Total Stability in Nonlinear Discrete Volterra Equations with Unbounded Delay

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 𝑥^′(𝑡)=𝑓(𝑡,𝑥(𝑡))+⁰_−∞𝐹(𝑡,𝑠,𝑥(𝑡+𝑠),𝑥(𝑡))𝑑𝑠,(1.1) 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 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛)),𝑛∈ℤ⁺,(1.2) 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 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑛,𝑥_𝑛).(1.3) To do this, we will employ to change Hamaya's results in [2] for the integrodifferential equation 𝑥^′(𝑡)=𝑓(𝑡,𝑥(𝑡))+⁰_−∞𝐹(𝑡,𝑠,𝑥(𝑡+𝑠),𝑥(𝑡))𝑑𝑠+ℎ(𝑡,𝑥_𝑡),(1.4) 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-dimensional Euclidean space.

Definition 2.1 (see [12]). A continuous function 𝑓∶ℤ×ℝ^𝑑→ℝ^𝑑 is said to be almost periodic in𝑛∈ℤ uniformly for𝑥∈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 |𝑓(𝑛+𝑝,𝑥)−𝑓(𝑛,𝑥)|<𝜀(2.1) 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 in𝑛 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 |𝐵(𝑛+𝜏,𝑗,𝑥,𝑦)−𝐵(𝑛,𝑗,𝑥,𝑦)|≤𝜀(2.2) 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
‖𝜙‖=sup_{𝑗∈ℤ−}|𝜙(𝑗)|<∞(2.3) for any 𝜙∈𝑙⁻(ℝ^𝑑).

Consider the discrete Volterra equation with unbounded delay of the form 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑛,𝑥_𝑛),𝑛∈ℤ⁺,(2.4) 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,𝜙⁰),𝜙⁰∈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 𝜌(𝜙,𝜓)=^∞_𝑞=0𝜌_𝑞(𝜙,𝜓)2^𝑞[1+𝜌_𝑞(𝜙,𝜓)],(2.5) where 𝜌_𝑞(𝜙,𝜑)=max_{−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 𝑓(𝑛+𝑛_𝑘,𝑥)→𝑔(𝑛,𝑥)(2.6) uniformly on ℤ×𝑆 for any compact set 𝑆⊂ℝ^𝑑, 𝐵(𝑛+𝑛_𝑘,𝑛+𝑙+𝑛_𝑘,𝑥,𝑦)→𝐷(𝑛,𝑛+𝑙,𝑥,𝑦)(2.7) uniformly on ℤ×𝑆^∗ for any compact set 𝑆^∗⊂𝑍^∗. We define the hull𝐻(𝑓,𝐵)={(𝑔,𝐷)∶(2.6)and(2.7)holdforsomesequence(𝑛_𝑘)⊂ℤ⁺with𝑛_𝑘→∞a𝑠𝑘→∞}.(2.8) 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 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑛,𝑥_𝑛),𝑛∈ℤ⁺.(3.1) 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 ^−𝑀_𝑗=−∞|𝐵(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))|≤𝜀(3.2) 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 |ℎ(𝑛,𝜙)|≤𝛼_𝑟(𝑛)(3.3) whenever |𝜙(𝑗)|≤𝑟 for all 𝑗∈ℤ⁻.(H4)Equation (3.1) has a bounded solution 𝑢(𝑛)=𝑢(𝑛,𝜙⁰) defined on ℤ⁺, through (0,𝜙⁰), 𝜙⁰∈𝑙⁻(ℝ^𝑑) such that for some 0≤𝑀<∞, |𝑢(𝑛,𝜙⁰)|≤𝑀.(3.4)

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 𝑥(𝑛+1)=𝑔(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐷(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛)),𝑛∈ℤ⁺,(3.5) 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,𝜙⁰), 𝜙⁰∈𝑙⁻(ℝ^𝑑) is a bounded solution of (3.1), passing through (0,𝜙⁰) 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 𝑛₀≥0,𝜌(𝑢_𝑛0,𝑥_𝑛0)<𝛿, and 𝑝(𝑛) is a function such that |𝑝(𝑛)|<𝛿 for all 𝑛≥𝑛₀, then 𝜌(𝑢_𝑛,𝑥_𝑛)<𝜀 for all 𝑛≥𝑛₀, where 𝑥(𝑛) is any solution of 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑛,𝑥_𝑛)+𝑝(𝑛)(3.6) such that 𝑥_𝑛0(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 𝜙₁ and a function 𝜙₂ defined on ℤ which tends to zero as 𝑛→∞, that is, 𝜙(𝑛)=𝜙₁(𝑛)+𝜙₂(𝑛) for all 𝑛∈ℤ.

It is known [15] that the decomposition 𝜙=𝜙₁+𝜙₂ 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 𝑥(𝑛+1)=𝑓(𝑛+𝑛_𝑘,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛+𝑛_𝑘,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑛+𝑛_𝑘,𝑥_𝑛)(3.7) 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 𝑘₁(𝜀)>0 such that 𝜌(𝑢^𝑘₀,𝑢^𝑚₀)<𝛿(𝜀) whenever 𝑘,𝑚≥𝑘₁(𝜀). Put 𝑝(𝑛)=𝑓(𝑛+𝑛_𝑚,𝑢^𝑚(𝑛))+⁰_𝑗=−∞𝐵(𝑛+𝑛_𝑚,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚(𝑛))+ℎ(𝑛+𝑛_𝑚,𝑢^𝑚_𝑛)−𝑓(𝑛+𝑛_𝑘,𝑢^𝑚−(𝑛))_0𝑗=−∞𝐵(𝑛+𝑛_𝑘,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚(𝑛))−ℎ(𝑛+𝑛_𝑘,𝑢^𝑚_𝑛).(3.8) Then 𝑢^𝑚(𝑛)=𝑢(𝑛+𝑛_𝑚) is a solution of 𝑥(𝑛+1)=𝑓(𝑛+𝑛_𝑘,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛+𝑛_𝑘,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑛+𝑛_𝑘,𝑥_𝑛)+𝑝(𝑛)(3.9) and 𝑢^𝑚(𝑛)∈𝐾 for all 𝑛∈ℤ. We will show that there exists a number 𝑘₂(𝜀)>0 such that |𝑝(𝑛)|<𝛿(𝜀) for all 𝑛≥0 whenever 𝑘,𝑚≥𝑘₂(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
^−𝑀_𝑗=−∞|𝐵(𝑛+𝑛_𝑚,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚1(𝑛))|≤5𝛿(𝜀),𝑛∈ℤ,_{−𝑀𝑗=−∞}|𝐵(𝑛+𝑛_𝑘,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚1(𝑛))|≤5𝛿(𝜀),𝑛∈ℤ.(3.10) From the almost periodicity of 𝑓(𝑛,𝑥) and 𝐵(𝑛,𝑗,𝑥,𝑦), respectively, there exists a number 𝑘₂(𝜀)≥𝑘₁(𝜀) for which |𝐵(𝑛+𝑛_𝑚,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚(𝑛))−𝐵(𝑛+𝑛_𝑘,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚<(𝑛))|𝛿(𝜀)5𝑀,𝑛∈ℤ,𝑗∈[−𝑀+1,0],(3.11)|𝑓(𝑛+𝑛_𝑚,𝑢^𝑚(𝑛))−𝑓(𝑛+𝑛_𝑘,𝑢^𝑚1(𝑛))|<5𝛿(𝜀),𝑛∈ℤ,(3.12) whenever 𝑘,𝑚≥𝑘₂(𝜀). Since ℎ(𝑛,𝜙)→0 as 𝑛→∞, we obtain that if 𝑘,𝑚≥𝑘₂(𝜀), then |ℎ(𝑛+𝑛_𝑚,𝑢^𝑚_𝑛)−ℎ(𝑛+𝑛_𝑘,𝑢^𝑚_𝑛1)|<5𝛿(𝜀),𝑛∈ℤ⁺.(3.13) Then, by (3.10), and (3.11), we have ⁰_𝑗=−∞|𝐵(𝑛+𝑛_𝑚,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚(𝑛))−𝐵(𝑛+𝑛_𝑘,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚≤(𝑛))|_{−𝑀𝑗=−∞}|𝐵(𝑛+𝑛_𝑚,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚+(𝑛))|_{−𝑀𝑗=−∞}|𝐵(𝑛+𝑛_𝑘,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚+(𝑛))|_{0𝑗=−𝑀+1}|𝐵(𝑛+𝑛_𝑚,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚(𝑛))−𝐵(𝑛+𝑛_𝑘,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚<1(𝑛))|51𝛿(𝜀)+5𝛿(𝜀)+𝛿(𝜀)𝑀=35𝑀5𝛿(𝜀).(3.14) Therefore we have |𝑝(𝑛)|≤|𝑓(𝑛+𝑛_𝑚,𝑢^𝑚(𝑛))−𝑓(𝑛+𝑛_𝑘,𝑢^𝑚+(𝑛))|_0𝑗=−∞|𝐵(𝑛+𝑛_𝑚,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚(𝑛))−𝐵(𝑛+𝑛_𝑘,𝑗,𝑢^𝑚(𝑛+𝑗),𝑢^𝑚(𝑛))|+|ℎ(𝑛+𝑛_𝑚,𝑢^𝑚_𝑛)−ℎ(𝑛+𝑛_𝑘,𝑢^𝑚_𝑛<1)|53𝛿(𝜀)+51𝛿(𝜀)+5𝛿(𝜀)=𝛿(𝜀),(3.15)𝑘,𝑚≥𝑘₂(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≥𝑘₂(𝜀). This implies that for all 𝑘,𝑚≥𝑘₂(𝜀), |𝑢(𝑛+𝑛_𝑘)−𝑢(𝑛+𝑛_𝑚)|≤sup_{−𝑀≤𝑠≤0}|𝑢(𝑛+𝑛_𝑘+𝑠)−𝑢(𝑛+𝑛_𝑚+𝑠)|<4𝜀(3.16) 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_𝑘𝑗)⊂(𝑛_𝑘) such that (𝑢(𝑛+𝑛_𝑘𝑗)) 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 𝑥^′(𝑡)=𝑓(𝑡,𝑥_𝑡),(3.17) the solution 𝑣(𝑡) of the limiting equation 𝑥^′(𝑡)=𝑔(𝑡,𝑥_𝑡),(𝑣,𝑔)∈𝐻(𝑢,𝑓)(3.18) 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 𝑥(𝑛+1)=𝑓(𝑛+𝑛_𝑘,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛+𝑛_𝑘,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑛+𝑛_𝑘,𝑥_𝑛),(3.19) such that 𝑢^𝑘₀(𝑗)∈𝐾 for all 𝑗≤0, where 𝑢^𝑘₀(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 ^−𝑀_𝑗=−∞|𝐵(𝑛+𝑛_𝑘1,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))|≤51𝛿(2𝜀𝛿(2)),(3.20) where 𝛿(⋅) is the number for the total stability of 𝑣(𝑛). Also, we have ^−𝑀_𝑗=−∞1|𝐷(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))|≤51𝛿(2𝜀𝛿(2))(3.21) 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 𝑘₀(𝜀) such that if 𝑘≥𝑘₀(𝜀), then |𝑓(𝑛+𝑛_𝑘,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛+𝑛_𝑘,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑛+𝑛_𝑘,𝑥_𝑛)−𝑔(𝑛,𝑥(𝑛))−⁰_𝑗=−∞1𝐷(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))|<𝛿(2𝜀𝛿(2)),(3.22)𝜌(𝑢^𝑘₀,𝑣₀1)<𝛿(2𝜀𝛿(2)).(3.23) Put 𝑟(𝑛)=𝑓(𝑛+𝑛_𝑘,𝑢^𝑘(𝑛))+⁰_𝑗=−∞𝐵(𝑛+𝑛_𝑘,𝑗,𝑢^𝑘(𝑛+𝑗),𝑢^𝑘(𝑛))+ℎ(𝑛+𝑛_𝑘,𝑢^𝑘_𝑛)−𝑔(𝑛,𝑢^𝑘(𝑛))−⁰_𝑗=−∞𝐷(𝑛,𝑗,𝑢^𝑘(𝑛+𝑗),𝑢^𝑘(𝑛)).(3.24) Then 𝑢^𝑘(𝑛) is a solution of 𝑥(𝑛+1)=𝑔(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐷(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+𝑟(𝑛)(3.25) such that 𝑢^𝑘₀(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 𝜌(𝑢^𝑘_𝑛,𝑣_𝑛1)<2𝜀𝛿(2)(3.26) for all 𝑛≥0.
Let 𝑚=𝑘₀(𝜀). To show that 𝑢(𝑛) is totally stable we will show that if 𝑛₀≥0, 𝜌(𝑢_𝑛0,𝑦_𝑛0)<(1/2)𝛿(𝜀/2), and |𝑝(𝑛)|<(1/2)𝛿(𝜀/2) for 𝑛≥𝑛₀, then 𝜌(𝑢_𝑛,𝑦_𝑛)<𝜀 for all 𝑛≥𝑛₀, where 𝑦(𝑛) is a solution of
𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑛,𝑥_𝑛)+𝑝(𝑛)(3.27) such that 𝑦_𝑛0(𝑗)∈𝐾 for all 𝑗≤0. Suppose that this is not the case. Then there exists an integer 𝜎>𝑛₀ such that 𝜌(𝑢_𝜎,𝑦_𝜎)=𝜀for𝜎>𝑛₀,𝜌(𝑢_𝑛,𝑦_𝑛)<𝜀for𝑛₀≤𝑛<𝜎.(3.28) We set 𝑧(𝑛)=𝑦(𝑛+𝑛_𝑚). Then 𝑧(𝑛) is a solution of 𝑥(𝑛+1)=𝑓(𝑛+𝑛_𝑚,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛+𝑛_𝑚,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑛+𝑛_𝑚,𝑥_𝑛)+𝑝(𝑛+𝑛_𝑚)(3.29) defined on [𝑛₀−𝑛_𝑚,𝜎−𝑛_𝑚] such that 𝑧_{𝑛0−𝑛𝑚}(𝑗)=𝑦_𝑛0(𝑗)∈𝐾 for all 𝑗≤0. Also, 𝑧(𝑛) is a solution of 𝑥(𝑛+1)=𝑔(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐷(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+𝑞(𝑛),(3.30) where 𝑞(𝑛)=𝑓(𝑛+𝑛_𝑚,𝑧(𝑛))+⁰_𝑗=−∞𝐵(𝑛+𝑛_𝑚,𝑗,𝑧(𝑛+𝑗),𝑧(𝑛))+ℎ(𝑛+𝑛_𝑚,𝑧_𝑛)+𝑝(𝑛+𝑛_𝑚)−𝑔(𝑛,𝑧(𝑛))−⁰_𝑗=−∞𝐷(𝑛,𝑗,𝑧(𝑛+𝑗),𝑧(𝑛)).(3.31) Note that |𝑧(𝑛)|≤𝑐 for all 𝑛≤𝜎−𝑛_𝑚, and |𝑝(𝑛+𝑛_𝑚)|<(1/2)𝛿(𝜀/2) for 𝑛≥𝑛₀−𝑛_𝑚. Thus |𝑞(𝑛)|<𝛿(𝜀/2) for 𝑛₀−𝑛_𝑚≤𝑛≤𝜎−𝑛_𝑚. Also, we have 𝜌(𝑢_𝑛0,𝑣_{𝑛0−𝑛𝑚}1)<2𝜀𝛿(2),𝜌(𝑢_𝑛0,𝑧_{𝑛0−𝑛𝑚})=𝜌(𝑢_𝑛0,𝑦_𝑛01)<2𝜀𝛿(2)(3.32) from (3.26). Thus we obtain 𝜌(𝑣_{𝑛0−𝑛𝑚},𝑧_{𝑛0−𝑛𝑚})≤𝜌(𝑣_{𝑛0−𝑛𝑚},𝑢_𝑛0)+𝜌(𝑢_𝑛0,𝑧_{𝑛0−𝑛𝑚}𝜀)<𝛿(2).(3.33) Since 𝑣(𝑛) is totally stable, we have 𝜌(𝑣_{𝜎−𝑛𝑚},𝑧_{𝜎−𝑛𝑚}𝜀)<2.(3.34) On the other hand, (3.26) implies that 𝜌(𝑢_𝑛,𝑣_{𝑛−𝑛𝑚}1)<2𝜀𝛿(2),𝑛≥𝑛₀.(3.35) Hence, if 𝑛₀≥0,𝜌(𝑢_𝑛0,𝑦_𝑛01)<(2𝜀)𝛿(2), and |𝑝(𝑛)|<(1/2)𝛿(𝜀/2) for 𝑛≥𝑛₀, then we obtain 𝜌(𝑢_𝜎,𝑦_𝜎)≤𝜌(𝑢_𝜎,𝑣_{𝜎−𝑛𝑚})+𝜌(𝑣_{𝜎−𝑛𝑚},𝑧_{𝜎−𝑛𝑚})<𝜀.(3.36) This contradicts (3.28). Therefore 𝜌(𝑢_𝑛,𝑦_𝑛)<𝜀 for all 𝑛≥𝑛₀ when 𝑛₀≥0, 𝜌(𝑢_𝑛0,𝑦_𝑛0)<𝛿^∗(𝜀) and |𝑝(𝑛)|<𝛿^∗(𝜀) for all 𝑛≥𝑛₀, 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 𝛿₀>0 such that for any 𝑛₀≥0 and any (𝑣,𝑔,𝐷)∈𝐻(𝑢,𝑓,𝐵),𝜌(𝑣_𝑛0,𝑥_𝑛0)<𝛿₀ implies 𝜌(𝑣_𝑛,𝑥_𝑛)→0 as 𝑛→∞, where 𝑥(𝑛) is a solution of (3.5) such that 𝑥_𝑛0(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 𝛿₀>0 and for any 𝜀>0 there exists an 𝜂(𝜀)>0 and a 𝑇(𝜀)>0 such that if 𝑛₀≥0,𝜌(𝑢_𝑛0,𝑥_𝑛0)<𝛿₀ and 𝑝(𝑛) is any function which satisfies |𝑝(𝑛)|<𝜂(𝜀) for 𝑛≥𝑛₀, then 𝜌(𝑢_𝑛,𝑥_𝑛)<𝜀 for all 𝑛≥𝑛₀+𝑇(𝜀), where 𝑥(𝑛) is a solution of 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑥,𝑥_𝑛)+𝑝(𝑛)(3.37) such that 𝑥_𝑛0(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 𝛿₀ be the number for the attracting of 𝑢(𝑛) in 𝐻(𝑓,𝐵) and let 𝛿^∗₀=𝛿(𝛿₀/2), 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≤(𝛿₀/4) and exist sequences (𝑗_𝑘),(𝑛_𝑘),(𝑝_𝑘), and (𝑥^𝑘) such that 𝑗_𝑘≥0,𝑛_𝑘≥𝑗_𝑘+2𝑘,𝜌(𝑢_𝑗𝑘,𝑥^𝑘_𝑗𝑘)<𝛿^∗₀ and 𝜌(𝑢_𝑛𝑘,𝑥^𝑘_𝑛𝑘)≥𝜀 for all 𝑘=1,2,…, where 𝑥^𝑘(𝑛) is a solution of 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐵(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))+ℎ(𝑥,𝑥_𝑛)+𝑝_𝑘(𝑛).(3.38) such that 𝑥^𝑘_𝑗𝑘(𝑗)∈𝐾 for all 𝑗≤0 and 𝑝_𝑘∶ℤ→ℝ^𝑑 with |𝑝_𝑘(𝑛)|<min{1/𝑘,𝛿^∗₀} for 𝑛≥𝑗_𝑘. Note that 𝜌(𝑢_𝑗𝑘,𝑥^𝑘_𝑗𝑘)<𝛿^∗₀ and |𝑝_𝑘(𝑛)|<𝛿^∗₀ for 𝑛≥𝑗_𝑘. Then we have 𝜌(𝑢_𝑛,𝑥^𝑘_𝑛1)<2𝛿₀(3.39) for all 𝑛≥𝑗_𝑘 and 𝑘=1,2,…, since 𝑢(𝑛) is totally stable. Also, there exists an integer number 𝑘₀(𝜀)>0 such that if 𝑘≥𝑘₀(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+2𝑘] if 𝑛≥𝑛₀. If we assume that 𝜌(𝑢_𝑛,𝑥^𝑘_𝑛)<𝛿(𝜀) on [𝑗_𝑘+𝑘,𝑗_𝑘+2𝑘], then 𝜌(𝑢_𝑛,𝑥^𝑘_𝑛)<𝜀 for 𝑛≥𝑗_𝑘+2𝑘 since 𝑢(𝑛) is totally stable. This contradicts 𝜌(𝑢_𝑛𝑘,𝑥^𝑘_𝑛𝑘)≥𝜀,𝑘=1,2,…, because 𝑛_𝑘≥𝑗_𝑘+2𝑘.
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 𝑦₀(𝑗)∈𝐾 for all 𝑗≤0. Moreover, we can show that 𝑦(𝑛) is a solution of
𝑥(𝑛+1)=𝑔(𝑛,𝑥(𝑛))+⁰_𝑗=−∞𝐷(𝑛,𝑗,𝑥(𝑛+𝑗),𝑥(𝑛))(3.40) such that 𝑦₀(𝑗)∈𝐾 for all 𝑗≤0, by the same method as in [13, Theorem 3.1] . We have 𝛿(𝜀)<𝜌(𝑢_{𝑛+𝑗𝑘+𝑘},𝑦^𝑘_𝑛𝛿)<⁰2,0≤𝑛≤𝑘,𝑘≥𝑘₀.(3.41) Then, by letting 𝑘→∞, we obtain 𝛿(𝜀)<𝜌(𝑣_𝑛,𝑦_𝑛𝛿)≤⁰2,𝑛≥0.(3.42) 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.