Abstract and Applied Analysis
Volume 2012 (2012), Article ID 963105, 13 pages
http://dx.doi.org/10.1155/2012/963105
Research Article

Existence of Three Solutions for a Nonlinear Fractional Boundary Value Problem via a Critical Points Theorem

Chuanzhi Bai

Department of Mathematics, Huaiyin Normal University, Jiangsu, Huaian 223300, China

Received 13 May 2012; Revised 8 July 2012; Accepted 9 July 2012

Academic Editor: Bashir Ahmad

Copyright © 2012 Chuanzhi Bai. 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

This paper is concerned with the existence of three solutions to a nonlinear fractional boundary value problem as follows: ( 𝑑 / 𝑑 𝑡 ) ( ( 1 / 2 ) 0 𝐷 𝑡 𝛼 − 1 ( 𝐶 0 𝐷 𝛼 𝑡 𝑢 ( 𝑡 ) ) − ( 1 / 2 ) 𝑡 𝐷 𝑇 𝛼 − 1 ( 𝐶 𝑡 𝐷 𝛼 𝑇 𝑢 ( 𝑡 ) ) ) + 𝜆 𝑎 ( 𝑡 ) 𝑓 ( 𝑢 ( 𝑡 ) ) = 0 , a . e . 𝑡 ∈ [ 0 , 𝑇 ] , 𝑢 ( 0 ) = 𝑢 ( 𝑇 ) = 0 , where 𝛼 ∈ ( 1 / 2 , 1 ] , and 𝜆 is a positive real parameter. The approach is based on a critical-points theorem established by G. Bonanno.