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

摘要

We study the problem of existence of harmonic solutions for some generalisations of the periodically perturbed Lienard 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 Poincare map. Applications are given to the case of the p-Laplacian and the prescribed curvature equation.