Existence Result and Uniqueness for Some Fractional Problem

Guotao Wang,Samah Horrigue,Samar Madian,Abdeljabbar Ghanmi

doi:10.3390/math7060516

Abstract

In this article, by the use of the lower and upper solutions method, we prove the existence of a positive solution for a Riemann–Liouville fractional boundary value problem. Furthermore, the uniqueness of the positive solution is given. To demonstrate the serviceability of the main results, some examples are presented.

Highlights

Where β, γ, and η are positive real numbers such that β − 2γη α−1 > 0, j is a nonnegative continuous function on [0, 1] × [0, ∞), and D0α+ is the fractional derivative in the sense of Riemann–Liouville
This type of equation is important in many disciplines such as chemistry, aerodynamics, polymer rheology, etc
Different techniques are used in such problems to obtain the existence of solutions, for example the variational method, the Adomian decomposition method, etc.; we refer the reader to [1,2,3,4,5,6,7,8] and references therein

Summary

Introduction

The aim of this work is to study the existence and uniqueness of the positive solution for the following problem:. Where β, γ, and η are positive real numbers such that β − 2γη α−1 > 0, j is a nonnegative continuous function on [0, 1] × [0, ∞), and D0α+ is the fractional derivative in the sense of Riemann–Liouville. Motivated by the above works, in this article, we will present a new method to study the given problem, that is we combine the lower and upper solution method with the fixed point theorem method in order to prove the existence and uniqueness of the positive solution.

Preliminaries

Proof of the Main Results

Examples