Periodic solutions of measure and neutral functional differential equations

• Ano de defesa: 2021
• Autor(a) principal: Silva, Márcia Richtielle da
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Equações diferenciais funcionais em medida; Equações diferenciais funcionais neutras; Equações diferenciais impulsivas; Fixed-point theorems; Impulsive differential equations; Measure functional differential equations; Neutral functional differential equations; Periodic solutions; Soluções periódicas; Teoremas de ponto fixo
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-20122021-161145/

In this work, we investigate the existence and uniqueness of periodic solutions for two classes of functional differential equations. At first, we consider measure functional differential equations of type: x(t) = x(0) + ∫t0 f(s, xs)ds + ∫t0 g(s, xs)du(s), defined for every t ∈ R, under suitable assumptions on f,g and u. The integrals on the righthand side of the equation exist in the senses of Perron and PerronStieltjes, respectively. Using a topological transversality theorem, we exhibit sufficient conditions to guarantee the existence of periodic solutions for this type of equation. As a consequence of the obtained results, we study the existence and uniqueness of periodic solutions for a class of functional differential equations with impulses. In addition, we present a periodicity study for the solutions of the following class of neutral functional differential equations: d/dt (x(t) – A(t, xt)) = f(t, xt), defined almost everywhere in R, under suitable assumptions on A and f . In this case, in order to guarantee the existence of periodic solutions, we apply a fixed-point theorem variation for condensing maps. As a consequence, we obtain the existence of periodic solutions for a class of impulsive neutral functional differential equations. Some applications are presented to illustrate the theory. The new results presented in this work gave rise to the following articles: (1) Periodic solutions of measure functional differential equations. See (AFONSO; BONOTTO; SILVA, a). (2) Periodic solutions of neutral functional differential equations. See (AFONSO; BONOTTO; SILVA, b).