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Asymptotic behaviour of solutions of difference equations in Banach spaces

Anna Kisiołek — 2008

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we consider the first order difference equation in a Banach space Δ x n = i = 0 a n i f ( x n + i ) . We show that this equation has a solution asymptotically equal to a. As an application of our result we study the difference equation Δ x n = i = 0 a n i g ( x n + i ) + i = 0 b n i h ( x n + i ) + y n and give conditions when this equation has solutions. In this note we extend the results from [8,9]. For example, in [9] the function f is a real Lipschitz function. We suppose that f has values in a Banach space and satisfies some conditions with respect to the measure of noncompactness...

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