Existence of limit cycles for some generalisation of the Liénard equations: the relativistic and the prescribed curvature cases

We study the problem of existence of periodic solutions for some generalisations of the relativistic Liénard equation and the prescribed curvature Liénard equation where the damping function depends both on the position and the velocity. In the associated phase-plane this corresponds to a term of the form f(x,y) instead of the standard dependence on x alone. By controlling the continuability of the solutions, we are able to prove the existence of at least a limit cycle in the associated phase-plane for both cases, moreover we provide results with a prefixed arbitrary number of limit cycles. Some examples are given to show the applicability of these results.