Positive solutions for nonlocal problems of nonlinear (p,q)- difference equations

Abstract

In order to improve the basic theory of boundary value problems for nonlinear quantum difference equations,in this paper,we study the solvability of nonlocal problems for second order three-point nonlinear (p,q)-difference equations.Firstly,the Green function of the boundary value problem of linear (p,q)-difference equation is calculated and the property of Green function is studied.Secondly,we obtain the existence and uniqueness of the positive solution for the problem by the Banach contraction mapping principle and the Guo-Krasnoselskii fixed point theorem in a cone.Next,we get the Lyapunov inequality for nonlocal problems of linear (p,q)-difference equations.Finally,two examples are given to illustrate the validity of the results.The results show that the existence and uniqueness of positive solutions for nonlocal problems of nonlinear (p,q)-difference equations are obtained,under the condition of nonlinear term f certain growth.The research results enrich the theory of solvability of quantum difference equations and provide important theoretical basis for the application of(p,q)-difference equation in mathematics,physics and other fields.

Keywords

• nonlinear functional analysis; nonlinear (p
• q)-difference equation; nonlocal problem; banach contraction mapping principle; guo-krasnoselskii fixed point theorem; positive solution