Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

Lily Li Liu,Lishan Liu,Xinqiu Zhang,Yonghong Wu

doi:10.1186/s13662-016-0876-5

Abstract

In this article, we study the existence of iterative positive solutions for a class of singular nonlinear fractional differential equations with Riemann-Stieltjes integral boundary conditions, where the nonlinear term may be singular both for time and space variables. By using the properties of the Green function and the fixed point theorem of mixed monotone operators in cones we obtain some results on the existence and uniqueness of positive solutions. We also construct successively some sequences for approximating the unique solution. Our results include the multipoint boundary problems and integral boundary problems as special cases, and we also extend and improve many known results including singular and non-singular cases.

Highlights

1 Introduction Fractional differential equations have attracted more and more attention in recent decades, which is partly due to their numerous applications in many branches of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, probability, etc
We apply the fixed point theorem of mixed monotone operators to get the existence and uniqueness of the iterative solutions for singular fractional differential equations ( . ) without using the above condition (b)
We present the Green function of the fractional differential equation boundary value problem

Summary

Introduction

Fractional differential equations have attracted more and more attention in recent decades, which is partly due to their numerous applications in many branches of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, probability, etc. Zhang et al [ ], by using the properties of the Green function and the fixed point index theory, considered the existence of a positive solution of the following nonlinear fractional differential equation with integral boundary conditions: Dα + x(t) + h(t)f (t, x(t)) = , < t < , x( ) = x ( ) = x ( ) = , x( ) = λ η ds, where. By cone expansion fixed point theorem, Li et al [ ], obtained the positive solutions of the following class of singular fractional differential equations:. We apply the fixed point theorem of mixed monotone operators to get the existence and uniqueness of the iterative solutions for singular fractional differential equations In Section , we give an example to demonstrate the application of our theoretical results

Preliminaries and lemmas

An example

Methods