Stability in discrete equations with variable delays

In this paper we study the stability of the zero solution of difference equations with variable delays. In particular we consider the scalar delay equation

Remark 1.1 In [7], the author and Islam showed that the zero solution of the equation is asymptotically stable with one of the assumptions being that However, as pointed out in [11], condition (1.3) cannot hold for (1.2) since b(n) = 1, for all n ∈ Z.The results we obtain in this paper overcome the requirement of (1.3).
Let D(n 0 ) denote the set of bounded sequences ψ : [m(n 0 ), n 0 ] → R with the maximum norm ||.||.Also, let (B, ||.||) be the Banach space of bounded sequences ϕ : [m(n 0 ), ∞) → R with the maximum norm.Define the inverse of n − τ i (n) by g i (n) if it exists and set where In this section we obtain conditions for the zero solution of (1.2) to be asymptotically stable.
We begin by rewriting (1.2) as where ∆ n represents that the difference is with respect to n.If we let Lemma 2.1 Suppose that Q(n) = 0 for all n ∈ Z + and the inverse function g j (n) of n − τ j (n) exists.Then x(n) is a solution of (2.2) if and only if Proof.By the variation of parameters formula we obtain a j (g j (s))x(s) .
We next state and prove our main results.
Theorem 2.1 Suppose that the inverse function g j (n) of n − τ j (n) exists, and assume there exists a constant α ∈ (0, 1) such that |a j (g j (u))| ≤ α. (2.5) Moreover, assume that there exists a positive constant M such that Then the zero solution of (1.2) is stable.
Define the mapping P : S → S by Clearly, Pϕ is continuous.We first show that P maps from S to S. By (2.6) Thus P maps from S into itself.We next show that P is a contraction.Let ζ, η ∈ S. Then This shows that P is a contraction.Thus, by the contraction mapping principle, P has a unique fixed point in S which solves (1.2) and for any ϕ ∈ S, ||P ϕ|| ≤ .This proves that the zero solution of (1. (2.7) Then the zero solution of (1.2) is asymptotically stable.
Define P : S * → S * by (2.6).From the proof of Theorem 2.2, the map P is a contraction and for every ϕ ∈ S * , ||(P ϕ)|| ≤ .We next show that (P ϕ)(n) → 0 as n → ∞.The first term on the right side of (2.6) goes to zero because of condition (2.7).It is clear from (2.5) and the fact that ϕ(n) → 0 as n → ∞ that N j=1 n−1 s=n−τ j (n) |a j (g j (s))||ϕ(s)| → 0 as n → ∞.Now we show that the last term on the right side of (2.6) goes to zero as