Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions

Daliang Zhao,Juan Mao

doi:10.3390/sym13010107

Abstract

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

Highlights

Differential equations of fractional order have received more and more attention in virtue of their various applications in the fields of science and engineering.Compared with integer-order differential equations, fractional-order models can provide more accurate characterizations of many natural phenomena and mathematical problems.Many good results about fractional differential equations have been obtained in some recent literature
From the proof of Lemma 5 and 6, we can see that singularity difficulty in proving operator T to be completely continuous is overcome by the special construction of cone P and the integral condition (Hypothesis 2) imposed on the nonlinear terms
We deal with the existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems with Riemann–Stieltjes coupled integral boundary value conditions

Summary

Introduction

Differential equations of fractional order have received more and more attention in virtue of their various applications in the fields of science and engineering. Asif et al [24] considered a integral (second)-order differential system with coupled boundary value conditions They found the existence of positive solutions by Krasnoselskii’s fixed point theorem with the singularity only at t = 0 and t = 1. Y. Cui et al [25] studied a class of second-order differential equations involving coupled integral boundary value conditions, and they derived the existence and uniqueness result by a mixed monotone method. C. Yuan et al [27] obtained some results about multiple positive solutions of a nonsingular fractional differential equation with coupled boundary value conditions in view of some fixed point theorems on cone.

Preliminaries

Main Results

An Example

Conclusions