Stability in delay Volterra di ﬀ erence equations of neutral type

Su ﬃ cient conditions for the zero solution of a certain class of neutral Volterra di ﬀ erence equations with variable delays to be asymptotically stable are obtained. The Banach’s ﬁ xed point theorem is employed in proving our results. 2000 Mathematics Subject


Introduction
The study of the stability of the zero solution of difference equations has gained the attention of many mathematicians lately, see [1], [2], [3], [5], [7], [9], [11] and [12].In this paper we consider the nonlinear difference equation with variable delays k j (n, s)f j (s, x(s)), (1.1) with the initial condition where ψ : [m(n 0 ), n 0 ] ∩ Z → R is a bounded sequence and for n 0 ≥ 0, Here ∆ denotes the forward difference operator.That is, ∆x(n) = x(n+ 1) − x(n) for any sequence {x(n) : n ∈ Z + }.We assume throughout this paper that a j : and τ j : Z + → Z + , for j = 1, ..., N. Special cases of (1.1) have been considered by a number of researchers in recent times.
For instance, Raffoul in [7] considered the equation where τ is a positive constant.The first author in [11], extended the results obtained in [7] to the equation The first author also in [12] considered the the following nonlinear Volterra difference equation k(n, s)q(x(s)).
(1.5)Moreover, Ardjouni and Djoudi in [2] considered the difference equations with variable delays Motivated by the above mentioned papers, we obtain in this paper sufficient conditions for the zero solution of (1.1) to be asymptotically stable.

Stability
Let n 0 ∈ Z ∩ [0, ∞), be fixed.We let D(n 0 ) be the set of bounded se- In this paper we assume that for j=1,...,N, for some positive constants L 1 and L 2 .Also, for j=1,...,N, Then x is a solution of equation (1.1) if and only if Stability in delay Volterra difference equations of neutral type 233 where ∆ n denotes the difference taken with respect to n.
The above equation is equivalent to Rewrite equation (2.6) as

Consequently, we have
Dividing both sides of (2.8) by Stability in delay Volterra difference equations of neutral type 235 Using the summation by parts formula, we obtain h j (r)x(r).(2.11) Substituting (2.10) and (2.11) into (2.9)gives the desired results.