| Author(s): |
Halas, Zdenek
;
Monteiro, Giselle A.
;
Tvrdy, Milan
Total Authors: 3
|
| Document type: | Journal article |
| Source: | MEMOIRS ON DIFFERENTIAL EQUATIONS AND MATHEMATICAL PHYSICS; v. 54, p. 23-pg., 2011-01-01. |
| Abstract | |
This contribution deals with systems of generalized linear differential equations of the form x(k) (t) = (x) over tilde (k) + integral(t)(a) d[A(k)(s)] x(k)(s) + f(k)(t) - f(k)(a), t is an element of [a, b], k is an element of N, where -infinity< a < b < infinity, X is a Banach space, L(X) is the Banach space of linear bounded operators on X, <(n)over tilde>(k) is an element of X, A(k) : [a, b] -> L(X) have bounded variations on [a, b], f(k) : [a, b] X are regulated on [a, b] and the integrals are understood in the Kurzweil Stieltjes sense. Our aim is to present new results on continuous dependence of solutions to generalized linear differential equations on the parameter k. We continue our research from [18], where we were assuming that Ak tends uniformly to A and fk tends uniformly to integral on [a, b]. Here we are interested in the cases when these assumptions are violated. Furthermore, we introduce a notion of a sequential solution to generalized linear differential equations as the limit of solutions of a properly chosen sequence of ODE's obtained by piecewise linear approximations of functions A and f. Theorems on the existence and uniqueness of sequential solutions are proved and a comparison of solutions and sequential solutions is given, as well. The convergence effects occurring in our contribution are, in some sense, very close to those described by Kurzweil and called by him emphatic convergence. (AU) | |
| FAPESP's process: | 11/06392-2 - Generalized Linear Differential Equations: continuous dependence and applications to Functional Differential Equations |
| Grantee: | Giselle Antunes Monteiro |
| Support Opportunities: | Scholarships in Brazil - Doctorate |