Existence and uniqueness of solutions for a singular system of higher-order nonlinear fractional differential equations with integral boundary conditions

Xinyi Lu,Xingqiu Zhang,Lin Wang

doi:10.15388/na.18.4.13975

Abstract

In this paper, we study the existence and uniqueness of solutions for a singular system of nonlinear fractional differential equations with integral boundary conditions. We obtain existence and uniqueness results of solutions by using the properties of the Green’s function, a nonlinear alternative of Leray–Schauder-type, Guo–Krasnoselskii’s fixed point theorem in a cone and the Banach fixed point theorem. Some examples are included to show the applicability of our results.

Highlights

Fractional differential equations have been of great interest recently
There are many papers deal with the existence and multiplicity of solution of nonlinear fractional differential equations
Inspired by the work of the above papers and many known results, in this paper, we study the existence of positive solutions of boundary value problems (BVP) (1)

Summary

Introduction

Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various fields of sciences and engineering such as control, porous media, electromagnetic, and other fields. The paper [1] considered the existence of positive solutions of singular coupled system Dsu = f (t, v), 0 < t < 1, Dpv = g(t, u), 0 < t < 1, where 0 < s, p < 1, and f, g : (0, 1] × [0, +∞) → [0, +∞) are two given continuous functions, limt→0+ f (t, ·) = +∞, limt→0+ g(t, ·) = +∞ and Ds, Dp are the standard fractional Riemann–Liouville’s derivatives. Existence and uniqueness of solutions for singular system with integral boundary conditions results of solutions are obtained by a nonlinear alternative of Leray–Schauder-type, Guo– Krasnoselskii’s fixed point theorem in a cone and the Banach fixed point theorem. The existence results of positive solutions are obtained by a nonlinear alternative of Leray–Schauder-type, Guo–Krasnoselskii’s fixed point theorem in a cone and the Banach fixed point theorem. We construct some examples to demonstrate the application of our main results

Background materials and Green’s function

Main results and proof