Existence and Uniqueness of Positive Solution for a Boundary Value Problem of Fractional Order

Abstract

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear fractional boundary value problem: $D_{0+}^{\alpha }u\left(t\right)+f(t,u(t\left)\right)=0$ , $0\le t\le 1$, $3<\alpha \le 4$, $u\left(0\right)=u^{\prime }\left(0\right)=u^{?}\left(0\right)=u^{?}\left(1\right)=0$, where $D_{0+}^{\alpha }$ denotes the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also given to illustrate the results.