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 $y^{\Delta }(t)=A(t)y(t)$ y Δ ( t ) = A ( t ) y ( t ) and $x^{\Delta }(t)=A(t)x(t)+f(t,x(t))$ x Δ ( t ) = A ( t ) x ( t ) + f ( t , x ( t ) ) in the sense that if $y(t)$ y ( t ) is a given solution of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions $x(t)$ x ( t ) for the perturbed system such that $\Vert y(t)-x(t) \Vert =o( \Vert y(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ y ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ , and conversely, given a solution $x(t)$ x ( t ) of the perturbed system, we give sufficient conditions for the existence of a family of solutions $y(t)$ y ( t ) for the unperturbed system, and such that $\Vert y(t)-x(t) \Vert =o( \Vert x(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ x ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ ; and in doing so, we have to extend Rodrigues inequality, the Lyapunov exponents, and the polynomial exponential trichotomy on time scales.