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2013 The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation
Yanan Li, Shurong Sun, Zhenlai Han, Hongling Lu
Abstr. Appl. Anal. 2013(SI05): 1-20 (2013). DOI: 10.1155/2013/301560

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.

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Yanan Li. Shurong Sun. Zhenlai Han. Hongling Lu. "The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation." Abstr. Appl. Anal. 2013 (SI05) 1 - 20, 2013. https://doi.org/10.1155/2013/301560

Information

Published: 2013
First available in Project Euclid: 26 February 2014

zbMATH: 1295.34086
MathSciNet: MR3108476
Digital Object Identifier: 10.1155/2013/301560

Rights: Copyright © 2013 Hindawi

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Vol.2013 • No. SI05 • 2013
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