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bilinear triple; Perron-Stieltjes integral
Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. \cite4). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. In \cite3 Ch. S. Honig presented a Stieltjes integral for Banach space valued functions. For Honig's integral the Dushnik interior integral presents the background. \endgraf It should be mentioned that abstract Stieltjes integration was recently used by O. Diekmann, M. Gyllenberg and H. R. Thieme in \cite1 and \cite2 for describing the behaviour of some evolutionary systems originating in problems concerning structured population dynamics.
[1] O. Diekmann M. Gyllenberg H. R. Thieme: Perturbing semigroups by solving Stieltjes renewal equations. Differential Integral Equations 6 (1993), 155-181. MR 1190171
[2] O. Diekmann M. Gyllenberg H. R. Thieme: Perturbing evolutionary systems by step responses on cumulative outputs. Differential Integral Equations 7 (1995). To appear. MR 1325554
[3] Ch. S. Hönig: Volterra Stieltjes-Integral Equations. North-Holland Publ. Comp., Amsterdam, 1975. MR 0499969
[4] J. Kurzweil: Nichtabsolut konvergente Integrate. Teubner Verlagsgesellschaft, Leipzig, Teubner-Texte zur Mathematik Bd. 26, 1980. MR 0597703
[5] W. Rudin: Functional Analysis. McGraw-Hill Book Company, New York, 1973. MR 0365062 | Zbl 0253.46001
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