Lyapunov-type inequalities for a relativistic second-order differential equation

A. M. Lyapunov proved the inequality that makes it possible to estimate the distance between two consecutive zeros a and b of solutions of a linear differential equation of the second order (x) over dot (t) +q(t)x(t) = 0 where q(t) is a continuous function for t is an element of[a, b]. In the present note, a similar problem is solved for a differential equation of the form at d/dt ((x) over dot/root 1 - (x) over dot) p(t) (x) over dot q(t)x = 0. The obtained inequality is applied to the estimate ok the period of a periodic solution of relativistic differential Van der Pol equation. (C) 2018 Elsevier Ltd. All rights reserved.