Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 379876, 25 pages doi:10.1155/2011/379876 Research Article Global Uniqueness Results for Fractional Order Partial Hyperbolic Functional Differential Equations Sa¨ ıd Abbas, 1 Mouffak Benchohra, 2 and Juan J. Nieto 3 1 Laboratoire de Math´ ematiques, Universit´ e de Sa¨ ıda, P.O. Box 138, Sa¨ ıda 20000, Algeria 2 Laboratoire de Math´ ematiques, Universit´ e de Sidi Bel-Abb` es, P.O. Box 89, Sidi Bel-Abb` es 22000, Algeria 3 Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain Correspondence should be addressed to Juan J. Nieto, juanjose.nieto.roig@usc.es Received 22 November 2010; Accepted 29 January 2011 Academic Editor: J. J. Trujillo Copyright q 2011 Sa¨ ıd Abbas 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. We investigate the global existence and uniqueness of solutions for some classes of partial hyperbolic differential equations involving the Caputo fractional derivative with finite and infinite delays. The existence results are obtained by applying some suitable fixed point theorems. 1. Introduction In this paper, we provide sufficient conditions for the global existence and uniqueness of some classes of fractional order partial hyperbolic differential equations. As a first problem, we discuss the global existence and uniqueness of solutions for an initial value problem IVP for short of a system of fractional order partial differential equations given by ( c D r 0 u )( x, y )  f ( x, y, u x,y ) ; if ( x, y ) ∈ J, 1.1 u ( x, y )  φ ( x, y ) ; if ( x, y ) ∈  J, 1.2 ux, 0 ϕx, u ( 0,y )  ψ ( y ) ; x, y ∈ 0, ∞, 1.3 where J 0, ∞ × 0, ∞,  J :−α, ∞ × −β, ∞ \ 0, ∞ × 0, ∞; α, β > 0, φ ∈ C  J, R n , c D r 0 is the Caputo’s fractional derivative of order r r 1 ,r 2  ∈ 0, 1 × 0, 1,f : J ×C→ R n , is a given function ϕ : 0, ∞ → R n , ψ : 0, ∞ → R n are given absolutely continuous functions