The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation

Abstract

We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative: ${}^C{D}^{\beta }(p(t{)}^C{D}^{\alpha }u(t))$ + $f(t,u(t-\tau ),u(t+\theta ))=\mathrm{0}$, $t\in (\mathrm{0,1})$, ${\hspace{0.17em}}^C{D}^{\alpha }u(\mathrm{0}){=\hspace{0.17em}}^C{D}^{\alpha }u(\mathrm{1})={({\hspace{0.17em}}^C{D}^{\alpha }u(\mathrm{0}))}^{\prime \prime }=\mathrm{0}$, $au(t)-b{u}^{\prime }(t)=\eta (t)$, $t\in [-\tau ,\mathrm{0}]$, $cu(t)+d{u}^{\prime }(t)=\xi (t)$, $t\in [\mathrm{1,1}+\theta ]$, where ${}^C{D}^{\alpha }$, ${\hspace{0.17em}}^C{D}^{\beta }$ denote the Caputo fractional derivatives, $f$ is a nonnegative continuous functional defined on $C([-\tau ,\mathrm{1}+\theta ],ℝ)$, , , , are suitably small, $a,b,c,d>\mathrm{0}$, and $\eta \in C([-\tau ,\mathrm{0}],[\mathrm{0},\mathrm{\infty }))$, $\xi \in C([\mathrm{1,1}+\theta ],[\mathrm{0},\mathrm{\infty }))$. By means of the Guo-Krasnoselskii fixed point theorem and the fixed point index theorem, some positive solutions are obtained, respectively. As an application, an example is presented to illustrate our main results.