Global structure of positive solutions for second-order difference equation with nonlinear boundary value condition

Yanqiong Lu,Ruyun Ma

doi:10.1186/1687-1847-2014-188

Abstract

This paper is devoted to the study of the global structure of the positive solution of a second-order nonlinear difference equation coupled with a nonlinear boundary value condition. The main result is based on Dancer’s bifurcation theorem.MSC:34B15, 39A12.

Highlights

The development in numerical analysis has propelled interest in difference equations and their relationship to their differential counterparts
We investigate the nonlinear discrete Sturm-Liouville problems coupled with a nonlinear boundary value condition, transform it into the equivalent operator equation, and use Dancer’s bifurcation theorem to obtain the existence of a positive solution
As far as we know, there is very little work to study the existence of positive solutions of second-order difference equation with nonlinear boundary value condition

Summary

Introduction

The development in numerical analysis has propelled interest in difference equations and their relationship to their differential counterparts. We investigate the nonlinear discrete Sturm-Liouville problems coupled with a nonlinear boundary value condition, transform it into the equivalent operator equation, and use Dancer’s bifurcation theorem to obtain the existence of a positive solution. It is worth to point out that Rodríguez and Abernathy [ ] studied the existence of solutions of the following boundary value problem of the difference equation: p(k – ) x(k – ) + q(k)x(k) + ψ x(k) = G x(k) , k ∈ {a + , .

Results

Conclusion