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Almost automorphic solutions of non-autonomous difference equations. (English) Zbl 1306.39009

Summary: 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 \(k\in\mathbb Z\), where \(A(k)\) is a given non-singular \(n\times n\) matrix with elements \(a_{ij}(k)\), \(1\leq i\), \(j\leq n\), \(f:\mathbb Z\to E^n\) is a given \(n\times 1\) vector function, \(g:\mathbb Z\times E^n\to E^n\) and \(u(k)\) is an unknown \(n\times 1\) vector with components \(u_i(k)\), \(1\leq i\leq n\). 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.

MSC:

39A30 Stability theory for difference equations
39A12 Discrete version of topics in analysis
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