The positive solutions of infinite-point boundary value problem of fractional differential equations on the infinite interval

Xiaochen Li,Luchao Zhang,Xiping Liu,Mei Jia

doi:10.1186/s13662-017-1185-3

Abstract

In this paper, we consider a class of infinite-point boundary value problems of fractional differential equations on the infinite interval [0,+infty) with a disturbance parameter. By using the method of upper and lower solutions, fixed point index theory and some fixed point theorems, the existence, multiplicity and nonexistence for the positive solution of the boundary value problem are obtained, respectively. The impact of the disturbance parameters on the existence of positive solutions is also given. Finally, some examples are presented to illustrate the wide range of potential applications of our main results.

Highlights

We are concerned with a class of infinite-point boundary value problems of fractional differential equations on the infinite interval with a disturbance parameter λ as follows:
The purpose of this paper is to investigate the existence of positive solutions for the infinite-point boundary value problem of fractional differential equations on the half-line
If u(t) ≥, t ∈ R+, we say that u is a positive solution of boundary value problem ( . )

Summary

Introduction

We are concerned with a class of infinite-point boundary value problems of fractional differential equations on the infinite interval with a disturbance parameter λ as follows:. In [ ], the authors studied the existence and nonexistence of the positive solutions for the fractional differential equation with two disturbance parameters. The authors studied the impact of the disturbance parameters a and b on the existence of positive solutions. Zhang studied the following m-point boundary value problem of fractional differential equations on the infinite interval:. By using the Leggett-Williams fixed point theorem, the existence of three positive solutions for the boundary value problem on the infinite interval was obtained. In [ ], authors investigated the integral boundary value problem of fractional differential equations on infinite intervals with two disturbance parameters. The purpose of this paper is to investigate the existence of positive solutions for the infinite-point boundary value problem of fractional differential equations on the half-line Some examples are presented to illustrate the main results

Preliminaries

Comparison principle

The properties of positive solutions

Examples