## Abstract and Applied Analysis

### 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: ${\mathrm{ }}^{C}{D}^{\beta }(p(t{)}^{C}{D}^{\alpha }u(t))$ + $f(t,u(t-\tau ),u(t+\theta ))=\mathrm{0}$, $t\in (\mathrm{0,1})$, ${ }^{C}{D}^{\alpha }u(\mathrm{0}){= }^{C}{D}^{\alpha }u(\mathrm{1})={({ }^{C}{D}^{\alpha }u(\mathrm{0}))}^{\prime \prime }=\mathrm{0}$, $au(t)-b{u}^{\prime }(t)=\eta (t)$, $t\in [-\tau ,\mathrm{0}]$, $cu(t)+d{u}^{\prime }(t)=\xi (t)$, $t\in [\mathrm{1,1}+\theta ]$, where ${\mathrm{ }}^{C}{D}^{\alpha }$, ${ }^{C}{D}^{\beta }$ denote the Caputo fractional derivatives, $f$ is a nonnegative continuous functional defined on $C([-\tau ,\mathrm{1}+\theta ],\Bbb R)$, $\mathrm{1}<\alpha \le \mathrm{2}$, $\mathrm{2}<\beta \le \mathrm{3}$, $\mathrm{0}<\tau$, $\theta <\mathrm{1}/\mathrm{4}$ 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 301560, 20 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393450394

Digital Object Identifier
doi:10.1155/2013/301560

Mathematical Reviews number (MathSciNet)
MR3108476

Zentralblatt MATH identifier
1295.34086

#### Citation

Li, Yanan; Sun, Shurong; Han, Zhenlai; Lu, Hongling. The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 301560, 20 pages. doi:10.1155/2013/301560. https://projecteuclid.org/euclid.aaa/1393450394