| Full text | |
| Author(s): |
Total Authors: 3
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| Affiliation: | [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
Total Affiliations: 2
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| Document type: | Journal article |
| Source: | CZECHOSLOVAK MATHEMATICAL JOURNAL; v. 68, n. 4, p. 889-920, DEC 2018. |
| Web of Science Citations: | 1 |
| Abstract | |
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) | |
| FAPESP's process: | 11/01316-6 - Linear generalized differential equations and applications to retarded functional differential equations |
| Grantee: | Rodolfo Collegari |
| Support Opportunities: | Scholarships in Brazil - Doctorate (Direct) |