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

Timoteo Carletti, Gabriele Villari, Fabio Zanolin

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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-Laplacian and the prescribed curvature equation.
Original languageEnglish
Pages (from-to)243-257
Number of pages15
JournalMonatshefte für Mathematik
Issue number2
Publication statusPublished - 12 Jan 2022


  • non-autonomous systems
  • generalised Liénard equation
  • prescribed curvature operator
  • relativistic acceleration
  • p-Laplacian
  • positively invariant sets
  • Brouwer fixed point
  • periodic orbits
  • Generalized Liénard equations
  • φ-Laplacian
  • Relativistic acceleration
  • Brouwer fixed point theorem
  • Non-autonomous systems
  • Prescribed curvature operator
  • Positively invariant sets


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