Continuous dependence of solutions of abstract generalized linear differential equations with potential converging uniformly with a weight
1 Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Praha, 115 67, Czech Republic
2 Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Praha, 115 67, Czech Republic
Boundary Value Problems 2014, 2014:71 doi:10.1186/1687-2770-2014-71
Dedicated to Professor Ivan Kiguradze for his merits in mathematical sciences.Published: 26 March 2014
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 , , , where , X is a Banach space, is the Banach space of linear bounded operators on X, , have bounded variations on , are regulated on . 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 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.