Asymptotic behavior of solutions to difference equations in Banach spaces

Abstract

We investigate the asymptotic properties of solutions to higher order nonlinear 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 topology on the space of all sequences on a given Banach space. We establish sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we are dealing with the problem of approximation of solutions. Our technique allows us to control the degree of approximation of solutions.

Keywords

• difference equation in banach space
• prescribed asymptotic behavior
• degree of approximation
• remainder operator
• regional topology