an:06370893
Zbl 1303.45001
Monteiro, Giselle; Tvrdý, Milan
Continuous dependence of solutions of abstract generalized linear differential equations with potential converging uniformly with a weight
EN
Bound. Value Probl. 2014, Paper No. 71, 18 p. (2014).
1687-2770
2014
j
45A05 45N05
abstract generalized differential equation; continuous dependence; time scale dynamics; linear integral equations; Kurzweil-Stieltjes integral
Summary: We continue our research from the paper by \textit{G. A. Monteiro} and \textit{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 \textit{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.
1268.45009; 0090.30002; 1248.34001