About the notion of non-$T$-resonance and applications to topological multiplicity results for ODEs on differentiable manifolds

TL;DR

This paper explores the concept of non-$T$-resonance and applies topological degree theory to establish multiplicity results for $T$-periodic solutions of parametrized ODEs on manifolds, with detailed analysis for real line and plane cases.

Contribution

It introduces the notion of non-$T$-resonance and enhances conditions for multiplicity results in periodic ODEs on manifolds, especially for elementary cases.

Findings

Established multiplicity results for $T$-periodic solutions using topological degree.

Provided detailed analysis and improved conditions for $M = \\mathbb{R}$ and $M = \\mathbb{R}^2$.

Clarified the role of $T$-resonance in the existence of solutions.

Abstract

By using topological methods, mainly the degree of a tangent vector field, we establish multiplicity results for $T$-periodic solutions of parametrized $T$-periodic perturbations of autonomous ODEs on a differentiable manifold $M$. In order to provide insights into the key notion of $T$-resonance, we consider the elementary situations $M = \mathbb{R}$ and $M = \mathbb{R}^2$. So doing, we provide more comprehensive analysis of those cases and find improved conditions.