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Asymptotic behaviour of solutions of difference equations in Banach spaces

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Kisiołek A.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2008

tom 28

nr 1

5-13

In this paper we consider the first order difference equation in a Banach space $Δx_{n} = ∑_{i=0}^∞ a^{i}_{n} f(x_{n+i})$. We show that this equation has a solution asymptotically equal to a. As an application of our result we study the difference equation $Δx_{n} = ∑_{i=0}^∞ a^i_{n}g(x_{n+i}) + ∑_{i=0}^∞ b^{i}_{n}h(x_{n+i}) + y_{n}$ 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 and measure of weak noncompactness.

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1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Kisiołek A.

1 2008

Wyniki wyszukiwania

Asymptotic behaviour of solutions of difference equations in Banach spaces