Positive Solutions for a Higher-Order Semipositone Nonlocal Fractional Differential Equation with Singularities on Both Time and Space Variable

Kemei Zhang

doi:10.1155/2019/7161894

Abstract

In this paper, we consider the following higher-order semipositone nonlocal Riemann-Liouville fractional differential equation D0+αx(t)+f(t,x(t),D0+βx(t))+e(t)=0, 0<t<1,D0+βx(0)=D0+β+1x(0)=⋯=D0+n+β-2x(0)=0, and D0+βx(1)=∑i=1m-2ηiD0+βx(ξi), where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives. The existence results of positive solution are given by Guo-krasnosel’skii fixed point theorem and Schauder’s fixed point theorem.

Highlights

We devote to the investigation of the following nonlinear fractional differential equation
In [3], the authors were concerned with the existence of monotone positive solutions to the following fractional-order multipoint boundary value problems
In [4], the authors investigated the existence of positive solutions of the following fractional differential equation multipoint boundary value problems with changing sign nonlinearity Journal of Function Spaces

Summary

Introduction

We devote to the investigation of the following nonlinear fractional differential equation. In [3], the authors were concerned with the existence of monotone positive solutions to the following fractional-order multipoint boundary value problems. In [4], the authors investigated the existence of positive solutions of the following fractional differential equation multipoint boundary value problems with changing sign nonlinearity. In [5], the authors established the uniqueness of a positive solution to the following higher-order fractional differential equation: Dα0+ u (t). By using the fixed point theorem for the mixed monotone operator, the existence of unique positive solutions for above singular nonlocal boundary value problems of fractional differential equations is established. In [11], the authors studied the existence of positive solutions for the following nonlocal fractional-order differential equations with sign-changing singular perturbation.

Preliminaries and Some Lemmas

Main Result I: f Is Singular with Respect to the Time Variables