥 𝑛 ) → 0 as 𝑛 → ∞ , where 𝑥 ( 𝑛 ) is a solution of (3.5) such that 𝑥 𝑛 0 ( 𝑗 ) ∈ 𝐾 for all 𝑗 ≤ 0 .

Definition 3.8. The bounded solution 𝑢 ( 𝑛 ) of (3.1) is said to be totally asymptotically stable if it is totally stable and there exists a 𝛿 0 > 0 and for any 𝜀 > 0 there exists an 𝜂 ( 𝜀 ) > 0 and a 𝑇 ( 𝜀 ) > 0 such that if 𝑛 0 ≥ 0 , 𝜌 ( 𝑢 𝑛 0 , 𝑥 𝑛 0 ) < 𝛿 0 and 𝑝 ( 𝑛 ) is any function which satisfies | 𝑝 ( 𝑛 ) | < 𝜂 ( 𝜀 ) for 𝑛 ≥ 𝑛 0 , then 𝜌 ( 𝑢 𝑛 , 𝑥 𝑛 ) < 𝜀 for all 𝑛 ≥ 𝑛 0 + 𝑇 ( 𝜀 ) , where 𝑥 ( 𝑛 ) is a solution of such that 𝑥 𝑛 0 ( 𝑗 ) ∈ 𝐾 for all 𝑗 ≤ 0 .

Note that the total asymptotic stability is equivalent to the uniform asymptotic stability whenever 𝑝 ( 𝑛 ) ≡ 0 .

Theorem 3.9. Under the assumptions (H1)–(H4), if the bounded solution 𝑢 ( 𝑛 ) of (3.1) is attracting in 𝐻 ( 𝑓 , 𝐵 ) and totally stable, then it is totally asymptotically stable.

Proof. Let 𝛿 0 be the number for the attracting of 𝑢 ( 𝑛 ) in 𝐻 ( 𝑓 , 𝐵 ) and let 𝛿 ∗ 0 = 𝛿 ( 𝛿 0 / 2 ) , where 𝛿 ( ⋅ ) is the number for the total stability of 𝑢 ( 𝑛 ) . Suppose that 𝑢 ( 𝑛 ) is not totally asymptotically stable. Then there exists a number 𝜀 > 0 with 𝜀 ≤ ( 𝛿 0 / 4 ) and exist sequences ( 𝑗 𝑘 ) , ( 𝑛 𝑘 ) , ( 𝑝 𝑘 ) , and ( 𝑥 𝑘 ) such that 𝑗 𝑘 ≥ 0 , 𝑛 𝑘 ≥ 𝑗 𝑘 + 2 𝑘 , 𝜌 ( 𝑢 𝑗 𝑘 , 𝑥 𝑘 𝑗 𝑘 ) < 𝛿 ∗ 0 and 𝜌 ( 𝑢 𝑛 𝑘 , 𝑥 𝑘 𝑛 𝑘 ) ≥ 𝜀 for all 𝑘 = 1 , 2 , … , where 𝑥 𝑘 ( 𝑛 ) is a solution of such that 𝑥 𝑘 𝑗 𝑘 ( 𝑗 ) ∈ 𝐾 for all 𝑗 ≤ 0 and 𝑝 𝑘 ∶ ℤ → ℝ 𝑑 with | 𝑝 𝑘 ( 𝑛 ) | < m i n { 1 / 𝑘 , 𝛿 ∗ 0 } for 𝑛 ≥ 𝑗 𝑘 . Note that 𝜌 ( 𝑢 𝑗 𝑘 , 𝑥 𝑘 𝑗 𝑘 ) < 𝛿 ∗ 0 and | 𝑝 𝑘 ( 𝑛 ) | < 𝛿 ∗ 0 for 𝑛 ≥ 𝑗 𝑘 . Then we have for all 𝑛 ≥ 𝑗 𝑘 and 𝑘 = 1 , 2 , … , since 𝑢 ( 𝑛 ) is totally stable. Also, there exists an integer number 𝑘 0 ( 𝜀 ) > 0 such that if 𝑘 ≥ 𝑘 0 ( 𝜀 ) , then | 𝑝 𝑘 ( 𝑛 ) | < 1 / 𝑘 < 𝛿 ( 𝜀 ) for all 𝑛 ≥ 𝑗 𝑘 .
We claim that 𝜌 ( 𝑢 𝑛 , 𝑥 𝑘 𝑛 ) ≥ 𝛿 ( 𝜀 ) on [ 𝑗 𝑘 + 𝑘 , 𝑗 𝑘 + 2 𝑘 ] if 𝑛 ≥ 𝑛 0 . If we assume that 𝜌 ( 𝑢 𝑛 , 𝑥 𝑘 𝑛 ) < 𝛿 ( 𝜀 ) on [ 𝑗 𝑘 + 𝑘 , 𝑗 𝑘 + 2 𝑘 ] , then 𝜌 ( 𝑢 𝑛 , 𝑥 𝑘 𝑛 ) < 𝜀 for 𝑛 ≥ 𝑗 𝑘 + 2 𝑘 since 𝑢 ( 𝑛 ) is totally stable. This contradicts 𝜌 ( 𝑢 𝑛 𝑘 , 𝑥 𝑘 𝑛 𝑘 ) ≥ 𝜀 , 𝑘 = 1 , 2 , … , because 𝑛 𝑘 ≥ 𝑗 𝑘 + 2 𝑘 .
Now, for the sequence ( 𝑗 𝑘 + 𝑘 ) , taking a subsequence if necessary, there exists a ( 𝑣 , 𝑔 , 𝐷 ) ∈ 𝐻 ( 𝑢 , 𝑓 , 𝐵 ) .
If we set 𝑦 𝑘 ( 𝑛 ) = 𝑥 𝑘 ( 𝑛 + 𝑗 𝑘 + 𝑘 ) , then 𝑦 𝑘 ( 𝑛 ) is the defined on [ − 𝑘 , 𝑘 ] . There exists a subsequence of ( 𝑦 𝑘 ( 𝑛 ) ) , which we denote by ( 𝑦 𝑘 ( 𝑛 ) ) again, and a function 𝑦 ( 𝑛 ) such that 𝑦 𝑘 ( 𝑛 ) → 𝑦 ( 𝑛 ) uniformly on any compact set in ℝ as 𝑘 → ∞ such that 𝑦 0 ( 𝑗 ) ∈ 𝐾 for all 𝑗 ≤ 0 . Moreover, we can show that 𝑦 ( 𝑛 ) is a solution of
such that 𝑦 0 ( 𝑗 ) ∈ 𝐾 for all 𝑗 ≤ 0 , by the same method as in [13, Theorem 3.1] . We have Then, by letting 𝑘 → ∞ , we obtain Since 𝑢 ( 𝑛 ) is attracting in 𝐻 ( 𝑓 , 𝐵 ) , we have 𝜌 ( 𝑣 𝑛 , 𝑦 𝑛 ) → 0 as 𝑛 → ∞ . This contradicts 𝜌 ( 𝑣 𝑛 , 𝑦 𝑛 ) ≥ 𝛿 ( 𝜀 ) . Hence 𝑢 ( 𝑛 ) is totally asymptotically stable. This completes the proof of the theorem.

Acknowledgment

The authors are thankful to the anonymous referees for their valuable comments and corrections to improve this paper.

References

  1. Z. S. Athanassov, “Total stability of sets for nonautonomous differential systems,” Transactions of the American Mathematical Society, vol. 295, no. 2, pp. 649–663, 1986.
  2. Y. Hamaya, “Total stability property in limiting equations of integrodifferential equations,” Funkcialaj Ekvacioj, vol. 33, no. 2, pp. 345–362, 1990.
  3. Y. Hino and S. Murakami, “Total stability and uniform asymptotic stability for linear Volterra equations,” Journal of the London Mathematical Society, vol. 43, no. 2, pp. 305–312, 1991.
  4. Y. Hino, “Total stability and uniformly asymptotic stability for linear functional-differential equations with infinite delay,” Funkcialaj Ekvacioj, vol. 24, no. 3, pp. 345–349, 1981.
  5. Y. Hino and T. Yoshizawa, “Total stability property in limiting equations for a functional-differential equation with infinite delay,” Československá Akademie Vĕd. Časopis Pro Pĕstování Matematiky, vol. 111, no. 1, pp. 62–69, 1986.
  6. Y. Hino and S. Murakami, “Total stability in abstract functional differential equations with infinite delay,” in Proceedings of the 6th Colloquium on the Qualitative Theory of Differential Equations (Szeged, 1999), Proc. Colloq. Qual. Theory Differ. Equ., pp. 1–9, Electron. J. Qual. Theory Differ. Equ., Szeged, Hungary, 2000.
  7. P. Anderson and S. R. Bernfeld, “Total stability of scalar differential equations determined from their limiting functions,” Journal of Mathematical Analysis and Applications, vol. 257, no. 2, pp. 251–273, 2001.
  8. T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, vol. 14 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1975.
  9. S. K. Choi and N. Koo, “Almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay,” Advances in Difference Equations, vol. 2008, Article ID 692713, 15 pages, 2008.
  10. Y. Hamaya, “Stability property for an integrodifferential equation,” Differential and Integral Equations, vol. 6, no. 6, pp. 1313–1324, 1993.
  11. Y. Hamaya, “Periodic solutions of nonlinear integro-differential equations,” The Tohoku Mathematical Journal, vol. 41, no. 1, pp. 105–116, 1989.
  12. Y. Song and H. Tian, “Periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay,” Journal of Computational and Applied Mathematics, vol. 205, no. 2, pp. 859–870, 2007.
  13. Y. Song, “Asymptotically almost periodic solutions of nonlinear Volterra difference equations with unbounded delay,” Journal of Difference Equations and Applications, vol. 14, no. 9, pp. 971–986, 2008.
  14. C. Corduneanu, Almost Periodic Functions, Chelsea, New York, NY, USA, 2nd edition, 1989.
  15. C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press, Beijing, China; Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
  16. J. K. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcialaj Ekvacioj, vol. 21, no. 1, pp. 11–41, 1978.