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^Δ(t) = A(t) y(t) and x^Δ(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 ∥ y(t) − x(t) ∥ = o(∥ y(t) ∥) , as t→ ∞ , 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 ∥ y(t) − x(t) ∥ = o(∥ x(t) ∥) , as t→ ∞ ; and in doing so, we have to extend Rodrigues inequality, the Lyapunov exponents, and the polynomial exponential trichotomy on time scales.