On \(q\)-Gevrey asymptotics for singularly perturbed \(q\)-difference-differential problems with an irregular singularity (Q410244)

Summary: We study a \(q\)-analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain which generalizes a previous result by the second author [J. Dyn. Control Syst. 17, No. 2, 243--271 (2011; Zbl 1219.35020)]. First, we construct solutions defined in open \(q\)-spirals to the origin. By means of a \(q\)-Gevrey version of the 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.