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On the Opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations. (English) Zbl 1413.34058

The author considers the continuous dependence on a parameter \(k\in\mathbb{N}\) of solutions to a sequence of initial value problems for systems of generalized linear differential equations of the form
\[x(t)=c+\int_{t_k}^t\text{d}A_k\,x+f_k(t)-f_k(t_0).\]
The Stieltjes integral there is based on the Lebesgue-Stieltjes integral, but it is constructed in such a way that, under the assumptions of the paper, it its equivalent to the Kurzweil-Stieltjes one. Main tool is the weighted convergence analogous to that applied by Z. Opial to systems of linear ordinary differential systems. In addition, effective sufficient conditions are given, as well.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems
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