An oscillation theorem for second order superlinear dynamic equations on time scales

In this paper, the oscillatory behavior of the second order superlinear dynamic equation (r(t)x(Delta)(t))(Delta) + p(t)x(alpha)(sigma(t)) = 0; alpha > 1, (0,1) is studied under the assumption integral(infinity) 1 Delta t/r(t) < infinity, where r; p is an element of C-rd(T, R), r(t) > 0; T in our main theorem is assumed to be a regular time scale, alpha is the quotient of odd positive integers. When the coefficient function p(t) is allowed to be negative for arbitrarily large values of t, we establish a sufficient condition for oscillation of all solutions of Eq. (0.1). As special cases, we get that the superlinear differential equation (r(t)x(')(t))(') + p(t)x(alpha)(t) = 0, alpha > 1 is oscillatory, if integral(infinity) R-alpha (t)p(t)dt = infinity, R(t) = integral(infinity)(t) ds/rds, and the superlinear difference equation Delta(r(n)Delta x(n))p(n)x(alpha) (n + 1) = 0; alpha > 1; is oscillatory, if Sigma(infinity) R-alpha (n + 1)p(n) = infinity; R(n) = Sigma(infinity)(k=n) 1/r(k). (C) 2013 Elsevier Inc. All rights reserved.