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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2014.tde-24042014-165405
Document
Author
Full name
Rodolfo Collegari
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2014
Supervisor
Committee
Federson, Márcia Cristina Anderson Braz (President)
Arita, Andréa Cristina Prokopczyk
Barbanti, Luciano
Frasson, Miguel Vinicius Santini
Gadotti, Marta Cilene
Title in Portuguese
Equações diferenciais ordinárias generalizadas lineares e aplicações às equações diferenciais funcionais lineares
Keywords in Portuguese
Equações diferenciais funcionais
Equações diferenciais ordinárias generalizadas
Fórmula da variação das cnstantes
Abstract in Portuguese
Neste trabalho, apresentamos uma fórmula da variação das constantes para EDOs generalizadas lineares em espaços de Banach. Mais especificamente, estamos interessados em estabelecer uma relação entre as soluções do problema de Cauchy para uma EDO generalizada linear 'dx SUP. d 'tau' =D[A(t )x], x('t IND. 0') = 'x SOB. ~' e as soluções do problema de Cauchy perturbado 'dx SUP. d 'tau' =D[A(t )x +F(x, t )], x('t IND. 0') = x('t IND. 0') = 'x SOB. ~' , em que as funções envolvidas são Perron integráveis e, portanto, admitem muitas descontinuidades e oscilações. Também provamos a existência de uma correspondência biunívoca entre o problema de Cauchy para uma EDF linear da forma { ' y PONTO' =L(t )'y IND. t' , 'y IND. t IND. 0 = \varphi', , em que L é um operador linear e limitado e 'varphi' é uma função regrada, e uma certa classe de EDOs generalizadas lineares. Como consequência, obtemos uma fórmula da variação das constantes relacionando as soluções da EDF linear e as soluções do problema perturbado { 'y PONTO' = L(t )'y IND.t' + f ('yIND. t' , 'y IND. t IND. 0' = '\varphi ', em que a aplicação 't SETA ' f ('y IND. t' , t) é Perron integrável, com t em um intervalo de R, para cada função regrada y
Title in English
Linear generalized ordinary differential equations and application to linear functional differential equations
Keywords in English
Functional differential equations
Generalized ordinary differential equations
Variation of constants formula
Abstract in English
In this work, we present a variation-of-constants formula for linear generalized ordinary differential equations in Banach spaces. More specifically, we are interested in establishing a relation between the solutions of the Cauchy problem for a linear generalized ordinary differential equation 'dx SUP. d \tau' =D[A(t )x], x('t IND. 0') = x ('t IND. 0') = 'x SOB. ~' and the solutions of the perturbed Cauchy problem 'dx SUP. 'd \tau' =D[A(t )x +F(x, t )], x('t IND. '0) = 'x SOB.~', where the functions involved are generalized Perron integrable and, hence, admit many discontinuities and oscillations. We also prove that there exists a one-to-one correspondence between the Cauchy problem for a linear functional differential equations of the form { 'y PONTO' = L(t) 'y IND. t, 'y IND> 0 = \varphi, where L is a bounded linear operator and " is a regulated function, and a certain class of linear generalized ordinary differential equations. As a consequence, we are able to obtain a variation-of-constants formula relating the solutions of the linear functional differential equation and the solutions of the perturbed problem { 'y PONTO' = L(T)'y IND.t´+ f ('y IND. t', t), 'y IND.t IND. 0' = \varphi, where the application t 'ARROW' f('y IND. t', t) is Perron integrable, with t in an interval of R, for each regulated function y
 
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Rodolfo_revisada.pdf (990.68 Kbytes)
Publishing Date
2014-04-25
 
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