Almost automorphic solutions of non-autonomous difference equations

In the present paper, we study the non-autonomous difference equations given by u ( k + 1 ) = A ( k ) u ( k ) + f ( k ) and u ( k + 1 ) = A ( k ) u ( k ) + g ( k , u ( k ) ) for k ∈ Z , where A ( k ) is a given non-singular n × n matrix with elements a i j ( k ) , 1 ≤ i , j ≤ n , f : Z → E n is a given n × 1 vector function, g : Z × E n → E n and u ( k ) is an unknown n × 1 vector with components u i ( k ) , 1 ≤ i ≤ n . We obtain the existence of a discrete almost automorphic solution for both the equations, assuming that A ( k ) and f ( k ) are discrete almost automorphic functions and the associated homogeneous system admits an exponential dichotomy. Also, assuming the function g satisfies a global Lipschitz type condition, we prove the existence and uniqueness of an almost automorphic solution of the nonlinear difference equation.