B. W. Helton has studied linear
equations of the form
(1)
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 ℰ be J. S. Mac Nerney’s reversible mapping from 𝒪𝒜 onto 𝒪ℳ. If
{V,W} is in ℰ, 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).
