Periodic solutions of relativistic Liénard-type equations

Abstract

In this paper, we prove that the relativistic Liénard-type equation \begin{equation*} \begin{split} \frac{d}{dt}\left(\frac{\dot{x}\left\vert \dot{x} \right\vert ^{p-2}}{\big( 1-\left\vert \dot{x}\right\vert ^{p}\big) ^{\frac{p-1}{p}}}\right) +f\left( x\right) \dot{x} +g\left( x\right) =0 \text{,}\qquad p>1, \end{split} \end{equation*} and its special case, relativistic Van der Pol-type equation, have a periodic solution. Our results are inspired by the results obtained by Mawhin and Villari [Nonlinear Anal. 160(2017), 16–24] and extend their results to this more general case.

Keywords

• closed orbits
• periodic solutions
• limit cycles
• relativistic liénard-type equations