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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2020.tde-07012020-090607
Document
Author
Full name
Alex Pereira da Silva
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2019
Supervisor
Committee
Silva, Paulo Leandro Dattori da (President)
Barostichi, Rafael Fernando
Bonotto, Everaldo de Mello
Rosado, José Antonio Langa
Rubio, Pedro Marin
Title in English
Resolubility of linear Cauchy problems on Fréchet spaces and a de- layed Kaldors model
Keywords in English
Delay differential equations
Fréchet spaces
Kaldors model.
Linear Cauchy problems
Pseudodifferential operators
Abstract in English
The long-run aim of this thesis is to solve delay differential equations with infinite delay of the type
d dt u(t) = Au(t) + ∫t-∞ u(s)k(t - s)ds+ f (t, u(t)),

on Fréchet spaces under an extended theory of groups of linear operators; where A is a linear operator, k(s) ≥ 0 satisfies ∫0 k(s)ds = 1 and f is a nonlinear map. In order to pursue such a goal we study a discrete delay model which explains the natural economic fluctuations considering how economic stability is affected by the role of the fiscal and monetary policies and a possible government inefficiency concerning its fiscal policy decision-making. On the other hand, we start to develop such an extended theory by considering linear Cauchy problems associated to a continuous linear operator on Fréchet spaces, for which we establish necessary and sufficient conditions for generation of a uniformly continuous group which provides the unique solution. Further consequences arises by considering pseudodifferential operators with constant coefficients defined on a particular Fréchet space of distributions, namely FL2loc, and special attention is given to the distributional solution of the heat equation on FL2loc for all time, which extends the standard solution on Hilbert spaces for positive time.
Title in Portuguese
Resolubilidade de problemas lineares de Cauchy em espaços de Fréchet e um modelo de Kaldor com retardo
Keywords in Portuguese
Equações diferenciais com retardo
Espaços de Fréchet
Modelo de Kaldor.
Operadores pseudodiferenciais
Problemas de Cauchy lineares
Abstract in Portuguese
O objetivo a longo prazo desta tese é resolver equações diferenciais da forma
d dt u(t) = Au(t) + ∫t-∞ u(s)k(t - s)ds+ f (t, u(t)),

em espaços de Fréchet estendendo a teoria de grupos de operadores lineares; sendo A um operador linear, k(s) ≥ 0 tal que ∫0 k(s)ds = 1 e f uma função não linear. Perseguindo tal fim, estudamos um modelo com retardo que explica as flutuações naturais da economia considerando como a estabilidade econômica é afetada pela atuação do governo, suas políticas fiscal e monetária e uma possível ineficiência do governo no que diz respeito à sua tomada de decisão na política fiscal. Por outro lado, damos início a referida extensão da teoria de grupos ao considerar problemas de Cauchy lineares associados a operadores lineares contínuos em espaços de Fréchet, para os quais estabelecemos condições necessárias e suficientes para a geração de um grupo uniformemente contínuo em tal espaço que fornece a única solução do problema. Consequências adicionais surgem quando se considera operadores pseudodiferenciais com coeficientes constantes definidos em um particular espaço de Fréchet de distribuições, a saber FL2loc, e uma atenção especial é dada à solução distribucional da equação do calor em FL2loc para todo tempo, a qual estende a solução usual em espaços de Hilbert para tempo positivo.
 
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Publishing Date
2020-01-10
 
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