LINEAR FDES IN THE FRAME OF GENERALIZED ODES: VARIATION-OF-CONSTANTS FORMULA

Autor(es)

Collegari, Rodolfo ^[1] ; Federson, Marcia ^[2] ; Frasson, Miguel ^[2]

Afiliação do(s) autor(es)

^[1] Univ Fed Uberlandia, Ave Joao Naves Avila 2121, BR-38400902 Uberlandia, MG - Brazil

^[2] Univ Sao Paulo, Ave Trab Sao Carlense 400, Parque Arnold Schimidt, BR-13566590 Sao Carlos, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: CZECHOSLOVAK MATHEMATICAL JOURNAL; v. 68, n. 4, p. 889-920, DEC 2018.
• Citações Web of Science: 1

Resumo

We present a variation-of-constants formula for functional differential equations of the form (y)over dot = L(t)yt + F(yt, t), yt(0) = phi, where L is a bounded linear operator and phi is a regulated function. Unlike the result by G. Shanholt ( 1972), where the functions involved are continuous, the novelty here is that the application t -> f (y t, t) is Kurzweil integrable with t in an interval of R, for each regulated function y. This means that t -> f (yt, t) may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula. Our main goal is achieved via theory of generalized ordinary differential equations introduced by J. Kurzweil ( 1957). As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable. We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type dx/d tau = D{[}A(t)x], x(t(0)) = (x) over tilde and the solutions of the perturbed Cauchy problem dx/dT = D{[}A(t)x + F(x, t)], x(t(0)) = (x) over tilde. Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form (y)over dot = L(t)yt, yt(0) = phi, where L is a bounded linear operator and phi is a regulated function. The main result comes as a consequence of such results. Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs. (AU)

• Processo FAPESP: 11/01316-6 - Equações diferenciais generalizadas lineares e aplicações às equações diferenciais funcionais com retardamento
• Beneficiário: Rodolfo Collegari
• Modalidade de apoio: Bolsas no Brasil - Doutorado Direto