Averaging for retarded functional differential equations

Federson, M.; Mesquita, Jaqueline G.

doi:10.1016/j.jmaa.2011.04.034

Cited by 8 publications

(8 citation statements)

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“…Less amount of work has been done in the field of averaging principle. For further read see Bainov and Milusheva (1982); Hale (1966); Hale and Verduyn Lunel (1990); Federson and Godoy (2010) and the reference therein. Recently, few authors studied the averaging principle for stochastic differential equation under some restrictive conditions with non-Lipschitz conditions Mao et al (2015); Tan and Lei (2013).…”

Section: Introductionmentioning

confidence: 99%

Some results in stochastic functional integro-differential equations with infinite delays

Vinodkumar

2017

IJDSDE

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Section: Introductionmentioning

confidence: 99%

Some results in stochastic functional integro-differential equations with infinite delays

Vinodkumar

2017

IJDSDE

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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%

“…Now, we present an original auxiliary lemma which will be essential to our purposes. Its proof is inspired in the proof of Lemma 3.1 in [35].…”

Section: Measure Fdes and Functional Dynamic Equations On Time Scalesmentioning

confidence: 99%

“…The proof of the next lemma is inspired in [35], Lemma 3.2. Such result is new and it can be found in [34].…”

Section: Measure Fdes and Functional Dynamic Equations On Time Scalesmentioning

confidence: 99%

“…Its proof follows the same basic ideas we used in the proof of Theorem 7.11. Certain details are inspired by the paper [35] which is devoted to non-periodic averaging. Our result is original and it is contained in [84].…”

Section: Measure Fdes and Functional Dynamic Equations On Time Scalesmentioning

confidence: 99%

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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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2021

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

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Averaging for retarded functional differential equations

Abstract: We consider retarded functional differential equations in the setting of Kurzweil-Henstock integrable functions and we state an averaging result for these equations. Our result generalizes previous ones.

Cited by 8 publications

References 8 publications

Some results in stochastic functional integro-differential equations with infinite delays

Some results in stochastic functional integro-differential equations with infinite delays

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

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