Positive properties of Green’s function for focal-type BVPs of singular nonlinear fractional differential equations and its application

Manli Jin,Yuguo Lin

doi:10.1186/1687-1847-2013-159

Abstract

In this paper, we consider the properties of Green’s function for the singular nonlinear fractional differential equation boundary value problem

Highlights

1 Introduction Fractional differential equations have been of great interest recently
Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in selfsimilar and porous structures, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science
If we assume that u ∈ C(, ) ∩ L(, ), the fractional differential equation

Summary

Introduction

Fractional differential equations have been of great interest recently. This is due to the intensive development of the theory of fractional calculus itself as well as its applications. Jiang and Yuan [ ] considered the nonlinear fractional differential equation Dirichlet-type boundary value problem and established the existence of positive solutions for the corresponding BVP. Some properties of Green’s function for the above BVP were obtained. We present Green’s function of the fractional differential equation boundary value problem For fixed s ∈ ( , ), let z(t) = (α)G(t, s) – (α – )sα– ( – s)α– tα– , t ∈ [s, ]. (α – )sα– ( – s)α– tα– ≤ (α)G(t, s) ≤ sα– ( – s)α– , s ≤ t ). As an application of properties of Green’s function, we will establish the existence of positive solutions for BVP BVP ( . ), ( . ) has at least one positive solution u with < u < r

For fixed n

Since a