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Asymptotic expansions and analytic dynamic equations
Author(s) -
Bohner M.,
Lutz D.A.
Publication year - 2006
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200410219
Subject(s) - mathematics , exponential function , dynamic equation , differential equation , perturbation (astronomy) , exponential growth , mathematical analysis , independent equation , monomial , term (time) , order (exchange) , nonlinear system , pure mathematics , physics , finance , quantum mechanics , economics
Abstract Time scales have been introduced in order to unify the theories of differential and difference equations and in order to extend these cases to many other so‐called dynamic equations. In this paper we consider a linear dynamic equation on a time scale together with a perturbed equation. We show that, if certain exponential dichotomy conditions are satisfied, then for any solution of the perturbed equation there exists a solution of the unperturbed equation that asymptotically differs from the solution of the perturbed equation no more than the order of the perturbation term. In order to show this perturbation theorem, we use many properties of the exponential function on time scales and derive several bounds for certain monomials that appear in the dynamic version of Taylor's formula.