Qualitative approximation of solutions to difference equations

TL;DR

This paper introduces a novel method for analyzing the asymptotic behavior of difference equations by replacing the traditional space of sequences converging to zero with various subspaces, utilizing the iterated remainder operator and regional topology.

Contribution

It proposes a new approach that generalizes asymptotic equivalence by employing subspaces of c_0, regional topology, and fixed point theorems, advancing the theoretical framework.

Findings

New framework for asymptotic analysis of difference equations

Application of regional topology and Schauder fixed point theorem

Enhanced understanding of solution behaviors in difference equations

Abstract

We present a new approach to the theory of asymptotic properties of solutions of difference equations. Usually, two sequences $x,y$ are called asymptotically equivalent if the sequence $x-y$ is convergent to zero i.e., $x-y\in c_0$, where $c_0$ denotes the space of all convergent to zero sequences. We replace the space $c_0$ by various subspaces of $c_0$. Our approach is based on using the iterated remainder operator. Moreover, we use the regional topology on the space of all real sequences and the `regional' version of the Schauder fixed point theorem.