Vol. 16, No. 2, 1966

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ISSN: 0030-8730
Integral equations and product integrals

Burrell Washington Helton

Vol. 16 (1966), No. 2, 297–322

H. S. Wall, J. S. MacNerney and T. H. Hildebrandt have shown interdependencies between the equations

       ∏x                     ∫ x
f(x) = a  (1 + dg) and f(x) = 1+ a f dg;

this paper extends and consolidates some of their results. Let S be a linearly ordered set, R be a normed ring, and OA0 and OM0 be classes of functions G from S ×S to R for which

∫         ∫
b|G (I)−   G| = 0
a         I


∫ b           ∏
|[1+ G (I)]−    (1 + G)| = 0,
a             I

respectively. We show the following. If G has bounded variation, G OA0 if and only if G OM0. For some rings, the existence of abG(I) and a b[1 + G(I)] imply that G OA0 and OM0, respectively. This is used to prove a product integral solution of integral equations such as

                 ∫ x
f(x) = f(a)+ (RL )  (fG + Hf ),

where f is a function from S to R and G and H are functions from S × S to R. Then these results are used (a) to show that each nonsingular m × m matrix of complex numbers has n distinct n-th roots, (b) to show that, with some restrictions, i=1Ai exists if and only if i=1(1 + Ai) exists and (c) to find solutions of integrals equations such as

x n
f (x) = f(a)+  a f dg.

Mathematical Subject Classification
Primary: 45.00
Received: 8 May 1964
Published: 1 February 1966
Burrell Washington Helton