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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Generalized Cauchy difference equations. II
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by Bruce Ebanks PDF
Proc. Amer. Math. Soc. 136 (2008), 3911-3919 Request permission


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:
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 3911-3919
  • MSC (2000): Primary 39B22
  • DOI:
  • MathSciNet review: 2425731