# Total Stability in Nonlinear Discrete Volterra Equations with Unbounded Delay

Authors: Sung Kyu Choi; Yoon Hoe Goo; Dong Man Im; Namjip Koo
Journal: Abstract and Applied Analysis
 (2009)

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## 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).


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## Body

## 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
(1.1)x′(t)=f̂(t,x(t))+∫-∞0F(t,s,x(t+s),x(t))ds,
where f̂:ℝ×ℝd→ℝd is continuous and is almost periodic in t uniformly for x∈ℝd, and F:ℝ×(-∞,0]×ℝd×ℝd→ℝd is continuous and is almost periodic in t uniformly for (s,x,y)∈R*=(-∞,0]×ℝd×ℝd. 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
(1.2)x(n+1)=f(n,x(n))+∑j=-∞0B(n,j,x(n+j),x(n)),n∈ℤ+,
where f:ℤ×ℝd→ℝd is continuous in x∈ℝd for every n∈ℤ, and for any j,n∈ℤ(j≤n), B:ℤ×ℤ×ℝd×ℝd→ℝd is continuous for x,y∈ℝd. 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
(1.3)x(n+1)=f(n,x(n))+∑j=-∞0B(n,j,x(n+j),x(n))+h(n,xn).
To do this, we will employ to change Hamaya's results in [2] for the integrodifferential equation
(1.4)x′(t)=f̂(t,x(t))+∫-∞0F(t,s,x(t+s),x(t))ds+ĥ(t,xt),
into results for the discrete Volterra equation (1.3).


## 2. Preliminaries

We denote byℤ,ℤ+,ℤ-, respectively, the set of integers, the set of nonnegative integers, and the set of nonpositive integers. Let ℝd denote d-dimensional Euclidean space.Definition 2.1 (see [12]).
A continuous functionf:ℤ×ℝd→ℝd is said to be almost periodic inn∈ℤuniformly forx∈ℝd if for every ε>0 and every compact set K⊂ℝd, there corresponds an integer N=N(ε,K)>0 such that among N consecutive integers there is one, here denoted by p, such that
(2.1)|f(n+p,x)-f(n,x)|<ε
for all n∈ℤ, uniformly for x∈ℝd.Definition 2.2 (see [12]).
LetB:ℤ×ℤ*→ℝd be continuous for x,y∈ℝd, for any n∈ℤ,j∈ℤ-, where ℤ*=ℤ-×ℝd×ℝd. B(n,j,x,y) is said to be almost periodic innuniformly 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
(2.2)|B(n+τ,j,x,y)-B(n,j,x,y)|≤ε
for all n∈ℤ and all (j,x,y)∈K*.
For the basic results of almost periodic functions, see [8, 14, 15].
Letl-(ℝd) denote the space of all ℝd-valued bounded functions on ℤ- with
(2.3)∥ϕ∥=supj∈ℤ-|ϕ(j)|<∞
for any ϕ∈l-(ℝd).Consider the discrete Volterra equation with unbounded delay of the form(2.4)x(n+1)=f(n,x(n))+∑j=-∞0B(n,j,x(n+j),x(n))+h(n,xn),n∈ℤ+,
under certain conditions for f,B, and h (see below). We assume that, given ϕ∈l-(ℝd), there is a solution x of (1.3) such that x(n)=ϕ(n) for n∈ℤ-, passing through (0,ϕ0),ϕ0∈l-(ℝd).LetK be any compact subset of ℝd such that ϕ(j)∈K for all j≤0 and x(n)∈K for all n≥1.For anyϕ,ψ∈l-(ℝd), we set
(2.5)ρ(ϕ,ψ)=∑q=0∞ρq(ϕ,ψ)2q[1+ρq(ϕ,ψ)],
where ρq(ϕ,φ)=max-q≤j≤0|ϕ(j)-ψ(j)|,q≥0. Then ρ defines a metric on the space l-(ℝd). 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(nk′)⊂ℤ+ with nk′→∞ as k→∞, there exists a subsequence (nk)⊂(nk′) such that
(2.6)f(n+nk,x)→g(n,x)
uniformly on ℤ×S for any compact set S⊂ℝd,
(2.7)B(n+nk,n+l+nk,x,y)→D(n,n+l,x,y)
uniformly on ℤ×S* for any compact set S*⊂Z*. We define the hull(2.8)H(f,B)={(g,D):(2.6)and(2.7)holdforsomesequence(nk)⊂ℤ+withnk→∞ask→∞}.
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(3.1)x(n+1)=f(n,x(n))+∑j=-∞0B(n,j,x(n+j),x(n))+h(n,xn),n∈ℤ+.
Throughout this paper we assume the following.(H1)
f:ℤ×ℝd→ℝd is continuous in x∈ℝd for every n∈ℤ and is almost periodic in n∈ℤ uniformly for x∈ℝd.(H2)
B:ℤ×ℤ*→ℝd is continuous in x,y∈ℝd for any n∈ℤ, j∈ℤ- and is almost periodic in n∈ℤ uniformly for (j,x,y)∈ℤ*=ℤ-×ℝd×ℝd. Moreover, for any ε>0 and any τ>0 there exists a number M=M(ε,τ)>0 such that
(3.2)∑j=-∞-M|B(n,j,x(n+j),x(n))|≤ε
for all n∈ℤ whenever |x(j)|≤τ for all j∈ℤ-.(H3)
h:ℤ×l-(ℝd)→ℝd is continuous in ϕ∈l-(ℝd) for every n∈ℤ. xn∈l-(ℝd) is defined as xn(j)=x(n+j) for j∈ℤ-. Furthermore, for any r>0, there exists a function αr:ℤ→ℝd with the property that αr(n)→0 as n→∞ and
(3.3)|h(n,ϕ)|≤αr(n)
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-(ℝd) such that for some 0≤M<∞,
(3.4)|u(n,ϕ0)|≤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
(3.5)x(n+1)=g(n,x(n))+∑j=-∞0D(n,j,x(n+j),x(n)),n∈ℤ+,
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 ℝd.Theorem 3.1 (see [12]).
Under the assumptions (H1)–(H4), ifu(n)=u(n,ϕ0), ϕ0∈l-(ℝd) 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 ofx′(t)=f̂(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 solutionu(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(un0,xn0)<δ, and p(n) is a function such that |p(n)|<δ for all n≥n0, then ρ(un,xn)<ε for all n≥n0, where x(n) is any solution of
(3.6)x(n+1)=f(n,x(n))+∑j=-∞0B(n,j,x(n+j),x(n))+h(n,xn)+p(n)
such that xn0(j)∈K for all j∈ℤ-.Definition 3.3.
A functionϕ:ℤ→ℝd 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 (τk′) with τk′→∞ as k→∞, there exists a subsequence (τk)⊂(τk′) for which ϕ(n+τk) converges uniformly for n∈ℤ as k→∞ [15].Theorem 3.4.
Under the assumptions (H1)–(H4), ifu(n) is a bounded and totally stable solution of (3.1), then it is asymptotically almost periodic.Proof.
Let(nk) be an integer sequence with nk→∞ as k→∞. Set uk(n)=u(n+nk),k=1,2,…. Then uk(n) is a solution of
(3.7)x(n+1)=f(n+nk,x(n))+∑j=-∞0B(n+nk,j,x(n+j),x(n))+h(n+nk,xn)
and uk(n)∈K for k=1,2,…. Clearly, uk(n) is totally stable with the same number (ε,δ) for the total stability of u(n). We can assume that (uk(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(u0k,u0m)<δ(ε) whenever k,m≥k1(ε). Put
(3.8)p(n)=f(n+nm,um(n))+∑j=-∞0B(n+nm,j,um(n+j),um(n))+h(n+nm,unm)-f(n+nk,um(n))-∑j=-∞0B(n+nk,j,um(n+j),um(n))-h(n+nk,unm).
Then um(n)=u(n+nm) is a solution of
(3.9)x(n+1)=f(n+nk,x(n))+∑j=-∞0B(n+nk,j,x(n+j),x(n))+h(n+nk,xn)+p(n)
and um(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 allx∈K, there exists a number c>0 such that |x|≤c. It is clear that |uk(n)|≤c and |um(n)|≤c for all n∈ℤ.
In view of (H2), there exists a numberM=M(c,ε)>0 such that
(3.10)∑j=-∞-M|B(n+nm,j,um(n+j),um(n))|≤15δ(ε),n∈ℤ,∑j=-∞-M|B(n+nk,j,um(n+j),um(n))|≤15δ(ε),n∈ℤ.
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
(3.11)|B(n+nm,j,um(n+j),um(n))-B(n+nk,j,um(n+j),um(n))|<δ(ε)5M,n∈ℤ,j∈[-M+1,0],(3.12)|f(n+nm,um(n))-f(n+nk,um(n))|<15δ(ε),n∈ℤ,
whenever k,m≥k2(ε). Since h(n,ϕ)→0 as n→∞, we obtain that if k,m≥k2(ε), then
(3.13)|h(n+nm,unm)-h(n+nk,unm)|<15δ(ε),n∈ℤ+.
Then, by (3.10), and (3.11), we have
(3.14)∑j=-∞0|B(n+nm,j,um(n+j),um(n))-B(n+nk,j,um(n+j),um(n))|≤∑j=-∞-M|B(n+nm,j,um(n+j),um(n))|+∑j=-∞-M|B(n+nk,j,um(n+j),um(n))|+∑j=-M+10|B(n+nm,j,um(n+j),um(n))-B(n+nk,j,um(n+j),um(n))|<15δ(ε)+15δ(ε)+δ(ε)5MM=35δ(ε).
Therefore we have
(3.15)|p(n)|≤|f(n+nm,um(n))-f(n+nk,um(n))|+∑j=-∞0|B(n+nm,j,um(n+j),um(n))-B(n+nk,j,um(n+j),um(n))|+|h(n+nm,unm)-h(n+nk,unm)|<15δ(ε)+35δ(ε)+15δ(ε)=δ(ε),k,m≥k2(ε), by (3.12), (3.13), and (3.14). Since uk(n) is totally stable, we obtain that ρ(unk,unm)<ε for all n≥0 if k,m≥k2(ε). This implies that for all k,m≥k2(ε),
(3.16)|u(n+nk)-u(n+nm)|≤sup-M≤s≤0|u(n+nk+s)-u(n+nm+s)|<4ε
for all ε≤(1/4) and all n≥0. It follows that for any (nk)⊂ℤ with nk→∞ as k→∞ there exists a subsequence (nkj)⊂(nk) such that (u(n+nkj)) 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
(3.17)x′(t)=f̂(t,xt),
the solution v(t) of the limiting equation
(3.18)x′(t)=ĝ(t,xt),(v,ĝ)∈H(u,f̂)
of (3.17) is asymptotically almost periodic if v(t) is totally stable. Here (v,ĝ)∈H(u,f̂) means that there exists a sequence (tk),tk→∞ as k→∞, such that f̂(t+tk,ϕ)→ĝ(t,ϕ)∈H(f̂) uniformly on any compact set in B and u(t+tk)→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 solutionv(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 (nk)⊂ℤ,nk→∞ as k→∞, such that f(n+nk,x)→g(n,x) uniformly on ℤ×K, B(n+nk,j,x,y)→D(n,j,x,y) uniformly on ℤ×S*×K×K for any compact set S*⊂ℤ-, and u(n+nk)→v(n) uniformly on any compact set in ℝ as k→∞. Set uk(n)=u(n+nk),k=1,2,…. Then it is clear that uk(n) is a solution of
(3.19)x(n+1)=f(n+nk,x(n))+∑j=-∞0B(n+nk,j,x(n+j),x(n))+h(n+nk,xn),
such that u0k(j)∈K for all j≤0, where u0k(j)=uk(0+j)=u(j+nk). 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
(3.20)∑j=-∞-M|B(n+nk,j,x(n+j),x(n))|≤15δ(12δ(ε2)),
where δ(·) is the number for the total stability of v(n). Also, we have
(3.21)∑j=-∞-M|D(n,j,x(n+j),x(n))|≤15δ(12δ(ε2))
for the same M since B(n+nk,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
(3.22)|f(n+nk,x(n))+∑j=-∞0B(n+nk,j,x(n+j),x(n))+h(n+nk,xn)-g(n,x(n))-∑j=-∞0D(n,j,x(n+j),x(n))|<δ(12δ(ε2)),(3.23)ρ(u0k,v0)<δ(12δ(ε2)).
Put
(3.24)r(n)=f(n+nk,uk(n))+∑j=-∞0B(n+nk,j,uk(n+j),uk(n))+h(n+nk,unk)-g(n,uk(n))-∑j=-∞0D(n,j,uk(n+j),uk(n)).
Then uk(n) is a solution of
(3.25)x(n+1)=g(n,x(n))+∑j=-∞0D(n,j,x(n+j),x(n))+r(n)
such that u0k(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
(3.26)ρ(unk,vn)<12δ(ε2)
for all n≥0.
Letm=k0(ε). To show that u(n) is totally stable we will show that if n0≥0, ρ(un0,yn0)<(1/2)δ(ε/2), and |p(n)|<(1/2)δ(ε/2) for n≥n0, then ρ(un,yn)<ε for all n≥n0, where y(n) is a solution of
(3.27)x(n+1)=f(n,x(n))+∑j=-∞0B(n,j,x(n+j),x(n))+h(n,xn)+p(n)
such that yn0(j)∈K for all j≤0. Suppose that this is not the case. Then there exists an integer σ>n0 such that
(3.28)ρ(uσ,yσ)=εforσ>n0,ρ(un,yn)<εforn0≤n<σ.
We set z(n)=y(n+nm). Then z(n) is a solution of
(3.29)x(n+1)=f(n+nm,x(n))+∑j=-∞0B(n+nm,j,x(n+j),x(n))+h(n+nm,xn)+p(n+nm)
defined on [n0-nm,σ-nm] such that zn0-nm(j)=yn0(j)∈K for all j≤0. Also, z(n) is a solution of
(3.30)x(n+1)=g(n,x(n))+∑j=-∞0D(n,j,x(n+j),x(n))+q(n),
where
(3.31)q(n)=f(n+nm,z(n))+∑j=-∞0B(n+nm,j,z(n+j),z(n))+h(n+nm,zn)+p(n+nm)-g(n,z(n))-∑j=-∞0D(n,j,z(n+j),z(n)).
Note that |z(n)|≤c for all n≤σ-nm, and |p(n+nm)|<(1/2)δ(ε/2) for n≥n0-nm. Thus |q(n)|<δ(ε/2) for n0-nm≤n≤σ-nm. Also, we have
(3.32)ρ(un0,vn0-nm)<12δ(ε2),ρ(un0,zn0-nm)=ρ(un0,yn0)<12δ(ε2)
from (3.26). Thus we obtain
(3.33)ρ(vn0-nm,zn0-nm)≤ρ(vn0-nm,un0)+ρ(un0,zn0-nm)<δ(ε2).
Since v(n) is totally stable, we have
(3.34)ρ(vσ-nm,zσ-nm)<ε2.
On the other hand, (3.26) implies that
(3.35)ρ(un,vn-nm)<12δ(ε2),n≥n0.
Hence, if n0≥0,ρ(un0,yn0)<(12)δ(ε2), and |p(n)|<(1/2)δ(ε/2) for n≥n0, then we obtain
(3.36)ρ(uσ,yσ)≤ρ(uσ,vσ-nm)+ρ(vσ-nm,zσ-nm)<ε.
This contradicts (3.28). Therefore ρ(un,yn)<ε for all n≥n0 when n0≥0, ρ(un0,yn0)<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 solutionu(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(vn0,xn0)<δ0 implies ρ(vn,xn)→0 as n→∞, where x(n) is a solution of (3.5) such that xn0(j)∈K for all j≤0.Definition 3.8.
The bounded solutionu(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(un0,xn0)<δ0 and p(n) is any function which satisfies |p(n)|<η(ε) for n≥n0, then ρ(un,xn)<ε for all n≥n0+T(ε), where x(n) is a solution of
(3.37)x(n+1)=f(n,x(n))+∑j=-∞0B(n,j,x(n+j),x(n))+h(x,xn)+p(n)
such that xn0(j)∈K for all j≤0.Note that the total asymptotic stability is equivalent to the uniform asymptotic stability wheneverp(n)≡0.Theorem 3.9.
Under the assumptions (H1)–(H4), if the bounded solutionu(n) of (3.1) is attracting in H(f,B) and totally stable, then it is totally asymptotically stable.Proof.
Letδ0 be the number for the attracting of u(n) in H(f,B) and let δ0*=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 (jk),(nk),(pk), and (xk) such that jk≥0,nk≥jk+2k,ρ(ujk,xjkk)<δ0* and ρ(unk,xnkk)≥ε for all k=1,2,…, where xk(n) is a solution of
(3.38)x(n+1)=f(n,x(n))+∑j=-∞0B(n,j,x(n+j),x(n))+h(x,xn)+pk(n).
such that xjkk(j)∈K for all j≤0 and pk:ℤ→ℝd with |pk(n)|<min{1/k,δ0*} for n≥jk. Note that ρ(ujk,xjkk)<δ0* and |pk(n)|<δ0* for n≥jk. Then we have
(3.39)ρ(un,xnk)<12δ0
for all n≥jk 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 |pk(n)|<1/k<δ(ε) for all n≥jk.
We claim thatρ(un,xnk)≥δ(ε) on [jk+k,jk+2k] if n≥n0. If we assume that ρ(un,xnk)<δ(ε) on [jk+k,jk+2k], then ρ(un,xnk)<ε for n≥jk+2k since u(n) is totally stable. This contradicts ρ(unk,xnkk)≥ε,k=1,2,…, because nk≥jk+2k.
Now, for the sequence(jk+k), taking a subsequence if necessary, there exists a (v,g,D)∈H(u,f,B).
If we setyk(n)=xk(n+jk+k), then yk(n) is the defined on [-k,k]. There exists a subsequence of (yk(n)), which we denote by (yk(n)) again, and a function y(n) such that yk(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
(3.40)x(n+1)=g(n,x(n))+∑j=-∞0D(n,j,x(n+j),x(n))
such that y0(j)∈K for all j≤0, by the same method as in [13, Theorem 3.1] . We have
(3.41)δ(ε)<ρ(un+jk+k,ynk)<δ02,0≤n≤k,k≥k0.
Then, by letting k→∞, we obtain
(3.42)δ(ε)<ρ(vn,yn)≤δ02,n≥0.
Since u(n) is attracting in H(f,B), we have ρ(vn,yn)→0 as n→∞. This contradicts ρ(vn,yn)≥δ(ε). Hence u(n) is totally asymptotically stable. This completes the proof of the theorem.



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*Source: 976369-2009-06-22.xml*