Hyers-Ulam-Rassias stability of Jensen’s equation and its application

Jung, Soon-Mo

doi:10.1090/S0002-9939-98-04680-2

Hyers-Ulam-Rassias stability of Jensen’s equation and its application
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by Soon-Mo Jung

Proc. Amer. Math. Soc. 126 (1998), 3137-3143

DOI: https://doi.org/10.1090/S0002-9939-98-04680-2

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Abstract:

The Hyers-Ulam-Rassias stability for the Jensen functional equation is investigated, and the result is applied to the study of an asymptotic behavior of the additive mappings; more precisely, the following asymptotic property shall be proved: Let $X$ and $Y$ be a real normed space and a real Banach space, respectively. A mapping $f: X \rightarrow Y$ satisfying $f(0)=0$ is additive if and only if $\left \| 2f\left [ (x+y)/2 \right ] - f(x) - f(y) \right \| \rightarrow 0$ as $\| x \| + \| y \| \rightarrow \infty$.

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