Consider the Lienard Equation \begin{equation*} x'' + g(x)x' + h(x) =0 . \end{equation*} We can rewrite this and gain the system \begin{equation*} \begin{pmatrix} x'\\y' \end{pmatrix} = \begin{pmatrix} y\\ -g(x)y-h(x) \end{pmatrix}. \end{equation*} In the book of Walter about Ordinary differential equations he states, that for $xh(x)>0$ and $g(x)\geq 0$ for $x\neq 0$, the solution $(x,y)^T\equiv (0,0)^T$ of our system is asymptotically stable iff we can find a sequence $(x_k)_k\subset \mathbb{R}$ with $x_k\to 0$ and $g(x_k)>0$.
For sure the solution is asymptotically stable if we have $g(x)>0$ for $x\neq 0$, because we can choose a Lyapunov-Function \begin{equation*} V(x,y)= \frac{1}{2}y^2 + H(x) \end{equation*} were $H(x)= \int_0^x h(s)\, ds$. Then $\dot{V}(x,y)=-y^2g(x)$ and the only invariant set of $N=\{ \dot{V} = 0 \}=\{xy=0\}$ is $\{(0,0) \}$ so we can apply well known theorems to conclude the proof. But I don't see how we can proof this similary if we don't have $g(x)>0$ in some pointet neighborhood of $(0,0)$.
Any hints would be helpful. Thanks a lot.
