Existence and Uniqueness Results for Fractional (p, q)-Difference Equations with Separated Boundary Conditions

Pheak Neang,Kamsing Nonlaopon,Bashir Ahmad,Sotiris Ntouyas,Jessada Tariboon

doi:10.3390/math10050767

Abstract

In this paper, we study the existence of solutions to a fractional (p, q)-difference equation equipped with separate local boundary value conditions. The uniqueness of solutions is established by means of Banach’s contraction mapping principle, while the existence results of solutions are obtained by applying Krasnoselskii’s fixed-point theorem and the Leary–Schauder alternative. Some examples illustrating the main results are also presented.

Highlights

For some interesting results on fractional differential equations ranging from the existence and uniqueness of solutions to the analytic and numerical methods for finding solutions, we refer the reader to the following articles:
To make this paper more smooth and convenient, we have investigated the existence and uniqueness of solutions for the local boundary value problem of fractional ( p, q)-difference equation with a new function obtained g ∈ C ([0, b] × R, R), given by the following: c
To obtain the sufficient condition of existence and uniqueness of solutions of (7)–(8), employing the following Lemmas of fractional ( p, q)-calculus play an important role in those main results

Summary

Introduction

Fractional calculus, dealing with the integrals and derivatives of arbitrary order, constitutes an important area of investigation in view of its extensive theoretical development and applications during the last few decades. In 2021, Neang et al [31] considered the nonlocal boundary value problem of nonlinear fractional ( p, q)-difference equations with taking care of solutions of existence and uniqueness results obtained by c. Qin and Sun [33] studied positive solutions for fractional ( p, q)-difference boundary value problems given by the following:. Even though Neang et al [31] investigated and proved the nonlocal boundary value problems by considering on existence results of a class of fractional ( p, q)difference equations, it still was a bit complicated with the domain of a function when the authors applied the fractional ( p, q)-integral operators. To make this paper more smooth and convenient, we have investigated the existence and uniqueness of solutions for the local boundary value problem of fractional ( p, q)-difference equation with a new function obtained g ∈ C ([0, b] × R, R), given by the following: c.

Preliminaries

Main Results

Examples

Conclusions