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Continuous dependence of solutions of abstract generalized linear differential equations with potential converging uniformly with a weight. (English) Zbl 1303.45001
Summary: We continue our research from the paper by G. A. Monteiro and M. Tvrdý [Discrete Contin. Dyn. Syst. 33, No. 1, 283–303 (2013; Zbl 1268.45009)] on continuous dependence on a parameter $$k$$ of solutions to linear integral equations of the form $$x(t) = \widetilde{x_k} + \int_a^t \mathbf d[A_k]x + f_k(t) - f_k(a)$$, $$t \in [a,b]$$, $$k \in \mathbb N$$, where $$-\infty < a<b< \infty$$, $$X$$ is a Banach space, $$L(X)$$ is the Banach space of linear bounded operators on $$X$$, $$\widetilde{x_k} \in X$$, $$A_k:[a,b] \to L(X)$$ have bounded variations on $$[a,b]$$, $$f_k:[a,b] \to X$$ are regulated on $$[a,b]$$. The integrals are understood as the abstract Kurzweil-Stieltjes integral and the studied equations are usually called generalized linear differential equations (in the sense of J. Kurzweil, cf. [Czech. Math. J. 7(82), 418–449 (1957; Zbl 0090.30002)] or [Generalized ordinary differential equations. Not absolutely continuous solutions. Series in Real Analysis 11. Hackensack, NJ: World Scientific (2012; Zbl 1248.34001]). In particular, we are interested in the situation when the variations $$\mathrm{var}_a^b A_k$$ need not be uniformly bounded. Our main goal here is the extension of Theorem 4.2 from Monteiro and Tvrdý [loc. cit.] to the nonhomogeneous case. Applications to second-order systems and to dynamic equations on time scales are included as well.

##### MSC:
 45A05 Linear integral equations 45N05 Abstract integral equations, integral equations in abstract spaces
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##### References:
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