NONLINEAR VOLTERRA DIFFERENCE EQUATIONS IN SPACE lp

Volterra difference equations arise in the mathematical modeling of some real phenomena, and also in numerical schemes for solving differential and integral equations (cf. [7, 8] and the references therein). One of the basic methods in the theory of stability and boundedness of Volterra difference equations is the direct Lyapunov method (see [1, 3, 4] and the references therein). But finding the Lyapunov functionals for Volterra difference equations is a difficult mathematical problem. In this paper, we derive estimates for the c0and lp-norms of solutions for a class of vector Volterra difference equations. These estimates give us explicit stability conditions. To establish the solution estimates, we will interpret the Volterra equations with matrix kernels as operator equations in appropriate spaces. Such an approach for Volterra difference equations has been used by Kolmanovskii and Myshkis [7], Kolmanovskii et al. [8], Kwapisz [9], Medina [10, 11], and Gil’ and Medina [6]. Under some restriction, our results generalize the main results from [6, 8, 11]. Let Cn be an n-dimensional complex Euclidean space with the Euclidean norm ‖ · ‖Cn . For a positive r ≤∞, put ωr = { h∈Cn : ‖ · ‖Cn ≤ r } . (1.1)


Introduction and statement of the main result
Volterra difference equations arise in the mathematical modeling of some real phenomena, and also in numerical schemes for solving differential and integral equations (cf.[7,8] and the references therein).
One of the basic methods in the theory of stability and boundedness of Volterra difference equations is the direct Lyapunov method (see [1,3,4] and the references therein).But finding the Lyapunov functionals for Volterra difference equations is a difficult mathematical problem.
In this paper, we derive estimates for the c 0 -and l p -norms of solutions for a class of vector Volterra difference equations.These estimates give us explicit stability conditions.To establish the solution estimates, we will interpret the Volterra equations with matrix kernels as operator equations in appropriate spaces.Such an approach for Volterra difference equations has been used by Kolmanovskii and Myshkis [7], Kolmanovskii et al. [8], Kwapisz [9], Medina [10,11], and Gil' and Medina [6].Under some restriction, our results generalize the main results from [6,8,11].
Let C n be an n-dimensional complex Euclidean space with the Euclidean norm As usual, c 0 = c 0 (C n ) is the Banach space of sequences of vectors from C n equipped with the norm and is the Banach space of sequences of vectors from C n equipped with the norm be n × n matrices dependent on j − 1 arguments.Consider the equation where the mappings G j : C ( j−1)n → C n have the properties Moreover, G 1 ∈ C n is given and In addition, it is assumed that ) (1.10) M. I. Gil' and R. Medina 303 To formulate the result, denote Now we are in a position to formulate the main result of the paper.

Proof of Theorem 1.1
First, assume that r = ∞.Then conditions (1.7) and (1.9) imply Define on l p = l p (R) the operator V by Here, [h] j means the jth coordinate of the element h ∈ l p (R).The operator V is a quasinilpotent one.So, due to the well-known lemma from the book by Dalec'kiȋ and Kreȋn (see [2, Lemma 3.2.1])(the comparison principle), where y j is a solution of the equation Rewrite this equation as (2.5) Lemma 2.1.Let conditions (1.7) and (1.9) hold.Then a solution y of (2.5) satisfies the inequality Proof.Rewrite (2.5) as M. I. Gil' and R. Medina 305 we have concluding the proof.
Lemma 2.2.Let conditions (1.7) and (1.9) hold.Then a solution y of (2.5) satisfies the inequality Proof.From (2.5) it follows that Proof of Theorem 1.1.If r = ∞, then the required result follows from Lemmas 2.1 and 2.2.Let now r < ∞.By a simple application of the Urysohn lemma and Lemma 2.2, we get the required result.