Asymptotic behavior of solutions to difference equations in Banach spaces

We investigate the asymptotic properties of solutions to higher order nonlin-ear difference equations in Banach spaces. We introduce a new technique based on a vector version of discrete L'Hospital's rule, remainder operator, and the regional topol-ogy on the space of all sequences on a given Banach space. We establish sufficient conditions for the existence of solutions with prescribed asymptotic behavior. More-over, we are dealing with the problem of approximation of solutions. Our technique allows us to control the degree of approximation of solutions.