On positive solutions for a m-point fractional boundary value problem on an infinite interval

Abstract

In this paper, by using a recent fixed point theorem, we study the existence and uniqueness of positive solutions for the following m-point fractional boundary value problem on an infinite interval

$$\begin{aligned} \left\{ \begin{array}{ll} D_{0^{+}}^{\alpha }x(t)+f(t,x(t))=0,&{}\quad 0<t<\infty ,\\ x(0)=x'(0)=0,&{}\quad D_{0^{+}}^{\alpha -1}x(+\infty )= \sum _{i=1}^{m-2}\beta _{i}x(\xi _{i}), \end{array} \right. \end{aligned}$$

where $2<\alpha <3$, $D_{0^{+}}^{\alpha }$ is the standard Riemann-Liouville fractional derivative,

$$\begin{aligned}&D_{0^{+}}^{\alpha -1}x(+\infty )=\displaystyle \lim _{t\rightarrow \infty }D_{0^{+}}^{\alpha -1}x(t),\\&0<\xi _{1}<\xi _{2}<\cdots<\xi _{m-2}<\infty \quad \text {and} \quad \beta _{i}\ge 0 \quad for \quad i=1,2,\ldots ,m-2. \end{aligned}$$

Moreover, we present an example illustrating our results.