AbstractThis 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.
Relative asymptotic equivalence of dynamic equations on time scales
