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Abstract and Applied Analysis
Volume 2013, Article ID 723040, 18 pages
http://dx.doi.org/10.1155/2013/723040
Research Article

On Parametric Gevrey Asymptotics for Singularly Perturbed Partial Differential Equations with Delays

1Departamento de Física y Matemática, University of Alcalá, Apartado de Correos 20, Alcalá de Henares, 28871 Madrid, Spain
2Laboratoire Paul Painlevé, University of Lille 1, 59655 Villeneuve d'Ascq Cedex, France

Received 17 July 2013; Accepted 16 September 2013

Academic Editor: Alexander Domoshnitsky

Copyright © 2013 Alberto Lastra and Stéphane Malek. 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 a family of singularly perturbed -difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter . Moreover, we achieve the existence of a common formal power series in which represents each actual solution and establish -Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya theorem regarding -Gevrey asymptotics. A particular Dirichlet like series is studied on the way.