OSCILLATION OF nTH ORDER SUPERLINEAR DYNAMIC EQUATIONS ON TIME SCALES

Consider the following nth order superlinear dynamic equation x Delta(n) (t) + p(t)x(alpha) (sigma(t)) = 0, alpha > 1, where p is an element of C-rd(T,R+), and T is an isolated time scale, alpha is a ratio of odd positive integers. We obtain an analog of the Kiguradze-Liao-Svec-type oscillation theorem for this dynamic equation. As an application, we obtain (i) when n is even, every solution x(k) of the difference equation Delta(n)x(k) + p(k)x(alpha) (k + 1) = 0, where p(k) >= 0 and alpha > 1 is oscillatory if and only if Sigma(infinity)(k=1)(k + 1)(n - 1) p(k) = infinity. (ii) when n is odd, every solution x(k) of this difference equation is either oscillatory or lim(k-->infinity) x(k) = 0 if and only if the above sum is infinite.