Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation

Stevo Stević

doi:10.1186/s13662-017-1227-x

Abstract

We present some interesting facts connected with the following second-order difference equation: xn+2−qnxn=fn,n∈N0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$x_{n+2}-q_{n}x_{n}=f_{n},\\quad n\\in \\mathbb{N}_{0}, $$\\end{document} where (q_{n})_{ninmathbb{N}_{0}} and (f_{n})_{ninmathbb {N}_{0}} are given sequences of numbers. We give some sufficient conditions for the existence of a unique bounded solution to the difference equation and present an elegant proof based on a combination of theory of linear difference equations and the Banach fixed point theorem. We also deal with the equation by using theory of solvability of difference equations. A global convergence result of solutions to a linear first-order difference equation is given. Some comments on an abstract version of the linear first-order difference equation are also given.

Highlights

Difference equations and systems of difference equations have been of a great interest in the last several decades
We will present two interesting ways how the problem of the existence of a unique bounded solution can be solved for the case of the difference equation ( ), wheren∈N andn∈N are two given bounded sequences
By a nice combination of the theory of linear difference equations and the Banach fixed point theorem we present some sufficient conditions for the existence of a unique bounded solution to equation ( )

Summary

Introduction

Difference equations and systems of difference equations have been of a great interest in the last several decades. Where x , x are given numbers, is solved and it has the following general solution: xn = c (√q)n + c (–√q)n, n ∈ N , ( ) We will present two interesting ways how the problem of the existence of a unique bounded solution can be solved for the case of the difference equation ( ), where (qn)n∈N and (fn)n∈N are two given bounded sequences.

Results

Conclusion