Lyapunov's inequality on timescales

The purpose of this work is to establish the timescale version of Lyapunov's inequality as follows: Let x (t) be a nontrivial solution of (r(t)x(Delta)(t))(Delta) + p(t)x(sigma)(t) = 0 on [a, b] satisfying x (a) = x (b) = 0. Then, under suitable conditions on p, r, a and b, we have [GRAPHICS] where p(+) (t) = max {p (t), 0}, f (t) = (t - a) (b - t) and d is an element of T satisfies \a+b/2 -d\= min{\a+b/2 -s\\s is an element of[a, b] boolean AND T} a+b/2 is an element of T. Here T is a timescale (see below). (C) 2005 Elsevier Ltd. All rights reserved.