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On integration by parts for Stieltjes-type integrals of Banach space-valued functions. (English) Zbl 1069.28008
In the paper integration by parts for several generalizations of the Riemann-Stieltjes integral are discussed. The author recalls known formulae for Stieltjes type integration of real functions starting with Pollard type integrals and finishing with variational and gauge integrals. Given three Banach spaces \(X,Y\) and \(Z\) and a bilinear mapping \(\bullet:X\times{Y}\to{Z}\) such that \(\| x\bullet{y}\| \leq\| x\| \| y\| \) for all \(x\in{X},y\in{Y},z\in{Z}\), the author formulates new results for the Henstock-Stieltjes type integration. The general form of the formula of integration by parts is the following: \[ \int_a^bf\bullet{dg}+\int_a^bdf\bullet{g}=f(b)\bullet{g(b)}-f(a)\bullet{g(a)}+c(f,g), \] where (depending on the assumptions on functions \(f:[a,b]\to{X},\;g:[a,b]\to{Y}\)) the summand \(c(f,g)\) depends in a simple way on the jumps of \(f\) and \(g\) at \(a\) and \(b\), provided one of the integrals exists. More detailed review would require formulating of several definitions.

28B05 Vector-valued set functions, measures and integrals
26A39 Denjoy and Perron integrals, other special integrals
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
26A45 Functions of bounded variation, generalizations
46G10 Vector-valued measures and integration
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