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

Abstract

We study a $q$-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 $q$-spirals to the origin. By means of a $q$-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 $q$-Gevrey asymptotic expansion (of certain type) of the actual solutions.