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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
DOI: https://doi.org/10.1090/S0002-9939-08-09379-9
Published electronically: May 20, 2008
MathSciNet review: 2425731
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Abstract: The main result is an improvement of previous results on the equation\[ f(x)+f(y)-f(x+y)=g[\phi (x)+\phi (y)-\phi (x+y)] \] 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\[ f(x)+f(y)-f(x+y)=g(H(x,y)) \] 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

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.