# Existence of harmonic solutions for some generalisation of the non-autonomous Liénard equations

Existence of harmonic solutions for some generalisation of the non-autonomous Liénard equations We study the problem of existence of harmonic solutions for some generalisations of the periodically perturbed Liénard equation, where the damping function depends both on the position and the velocity. In the associated phase-space this corresponds to a term of the form f(x, y) instead of the standard dependence on x alone. We introduce suitable autonomous systems to control the orbits behaviour, allowing thus to construct invariant regions in the extended phase-space and to conclude about the existence of the harmonic solution, by invoking the Brouwer fixed point Theorem applied to the Poincaré map. Applications are given to the case of the p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${ p}$$\end{document}-Laplacian and the prescribed curvature equation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte für Mathematik Springer Journals

# Existence of harmonic solutions for some generalisation of the non-autonomous Liénard equations

, Volume OnlineFirst – Jan 12, 2022
15 pages

/lp/springer-journals/existence-of-harmonic-solutions-for-some-generalisation-of-the-non-00F24GBE16
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021
ISSN
0026-9255
eISSN
1436-5081
DOI
10.1007/s00605-021-01652-3
Publisher site
See Article on Publisher Site

### Abstract

We study the problem of existence of harmonic solutions for some generalisations of the periodically perturbed Liénard equation, where the damping function depends both on the position and the velocity. In the associated phase-space this corresponds to a term of the form f(x, y) instead of the standard dependence on x alone. We introduce suitable autonomous systems to control the orbits behaviour, allowing thus to construct invariant regions in the extended phase-space and to conclude about the existence of the harmonic solution, by invoking the Brouwer fixed point Theorem applied to the Poincaré map. Applications are given to the case of the p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${ p}$$\end{document}-Laplacian and the prescribed curvature equation.

### Journal

Monatshefte für MathematikSpringer Journals

Published: Jan 12, 2022

Keywords: Non-autonomous systems; Generalized Liénard equations; Prescribed curvature operator; Relativistic acceleration; φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document}-Laplacian; Positively invariant sets; Brouwer fixed point theorem; 35C25; 34L30; 34A26

### References

Access the full text.