Abstract and Applied Analysis
Volume 2012 (2012), Article ID 860716, 35 pages
http://dx.doi.org/10.1155/2012/860716
Research Article

On -Gevrey Asymptotics for Singularly Perturbed -Difference-Differential Problems with an Irregular Singularity

Alberto Lastra1 and Stéphane Malek2

1Facultad de Ciencias, Universidad de Valladolid, Calle del Doctor Mergelina s/n, 47011 Valladolid, Spain
2UFR de Mathématiques, Université Lille 1, Cité Scientifique M2, 59655 Villeneuve d'Ascq Cedex, France

Received 30 September 2011; Revised 28 November 2011; Accepted 28 November 2011

Academic Editor: Allan C. Peterson

Copyright © 2012 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 -analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain which generalizes a previous result by Malek in (2011). First, we construct solutions defined in open -spirals to the origin. By means of a -Gevrey version of Malgrange-Sibuya theorem we show the existence of a formal power series in the perturbation parameter which turns out to be the -Gevrey asymptotic expansion (of certain type) of the actual solutions.