Skip to main content
×
×
Home

GENERAL STABILITY OF THE EXPONENTIAL AND LOBAČEVSKIǏ FUNCTIONAL EQUATIONS

  • JAEYOUNG CHUNG (a1)
Abstract

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 $\unicode[STIX]{x1D713}$
$$\begin{eqnarray}\displaystyle \biggl|f\biggl(\frac{x+y}{2}\biggr)^{2}-f(x)f(y)\biggr|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$

Copyright
References
Hide All
[1] Baker, J. A., ‘The stability of the cosine functional equation’, Proc. Amer. Math. Soc. 80 (1980), 411416.
[2] Baker, J. A., Lawrence, J. and Zorzitto, F., ‘The stability of the equation f (x + y) = f (x)f (y)’, Proc. Amer. Math. Soc. 74 (1979), 242246.
[3] Chung, J. and Chung, S.-Y., ‘Stability of exponential functional equations with involutions’, J. Funct. Spaces Appl. 2014 (2014), Article ID 619710, 9 pages.
[4] Gǎvrutǎ, P., ‘An answer to a question of Th. M. Rassias and J. Tabor on mixed stability of mappings’, Bul. Ştiinţ. Univ. Politeh. Timiş. Ser. Mat. Fiz. 42(56) (1997), 16.
[5] Hyers, D. H., ‘On the stability of the linear functional equation’, Proc. Natl. Acad. Sci. USA 27 (1941), 222224.
[6] Hyers, D. H., Isac, G. and Rassias, Th. M., Stability of Functional Equations in Several Variables (Birkhäuser, Boston, MA, 1998).
[7] Jung, S.-M., Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis (Springer, New York, 2011).
[8] Székelyhidi, L., ‘On a theorem of Baker, Lawrence and Zorzitto’, Proc. Amer. Math. Soc. 84 (1982), 9596.
[9] Ulam, S. M., A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, 8 (Interscience, New York, 1960).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed