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:  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, CDβdenote the Caputo fractional derivatives,fis a nonnegative continuous functional defined onC([-τ,1+θ],ℝ),1<α≤2,2<β≤3,0<τ,θ<1/4are suitably small,a,b,c,d>0, andη∈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.