On asymptotic relationships between two higher order dynamic equations on time scales

We consider the nth order dynamic equations xΔn+p1(t)xΔn−1+⋯+pn(t)x=0 and yΔn+p1(t)yΔn−1+⋯+pn(t)y=f(t,y(τ(t))) on a time scale T, where τ is a composition of the forward jump operators, pi are real rd-continuous functions and f is a continuous function; T is assumed to be unbounded above. We establish conditions that guarantee asymptotic equivalence between some solutions of these equations. No restriction is placed on whether the solutions are oscillatory or nonoscillatory. Applications to second order Emden-Fowler type dynamic equations and Euler type dynamic equations are shown.