Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales

The purpose of this paper to establish oscillation criteria for second order nonlinear dynamic equation (r(t)(x(Delta)(t))(gamma))(Delta) + f(t, x(g(t))) - 0; on an arbitrary time scale T, where gamma is a quotient of odd positive integers and r is a positive rd-continuous function on T. The function g : T -> T satisfies g(t) >= t and lim(t -> g)(t) = infinity and f is an element of C(T x R, R). We establish some new sufficient conditions such that the above equation is oscillatory by using generalized Riccati transformation. Our results in the special cases when T = R and T = N involve and improve some oscillation results for second-order differential and difference equations; and when T = hN; T = q(N0) and T = N-2 our oscillation results are essentially new. Some examples illustrating the importance of our results are included. (C) 2010 Elsevier Inc. All rights reserved.