Positive Solutions for Multipoint Boundary Value Problem of Fractional Differential Equations

Abstract

We study the existence and multiplicity of positive solutions for the fractional $m$-point boundary value problem $D_{0+}^{\alpha }u\left(t\right)+f(t,u(t\left)\right)=0$, $0<t<1$, $u\left(0\right)=u\text{'}\left(0\right)=0$, $u\text{'}\left(1\right)={{\displaystyle \sum }}_{i=1}^{m-2}{a}_{i}u\text{'}\left({\xi }_i\right)$, where $2<\alpha <3$, $D_{0+}^{\alpha }$ is the standard Riemann-Liouville fractional derivative, and $f:[0,1]\times [0,\infty )\mapsto [0,\infty )$ is continuous. Here, $a_i\ge 0$ for $i=1,\dots ,m-2$, $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<1$, and $\rho ={{\displaystyle \sum }}_{i=1}^{m-2}{a}_i{\xi }_i^{\alpha -2}$ with $\rho <1$. In light of some fixed point theorems, some existence and multiplicity results of positive solutions are obtained.