Zdenĕk Halas, Giselle A. Monteiro, and Milan Tvrdý

Emphatic Convergence and Sequential Solutions of Generalized Linear Differential Equations

abstract:
This contribution deals with systems of generalized linear differential equations of the form
$$x_k(t)=\wt{x}_k+\int\nolimits_a^t \mbox{\rm d} [A_k(s)]\,x_k(s)+f_k(t)-f_k(a),\quad t\in [a,b], \;\;k\in\N,$$
where $-\infty<a<b<\infty,$ $X$ is a Banach space, $L(X)$ is the Banach space of linear bounded operators on $X,$ $\wt{x}_k\in X,$ $A_k:[a,b] \to L(X)$ have bounded variations on $[a,b]$, $f_k: [a,b] \to X$ 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  [G. Monteiro and M. Tvrdý, Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Preprint, Institute of Mathematics, AS CR, Prague, 2011-1-17], where we were assuming that $A_k$ tends uniformly to $A$ and $f_k$ tends uniformly to $f$ 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.

Mathematics Subject Classification: 34A37, 45A05, 34A30

Key words and phrases: Generalized linear differential equation, sequential solution, emphatic convergence