ARTICLE IN PRESS Nonlinear Analysis: Hybrid Systems ( ) – Contents lists available at ScienceDirect Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs Darboux problem for perturbed partial differential equations of fractional order with finite delay Saïd Abbas a , Mouffak Benchohra b,∗ a Laboratoire de Mathématiques, Université de Saïda, B.P. 138, 20000, Saïda, Algérie b Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, B.P. 89, 22000, Sidi Bel-Abbès, Algérie article info Article history: Received 1 March 2009 Accepted 1 May 2009 Keywords: Darboux problem Functional differential equations Fractional order Solution Left-sided mixed Riemann–Liouville integral Caputo fractional order derivative Fixed point abstract In this paper we investigate the existence of solutions for functional partial perturbed hyperbolic differential equations with fractional order. These results are based upon a fixed point theorem for the sum of contraction and compact operators. © 2009 Published by Elsevier Ltd 1. Introduction This paper deals with the existence of solutions to fractional order initial value problems (IVP for short), for the system ( c D r 0 u)(x, y) = f (x, y, u (x,y) ) + g (x, y, u (x,y) ), if (x, y) ∈ J , (1) u(x, y) = φ(x, y), if (x, y) ∈ ˜ J , (2) u(x, 0) = ϕ(x), u(0, y) = ψ(y), (x, y) ∈ J , (3) where φ(0) = ψ(0), J =[0, a]×[0, b], ˜ J = [−α, a] × [−β, b]\[0, a]×[0, b], a, b,α,β> 0, c D r 0 is the fractional Caputo derivative of order r = (r 1 , r 2 ) ∈ (0, 1]× (0, 1], f , g : J × C ([−α, 0] × [−β, 0], R) → R,φ ∈ C ([−α, 0] × [−β, 0], R),ϕ : [0, a]→ R,ψ :[0, b]→ R are given functions and C ([−α, 0] × [−β, 0], R) is the space of real continuous functions on [−α, 0] × [−β, 0]. We denote by u (x,y) the element of C ([−α, 0] × [−β, 0], R) defined by u (x,y) (s, t ) = u(x + s, y + t ),(s, t ) ∈ [−α, 0] × [−β, 0]. Here u (x,y) (.,.) represents the history of the state from time x − α up to the present time x and from time y − β up to the present time y. ∗ Corresponding address: Laboratoire de Mathematiques, University of Sidi Bel Abbes, PO Box 89, Sidi Bel-Abbes, 22000 Algeria. Fax: +213 48 54 43 44. E-mail addresses: abbas_said_dz@yahoo.fr (S. Abbas), benchohra@univ-sba.dz (M. Benchohra). 1751-570X/$ – see front matter © 2009 Published by Elsevier Ltd doi:10.1016/j.nahs.2009.05.001 Please cite this article in press as: S. Abbas, M. Benchohra, Darboux problem for perturbed partial differential equations of fractional order with finite delay, Nonlinear Analysis: Hybrid Systems (2009), doi:10.1016/j.nahs.2009.05.001