Time averaging for functional differential equations

Lakrib, Mustapha

doi:10.1155/s1110757x03203077

Cited by 8 publications

(6 citation statements)

References 15 publications

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“…For interested readers, Nonstandard Analysis in Practice [3] is a recommended book deal-ing with at least nine fields where non standard analysis has been used. One can read about the perturbation theory of ordinary differential equations in [2,10,14,18,19,25]. To the binary undefined predicate ∈ of the classical Set Theory ZFC (Zermelo-Fraenkel plus axiom of choice), a new unary undefined predicate standard (st) is joined.…”

Section: External Formulationsmentioning

confidence: 99%

Singular perturbations on the infinite time interval

Yadi

2008

Revue Africaine De Recherche en Informatique Et Mathématiques Appliquées

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International audience We consider the slow and fast systems that belong to a small neighborhood of an unperturbed problem. We study the general case where the slow equation has a compact positively invariant subset which is asymptotically stable, and meanwhile the fast equation has asymptotically stable equilibria (Tykhonov’s theory) or asymptotically stable periodic orbits (Pontryagin–Rodygin’s theory). The description of the solutions is by this way given on infinite time interval. We investigate the stability problems derived from this results by introducing the notion of practical asymptotic stability. We show that some particular subsets of the phase space of the singularly perturbed systems behave like asymptotically stable sets. Our results are formulated in classical mathematics. They are proved within Internal Set Theory which is an axiomatic approach to Nonstandard Analysis. On considère les systèmes lents-rapides appartenant à un petit voisinage d’un problème non perturbé. On étudie le cas général où l’équation lente admet un sous-ensemble compact positivement invariant qui soit asymptotiquement stable tandis que l’équation rapide a des équilibres asymptotiquement stables (théorie de Tykhonov) ou des cycles limites stables (théorie de Pontryagin). La description des solutions est de ce fait donnée sur des intervalles de temps infinis. On examine les problèmes de stabilité découlant de ces résultats en introduisant la notion de stabilité pratique. On montre que certains sous-ensembles de l’espace de phases des systèmes singulièrement perturbés se comportent comme des ensembles asymptotiquement stables. Les résultats sont formulés classiquement mais sont démontrés dans le cadre de la théorie IST, une approche axiomatique de l’Analyse Non Standard.

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Section: External Formulationsmentioning

confidence: 99%

Singular perturbations on the infinite time interval

Yadi

2008

Revue Africaine De Recherche en Informatique Et Mathématiques Appliquées

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“…In [6] and [7], the authors stated very nice averaging results for retarded functional differential equations employing the tools of non-standard analysis. Their results encompass, for instance, the results by J. Hale and S. Verduyn Lunel in [3].…”

Section: Introductionmentioning

confidence: 99%

“…The conditions we assume on the right-hand sides of the RFDEs are more general than those considered in [3,6] or [7]. Indeed, we consider that the right-hand sides of the equations are Kurzweil-Henstock integrable functions.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, in [6] and [7], the authors assume that f (φ, t) is continuous and Lipschitzian in φ. In [3], the authors assume that f (φ, t) is almost periodic in t, uniformly with respect to φ in compact subsets of C([−r, 0], R n ) and admits continuous Fréchet derivative in φ. Hereafter C([−r, 0], R n ) stands for the Banach space of continuous functions from [a, b] to R n equipped with the usual supremum norm, ∞ .…”

Section: Introductionmentioning

confidence: 99%

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Averaging for retarded functional differential equations

Federson

Mesquita

2011

Journal of Mathematical Analysis and Applications

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“…Recently, in [74] and [75], the authors stated averaging results for FDEs employing the tools of non-standard analysis. Also, based in such papers, the authors of [35] improved the results from [74] and [75] using standard analysis though.…”

Section: Non-periodic Averaging Principlesmentioning

confidence: 99%

Equações diferenciais funcionais em medida e equações dinâmicas funcionais impulsivas em escalas temporais

Mesquita¹

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The aim of this work is to investigate and develop the theory of impulsive functional dynamic equations on time scales. We prove that these equations represent a special case of impulsive measure functional differential equations. Moreover, we present a relation between these equations and measure functional differential equations and, also, a correspondence between them and generalized ordinary differential equations. Also, we clarify the relation between measure functional differential equations and functional dynamic equations on time scales. We obtain results on the existence and uniqueness of solutions, continuous dependence on parameters, non-periodic and periodic averaging principles and stability results for all these types of equations. Moreover, we prove some properties concerning regulated functions and equiregulated sets in a Banach space which were essential to our purposes. The new results presented in this work are contained in 7 papers, two of which have already been published and one accepted. See [16], [32], [34], [36], [37], [38] and [84]. viii Resumo O objetivo deste trabalho é investigar e desenvolver a teoria de equações dinâmicas funcionais impulsivas em escalas temporais. Mostramos que estas equações representam um caso especial de equações diferenciais funcionais em medida impulsivas. Também, apresentamos uma relação entre estas equações e as equações diferenciais funcionais em medida e, ainda, mostramos uma relação entre elas e as equações diferenciais ordinárias generalizadas. Relacionamos, também, as equações diferenciais funcionais em medida e as equações dinâmicas funcionais em escalas temporais. Obtemos resultados sobre existência e unicidade de soluções, dependência contínua, método da média periódico e não-periódico bem como resultados de estabilidade para todos os tipos de equações descritos anteriormente. Também, provamos algumas propriedades relativas às funções regradas e aos conjuntos equiregrados em espaços de Banach, que foram essenciais para os nossos propósitos. Os resultados novos apresentados neste trabalho estão contidos em 7 artigos, dos quais dois já foram publicados e um aceito. Veja [16], [32], [34], [36], [37], [38] e [84]. x With the correspondences between the solutions of a functional dynamic equation on time scales and the solutions of a measure functional differential equation and these last ones and the solutions of a generalized ordinary differential equation, we were able to prove several interesting results for these types of equations, such as local existence and uniqueness of solutions, continuous dependence of solutions on parameters and periodic averaging principles. All these results can be found in [36] and they will be described with more details in Chapters 5, 6 and 7. After this, we decided to improve the class of equations on time scales which we were dealing with. Since impulsive differential equations are powerful tools for applications and, Chapter 7 is devoted to averaging principles. It is divided in three sections. The first one concerns n...

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Time averaging for functional differential equations

Abstract: We present a result on the averaging for functional differential equations on finite time intervals. The result is formulated in both classical mathematics and nonstandard analysis; its proof uses some methods of nonstandard analysis.

Cited by 8 publications

References 15 publications

Singular perturbations on the infinite time interval

Singular perturbations on the infinite time interval

Averaging for retarded functional differential equations

Equações diferenciais funcionais em medida e equações dinâmicas funcionais impulsivas em escalas temporais

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