The use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4, 30] that there is an equivalence between a Poincaré inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by • showing that...

The Lyapunov stability method is the most popular and applicable stability analysis tool of nonlinear dynamic systems. However, there are some bottlenecks in the Lyapunov method, such as need for negative definiteness of the Lyapunov function derivative in the direction of the system’s solutions. In this paper, we develop a new theorem to dispense the need for negative definite-ness of Lyapunov...

In this paper, we investigate new Lyapunov-type inequalities for Hamiltonian-type problem {w? = ?(x)w + |y| p?2 y /(1 p) p?1/p y? ?? (x) |w|??2 w /(1+|w|?)??1/? ? ?(x) involving the p-prescribed curvature operator ?p (v) |v|p?2 v |v|p) p?1/p, p > 1, under Dirichlet boundary condition.