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GENERAL STABILITY OF THE EXPONENTIAL AND LOBAČEVSKIǏ FUNCTIONAL EQUATIONS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 94 / Issue 2 / October 2016
- Published online by Cambridge University Press:
- 08 March 2016, pp. 278-285
- Print publication:
- October 2016
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- Article
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- You have access
- Export citation
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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 In particular, if
$$\begin{eqnarray}\displaystyle |f(x+y)-f(x)f(y)|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$
$S$ is a uniquely 
$2$-divisible semigroup with an identity, we obtain the general superstability of Lobačevskiǐ’s functional equation with perturbation 
$\unicode[STIX]{x1D713}$
$$\begin{eqnarray}\displaystyle \biggl|f\biggl(\frac{x+y}{2}\biggr)^{2}-f(x)f(y)\biggr|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$
