q-Nagumo norms and formal solutions to singularly perturbed q-difference equations

TL;DR

This paper develops a framework for analyzing formal solutions to q-analogues of doubly-singular equations, establishing their existence, uniqueness, and q-Gevrey properties using novel Nagumo norms.

Contribution

It introduces a new family of Nagumo norms tailored for q-differences and identifies optimal divergence types for these equations.

Findings

Proved existence and uniqueness of formal solutions.

Established q-Gevrey regularity of solutions.

Illustrated results with relevant examples.

Abstract

The aim of this work is to establish the existence, uniqueness and q-Gevrey character of formal power series solutions of q-analogues of analytic doubly-singular equations. Using a new family of Nagumo norms adapted for q-differences we find new types of optimal divergence associated with these problems. We also provide some examples to illustrate our results.