**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