The Existence of Positive Solution to Three-Point Singular Boundary Value Problem of Fractional Differential Equation

Abstract

We investigate the existence of positive solution to nonlinear fractional differential equation three-point singular boundary value problem: $D^{q}u\left(t\right)+f\left(t,u\left(t\right)\right)=0$, $0<t<1$, $u\left(0\right)=0$, $u\left(1\right)=\alpha D^{(q-1)/2}u\left(t\right){|}_{t=\xi }$, where $1<q\le 2$ is a real number, $\xi \in \left(0,1/2\right]$, $\alpha \in \left(0,+\infty \right)$ and $\alpha \Gamma \left(q\right){\xi }^{\left(q-1\right)/2}<\Gamma \left((q+1)/2\right),D^q$ is the standard Riemann-Liouville fractional derivative, and $f\in C\left(\left(0,1\right]\times \left[0,+\infty \right),\left[0,+\infty \right)\right),{\lim }_{t\to +0}f\left(t,\cdot \right)=+\infty$ (i.e., $f$ is singular at $t=0$). By using the fixed-point index theory, the existence result of positive solutions is obtained.