Relative asymptotic equivalence of dynamic equations on time scales

This paper aims to study the relative equivalence of the solutions of the following dynamic equations y(Delta) (t) = A(t)y(t) and x(Delta) (t) = A(t)x(t) + f(t, x(t)) in the sense that if y(t) is a given solution of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions x(t) for the perturbed system such that parallel to y(t) - x(t)parallel to = o(parallel to y (t)parallel to), as t -> infinity, and conversely, given a solution x(t) of the perturbed system, we give sufficient conditions for the existence of a family of solutions y(t) for the unperturbed system, and such that parallel to y(t) -x(t)parallel to = o(parallel to x(t)parallel to), as t -> infinity; and in doing so, we have to extend Rodrigues inequality, the Lyapunov exponents, and the polynomial exponential trichotomy on time scales.