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

We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative:   CDβ(p(t)CDαu(t)) + f(t,u(t-τ),u(t+θ))=0, t∈(0,1),  CDαu(0)= CDαu(1)=( CDαu(0))=0, au(t)-bu′(t)=η(t), t∈[-τ,0], cu(t)+du′(t)=ξ(t), t∈[1,1+θ], where   CDα,  CDβ denote the Caputo fractional derivatives, f is a nonnegative continuous functional defined on C([-τ,1+θ],ℝ), 1<α≤2, 2<β≤3, 0<τ, θ<1/4 are suitably small, a,b,c,d>0, and η∈C([-τ,0],[0,∞)), ξ∈C([1,1+θ],[0,∞)). 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