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Master's Dissertation
DOI
10.11606/D.55.2009.tde-10052010-085321
Document
Author
Full name
Jaqueline Bezerra Godoy
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2009
Supervisor
Committee
Godoy, Sandra Maria Semensato de (President)
Barbanti, Luciano
Marconato, Suzinei Aparecida Siqueira
Title in Portuguese
Método da média para equações diferenciais funcionais retardadas impulsivas via equações diferenciais generalizadas
Keywords in Portuguese
Equações diferenciais funcionais retardadas
Equações diferenciais generalizadas
Equações diferenciais impulsivas
Método da média
Abstract in Portuguese
Neste trabalho, nós consideramos o seguinte problema de valor inicial para uma equação diferencial funcional retardada com impulsos { 'x PONTO' = 'varepsilon' f (t, 'x IND.t'), t ' DIFERENTE' 't IND. k', 'DELTA' x('t IND. k') = 'varepsilon' ' I IND. k' (x ( 't IND.k')), k = 0, 1, 2, ... 'x IND. t IND.0' = ' phi', onde f está definida em um aberto ' OMEGA' de R x ' G POT. -' ([- r, 0], ' R POT. n') e assume valores em 'R POT. n', ' 'varepsilon' 'G POT. - ([ - r, 0], 'R POT.n'), r .0, onde ' G POT -' ([ - r, 0], ' R POT. n') denota o espaço das funções de [ - r, 0] em ' R POT. n' que estão regradas e contínuas à esquerda. Além disso, ' t IND.0 < ' t IND. 1'< ... 't IND. k' < ... são momentos pré determinados de impulsos tais que 'lim SOBRE k SETA + ' INFINITO' 't IND. k = + ' INFINITO' e 'DELTA'x (' t IND.k') = x ( 't POT. + IND > k) - x ('t IND. k). Os operadores de impulso ' I IND. k', k = 0, 1, ... são funções contínuas de 'R POT. n' em ' R POT. n'. Consideramos, também, que para cada x 'varepsilon' ' G POT. -' ([- r, ' INFINITO'), 'R POT. n'), t 'SETA' f (t, 'x IND. t') é uma função localmente Lebesgue integrável e sua integral indefinida satisfaz uma condição do tipo Carathéodory. Além disso, f é Lipschitziana na segunda variável. Definimos ' f IND. 0' ( 'phi') = ' lim SOBRE T ' SETA' ' INFINITO' '1 SUP. T ' INT. SUP. T INF. ' T IND.0' f (t, ' PSI') dt e ' I IND. 0(x) = ' lim SOBRE T 'SETA' ' INFINITO' ' 1 SUP. T' ' SIGMA' IND. 0 < ou = ' t IND. i' < T onde ' psi' 'varepsilon' ' G POT. -' ([ - r, 0], ' R POT. n', e consideremos a seguinte equação diferencial funcioonal autônoma " média" y PONTO = ' varepsilon' [ ' f IND. 0' (' y IND. t' + ' I IND> 0' (y (t))], 'y IND. t IND. 0 = ' phi'. Então provamos que, sob certas condições, a solução x(t) de (1) se aproxima da solução y(t) de (2) em tempo assintoticamente grande
Title in English
Averaging method for retarded functional differential equations with impulses by generalized ordinary differential equations
Keywords in English
Averaging
Generalized ordinary differential equations
Impulsive differential equations
Retardedf functional differential equations
Abstract in English
In this present work, we condider the following initial value problem for a retarded functional differential equation with impulses { 'x POINT' = 'varepsilon' f (t, 'x IND.t'), t ' DIFFERENT' 't IND. k', 'DELTA' x('t IND. k') = 'varepsilon' ' I IND. k' (x ( 't IND.k')), k = 0, 1, 2, ... 'x IND. t IND.0' = ' phi', where f está defined in a open set ' OMEGA' de R x ' G POT. -' ([- r, 0], ' R POT. n'), r >0, and takes values in 'R POT. n', ' 'varepsilon' 'G POT. - ([ - r, 0], 'R POT.n'), r .0, where ' G POT -' ([ - r, 0], ' R POT. n') denotes the space of regulated functions from [ - r, 0] to ' R POT. n' which are left continuous. Furthermore, ' t IND.0 < ' t IND. 1'< ... 't IND. k' < ... are pre-assigned moments of impulse effects such that 'lim ON k ARROW + ' THE INFINITE' 't IND. k = + ' THE INFINITE' e 'DELTA'x (' t IND.k') = x ( 't POT. + IND>k) - x ('t IND. k). The impulse operators ' I IND. k', k = 0, 1, ... are continuous mappings from 'R POT. n' to ' R POT. n'. For each x 'varepsilon' ' G POT. -' ([- r, ' THE INFINITE'), 'R POT. n'), t 'ARROW' f (t, 'x IND. t') is locally Lebesgue integrable and its indefinite integral satisfies a Carathéodory. Moreover, f é Lipschitzian with respect to the second variable. We define ' f IND. 0' ( 'phi') = ' lim ON T ' ARROW' ' THE INFINITE' '1 SUP. T ' INT. SUP. T INF. ' T IND.0' f (t, ' PSI') dt and ' I IND. 0(x) = ' lim ON T 'ARROW' ' THE INFINITE' ' 1 SUP. T' ' SIGMA' IND. 0 < or = ' t IND. i' < T where ' psi' 'varepsilon' ' G POT. -' ([ - r, 0], ' R POT. n', and consider the "averaged" autonomous functional differential equation 'y PONTO = ' varepsilon' [ ' f IND. 0' (' y IND. t' + ' I IND> 0' (y (t))], 'y IND. t IND. 0 = ' phi'. Then we prove that, under certain conditions, the solution x(t) of (1) in aproximates the solution y(t) de (2) in an asymptotically large time interval
 
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Publishing Date
2010-05-12
 
WARNING: The material described below relates to works resulting from this thesis or dissertation. The contents of these works are the author's responsibility.
  • FEDERSON, M., and MESQUITA, J.G.. Averaging for retarded functional differential equations [doi:10.1016/j.jmaa.2011.04.034]. Journal of Mathematical Analysis and Applications [online], 2011, vol. 382, n. 1, p. 77-85.
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