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On the existence of almost automorphic solutions of Volterra difference equations. (English) Zbl 1261.39007

In the first part, the theory of almost automorphic functions of a real variable is utilized to establish properties of almost automorphic sequences (in a Banach space). Further, the authors consider the linear Volterra difference equation (in a Banach space) of the form \(u(n+1)=\lambda\sum_{j=-\infty}^{n}a(n-j)u(j)+f(n)\), \(n\in\mathbb{Z}\), where \(\lambda\) is a complex number or, more generally, a bounded linear operator, \(a:\mathbb{N}\to\mathbb{C}\) is summable, and \(f\) is almost automorphic. The existence of an almost automorphic solution to this equation is proved. The authors also show the existence of an almost automorphic solution to a semi-linear version of the Volterra equation where \(f(n)\) is replaced by \(f(n,u(n))\); \(f\) satisfying a Lipschitz type condition. A certain functional version of the Volterra equation along with the existence of its almost automorphic solution is discussed as well.
Reviewer: Pavel Rehak (Brno)

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
45D05 Volterra integral equations
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