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We consider the existence of multiple positive solutions for the following nonlinear fractional differential equations of nonlocal boundary value problems: {D0+αu(t)+f(t,u(t))=0,0<t<1,u(0)=0,D0+βu(0)=0,D0+βu(1)=∑i=1∞ξiD0+βu(ηi),$$ \left \{ \textstyle\begin{array}{l} D_{0{+}}^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t< 1, \\ u(0)=0,\qquad D_{0{+}}^{\beta}u(0)=0,\qquad D_{0{+}}^{\beta}u(1)=\sum_{i=1}^{\infty} \xi_{i} D_{0{+}}^{\beta}u(\eta_{i}), \end{array}\displaystyle \right . $$ where 2<α≤3$2<\alpha\leq3$, 1≤β≤2$1\leq\beta\leq2$, α−β≥1$\alpha-\beta\geq1$, 0<ξi,ηi<1$0<\xi_{i}, \eta_{i}<1$ with ∑i=1∞ξiηiα−β−1<1$\sum_{i=1}^{\infty} \xi_{i}\eta_{i}^{\alpha -\beta-1}<1$. 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.

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