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.


45A05 Linear integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
Full Text: DOI


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