Existence of positive solutions for a singular nonlinear fractional differential equation with integral boundary conditions involving fractional derivatives

Xiaoqian Liu,Yonghong Wu,Lishan Liu

doi:10.1186/s13661-018-0943-9

Xiaoqian Liu, Yonghong Wu + Show 1 more

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https://doi.org/10.1186/s13661-018-0943-9

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Journal: Boundary Value Problems	Publication Date: Feb 23, 2018
Citations: 34	License type: open-access

Affiliation: Qufu Normal University, Curtin University

Abstract

In this article, by using the spectral analysis of the relevant linear operator and Gelfand’s formula, some properties of the first eigenvalue of a fractional differential equation are obtained. Based on these properties and through the fixed point index theory, the singular nonlinear fractional differential equations with Riemann–Stieltjes integral boundary conditions involving fractional derivatives are considered under some appropriate conditions, and the nonlinearity is allowed to be singular in regard to not only time variable but also space variable and it includes fractional derivatives. The existence of positive solutions for boundary conditions involving fractional derivatives is established. Finally, an example is given to demonstrate the validity of our main results.

Highlights

Fractional differential equations have drawn more and more attention of the research community due to their numerous applications in various fields of science such as engineering, chemistry, physics, mechanics, etc. [1,2,3,4]
We consider the existence of positive solutions for the following integral boundary value problems of singular nonlinear fractional differential equations:
Zhang et al [15] studied the existence of positive solutions of the following singular nonlinear fractional differential equation with integral boundary value conditions:

Summary

Introduction

Fractional differential equations have drawn more and more attention of the research community due to their numerous applications in various fields of science such as engineering, chemistry, physics, mechanics, etc. [1,2,3,4]. We consider the existence of positive solutions for the following integral boundary value problems of singular nonlinear fractional differential equations: Zhang et al [15] studied the existence of positive solutions of the following singular nonlinear fractional differential equation with integral boundary value conditions: Through the spectral analysis and fixed point index theorem, the author obtained the existence of positive solutions.

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