### Asymptotic behaviour of solutions of difference equations in Banach spaces

In this paper we consider the first order difference equation in a Banach space $\Delta {x}_{n}={\sum}_{i=0}^{\infty}{a}_{n}^{i}f\left({x}_{n+i}\right)$. We show that this equation has a solution asymptotically equal to a. As an application of our result we study the difference equation $\Delta {x}_{n}={\sum}_{i=0}^{\infty}{a}_{n}^{i}g\left({x}_{n+i}\right)+{\sum}_{i=0}^{\infty}{b}_{n}^{i}h\left({x}_{n+i}\right)+{y}_{n}$ 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...