Abstract and Applied Analysis
Volume 2012 (2012), Article ID 797398, 16 pages
http://dx.doi.org/10.1155/2012/797398
Research Article

Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives

Tunhua Wu,1 Xinguang Zhang,2 and Yinan Lu3

1School of Information and Engineering, Wenzhou Medical College, Zhejiang, Wenzhou 325035, China
2School of Mathematical and Informational Sciences, Yantai University, Shandong, Yantai 264005, China
3Information Engineering Department, Anhui Xinhua University, Anhui, Hefei 230031, China

Received 24 April 2012; Accepted 29 May 2012

Academic Editor: Yonghong Wu

Copyright © 2012 Tunhua Wu et al. 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 singular fractional-order boundary-value problem with a sign-changing nonlinear term − 𝒟 𝛼 𝑥 ( 𝑡 ) = 𝑝 ( 𝑡 ) 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) , 𝒟 𝜇 1 𝑥 ( 𝑡 ) , 𝒟 𝜇 2 𝑥 ( 𝑡 ) , … , 𝒟 𝜇 𝑛 − 1 𝑥 ( 𝑡 ) ) , 0 < 𝑡 < 1 , 𝒟 𝜇 𝑖 𝑥 ( 0 ) = 0 , 1 ≤ 𝑖 ≤ 𝑛 − 1 , 𝒟 𝜇 𝑛 − 1 + 1 𝑥 ( 0 ) = 0 , 𝒟 𝜇 𝑛 − 1 ∑ 𝑥 ( 1 ) = 𝑝 − 2 𝑗 = 1 𝑎 𝑗 𝒟 𝜇 𝑛 − 1 𝑥 ( 𝜉 𝑗 ) , where 𝑛 − 1 < 𝛼 ≤ 𝑛 , 𝑛 ∈ ℕ and 𝑛 ≥ 3 with 0 < 𝜇 1 < 𝜇 2 < ⋯ < 𝜇 𝑛 − 2 < 𝜇 𝑛 − 1 and 𝑛 − 3 < 𝜇 𝑛 − 1 < 𝛼 − 2 , 𝑎 𝑗 ∈ ℝ , 0 < 𝜉 1 < 𝜉 2 < ⋯ < 𝜉 𝑝 − 2 < 1 satisfying ∑ 0 < 𝑝 − 2 𝑗 = 1 𝑎 𝑗 𝜉 𝛼 − 𝜇 𝑛 − 1 𝑗 − 1 < 1 , 𝒟 𝛼 is the standard Riemann-Liouville derivative, 𝑓 ∶ [ 0 , 1 ] × ℝ 𝑛 → ℝ is a sign-changing continuous function and may be unbounded from below with respect to 𝑥 𝑖 , and 𝑝 ∶ ( 0 , 1 ) → [ 0 , ∞ ) is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field.