The Existence of Solutions for a Fractional 2 m -Point Boundary Value Problems

Abstract

By using the coincidence degree theory, we consider the following 2$m$-point boundary value problem for fractional differential equation $D_{0+}^{\alpha }u\left(t\right)=f\left(t,u\left(t\right),D_{0+}^{\alpha -1}u\left(t\right),D_{0+}^{\alpha -2}u\left(t\right)\right)+e\left(t\right)$, , ${I_{0+}^{3-\alpha }u\left(t\right)|}_{t=0}=0,D_{0+}^{\alpha -2}u\left(1\right)={\sum }_{i=1}^{m-2}{a}_i{D}_{0+}^{\alpha -2}u\left({\xi }_i\right),u\left(1\right)={\sum }_{i=1}^{m-2}{b}_{i}u\left({\eta }_i\right),$ where $D_{0+}^{\alpha }$ and $I_{0+}^{\alpha }$ are the standard Riemann-Liouville fractional derivative and fractional integral, respectively. A new result on the existence of solutions for above fractional boundary value problem is obtained.