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GENERAL STABILITY OF THE EXPONENTIAL AND LOBAČEVSKIǏ FUNCTIONAL EQUATIONS

Published online by Cambridge University Press:  08 March 2016

JAEYOUNG CHUNG
JAEYOUNG CHUNG*
Affiliation:
Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea email jychung@kunsan.ac.kr
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Abstract

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Let $S$ be a semigroup possibly with no identity and $f:S\rightarrow \mathbb{C}$. We consider the general superstability of the exponential functional equation with a perturbation $\unicode[STIX]{x1D713}$ of mixed variables

$$\begin{eqnarray}\displaystyle |f(x+y)-f(x)f(y)|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$
In particular, if $S$ is a uniquely $2$-divisible semigroup with an identity, we obtain the general superstability of Lobačevskiǐ’s functional equation with perturbation $\unicode[STIX]{x1D713}$
$$\begin{eqnarray}\displaystyle \biggl|f\biggl(\frac{x+y}{2}\biggr)^{2}-f(x)f(y)\biggr|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$

Keywords

exponential functional equationLobačevskiǐ functional equationstability

MSC classification

Primary: 39B82: Stability, separation, extension, and related topics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 2 , October 2016 , pp. 278 - 285
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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