Integral equations and product integrals

Abstract

shown interdependencies between the equations /() = <J (1 + dg) and f(x) = 1 + [fdg Ja this paper extends and consolidates some of their results. Let S be a linearly ordered set, R be a normed ring, and OA and OM be classes of functions G from S x S to R for which -G = o and respectively. We show the following. If G has bounded variation, GeOA 0 if and only if GeOM 0 . For some rings, the existence of G(I) and b [1 + G(/)] imply that GeOA 0 and Ja OM, respectively. This is used to prove a product integral solution of integral equations such as fix) = f() + {RL)\\fG + Hf), Ja where / is a function from S to R and G and H are functions from S x S to R. Then these results are used (a) to show that each nonsingular mx m matrix of complex numbers has n distinct wth roots, (b) to show that, with some restrictions, =i AX exists if and only if =i ^ + A) exists and (c) to find solutions of integrals equations such as f(x)=f(a)+ In his recent paper, Integral Equations and Semigroups [7], J. S. MacNerney develops some of the interdependencies between additive and multiplicative integration processes for rings, defines two classes OA and OM of functions V and W such that the integral-like formulas Via, b) = [\ W -1) and W(a, b) = I (1 + V) Ja are mutually reciprocal, shows a one-to-one correspondence between the