Relative asymptotic equivalence of dynamic equations on time scales

Cosme Duque; Hugo Leiva; Abdessamad Tridane

doi:10.1186/s13662-022-03678-9

Relative asymptotic equivalence of dynamic equations on time scales

Cosme Duque, Hugo Leiva, Abdessamad Tridane

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

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.

Original language	English
Article number	4
Journal	Advances in Continuous and Discrete Models
Volume	2022
Issue number	1
DOIs	https://doi.org/10.1186/s13662-022-03678-9
Publication status	Published - Dec 2022

Keywords

Contraction mapping theorem
Dynamic equations on time scales
Lyapunov exponent
Polynomial exponential trichotomy
Relative asymptotic equivalence
Rodrigues inequality

ASJC Scopus subject areas

Algebra and Number Theory
Analysis
Applied Mathematics

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10.1186/s13662-022-03678-9

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title = "Relative asymptotic equivalence of dynamic equations on time scales",

abstract = "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.",

keywords = "Contraction mapping theorem, Dynamic equations on time scales, Lyapunov exponent, Polynomial exponential trichotomy, Relative asymptotic equivalence, Rodrigues inequality",

author = "Cosme Duque and Hugo Leiva and Abdessamad Tridane",

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year = "2022",

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doi = "10.1186/s13662-022-03678-9",

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AU - Leiva, Hugo

AU - Tridane, Abdessamad

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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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Relative asymptotic equivalence of dynamic equations on time scales

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