Positive properties of Green's function for three-point boundary value problems of nonlinear fractional differential equations and its applications

Abstract

In this article, we consider the properties of Green's function for the nonlinear fractional differential equation boundary value problem

where 1 < α < 2, 0 < β, η < 1, is the standard Riemann–Liouville derivative. Here our nonlinearity f may be singular at u = 0. As an application of Green's function, we establish some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray–Schauder nonlinear alternative, a fixed-point theorem on cones, and also we give uniqueness of a solution for a singular problem by a mixed monotone method.

Keywords:

• multiple positive solutions
• singular fractional differential equation
• semipositone
• three-point boundary value problem

AMS Subject Classification:

• 34B15

Acknowledgements

This study was supported by NSFC of China (No. 10971021), the Ministry of Education of China (No. 109051) the Ph.D. Programs Foundation of Ministry of China (No. 200918).