# Article

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Keywords:
linear Stieltjes integral equations; generalized linear differential equation; Banach space; equation in Banach space
Summary:
Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in \cite5. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. \cite3). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. Here basic results concerning equations of the form x(t) = x(a) +\int_a^t \dd[A(s)]x(s) +f(t) - f(a) are presented on the basis of the Kurzweil type Stieltjes integration. We are looking for generally discontinuous solutions which belong to the space of Banach space-valued regulated functions in the case that $A$ is a suitable operator-valued function and $f$ is regulated.
References:
[1] Dunford N., Schwartz J. T.: Linear Operators I. Interscience Publishers, New York, London, 1958. MR 0117523 | Zbl 0084.10402
[2] Hönig, Ch. S.: Volterra-Stieltjes Integral Equations. North-Holland Publ. Comp., Amsterdam, 1975. MR 0499969
[3] Kurzweil J.: Nichtabsolut konvergente Integrale. B. G.Teubner Verlagsgesellschaft, Leipzig, 1980. MR 0597703 | Zbl 0441.28001
[4] Rudin W.: Functional Analysis. McGraw-Hill Book Company, New York, 1973. MR 0365062 | Zbl 0253.46001
[5] Schwabik Š.: Abstract Perron-Stieltjes integral. Math. Bohem. 121 (1996), 425-447. MR 1428144 | Zbl 0879.28021
[6] Schwabik Š.: Generalized Ordinary Differential Equations. World Scientific, Singapore, 1992. MR 1200241 | Zbl 0781.34003
[7] Schwabik Š., Tvrdý M., Vejvoda O.: Differential and Integral Equations. Academia & Reidel, Praha & Dordrecht, 1979. MR 0542283

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