Research of the three-point boundary value problems for nonlinear second-order difference equation

Abstract

In order to extend the basic theory of nonlinear discrete boundary value problems,this paper studied the sufficient conditions for the existence of positive solutions for a class of nonlinear second-order difference equations with three-point boundary value problems.Firstly,the expressions of the solutions for the corresponding three-point boundary value problems for second-order difference equations were given and their properties were proved; Secondly,by constructing suitable cone and integral operator in Banach space and utilizing Krasnoselskii's fixed point theorem in cones,the sufficient conditions for the existence of positive solutions of three-point boundary value problems for nonlinear second-order difference equations were obtained under the condition that the nonlinear term was allowed to change sign.Finally,two examples were given to illustrate the validity of the main theorems and results.The results show that the conditions of the theorem are proved and the discrete boundary value problems satisfies the existence condition of positive solutions.The method is effective in the theoretical proof of the second-order discrete boundary value problem,and has reference for the study of the nonlinear high-order multi-point discrete boundary value problems.

Keywords

• difference equation; discrete boundary value problem; fixed point theorem; cone; positive solution; existence