Asymptotic behaviour of solutions of difference equations in Banach spaces

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