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Almost automorphic solutions of difference equations. (English) Zbl 1177.39002

Summary: We study discrete almost automorphic functions (sequences) defined on the set of integers with values in a Banach space \(X\). Given a bounded linear operator \(T\) defined on \(X\) and a discrete almost automorphic function \(f(n)\), we give criteria for the existence of discrete almost automorphic solutions of the linear difference equation \(\Delta u(n)=Tu(n)+f(n)\). We also prove the existence of a discrete almost automorphic solution of the nonlinear difference equation \(\Delta u(n)=Tu(n)+g(n,u(n))\) assuming that \(g(n,x)\) is discrete almost automorphic in \(n\) for each \(x\in X\), satisfies a global Lipschitz type condition, and takes values on \(X\).

MSC:

39A10 Additive difference equations
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