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ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION

Published online by Cambridge University Press:  27 October 2015

JAEYOUNG CHUNG  and
JOHN MICHAEL RASSIAS
JAEYOUNG CHUNG*
Affiliation:
Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea email jychung@kunsan.ac.kr
JOHN MICHAEL RASSIAS
Affiliation:
Section of Mathematics and Informatics, Pedagogical Department E. E., National and Kapodistrian University of Athens, Greece email jrassias@primedu.uoa.gr
*
✉jychung@kunsan.ac.kr
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Abstract

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Let $G$ be a commutative group, $Y$ a real Banach space and $f:G\rightarrow Y$. We prove the Ulam–Hyers stability theorem for the cyclic functional equation

$$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$
for all $x,y\in {\rm\Omega}$, where $H$ is a finite cyclic subgroup of $\text{Aut}(G)$ and ${\rm\Omega}\subset G\times G$ satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation
$$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$
 for all $(z,{\it\zeta})\in {\rm\Omega}$, where $f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$ and ${\rm\Omega}\subset \mathbb{C}^{2}$ has four-dimensional Lebesgue measure $0$.

Keywords

asymptotic behaviourBaire category theoremCauchy functional equationcyclic functional equationfirst categoryLebesgue measurequadratic functional equation

MSC classification

Primary: 39B82: Stability, separation, extension, and related topics
Secondary: 39B52: Equations for functions with more general domains and/or ranges
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 2 , April 2016 , pp. 272 - 282
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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