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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Generalized Cauchy difference equations. II


Author: Bruce Ebanks
Journal: Proc. Amer. Math. Soc. 136 (2008), 3911-3919
MSC (2000): Primary 39B22
Published electronically: May 20, 2008
MathSciNet review: 2425731
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Abstract | References | Similar Articles | Additional Information

Abstract: The main result is an improvement of previous results on the equation

$\displaystyle f(x)+f(y)-f(x+y)=g[\phi(x)+\phi(y)-\phi(x+y)] $

for a given function $ \phi$. We find its general solution assuming only continuous differentiability and local nonlinearity of $ \phi$. We also provide new results about the more general equation

$\displaystyle f(x)+f(y)-f(x+y)=g(H(x,y)) $

for a given function $ H$. Previous uniqueness results required strong regularity assumptions on a particular solution $ f_{0},g_{0}$. Here we weaken the assumptions on $ f_{0},g_{0}$ considerably and find all solutions under slightly stronger regularity assumptions on $ H$.


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Additional Information

Bruce Ebanks
Affiliation: Department of Mathematics and Statistics, P.O. Box MA, Mississippi State University, Mississippi State, Mississippi 39762
Email: ebanks@math.msstate.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09379-9
PII: S 0002-9939(08)09379-9
Keywords: Cauchy difference, cocycle equation, functional independence, Pexider equation, implicit function theorem, philandering, regularity properties, functional equations
Received by editor(s): June 28, 2006
Received by editor(s) in revised form: September 20, 2007
Published electronically: May 20, 2008
Communicated by: David Preiss
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.