On the Asymptotic Stability of the Nonlinear Difference Equation System

Abstract

In this paper, we obtain some new results on the equi-boundedness of solutions and asymptotic stability for a class of nonlinear difference systems with variable delay of the form x(n+1)=ax(n)+B(n)F(x(n−m(n))), n=0,1,2,...x(n+1)=ax(n)+B(n)F(x(n−m(n))),\ \ \ \ \ \ n=0,1,2,... where FF is the real valued vector function, m:Z→Z+,m:Z→Z+, which is bounded function and maximum value of mm is kk and is a k×kk×k variable coefficient matrix. We carry out the proof of our results by using the Banach fixed point theorem and we use these results to determine the asymptotic stability conditions of an example.