OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR DELAY DYNAMIC EQUATIONS ON TIME SCALES

In this paper, we study the oscillation criteria of the following higher order nonlinear delay dynamic equation R-n(Delta) (t, x(t)) + b(t)vertical bar R-n-1(Delta)(t, x(t))vertical bar(gamma-1) R-n-1(Delta) (t, x(t)) + q(t) f (vertical bar(tau(t))vertical bar(gamma-1) x(tau(t))) = 0 on an arbitrary time scale T with supT = infinity, where n >= 2, gamma > 0 is a constant, tau : T -> T with tau(t) <= t and lim(t ->infinity) tau(t) = infinity and R-k(t, x(t)) = {x(t), if k = 0, r(k)(t)R-k-1(Delta) (t, x(t), if 1 <= k <= n - 1, r(n)(t)vertical bar R-n-1(Delta) (t, x(t))vertical bar(gamma-1) R-n-1(Delta) (t, x(t)), if k = n, with r(k)(t) (1 <= k <= n) are positive rd-continuous functions. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero. (C) 2016 Mathematical Institute Slovak Academy of Sciences