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On a Cauchy-Jensen functional equation and its stability. (English) Zbl 1104.39027
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:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
39B72Systems of functional equations and inequalities
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References:
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