Abstract and Applied Analysis

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

Yanan Li, Shurong Sun, Zhenlai Han, and Hongling Lu

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Abstract

We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative:    C D β ( p ( t ) C D α u ( t ) ) + f ( t , u ( t - τ ) , u ( t + θ ) ) = 0 , t ( 0,1 ) , C D α u ( 0 ) = C D α u ( 1 ) = ( C D α u ( 0 ) ) = 0 , a u ( t ) - b u ( t ) = η ( t ) , t [ - τ , 0 ] , c u ( t ) + d u ( t ) = ξ ( t ) , t [ 1,1 + θ ] , where    C D α , C D β denote the Caputo fractional derivatives, f is a nonnegative continuous functional defined on C ( [ - τ , 1 + θ ] , ) , 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.

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

Permanent link to this document
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


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