Stability analysis of nonlinear retarded difference equations in Banach spaces

In this paper, we give new sufficient conditions for asymptotic stability and instability of nonlinear difference equations with delays in infinite-dimensional spaces x(k + 1) = f(k,x(k), x(k−1), …x(k−r))), x(k) ε X, K = 0, 1,2,…, where X is a Banach space. Our results are obtained using a general comparison condition on the right-hand side function f(k,.), which generalizes the stability and instability results obtained by Yang, Miminis, Naulin, and Vanegas. An application to the stabilizability problem of retarded control systems and illustrative examples of the obtained results are given.