Báo cáo hóa học: Research Article Global Uniqueness Results for Fractional Order Partial Hyperbolic Functional

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 Global Uniqueness Results for Fractional Order Partial Hyperbolic Functional
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Báo cáo hóa học: " Research Article Global Uniqueness Results for Fractional Order Partial Hyperbolic Functional "Hindawi Publishing CorporationAdvances in Difference EquationsVolume 2011, Article ID 379876, 25 pagesdoi:10.1155/2011/379876Research ArticleGlobal Uniqueness Results forFractional Order Partial HyperbolicFunctional Differential Equations Sa¨d Abbas,1 Mouffak Benchohra,2 and Juan J. Nieto3 ı ı ı 1 Laboratoire de Math´ matiques, Universit´ de Sa¨da, P.O. Box 138, Sa¨da 20000, Algeria e e 2 Laboratoire de Math´ matiques, Universit´ de Sidi Bel-Abb` s, P.O. Box 89, Sidi Bel-Abb` s 22000, Algeria e e e e 3 Departamento de An´ lisis Matem´ tico, Facultad de Matem´ ticas, Universidad de Santiago de Compostela, a a a 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¨d Abbas et al. This is an open access article distributed under the Creative ı 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. IntroductionIn this paper, we provide sufficient conditions for the global existence and uniqueness ofsome 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 IVPfor short of a system of fractional order partial differential equations given by if x, y ∈ J, c r D0 u x , y f x, y, u x,y ; 1.1 if x, y ∈ J , u x, y φ x, y ; 1.2 x, y ∈ 0, ∞ , u x, 0 ϕx, u 0, y ψy; 1.3 0, ∞ × 0, ∞ , J : −α, ∞ × −β, ∞ \ 0, ∞ × 0, ∞ ; α, β > 0, φ ∈ C J , Rn , c D0 rwhere J r1 , r2 ∈ 0, 1 × 0, 1 , f : J × C → R , is a nis the Caputo’s fractional derivative of order rgiven function ϕ : 0, ∞ → Rn , ψ : 0, ∞ → Rn are given absolutely continuous functions Advances in Difference Equations2 φ 0, y for each x, y ∈ 0, ∞ , and C : C −α, 0 × −β, 0 , Rn iswith ϕ x φ x, 0 , ψ ythe space of continuous functions on −α, 0 × −β, 0 . If u ∈ C −α, ∞ × −β, ∞ , Rn , then for any x, y ∈ J define u x,y by for s, t ∈ −α, 0 × −β, 0 . u x,y s, t ux s, y t, 1.4Next we consider the following initial value problem for partial neutral functional differentialequations with finite delay of the form D0 u x, y − g x, y, u x,y if x, y ∈ J, c r f x, y, u x,y ; 1.5 if x, y ∈ J , 1.6 u x, y φ x, y ; x, y ∈ 0, ∞ , u x, 0 ϕx, u 0, y ψy; 1.7where f , φ, ϕ, ψ are as in problem 1.1 – 1.3 , and g : J × C → Rn is a given function. The third result deals with the existence of solutions to fractional order partialhyperbolic functional differential equations with infinite delay of the form if x, y ∈ J, c r D0 u x , y f x, y, u x,y ; 1.8 if x, y ∈ J , u x, y φ x, y ; 1.9 x, y ∈ 0, ∞ , u x, 0 ϕx, u 0, y ψy; 1.10 R2 \ 0, ∞ × 0, ∞ , f : J × B → Rn ,where ϕ, ψ are as in problem 1.1 – 1.3 and Jφ ∈ C J , R , and B is called a phase space that will be specified in Section 4. n We denote by u x,y the element of B defined by s, t ∈ −∞, 0 × −∞, 0 . u x,y s, t ux s, y t; ...