OSCILLATORY BEHAVIOR OF SOLUTIONS OF DYNAMIC EQUATIONS OF HIGHER ORDER ON TIME SCALES

We study the n-th-order nonlinear dynamic equationsx([n]) (t) + p (t)phi(alpha n-1) [(x([)n-2] (t))(Delta sigma)] + q (t) phi(gamma) (x(g(t))) = 0on an unbounded time scale T, where n = 2 and for i = 1, ..., n - 1x([i]) (t) := r(i) (t) phi(alpha i) [x([i-1])(t))(Delta)],with r(n) = alpha(n) = 1 and x([)(0]) = x; here the constants alpha(i) and the functions r(i), i = 1, ..., n - 1, are positive and p, q are nonnegative functions. Criteria are established for the oscillation of solutions for both even- and odd-order cases. The results improve several known results in the literature on second-order, third-order, and higher-order linear and nonlinear dynamic equations. In particular our results can be applied when g is not (delta) differentiable and the forward jump operator sigma and g do not commute.