Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order

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Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order Xuhuan Wang · Xiuqing Guo · Guosheng Tang

Received: 20 September 2012 / Published online: 6 November 2012 © Korean Society for Computational and Applied Mathematics 2012

Abstract In this paper, the existence of solutions of an anti-periodic fractional boundary value problem for nonlinear fractional differential equations is investigated. The contraction mapping principle and Leray-Schauder’s fixed point theorem are applied to establish the results. Keywords Fractional differential equations · Anti-periodic · Fractional boundary value problems · Fixed point Mathematics Subject Classification (2000) 34A60 · 34A12 · 34A40 1 Introduction Fractional calculus is more than 300 years old, but it did not attract enough interest at the early stage of development. In the last three decades, fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electro-dynamics of complex medium, polymer rheology, etc. [1–3]. For some recent development on the topic, see [4–13] and the references therein. Project supported by NNSF of China Grant No. 11271087 and No. 61263006. X. Wang () · X. Guo Department of Mathematics, Baoshan College, Baoshan, Yunnan 678000, P.R. China e-mail: [email protected] X. Guo e-mail: [email protected] G. Tang Zhangjiagang Campus, Jiangsu University of Science and Technology, Zhangjiagang, Jiangsu 215600, P.R. China e-mail: [email protected]

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X. Wang et al.

In [14], Wang considered the following anti-periodic fractional boundary value problems   c α D x(t) = f t, x(t), D q x(t) , t ∈ [0, T ], (1) x(0) = −x(T ), c D p x(0) = −c D p x(T ), where c D α denotes the Caputo fractional derivative of order α, 1 < α ≤ 2, 0 < p, q < 1, α − q ≥ 1 and f is a given continuous function. The results are based on contraction mapping principle and Schauder’s fixed point theorem. In [15], Cernea investigate the following anti-periodic fractional boundary value problem given by      α Dc x(t) ∈ f t, x(t) , a.e. [0, T ] , (2) x(0) = −x(T ), x  (0) = −x  (T ), x  (0) = −x  (T ), where Dcα denotes the Caputo fractional derivative of order α, 2 < α ≤ 3 and F : I × R → P(R) is a set-valued map. The results are obtained by using suitable fixed point theorems. There has been a great deal of research into the questions of existence and uniqueness of solutions to anti-periodic boundary value problems for differential equations. They include first, second and higher-order differential equations with anti-periodic boundary value conditions have been considered in papers [16–24]. However, to the best of our knowledge, few paper has concerned the existence of solutions for antiperiodic boundary value problems for fractional differential equations, see [25–31] and the references therein. Motivated by [14, 15], we investigate some existence