Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation

Abstract

We are concerned with the existence and nonexistence of positive solutions for the nonlinear fractional boundary value problem: $D_{0+}^{\alpha }u\left(t\right)+\lambda a\left(t\right)\mathrm{\hspace{0.17em}f}\left(u\left(t\right)\right)=0,\hspace{0.17em}$ $0<t<1,$ $u\left(0\right)=u^{\prime }\left(0\right)=u^{\prime }\left(1\right)=0,$ where $2<\alpha <3$ is a real number and $D_{0+}^{\alpha }$ is the standard Riemann-Liouville fractional derivative. Our analysis relies on Krasnoselskiis fixed point theorem of cone preserving operators. An example is also given to illustrate the main results.