New unique existence criteria for higher-order nonlinear singular fractional differential equations

Hui Wang,Lingling Zhang,Xiaoqiang Wang

doi:10.15388/na.2019.1.6

Abstract

In this paper, a nonlinear three-point boundary value problem of higher-order singular fractional differential equations is discussed. By applying the properties of Green function and some fixed point theorems for sum-type operator on cone, some new criteria on the existence and uniqueness of solutions are obtained. Moreover, two iterative sequences are given for uniformly approximating the positive solution, which are important for practical application. At last, we give two examples to illustrate the main results.

Highlights

As we all know, fractional calculus describes many phenomena in various fields of science and engineering such as physics, biology, chemistry, control, economics, etc.; see [7, 17, 19,20,21]
While the theory of fractional differential equations has developed in a variety of directions, we notice that the research are mainly focused on existence, uniqueness, and multiplicity of positive solutions for fractional differential equations under nonlinear boundary value conditions, see, for instances, [1, 2, 5, 6, 8,9,10,11,12,13,14,15,16, 18, 24, 26,27,28] and the references given there
Note that the results dealing with the existence and uniqueness of positive solutions of multi-point boundary value problems for higher-order singular fractional differential equations are relatively scarce

Summary

Introduction

Fractional calculus describes many phenomena in various fields of science and engineering such as physics, biology, chemistry, control, economics, etc.; see [7, 17, 19,20,21]. While the theory of fractional differential equations has developed in a variety of directions, we notice that the research are mainly focused on existence, uniqueness, and multiplicity of positive solutions for fractional differential equations under nonlinear boundary value conditions, see, for instances, [1, 2, 5, 6, 8,9,10,11,12,13,14,15,16, 18, 24, 26,27,28] and the references given there These results are obtained by applying some efficient. Note that the results dealing with the existence and uniqueness of positive solutions of multi-point boundary value problems for higher-order singular fractional differential equations are relatively scarce.

Preliminaries

Main results

Examples

Conclusions