Stability of Volterra difference delay equations

We study the asymptotic stability of the zero solution of the Volterra difference delay equation


Introduction
Fixed point theorems have been used extensively in recent times to study some of the qualitative properties of solutions of difference and differential equations.In the current paper we use Krasnoselskii's fixed point theorem to study the asymptotic stability of the zero solution of the difference equation, k(n, s)h(x(s)) (1.1) where a(n), c(n) : Z → R, k : Z × Z → R, h : Z → R, and g(n) : Z → Z + .The operator ∆ is defined as ∆x(n) = x(n + 1) − x(n).
We assume that a(n) and c(n) are bounded discrete functions whereas 0 ≤ g(n) ≤ g 0 for some integer g 0 .We also assume that h(0) = 0 and for some positive constant L.

EJQTDE, 2006
No. 20, p. 1 We refer to [1], [12] and [17] for some of the reasons why fixed point theorems are used to study stability.Equation (1.1) is the discrete version of one of the differential equations studied in [17].Continuous versions of (1.1) are generally known as neutral Volterra differential equations.
For any integer n 0 ≥ 0, we define Z 0 to be the set of integers in [−g 0 , n 0 ].Let ψ(n) : Z 0 → R be an initial discrete bounded function.

Asymptotic Stability
Lemma 2.1.Suppose that a(n) = 0 for all n ∈ Z .Then x(n) is a solution of equation (1.1) if and only if where Φ(r) = c(r) − c(r − 1)a(r).
Proof.The first term on the right of (2.4) goes to zero because of condition (2.5).
The second term on the right goes to zero because of condition (2.6) and the fact that ϕ ∈ S.
Finally we show that the last term k(r, u)h(ϕ(u)) n−1 s=r+1 a(s) k(r, u) This completes the proof of lemma 2.2.
We state below Krasnoselskii's fixed point Theorem which is the main mathematical tool in this paper and we refer to [19] for the proof.
Theorem 2.3 ( Krasnoselskii's ) Let M be a closed convex nonempty subset of a Banach space (B, ||.||).Suppose that A and Then there exists z ∈ M with z = Az + Qz.
The application of the above Theorem requires the construction of two mappings which we obtain by expressing (2.4) as EJQTDE, 2006 No. 20, p. 7 where A, Q : S → S are given by (2.9) Lemma 2.4 Suppose that (1.2) and (2.7) hold.Also suppose that there exist a constant γ such that, Then the mapping A defined by (2.9) is continuous and compact.
EJQTDE, 2006 No. 20, p. 8 We next show that A is compact.Let {ϕ n } ⊂ S denote a sequence of uniformly bounded functions with ||ϕ n || ≤ λ, where n is a positive integer and λ > 0. Thus, Thus showing that ||(Aϕ n )|| ≤ ι for some positive constant ι. Also, For some positive constant β.Thus the sequence (Aϕ n ) is uniformly bounded and equi-continuous.The Arzela-Ascoli theorem implies that (Aϕ n k ) converges uniformly to a function ϕ * .Thus A is compact.Theoem 2.6.Suppose the hypotheses of lemma 2.2, lemma 2.4, and lemma 2.5 hold.Also, suppose that there is a positive constant ρ such that and there exist an α ∈ (0, 1) Then the zero solution of (1.1) is asymptotically stable.
Let ψ(n) be any given initial function such that |ψ(n)| < δ.Therefore showing that the zero solution of (1.1) is stable.In addition ||x|| → 0 as n → ∞ by the fact that x(n) ∈ S. Therefore the zero solution of (1.1) is asymptotically stable.
Therefore the zero solution of (2.14) is asymptotically stable.