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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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[1] Kurzweil, J; Vorel, Z, Continuous dependence of solutions of differential equations on a parameter, Czechoslov. Math. J, 7, 568-583, (1957) · Zbl 0090.30001
[2] Krasnoselskij, MA; Krein, SG, On the averaging principle in nonlinear mechanics, Usp. Mat. Nauk, 10, 147-152, (1955)
[3] Kurzweil, J, Generalized ordinary differential equation and continuous dependence on a parameter, Czechoslov. Math. J, 7, 418-449, (1957) · Zbl 0090.30002
[4] Kurzweil, J, Series in real analysis 11, (2012), Singapore
[5] Schwabik Š: Generalized Ordinary Differential Equations. World Scientific, Singapore; 1992. · Zbl 0781.34003
[6] Schwabik Š, Tvrdý M, Vejvoda O: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia, Praha; 1979. · Zbl 0417.45001
[7] Ashordia, M, On the correctness of linear boundary value problems for systems of generalized ordinary differential equations, Proc. Georgian Acad. Sci., Math, 1, 385-394, (1993) · Zbl 0797.34015
[8] Schwabik, Š, Abstract Perron-Stieltjes integral, Math. Bohem, 121, 425-447, (1996) · Zbl 0879.28021
[9] Schwabik, Š, Linear Stieltjes integral equations in Banach spaces, Math. Bohem, 124, 433-457, (1999) · Zbl 0937.34047
[10] Schwabik, Š, Linear Stieltjes integral equations in Banach spaces II. operator valued solutions, Math. Bohem, 125, 431-454, (2000) · Zbl 0974.34057
[11] Monteiro, GA; Tvrdý, M, On kurzweil-Stieltjes integral in Banach space, Math. Bohem, 137, 365-381, (2012) · Zbl 1274.26014
[12] Monteiro, GA; Tvrdý, M, Generalized linear differential equations in a Banach space: continuous dependence on a parameter, Discrete Contin. Dyn. Syst, 33, 283-303, (2013) · Zbl 1268.45009
[13] Opial, Z, Continuous parameter dependence in linear systems of differential equations, J. Differ. Equ, 3, 571-579, (1967) · Zbl 0219.34004
[14] Dudley, RM; Norvaiša, R, Springer monographs in mathematics, (2011), New York · Zbl 1218.46003
[15] Kiguradze, IT, Boundary value problems for systems of ordinary differential equations, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Noveishie Dostizh, 30, 3-103, (1987)
[16] Hakl, R; Lomtatidze, A; Stavrolaukis, IP, On a boundary value problem for scalar linear functional differential equations, Abstr. Appl. Anal, 2004, 45-67, (2004) · Zbl 1073.34078
[17] Hönig, CS, Mathematics studies 16, (1975), Amsterdam
[18] Thorp, BLD, On the equivalence of certain notions of bounded variation, J. Lond. Math. Soc, 43, 247-252, (1968) · Zbl 0157.21001
[19] Meng, G; Zhang, M, Dependence of solutions and eigenvalues of measure differential equations on measures, J. Differ. Equ, 254, 2196-2232, (2013) · Zbl 1267.34014
[20] Meng, G, Zhang, M: Measure differential equations I. Continuity of solutions in measures with weak\^{∗} topology. Preprint, Tsinghua University (2009)
[21] Glivenko VI: The Stieltjes Integral. ONTI, Moscow; 1936. (in Russian)
[22] Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001. · Zbl 0978.39001
[23] Slavík, A, Dynamic equations on time scales and generalized ordinary differential equations, J. Math. Anal. Appl, 385, 534-550, (2012) · Zbl 1235.34247
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