AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation97636910.1155/2009/976369976369Research ArticleTotal Stability in Nonlinear Discrete Volterra Equations with Unbounded DelayChoiSung Kyu1GooYoon Hoe2ImDong Man3KooNamjip1LitsynElena1Department of MathematicsChungnam National UniversityDaejeon 305-764South Koreacnu.ac.kr2Department of MathematicsHanseo UniversitySeosan 352-820South Koreahanseo.ac.kr3Department of Mathematics EducationCheongju UniversityCheongju 360-764South Koreacju.ac.kr20093103200920091512200820022009240320092009Copyright © 2009This 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.

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 . Many results have been obtained concerning total stability .

In , Hamaya discussed the relationship between total stability and stability under disturbances from hull for the integrodifferential equation x(t)=f̂(t,x(t))+-0F(t,s,x(t+s),x(t))ds, where f̂:×dd is continuous and is almost periodic in t uniformly for xd, and F:×(-,0]×d×dd is continuous and is almost periodic in t uniformly for (s,x,y)R*=(-,0]×d×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 .

Song and Tian  studied periodic and almost periodic solutions of discrete Volterra equations with unbounded delay of the form x(n+1)=f(n,x(n))+j=-0B(n,j,x(n+j),x(n)),n+, where f:×dd is continuous in xd for every n, and for any j,n    (jn), B:××d×dd is continuous for x,yd. 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 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  for the integrodifferential equation 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 by ,+,-, respectively, the set of integers, the set of nonnegative integers, and the set of nonpositive integers. Let d denote d-dimensional Euclidean space.

Definition 2.1 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

A continuous function f:×dd is said to be almost periodic inn uniformly forxd if for every ε>0 and every compact set Kd, 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 for all n, uniformly for xd.

Definition 2.2 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let B:×*d be continuous for x,yd, for any n,j-, where *=-×d×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 for all n and all (j,x,y)K*.

For the basic results of almost periodic functions, see [8, 14, 15].

Let l-(d) denote the space of all d-valued bounded functions on - with

ϕ=supj-|ϕ(j)|< for any ϕl-(d).

Consider the discrete Volterra equation with unbounded delay of the form x(n+1)=f(n,x(n))+j=-0B(n,j,x(n+j),x(n))+h(n,xn),n+, under certain conditions for f,B, and h (see below). We assume that, given ϕl-(d), there is a solution x of (1.3) such that x(n)=ϕ(n) for n-, passing through (0,ϕ0),ϕ0l-(d).

Let K be any compact subset of d such that ϕ(j)K for all j0 and x(n)K for all n1.

For any ϕ,ψl-(d), we set ρ(ϕ,ψ)=q=0ρq(ϕ,ψ)2q[1+ρq(ϕ,ψ)], where ρq(ϕ,φ)=max-qj0|ϕ(j)-ψ(j)|,q0. Then ρ defines a metric on the space l-(d). Note that the induced topology by ρ is the same as the topology of convergence on any finite subset of - .

In view of almost periodicity, for any sequence (nk)+ with nk as k, there exists a subsequence (nk)(nk) such that f(n+nk,x)g(n,x) uniformly on ×S for any compact set Sd, B(n+nk,n+l+nk,x,y)D(n,n+l,x,y) uniformly on ×S* for any compact set S*Z*. We define the hullH(f,B)={(g,D):(2.6)  and  (2.7)  hold  for  some  sequence(nk)+    with    nk    as    k}. 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 .

3. Main Results

We deal with the discrete Volterra equation with unbounded delay of the form x(n+1)=f(n,x(n))+j=-0B(n,j,x(n+j),x(n))+h(n,xn),n+. Throughout this paper we assume the following.

f:×dd is continuous in xd for every n and is almost periodic in n uniformly for xd.

B:×*d is continuous in x,yd for any n, j- and is almost periodic in n uniformly for (j,x,y)*=-×d×d. Moreover, for any ε>0 and any τ>0 there exists a number M=M(ε,τ)>0 such that j=--M|B(n,j,x(n+j),x(n))|ε for all n whenever |x(j)|τ for all j-.

h:×l-(d)d is continuous in ϕl-(d) for every n. xnl-(d) is defined as xn(j)=x(n+j) for j-. Furthermore, for any r>0, there exists a function αr:d with the property that αr(n)0 as n and |h(n,ϕ)|αr(n) whenever |ϕ(j)|r for all j-.

Equation (3.1) has a bounded solution u(n)=u(n,ϕ0) defined on +, through (0,ϕ0), ϕ0l-(d) such that for some 0M<, |u(n,ϕ0)|M.

Note that for any (g,D)H(f,B),D(n,j,x,y) satisfies (H2) with B=D . The limiting equation of (3.1) is defined as x(n+1)=g(n,x(n))+j=-0D(n,j,x(n+j),x(n)),n+, where (g,D)H(f,B). We assume that for any solution v(n) of (3.5), v(n)K for all n1, where K is the above-mentioned compact set in d.

Theorem 3.1 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Under the assumptions (H1)–(H4), if u(n)=u(n,ϕ0), ϕ0l-(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 .

Total stability requires that the solution of x(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 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 n00,    ρ(un0,xn0)<δ, and p(n) is a function such that |p(n)|<δ for all nn0, then ρ(un,xn)<ε for all nn0, where x(n) is any solution of 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-.

Definition 3.3.

A function ϕ:d is called asymptotically almost periodic if it is a sum of an almost periodic function ϕ1 and a function ϕ2 defined on which tends to zero as n, that is, ϕ(n)=ϕ1(n)+ϕ2(n) for all n.

It is known  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 as k .

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 (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 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 - 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,mk1(ε). Put 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 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. We will show that there exists a number k2(ε)>0 such that |p(n)|<δ(ε) for all n0 whenever k,mk2(ε).

Note that for all xK, there exists a number c>0 such that |x|c. It is clear that |uk(n)|c and |um(n)|c for all n.

In view of (H2), there exists a number M=M(c,ε)>0 such that

j=--M|B(n+nm,j,um(n+j),um(n))|15δ(ε),n,j=--M|B(n+nk,j,um(n+j),um(n))|15δ(ε),n. 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+nm,j,um(n+j),um(n))-B(n+nk,j,um(n+j),um(n))|<δ(ε)5M,n,  j[-M+1,0],|f(n+nm,um(n))-f(n+nk,um(n))|<15δ(ε),n, whenever k,mk2(ε). Since h(n,ϕ)0 as n, we obtain that if k,mk2(ε), then |h(n+nm,unm)-h(n+nk,unm)|<15δ(ε),n+. Then, by (3.10), and (3.11), we have 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 |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,mk2(ε), by (3.12), (3.13), and (3.14). Since uk(n) is totally stable, we obtain that ρ(unk,unm)<ε for all n0 if k,mk2(ε). This implies that for all k,mk2(ε), |u(n+nk)-u(n+nm)|sup-Ms0|u(n+nk+s)-u(n+nm+s)|<4ε for all ε(1/4) and all n0. It follows that for any (nk) with nk as k there exists a subsequence (nkj)(nk) such that (u(n+nkj)) converges uniformly on + as j, that is, u(n) is asymptotically almost periodic. This completes the proof.

Remark 3.5.

Hino et al.  showed that for the functional differential equation x(t)=f̂(t,xt), the solution v(t) of the limiting equation 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 .

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 (nk),nk as k, such that f(n+nk,x)g(n,x) uniformly on ×K, B(n+nk,j,x,y)D(n,j,x,y) uniformly on ×S*×K×K for any compact set S*-, and u(n+nk)v(n) uniformly on any compact set in as k. Set uk(n)=u(n+nk),k=1,2,. Then it is clear that uk(n) is a solution of 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 j0, where u0k(j)=uk(0+j)=u(j+nk). Note that for all xK,|x|c for some c>0. Let x(τ) be a function such that x(τ)K for all τn. By (H2), there exists a number M=M(c,ε)>0 such that 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 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 kk0(ε), then |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)),ρ(u0k,v0)<δ(12δ(ε2)). Put 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 x(n+1)=g(n,x(n))+j=-0D(n,j,x(n+j),x(n))+r(n) such that u0k(j)K for j0. Note that |r(n)|<δ((1/2)δ(ε/2)) for n0 by (3.22). From (3.22) and the fact that v(n) is totally stable, we have ρ(unk,vn)<12δ(ε2) for all n0.

Let m=k0(ε). To show that u(n) is totally stable we will show that if n00, ρ(un0,yn0)<(1/2)δ(ε/2), and |p(n)|<(1/2)δ(ε/2) for nn0, then ρ(un,yn)<ε for all nn0, where y(n) is a solution of

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 j0. Suppose that this is not the case. Then there exists an integer σ>n0 such that ρ(uσ,yσ)=εfor  σ>n0,ρ(un,yn)<εfor  n0n<σ. We set z(n)=y(n+nm). Then z(n) is a solution of 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 j0. Also, z(n) is a solution of x(n+1)=g(n,x(n))+j=-0D(n,j,x(n+j),x(n))+q(n), where 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 nσ-nm, and |p(n+nm)|<(1/2)δ(ε/2) for nn0-nm. Thus |q(n)|<δ(ε/2) for n0-nmnσ-nm. Also, we have ρ(un0,vn0-nm)<12δ(ε2),ρ(un0,zn0-nm)=ρ(un0,yn0)<12δ(ε2) from (3.26). Thus we obtain ρ(vn0-nm,zn0-nm)ρ(vn0-nm,un0)+ρ(un0,zn0-nm)<δ(ε2). Since v(n) is totally stable, we have ρ(vσ-nm,zσ-nm)<ε2. On the other hand, (3.26) implies that ρ(un,vn-nm)<12δ(ε2),nn0. Hence, if n00,ρ(un0,yn0)<(12)δ(ε2), and |p(n)|<(1/2)δ(ε/2) for nn0, then we obtain ρ(uσ,yσ)ρ(uσ,vσ-nm)+ρ(vσ-nm,zσ-nm)<ε. This contradicts (3.28). Therefore ρ(un,yn)<ε for all nn0 when n00, ρ(un0,yn0)<δ*(ε) and |p(n)|<δ*(ε) for all nn0, 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 .

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 n00 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 j0.

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 n00,ρ(un0,xn0)<δ0 and p(n) is any function which satisfies |p(n)|<η(ε) for nn0, then ρ(un,xn)<ε for all nn0+T(ε), where x(n) is a solution of 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 j0.

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 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 jk0,  nkjk+2k,  ρ(ujk,xjkk)<δ0* and ρ(unk,xnkk)ε for all k=1,2,, where xk(n) is a solution of 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 j0 and pk:d with |pk(n)|<min{1/k,δ0*} for njk. Note that ρ(ujk,xjkk)<δ0* and |pk(n)|<δ0* for njk. Then we have ρ(un,xnk)<12δ0 for all njk and k=1,2,, since u(n) is totally stable. Also, there exists an integer number k0(ε)>0 such that if kk0(ε), then |pk(n)|<1/k<δ(ε) for all njk.

We claim that ρ(un,xnk)δ(ε) on [jk+k,jk+2k] if nn0. If we assume that ρ(un,xnk)<δ(ε) on [jk+k,jk+2k], then ρ(un,xnk)<ε for njk+2k since u(n) is totally stable. This contradicts ρ(unk,xnkk)ε,  k=1,2,, because nkjk+2k.

Now, for the sequence (jk+k), taking a subsequence if necessary, there exists a (v,g,D)H(u,f,B).

If we set yk(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 as k such that y0(j)K for all j0. Moreover, we can show that y(n) is a solution of

x(n+1)=g(n,x(n))+j=-0D(n,j,x(n+j),x(n)) such that y0(j)K for all j0, by the same method as in [13, Theorem 3.1] . We have δ(ε)<ρ(un+jk+k,ynk)<δ02,0nk,  kk0. Then, by letting k, we obtain δ(ε)<ρ(vn,yn)δ02,n0. 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.

Acknowledgment

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

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