Weakened Markus--Yamabe conditions for generalized $\varphi$-Laplacian Li\'enard systems

Abstract

For a general autonomous planar polynomial differential system it is difficult to find conditions that are easily verifiable and which guarantee global asymptotic stability. We consider the following three conditions \begin{itemize} \item[(c1)] the divergence of the differential system is negative in the whole $\R^2$; \item[(c2)] the differential system has a unique equilibrium point $p\in \R^2$; and \item[(c3)] the equilibrium point $p$ is locally asymptotically stable; \end{itemize} and the generalized $\varphi$-Laplacian Li\'enard systems \[ x' = y, \quad y' = -g(x) - f(x) \varphi(y), \] where $g(x)$, $f(x)$ and $\varphi(x)$ are polynomials of degrees $n$, $m$ and $\ell$, respectively. We prove that the equilibrium point $p$ is \emph{globally} asymptotically stable for all generalized $\varphi$-Laplacian Li\'enard systems satisfying conditions {\rm (c1)}-{\rm (c3)}. We note that these three conditions are weaker than the Markus--Yamabe conditions.