Existence of multiple positive solutions for $m$-point fractional boundary value problems on an infinite interval

Abstract

In this paper we consider the following $m$-point fractional boundary value problem on infinite interval

$$D_{0+}^{\alpha }u\left(t\right)+a\left(t\right)f(t,u\left(t\right))=0\text{,}0<t<+\infty \text{,}$$$$u\left(0\right)=u^{\prime }\left(0\right)=0\text{,}{D}^{\alpha -1}u(+\infty )=\sum _{i=1}^{m-2}{\beta }_{i}u\left({\xi }_i\right)\text{,}$$

where $2<\alpha <3$, $D_{0+}^{\alpha }$ is the standard Riemann–Liouville fractional derivative, $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<+\infty$, ${\beta }_i\ge 0$, $i=1,2,\dots ,m-2$ satisfies $0<{\sum }_{i=1}^{m-2}{\beta }_i{\xi }_i^{\alpha -1}<\Gamma \left(\alpha \right)$. Using a fixed point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem. As applications, examples are presented to illustrate the main results.