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∈ C : ‖ · ‖Cn ≤ r } . (1.1)