Advances in Difference Equations
Volume 2011 (2011), Article ID 690653, 12 pages
doi:10.1155/2011/690653
Research Article

Existence of Solutions to Fractional Mixed Integrodifferential Equations with Nonlocal Initial Condition

A. Anguraj,1 P. Karthikeyan,2 and J. J. Trujillo3

1Department of Mathematics, PSG College of Arts and Science, Coimbatore 641 014, India
2Department of Mathematics, KSR College of Arts and Science, Tiruchengode 637 215, India
3Departamento de Análisis Matemático, Universidad de La Laguna, Tenerife 38271 La Laguna, Spain

Received 14 November 2010; Accepted 6 January 2011

Academic Editor: Dumitru Baleanu

Copyright © 2011 A. Anguraj 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.

Abstract

We study the existence and uniqueness theorem for the nonlinear fractional mixed Volterra-Fredholm integrodifferential equation with nonlocal initial condition 𝑑 𝛼 𝑥 ( 𝑡 ) / 𝑑 𝑡 𝛼 ∫ = 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) , 𝑡 0 ∫ 𝑘 ( 𝑡 , 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝑠 , 1 0 ∫ ℎ ( 𝑡 , 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝑠 ) , 𝑥 ( 0 ) = 1 0 𝑔 ( 𝑠 ) 𝑥 ( 𝑠 ) 𝑑 𝑠 , where 𝑡 ∈ [ 0 , 1 ] , 0 < 𝛼 < 1 , and 𝑓 is a given function. We point out that such a kind of initial conditions or nonlocal restrictions could play an interesting role in the applications of the mentioned model. The results obtainded are applied to an example.