Vol. 60, No. 1, 1975

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Nonlinear integral equations and product integrals

Alvin John Kay

Vol. 60 (1975), No. 1, 203–222

B. W. Helton has studied linear equations of the form

f (x) = f(a)+ (RL ) x (Kf + M f );

this paper extends some of his results to a nonlinear setting. Let S be a linearly ordered set, {G,+, ∥} a complete normed abelian group, H the set of functions from G to G that take 0 to 0, 𝒪𝒜 and 𝒪ℳ classes of functions from SXS to H that are order-additive and order-multiplicative respectively and satisfy a Lipschitz-type condition, and be J. S. Mac Nerney’s reversible mapping from 𝒪𝒜 onto 𝒪ℳ. If {V,W} is in , we show the collection of all functions that are differentially equivalent to V is the same as the collection of functions that are differentially equivalent to W 1. This analysis is used to prove existence theorems for product integrals which we show solve (1).

Received: 29 May 1974
Revised: 9 September 1974
Published: 1 September 1975
Alvin John Kay