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Zbl 1104.39027
Park, Won-Gil; Bae, Jae-Hyeong
On a Cauchy-Jensen functional equation and its stability.
(English)
[J] J. Math. Anal. Appl. 323, No. 1, 634-643 (2006). ISSN 0022-247X

A mapping $f$ is called Cauchy-Jensen if it satisfies the system of functional equations $f(x+y, z)=f(x, z)+f(y, z)$ and $2f(x, \frac{y+z}{2})=f(x, y) + f(x, z)$. The authors show that a mapping $f$ is Cauchy-Jensen if and only if $2f(x+y, \frac{z+w}{2})= f(x, z)+f(x, w)+f(y, z)+f(y, w)$. They also prove the generalized Hyers-Ulam-Rassias stability of the latter functional equation and the above system of equations.
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains
39B72 Functional inequalities involving unknown functions

Keywords: Cauchy-Jensen mapping; system of functional equations; Hyers-Ulam-Rassias stability

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