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In this paper we consider the existence, multiplicity, and nonexistence of positive solutions to fractional differential equation with integral boundary conditions. Our analysis relies on the fixed point index.

Motivated by Vityuk and Golushkov (2004), using the Schauder Fixed Point Theorem and the Contraction Principle, we consider existence and uniqueness of positive solution of a singular partial fractional differential equation in a Banach space concerning with fractional derivative.

Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance $$x^{\left(n\right)}\left(t\right)=f(t,x\left(t\right),x^{\text{'}}\left(t\right),\cdots ,x^{(n-1)}\left(t\right)),t\in (0,1),x\left(0\right)=\sum _{i=1}^m{\alpha }_{i}x\left({\xi }_i\right),x^{\text{'}}\left(0\right)=\cdots =x^{(n-2)}\left(0\right)=0,x^{(n-1)}\left(1\right)=\sum _{j=1}^l{\beta }_j{x}^{(n-1)}\left({\eta }_j\right),$$ where $f:[0,1]\times {ℝ}^n\to ℝ$ is a Carathéodory function, $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_m<1$, ${\alpha }_i\in ℝ$, $i=1,2,\cdots ,m$, $m\ge 2$ and $0<{\eta }_1<\cdots <{\eta }_l<1$, ${\beta }_j\in ℝ$, $j=1,\cdots ,l$, $l\ge 1$. In this paper, two of the boundary value conditions are responsible for resonance.