Unique iterative solution for nonlinear fractional [BF](p,q)[KG-*2]-[BFQ][KG-*4]difference equation based on [BF]ψ-(h,r)[KG-*2]-[BFQ][KG-*4]concave operators

Abstract

In order to enrich the basic theory of boundary value problems of fractional (p,q)[KG-*2]-[KG-*3]difference equations,the solvability of nonlocal problems for a class of nonlinear fractional (p,q)[KG-*2]-[KG-*3]difference equations was investigated.Firstly,the Green function of the boundary value problem of linear fractional (p,q)[KG-*2]-[KG-*3]difference equation was calculated and its properties were studied.Secondly,a fixed point theorem based on the increasing ψ-(h,r)[KG-*2]-[KG-*3]concave operators defined on ordered sets was used to prove the existence and uniqueness of solutions for fractional (p,q)[KG-*2]-[KG-*3]difference equations.Thirdly,by selecting the initial value,the monotone iterative sequence was constructed to obtain the unique iterative solution of the boundary value problem.Finally,an example was provided to illustrate the correctness of the research results.The results show that when certain conditions are given to the nonlinear term f,the nonlinear fractional (p,q)[KG-*2]-[KG-*4]difference equation has a unique non-trivial solution.The research results extend the solvability theory of fractional quantum difference equations and provide a powerful theoretical basis for further application of fractional (p,q)[KG-*2]-[KG-*4]difference equation.

Keywords

• quantum difference equation; fractional [bf](p
• q)[kg-*2]-[bfq][kg-*4]difference equation; iterative solution; [bf]ψ-(h
• r)[kg-*2]-[bfq][kg-*4]concave operator; fixed point theorem