Multiple positive solutions for nonlocal boundary value problems of singular fractional differential equations

Abstract

We consider the existence of multiple positive solutions for the following nonlinear fractional differential equations of nonlocal boundary value problems: $$ \left \{ \textstyle\begin{array}{l} D_{0{+}}^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t< 1, u(0)=0,\qquad D_{0{+}}^{\beta}u(0)=0,\qquad D_{0{+}}^{\beta}u(1)=\sum_{i=1}^{\infty} \xi_{i} D_{0{+}}^{\beta}u(\eta_{i}), \end{array}\displaystyle \right . $$ where $2<\alpha\leq3$ , $1\leq\beta\leq2$ , $\alpha-\beta\geq1$ , $0<\xi_{i}, \eta_{i}<1$ with $\sum_{i=1}^{\infty} \xi_{i}\eta_{i}^{\alpha -\beta-1}<1$ . Existence result of at least two positive solutions is given via fixed point theorem on cones. The nonlinearity f may be singular both on the time and the space variables.