Positive Solutions for Boundary Value Problems of Singular Fractional Differential Equations

Abstract

In this paper, by using a fixed point theorem, we investigate the existence of a positive solution to the singular fractional boundary value problem ${D_{C}}_{0+}^{\alpha }u+f(t,u,{D_{C}}_{0+}^{\nu }u,{D_{C}}_{0+}^{\mu }u)+g(t,u,{D_{C}}_{0+}^{\nu }u,{D_{C}}_{0+}^{\mu }u)=0$, $u(0)={u}^{\prime }(0)={u}^{\prime \prime }(0)={u}^{\prime \prime \prime }(0)=0$, where $3<\alpha <4$, $0<\nu <1$, $1<\mu <2$, ${D_{C}}_{0+}^{\alpha }$ is Caputo fractional derivative, $f(t,x,y,z)$ is singular at the value 0 of its arguments $x,y,z$, and $g(t,x,y,z)$ satisfies the Lipschitz condition.