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

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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 \({ p}\)-Laplacian and the prescribed curvature equation.

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Authors and Affiliations

  1. naXys, Namur Institute for Complex Systems, University of Namur, Rue de Bruxelles 61, 5000, Namur, Belgium

    Timoteo Carletti

  2. Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, viale Morgagni 67/A, 50137, Firenze, Italy

    Gabriele Villari

  3. Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, via delle Scienze 206, 33100, Udine, Italy

    Fabio Zanolin

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  1. Timoteo Carletti
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  2. Gabriele Villari
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  3. Fabio Zanolin
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Correspondence to Gabriele Villari.

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Communicated by Adrian Constantin.

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Carletti, T., Villari, G. & Zanolin, F. Existence of harmonic solutions for some generalisation of the non-autonomous Liénard equations. Monatsh Math 199, 243–257 (2022). https://doi.org/10.1007/s00605-021-01652-3

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