# Article

Full entry | PDF   (5.2 MB)
Keywords:
linear Stieltjes integral equations; generalized linear differential equation; equation in Banach space
Summary:
This paper is a continuation of \cite9. In \cite9 results concerning equations of the form x(t) = x(a) +\int_a^t \dd[A(s)]x(s) +f(t) - f(a) were presented. The Kurzweil type Stieltjes integration in the setting of \cite6 for Banach space valued functions was used. Here we consider operator valued solutions of the homogeneous problem \Phi(t) = I +\int_d^t \dd[A(s)]\Phi(s) as well as the variation-of-constants formula for the former equation.
References:
[1] Ju. L. Daletskij M. G. Krejn: Stability of Solutions of Differential Equations in Banach Spaces. Nauka, Moskva, 1970. (In Russian.) MR 0352638
[2] N. Dunford J. T Schwartz: Linear Operators I. Interscience Publishers, New York, 1958. MR 0117523
[3] Ch. S. Hönig: Volterra-Stieltjes Integral Equations. North-Holland Publ. Comp., Amsterdam, 1975. MR 0499969
[4] J. Kurzweil: Nichtabsolut konvergente Integrale. B. G. Teubner Verlagsgesellschaft, Leipzig, 1980. MR 0597703 | Zbl 0441.28001
[5] W. Rudin: Functional Analysis. McGraw-Hill Book Company, New York, 1973. MR 0365062 | Zbl 0253.46001
[6] Š. Schwabik: Abstract Perron-Stieltjes integral. Math. Bohem. 121 (1996), 425-447. MR 1428144 | Zbl 0879.28021
[7] Š. Schwabik: Generalized Ordinary Differential Equations. World Scientific, Singapore, 1992. MR 1200241 | Zbl 0781.34003
[8] Š. Schwabik M. Tvrdý O. Vejvoda: Differential and Integral Equations. Academia & Reidel, Praha & Dordrecht, 1979. MR 0542283
[9] Š. Schwabik: Linear Stieltjes integral equations in Banach spaces. Math. Bohem. 124 (1999), 433-457. MR 1722877

Partner of