Asymptotic Behavior of Solutions of Higher-Order Dynamic Equations on Time Scales

We investigate the asymptotic behavior of solutions of the following higher-order dynamic equation x(Delta n)(t) + f (t, x(t), x(Delta)(t), ... , x(Delta n-1) (t)) = 0, on an arbitrary time scale T, where the function f is defined on T x R-n. We give sufficient conditions under which every solution x of this equation satisfies one of the following conditions: (1) lim(t ->infinity)x(Delta n-1)(t) = 0; (2) there exist constants a(i) (0 <= i <= n - 1) with a(0) not equal 0, such that lim(t ->infinity)x(t)/Sigma(n-1)(i=0)a(i)h(n-i-1)(t, t(0)) = 1, where h(i)(t, t(0)) (0 <= i <= n - 1) are as in Main Results.