On q-Gevrey asymptotics for logarithmic type solutions in singularly perturbed q-difference-differential equations

TL;DR

This paper investigates q-Gevrey asymptotics for solutions with logarithmic terms in singularly perturbed q-difference-differential equations, linking formal and analytic solutions through asymptotic expansions.

Contribution

It introduces a novel approach to construct solutions with logarithmic terms and describes their asymptotic behavior using q-Gevrey expansions, employing a new Banach space framework.

Findings

Constructed holomorphic solutions with logarithmic terms.

Established q-Gevrey asymptotic expansions relating formal and analytic solutions.

Developed a Banach space product in the Borel plane for fixed point analysis.

Abstract

A family of singularly perturbed q-difference-differential equations under the action of a small complex perturbation parameter is studied. The action of the formal monodromy around the origin is present in the equation, which suggests the construction of holomorphic solutions holding logarithmic terms in both, the formal and the analytic level. We provide both solutions and describe the asymptotic behavior relating them by means of $q-$gevrey asymptotic expansions of some positive order, with respect to the perturbation parameter. On the way, the development of a space product of Banach spaces in the Borel plane is needed to provide a fixed point for a coupled system of equations.