Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems

Abstract

We are concerned with the following nonlinear three-point fractional boundary value problem: $D_{0+}^{\alpha }u\left(t\right)+\lambda a\left(t\right)f\left(t,u\left(t\right)\right)=0$, , $u\left(0\right)=0$, and $u\left(1\right)=\beta u\left(\eta \right)$, where , , , $D_{0+}^{\alpha }$ is the standard Riemann-Liouville fractional derivative, $a\left(t\right)>0$ is continuous for $0\le t\le 1$, and $f\ge 0$ is continuous on $\left[\mathrm{0,1}\right]\times \left[0,\left.\infty \right)\right.$. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.