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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 632681, 14 pages
http://dx.doi.org/10.1155/2012/632681
Research Article

Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

1Department of Mathematics, Faculty of Basic Science, Babol University of Technology, Babol 47148-71167, Iran
2Department of Mathematics and Computer Science, Cankaya University, Turkey
3Institute of Space Science, 077125 Bucharest-Magurele, Romania

Received 19 February 2012; Accepted 12 April 2012

Academic Editor: Juan J. Trujillo

Copyright © 2012 Azizollah Babakhani and Dumitru Baleanu. 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 discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations ( π’Ÿ 𝛼 βˆ’ 𝜌 𝑑 π’Ÿ 𝛽 ) π‘₯ ( 𝑑 ) = 𝑓 ( 𝑑 , π‘₯ ( 𝑑 ) , π’Ÿ 𝛾 π‘₯ ( 𝑑 ) ) , 𝑑 ∈ ( 0 , 1 ) with boundary conditions π‘₯ ( 0 ) = π‘₯ 0 , π‘₯ ( 1 ) = π‘₯ 1 or satisfying the initial conditions π‘₯ ( 0 ) = 0 , π‘₯ ξ…ž ( 0 ) = 1 , where π’Ÿ 𝛼 denotes Caputo fractional derivative, 𝜌 is constant, 1 < 𝛼 < 2 , and 0 < 𝛽 + 𝛾 ≀ 𝛼 . Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions on 𝑓 .