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The existence of positive solutions for boundary value problem of the fractional Sturm-Liouville functional differential equation. (English) Zbl 1295.34086
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.

##### MSC:
 34K37 Functional-differential equations with fractional derivatives 34K10 Boundary value problems for functional-differential equations
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