TY - JOUR

T1 - Relative asymptotic equivalence of dynamic equations on time scales

AU - Duque, Cosme

AU - Leiva, Hugo

AU - Tridane, Abdessamad

N1 - Funding Information:
The authors would like to thank the anonymous reviewer for his/her valuable suggestions, comments, and criticism for improving the quality of this paper.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/12

Y1 - 2022/12

N2 - This paper aims to study the relative equivalence of the solutions of the following dynamic equations yΔ(t) = A(t) y(t) and xΔ(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.

AB - This paper aims to study the relative equivalence of the solutions of the following dynamic equations yΔ(t) = A(t) y(t) and xΔ(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.

KW - Contraction mapping theorem

KW - Dynamic equations on time scales

KW - Lyapunov exponent

KW - Polynomial exponential trichotomy

KW - Relative asymptotic equivalence

KW - Rodrigues inequality

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U2 - 10.1186/s13662-022-03678-9

DO - 10.1186/s13662-022-03678-9

M3 - Article

AN - SCOPUS:85123012220

SN - 2731-4235

VL - 2022

JO - Advances in Continuous and Discrete Models

JF - Advances in Continuous and Discrete Models

IS - 1

M1 - 4

ER -