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ON EXTENSIONS OF THE GENERALISED JENSEN FUNCTIONS ON SEMIGROUPS

Published online by Cambridge University Press:  13 February 2017

JANUSZ BRZDĘK  and
ELIZA JABŁOŃSKA
JANUSZ BRZDĘK
Affiliation:
Department of Mathematics, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland email [email protected]
ELIZA JABŁOŃSKA*
Affiliation:
Department of Discrete Mathematics, Rzeszów University of Technology, Powstańców Warszawy 12, 35-959 Rzeszów, Poland email [email protected]
*
✉[email protected]
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Abstract

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Assume that $(G,+)$ is a commutative semigroup, $\unicode[STIX]{x1D70F}$ is an endomorphism of $G$ and an involution, $D$ is a nonempty subset of $G$ and $(H,+)$ is an abelian group uniquely divisible by two. We prove that if $D$ is ‘sufficiently large’, then each function $g:D\rightarrow H$ satisfying $g(x+y)+g(x+\unicode[STIX]{x1D70F}(y))=2g(x)$ for $x,y\in D$ with $x+y,x+\unicode[STIX]{x1D70F}(y)\in D$ can be extended to a unique solution $f:G\rightarrow H$ of the generalised Jensen functional equation $f(x+y)+f(x+\unicode[STIX]{x1D70F}(y))=2f(x)$.

Keywords

Jensen functionrestricted domainabelian semigroupinvolutionfilter

MSC classification

Primary: 39B52: Equations for functions with more general domains and/or ranges
Secondary: 39B82: Stability, separation, extension, and related topics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 1 , August 2017 , pp. 110 - 116
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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