Tesis Doctoral
DOI
https://doi.org/10.11606/T.55.2020.tde-07012020-090607
Documento
Autor
Nombre completo
Alex Pereira da Silva
Dirección Electrónica
Área de Conocimiento
Fecha de Defensa
Publicación
São Carlos, 2019
Director
Tribunal
Silva, Paulo Leandro Dattori da (Presidente)
Barostichi, Rafael Fernando
Bonotto, Everaldo de Mello
Rosado, José Antonio Langa
Rubio, Pedro Marin
Título en inglés
Resolubility of linear Cauchy problems on Fréchet spaces and a de- layed Kaldors model
Palabras clave en inglés
Delay differential equations
Fréchet spaces
Kaldors model.
Linear Cauchy problems
Pseudodifferential operators
Resumen en inglés
The long-run aim of this thesis is to solve delay differential equations with infinite delay of the type
$\frac{d}{\mathrm{dt}}$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.
Título en portugués
Resolubilidade de problemas lineares de Cauchy em espaços de Fréchet e um modelo de Kaldor com retardo
Palabras clave en portugués
Equações diferenciais com retardo
Espaços de Fréchet
Modelo de Kaldor.
Problemas de Cauchy lineares
Resumen en portugués
O objetivo a longo prazo desta tese é resolver equações diferenciais da forma
$\frac{d}{\mathrm{dt}}$u(t) = Au(t) + ∫t-∞ u(s)k(t - s)ds+ f (t, u(t)),