Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Existence of Solutions to Nonlinear Langevin Equation Involving Two Fractional Orders with Boundary Value Conditions | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 516481 17 pages doi 2011 516481 Research Article Existence of Solutions to Nonlinear Langevin Equation Involving Two Fractional Orders with Boundary Value Conditions Anping Chen1 2 and Yi Chen2 1 Department of Mathematics Xiangnan University Chenzhou Hunan 423000 China 2 School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan 411005 China Correspondence should be addressed to Anping Chen chenap@ Received 30 September 2010 Revised 21 January 2011 Accepted 26 February 2011 Academic Editor Kanishka Perera Copyright 2011 A. Chen and Y. Chen. 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. We study a boundary value problem to Langevin equation involving two fractional orders. The Banach fixed point theorem and Krasnoselskii s fixed point theorem are applied to establish the existence results. 1. Introduction Recently the subject of fractional differential equations has emerged as an important area of investigation. Indeed we can find numerous applications in viscoelasticity electrochemistry control electromagnetic porous media and so forth. In consequence the subject of fractional differential equations is gaining much importance and attention. For some recent developments on the subject see 1-8 and the references therein. Langevin equation is widely used to describe the evolution of physical phenomena in fluctuating environments. However for systems in complex media ordinary Langevin equation does not provide the correct description of the dynamics. One of the possible generalizations of Langevin equation is to replace the ordinary derivative by a fractional derivative in it. This gives rise to fractional Langevin equation see for instance 9-12 and the references therein. In this paper we .
