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STABILITY OF AN EXPONENTIAL-MONOMIAL FUNCTIONAL EQUATION

  • CHANG-KWON CHOI (a1)

Abstract

Let $N$ be a fixed positive integer and $f:\mathbb{R}\rightarrow \mathbb{C}$ . As a generalisation of the superstability of the exponential functional equation we consider the functional inequalities

$$\begin{eqnarray}\displaystyle & \displaystyle \big|f\big(\!\sqrt[N]{x^{N}+y^{N}}\big)-f(x)f(y)\big|\leq \unicode[STIX]{x1D719}(x), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \big|f\big(\!\sqrt[N]{x^{N}+y^{N}}\big)-f(x)f(y)\big|\leq \unicode[STIX]{x1D713}(x,y) & \displaystyle \nonumber\end{eqnarray}$$
for all $x,y\in \mathbb{R}$ , where $\unicode[STIX]{x1D719}:\mathbb{R}\rightarrow \mathbb{R}^{+}$ is an arbitrary function and $\unicode[STIX]{x1D713}:\mathbb{R}^{2}\rightarrow \mathbb{R}^{+}$ satisfies a certain condition.

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[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.
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[7] Hyers, D. H., ‘On the stability of the linear functional equation’, Proc. Natl. Acad. Sci. USA 27 (1941), 222224.
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STABILITY OF AN EXPONENTIAL-MONOMIAL FUNCTIONAL EQUATION

  • CHANG-KWON CHOI (a1)

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