Abstract and Applied Analysis
Volume 2012 (2012), Article ID 648635, 21 pages
http://dx.doi.org/10.1155/2012/648635
Research Article

Existence and Multiplicity of Solutions for Some Fractional Boundary Value Problem via Critical Point Theory

Jing Chen1,2 and X. H. Tang1

1School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China
2School of Mathematics and Computing Sciences, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China

Received 18 October 2011; Accepted 27 November 2011

Academic Editor: Kanishka Perera

Copyright © 2012 Jing Chen and X. H. Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the existence and multiplicity of solutions for the following fractional boundary value problem: ( 𝑑 / 𝑑 𝑡 ) ( ( 1 / 2 ) 0 𝐷 𝑡 − 𝛽 ( 𝑢 ′ ( 𝑡 ) ) + ( 1 / 2 ) 𝑡 𝐷 𝑇 − 𝛽 ( 𝑢 ′ ( 𝑡 ) ) ) + ∇ 𝐹 ( 𝑡 , 𝑢 ( 𝑡 ) ) = 0 , a . e . 𝑡 ∈ [ 0 , 𝑇 ] , 𝑢 ( 0 ) = 𝑢 ( 𝑇 ) = 0 , where 𝐹 ( 𝑡 , ⋅ ) are superquadratic, asymptotically quadratic, and subquadratic, respectively. Several examples are presented to illustrate our results.