On the stable manifolds of difference equations with infinite delay

This paper is concerned with difference equation $ y(n+1) = A_ny_n+f(n, \, y_{n+\delta_n}, \lambda) infinite delay in a Banach space X $, where \delta_n ($ n\in {\mathbb Z} $) given sequence taking values \{0, 1\} and \lambda parameter. First, we construct Lipschitz invariant manifold {\mathcal M}^{\lambda} of the which consists all bounded forward solutions. Then prove that contains unique complete solution \gamma^{\lambda} \gamma^{\lambda}(n) depends on continuously. To understand dynamical behavior establish discrete inequality delay. By applying this finally show attracts (exponentially attracts) solutions provided constant f sufficiently small. Smoothness M}^\lambda also addressed.