Integration by parts for Perron type integrals of order 1 and 2 in Riesz spaces

In the present paper we investigate some fundamental properties of the Perron integrals of order 1 and 2 in the Riesz-space-valued case and obtain a version of integration by parts formulae for these integrals. For technical reasons, in this paper the involved major and minor functions of order 1 and 2 are taken to be regular enough. This gives an opportunity to replace, in the definition of the second order integral on [a, b] ⊂ R, the “boundary”-type conditions at the points a and b, by the “initial”-type conditions at the point a. In this respect our definition is similar to the one used by P. S. Bullen in the real-valued case and is slightly different from the one adopted by R. D. James. In the real-valued case it is known that some of regularity assumptions we are imposing here on major and minor functions do not make the class of the integrable functions smaller . In the case of general Riesz spaces even the problem whether the Perron integral of order 1, defined by continuous major and minor functions, is equivalent to the one defined without the continuity conditions, is still open. An important tool used here is the Maeda–Ogasawara–Vulikh Theorem on representation of Archimedean Riesz spaces as suitable spaces of continuous functions. Another important notion on which our definition of the major and minor functions is based, is the one of the global limit studied in Section 3. The main results of the paper, related to the integration by parts formulae are obtained in Section 8.-