Vol. 16, No. 2, 1966

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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