Summary: We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative \[ \begin{gathered} {^CD^\beta}(p(t)^\beta(p(t)^C D^\alpha u(t))+ f(t,u(t-\tau), u(t+\beta))= 0,\;t\in (0,1),\\ {^CD^\alpha} u(1)= ({^CD^\alpha} u(0))''= 0,\\ au(t)- bu'(t)= \eta(t),\;t\in [-\tau,0]\;cu(t)+ du'(t)= \xi(t),\;t\in [1,1+\theta],\end{gathered} \] where \({^CD^\alpha}\), \({^CD^\beta}\) denote the Caputo fractional derivatives, \(f\) is a nonnegative continuous functional defined on \(C([-\tau,1+\theta],\mathbb{R})\), \(1<\alpha\leq 2\), \(2< \beta\leq 3\), \(0<\tau\), \(\theta< 1/4\) are suitably small, \(a,b,c,d>0\), and \(\eta\in C([-\tau,0], [0,\infty))\), \(\xi\in C([1,1+\theta], [0,\infty))\). 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.