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

 

Extension of a generalized Pexider equation


Author: János Aczél
Journal: Proc. Amer. Math. Soc. 133 (2005), 3227-3233
MSC (2000): Primary 39B22
Published electronically: June 20, 2005
MathSciNet review: 2161144
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Abstract | References | Similar Articles | Additional Information

Abstract: The equations $k(s+t)=\ell(s)+n(t)$ and $k(s+t)=m(s)n(t)$, called Pexider equations, have been completely solved on $\mathbb{R} ^2.$ If they are assumed to hold only on an open region, they can be extended to $\mathbb{R} ^2$ (the second when $k$is nowhere 0) and solved that way. In this paper their common generalization $k(s+t)=\ell(s)+m(s)n(t)$ is extended from an open region to $\mathbb{R} ^2$ and then completely solved if $k$ is not constant on any proper interval. This equation has further interesting particular cases, such as $k(s+t)=\ell(s)+m(s)k(t)$ and $k(s+t)=k(s)+m(s)n(t),$ that arose in characterization of geometric and power means and in a problem of equivalence of certain utility representations, respectively, where the equations may hold only on an open region in $\mathbb{R} ^2.$ Thus these problems are solved too.


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Additional Information

János Aczél
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: jdaczel@math.uwaterloo.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08039-1
PII: S 0002-9939(05)08039-1
Keywords: Functional equations, extensions, generalized Pexider equation
Received by editor(s): February 25, 2004
Published electronically: June 20, 2005
Additional Notes: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada grant OGP 0002972. The author is grateful for an observation by Fulvia Skof.
Communicated by: M. Gregory Forest
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.