A Lyapunov-type stability criterion using $L^\alpha$ norms

Let $q(t)$ be a $T$-periodic potential such that $\int_0^T q(t)\,dt< 0$. The classical Lyapunov criterion to stability of Hill's equation $-\ddot x+ q(t) x=0$ is $\Vert q_-\Vert _1=\int_0^T\vert q_-(t)\vert dt \le 4/T$, where $q_-$is the negative part of $q$. In this paper, we will use a relation between the (anti-)periodic and the Dirichlet eigenvalues to establish some lower bounds for the first anti-periodic eigenvalue. As a result, we will find the best Lyapunov-type stability criterion using $L^\alpha$ norms of $q_-$, $1\le\alpha\le\infty$. The numerical simulation to Mathieu's equation shows that the new criterion approximates the first stability region very well.