Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions

Limin Guo,Yonghong Wu,Lishan Liu

doi:10.15388/na.2016.5.5

Abstract

In this paper, we investigate the existence of at least three positive solutions to a singular boundary value problem of Caputo's fractional differential equations with a boundary condition involving values at infinite number of points. Firstly, we establish Green's function and its properties. Then, the existence of multiple positive solutions is obtained by Avery–Peterson's fixed point theorem. Finally, an example is given to demonstrate the application of our main results.

Highlights

In this paper, we consider the following infinite-point fractional differential equations boundary value problem: cD0α+u(t) + f t, u(t), u (t) = 0, 0 < t < 1, ∞ (1)u(0) = u (0) = 0, u (1) = ηju(ξj), j=1c Vilnius University, 2016where 2 < α 3, ηj 0, 0 < ξ1 < ξ2 < · · · < ξj−1 < ξj < · · · < 1 (j =1, 2 . . . ), 0 < ∆ = 1 − ∞ j=1 ηj ξj, f (t, x, y) may be singular at t = 0, and cD0α+
In [9], the authors investigated the existence of multiple positive solutions of the following fractional differential equation boundary value problem: cD0α+u(t) + f t, u(t), u (t), . . . , u(i)(t) = 0, 0 < t < 1, u(0) = u (0) = · · · = u(i−1)(0) = u(i+1)(0) = · · · = u(n−1)(0) = 0, u(i)(1) = 0, http://www.mii.lt/NA
Motivated by the results above, in this paper, we investigate the existence of positive solutions for a class of singular fractional differential equations subject to infinite-point boundary conditions

Summary

Introduction

We consider the following infinite-point fractional differential equations boundary value problem: cD0α+u(t) + f t, u(t), u (t) = 0, 0 < t < 1,. In [9], the authors investigated the existence of multiple positive solutions of the following fractional differential equation boundary value problem: cD0α+u(t) + f t, u(t), u (t), . The authors obtained the existence result of at least three positive solutions for a two-point boundary value problem, in which the nonlinear terms contain derivatives up to order i by using Avery–Peterson’s fixed point theorem. Motivated by the results above, in this paper, we investigate the existence of positive solutions for a class of singular fractional differential equations subject to infinite-point boundary conditions. The main tool used in this paper is Avery–Peterson’s fixed point theorem

Preliminaries and lemmas

Main result

An example