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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.