Continuous dependence of solutions of abstract generalized linear differential equations with potential converging uniformly with a weight

Abstract

In this paper we continue our research from (Monteiro and Tvrdý in Discrete Contin. Dyn. Syst. 33(1):283-303, 2013) on continuous dependence on a parameter k of solutions to linear integral equations of the form x(t)= x ˜ k + ∫ a t d[ A k ]x+ f k (t)− f k (a), t∈[a,b], k∈N, where −∞<a<b<∞, X is a Banach space, L(X) is the Banach space of linear bounded operators on X, x ˜ k ∈X, A k :[a,b]→L(X) have bounded variations on [a,b], f k :[a,b]→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 Kurzweil, cf. (Kurzweil in Czechoslov. Math. J. 7(82):418-449, 1957) or (Kurzweil in Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions, 2012)). In particular, we are interested in the situation when the variations 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 in Discrete Contin. Dyn. Syst. 33(1):283-303, 2013) to the nonhomogeneous case. Applications to second-order systems and to dynamic equations on time scales are included as well.MSC:45A05, 34A30, 34N05.