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

Wang Fang, Liu, Liu Zhenhai

Description

By using Schauder’s fixed point theorem and the contraction mapping principle, we discuss the existence of solutions for nonlinear fractional differential equations with fractional anti-periodic boundary conditions. Some examples are given to illustrate the main results. By using Schauder’s fixed point theorem and the contraction mapping principle, we discuss the existence of solutions for nonlinear fractional differential equations with fractional anti-periodic boundary conditions. Some examples are given to illustrate the main results.

Sujets

• Fractional calculus
• Boundary value problem
• Fixed point theorem

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 4
• Langue: English

Extrait

Wang and LiuAdvances in Difference Equations2012,2012:116 http://www.advancesindifferenceequations.com/content/2012/1/116

R E S E A R C HOpen Access Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order 1,2* 3 Fang Wangand Zhenhai Liu

* Correspondence: [e-mail] 1 School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410075, P.R. China 2 School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410076, P.R. China Full list of author information is available at the end of the article

Abstract By using Schauder’s fixed point theorem and the contraction mapping principle, we discuss the existence of solutions for nonlinear fractional differential equations with fractional anti-periodic boundary conditions. Some examples are given to illustrate the main results. Keywords:fractional differential equations; boundary value problem; anti-periodic; fixed point theorem

1 Introduction Fractional calculus has been recognized as an effective modeling methodology by re- searchers. Fractional differential equations are generalizations of classical differential equations to an arbitrary order. They have broad application in engineering and sciences such as physics, mechanics, chemistry, economics and biology,etc.[–]. For some recent development on the topic, see [–] and the references therein. In [], Ahmadet al.considered the following anti-periodic fractional boundary value problems:

    c q D x(t) =f t,x(t) ,t∈[,T],T> ,  <q≤,  c pc p x() = –x(T),D x() = –D x(T), <p< ,

()

c q whereDdenotes the Caputo fractional derivative of orderq, andfis a given continuous function. The results are based on some standard fixed point principles. In recent years, 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 equa- tions. First, second and higher-order differential equations with anti-periodic boundary value conditions have been considered in papers [–]. The existence of solutions for anti-periodic boundary value problems for fractional differential equations was studied in [–].

©2012 Wang and Liu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.