Regional topology and approximative solutions of difference and differential equations

TL;DR

This paper introduces a new regional topology on function spaces, extends classical theorems, and uses these to establish existence of solutions with specific asymptotic behavior for certain difference and differential equations.

Contribution

It develops a novel regional topology and new versions of Schauder's and Ascoli's theorems, enabling the analysis of solutions with prescribed asymptotics.

Findings

Established conditions for existence of solutions with prescribed asymptotics

Extended classical theorems to new topological settings

Applied to difference and differential equations

Abstract

We introduce a topology, which we call the regional topology, on the space of all real functions on a given locally compact metric space. Next we obtain a new versions of Schauder's fixed point theorem and Ascoli's theorem. We use these theorems and the properties of the iterated remainder operator to establish conditions under which there exist solutions, with prescribed asymptotic behavior, of some difference and differential equations.