## 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 |
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Article number | 4 |

Journal | Advances in Continuous and Discrete Models |

Volume | 2022 |

Issue number | 1 |

DOIs | |

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