On a coupled system of fractional sum-difference equations with p-Laplacian operator

Saowaluck Chasreechai,Pimchana Siricharuanun,Thanin Sitthiwirattham

doi:10.1186/s13662-020-02826-3

Abstract

In this paper, we propose a nonlocal fractional sum-difference boundary value problem for a coupled system of fractional sum-difference equations with p-Laplacian operator. The problem contains both Riemann–Liouville and Caputo fractional difference with five fractional differences and four fractional sums. The existence and uniqueness result of the problem is studied by using the Banach fixed point theorem.

Highlights

Discrete fractional calculus and fractional difference equations have been widely studied
Discrete fractional calculus can be applied in queuing problems, economics, logistic map, and electrical networks, see [2–4]
We observe that the boundary value problem of a coupled system of nonlinear fractional difference equations with p-Laplacian operator has not been studied

Summary

Introduction

Discrete fractional calculus and fractional difference equations have been widely studied. The boundary value problem for fractional differential equations and the system of equations with p-Laplacian operator were presented in [33–39] and [40–44], respectively. The boundary value problem for fractional difference equations with p-Laplacian operator was presented in [45–47]. We observe that the boundary value problem of a coupled system of nonlinear fractional difference equations with p-Laplacian operator has not been studied. This result is the motivation for this research. We aim to study the coupled system of nonlinear fractional sum-difference equations with p-Laplacian operator α1 C φp β1 C u1. We find a solution of the linear variant of boundary value problem (1.1)–(1.2) as shown in the following lemma. After substituting C11 and C12 into (2.11), we obtain (2.4) and (2.5)

Existence and uniqueness result

C φ5 2

Conclusions