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)u(k+1)=A(k)u(k)+f(k) and u(k+1)=A(k)u(k)+g(k,u(k))u(k+1)=A(k)u(k)+g(k,u(k)) for k∈Zk∈Z, where A(k)A(k) is a given non-singular n×nn×n matrix with elements aij(k),1≤i,j≤n, f:Z→Enf:Z→En is a given n×1n×1 vector function, g:Z×En→Eng:Z×En→En and u(k)u(k) is an unknown n×1n×1 vector with components ui(k)ui(k), 1≤i≤n1≤i≤n. We obtain the existence of a discrete almost automorphic solution for both the equations, assuming that A(k)A(k) and f(k)f(k) are discrete almost automorphic functions and the associated homogeneous system admits an exponential dichotomy. Also, assuming the function gg satisfies a global Lipschitz type condition, we prove the existence and uniqueness of an almost automorphic solution of the nonlinear difference equation.