Vol. 58, No. 1, 1975

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Product integrals and the solution of integral equations

Jon Craig Helton

Vol. 58 (1975), No. 1, 87–103
Abstract

Functions are from R to N or R × R to N, where R denotes the set of real numbers and N denotes a normed complete ring. If β > 0,H and G are functions from R × R to N,f and h are functions from R to N, each of H,G and dh has bounded variation on [a,b] and |H| < 1 β on [a,b], then the following statements are equivalent:

(1) f is bounded on [a,b], each of abH, abG and (LR) ab(fG + fH) exists and

                 ∫
x
f(x) = h(x)+ (LR ) a (fG + fH)

for a x b, and

(2) each of x y(1 + j=1Hj), x y(1 + G) and

   ∫ y       ∞       y            ∞
(R)   dh(1+ ∑  Hj )⋅s∏  (1 + G)(1+ ∑  Hj )
x       j=1                  j=1

exists for a x < y b and

f(x) = h(a)a x(1 + G)(1 + j=1Hj)
+ (R) axdh(1 + =1Hj) s x(1 + G)(1 + j=1Hi)
for a x b. This result is obtained without requiring the existence of integrals of the form
∫ b    ∫           ∫ b        ∏
a |G −   G | = 0 and a |1+ G −  (1 +G )| = 0.

Mathematical Subject Classification 2000
Primary: 45D05
Milestones
Received: 13 December 1973
Revised: 19 February 1974
Published: 1 May 1975
Authors
Jon Craig Helton