Twin Rotor System
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 512127, 16 pages doi:10.1155/2012/512127 Research Article
Positive Solutions of Eigenvalue
Problems for a Class of Fractional Differential
Equations with Derivatives
Xinguang Zhang,1 Lishan Liu,2
Benchawan Wiwatanapataphee,3 and Yonghong Wu4
1
School of Mathematical and Informational Sciences, Yantai University, Yantai, Shandong 264005, China
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
3
Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
4
Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia
2
Correspondence should be addressed to Xinguang Zhang, zxg123242@sohu.com and Benchawan Wiwatanapataphee, scbww@mahidol.ac.th
Received 2 January 2012; Accepted 15 March 2012
Academic Editor: Shaoyong Lai
Copyright q 2012 Xinguang Zhang 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.
By establishing a maximal principle and constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of a class of fractional differential equations is discussed. Some sufficient conditions for the existence of positive solutions are established.
1. Introduction
In this paper, we discuss the existence of positive solutions for the following eigenvalue problem of a class fractional differential equation with derivatives
−Dt α x t
λf t, x t , Dt β x t ,
β
γ
t ∈ 0, 1 ,
1.1
p −2
Dt x 0
0,
γ
Dt x 1
aj Dt x ξj , j1 where λ is a parameter, 1 < α ≤ 2, α − β > 1, 0 < β ≤ γ < 1, 0 < ξ1 < ξ2 < · · · < ξp−2 < 1, aj ∈ 0, ∞ with c
p −2 j1 α−γ −1
aj ξj
< 1, and Dt is the standard Riemann-Liouville derivative.
2
Abstract and Applied Analysis
f : 0, 1 × 0, ∞ × 0, ∞ → 0, ∞
Abstract and Applied Analysis
Volume 2012, Article ID 512127, 16 pages doi:10.1155/2012/512127 Research Article
Positive Solutions of Eigenvalue
Problems for a Class of Fractional Differential
Equations with Derivatives
Xinguang Zhang,1 Lishan Liu,2
Benchawan Wiwatanapataphee,3 and Yonghong Wu4
1
School of Mathematical and Informational Sciences, Yantai University, Yantai, Shandong 264005, China
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
3
Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
4
Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia
2
Correspondence should be addressed to Xinguang Zhang, zxg123242@sohu.com and Benchawan Wiwatanapataphee, scbww@mahidol.ac.th
Received 2 January 2012; Accepted 15 March 2012
Academic Editor: Shaoyong Lai
Copyright q 2012 Xinguang Zhang 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.
By establishing a maximal principle and constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of a class of fractional differential equations is discussed. Some sufficient conditions for the existence of positive solutions are established.
1. Introduction
In this paper, we discuss the existence of positive solutions for the following eigenvalue problem of a class fractional differential equation with derivatives
−Dt α x t
λf t, x t , Dt β x t ,
β
γ
t ∈ 0, 1 ,
1.1
p −2
Dt x 0
0,
γ
Dt x 1
aj Dt x ξj , j1 where λ is a parameter, 1 < α ≤ 2, α − β > 1, 0 < β ≤ γ < 1, 0 < ξ1 < ξ2 < · · · < ξp−2 < 1, aj ∈ 0, ∞ with c
p −2 j1 α−γ −1
aj ξj
< 1, and Dt is the standard Riemann-Liouville derivative.
2
Abstract and Applied Analysis
f : 0, 1 × 0, ∞ × 0, ∞ → 0, ∞
References: 1 I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. 2 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. 3 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. Netherlands, 2006. 1038–1044, 2010. Theory, Methods & Applications A, vol. 72, no. 2, pp. 710–719, 2010. no. 17, pp. 8526–8536, 2012.