Some existence, uniqueness results on positive solutions for a fractional differential equation with infinite-point boundary conditions

Chengbo Zhai,Li Wang

doi:10.15388/na.2017.4.10

Abstract

We investigate a class of Riemann–Liouville's fractional differential equation with infinite-point boundary conditions. We give some new properties of the Green's function associated with the fractional differential equation boundary value problem. Based upon these new properties and by using Schauder's fixed point theorem, we establish some existence results on positive solutions for the boundary value problem. Further, by using a fixed point theorem of general concave operators, we also present an existence and uniqueness result on positive solutions for the boundary value problem.

Highlights

There are several researchers who studied fractional differential equation with infinite-point boundary conditions, see [2, 4,5,6,7, 19, 21] and some references therein
A function u ∈ C[0, 1] is said to be a positive solution of problem (1) if u(t) > 0 on (0, 1) and u satisfies (1) on [0, 1]
We use a fixed point theorem of general concave operators to obtain an existence and uniqueness result on positive solutions for problem (1)

Summary

Introduction

There are several researchers who studied fractional differential equation with infinite-point boundary conditions, see [2, 4,5,6,7, 19, 21] and some references therein. A function u ∈ C[0, 1] is said to be a positive solution of problem (1) if u(t) > 0 on (0, 1) and u satisfies (1) on [0, 1]. We will give some new properties of the Green’s function G(t, s) and use these new properties to study the existence, uniqueness of positive solutions for problem (1). By using Schauder’s fixed point theorem, we first establish some existence results on positive solutions for problem (1).

Results

Conclusion