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Stability of mixed additive-quadratic Jensen type functional equation in various spaces. (English) Zbl 1235.39024
The authors consider the Jensen type functional equation $$2f(\frac{x+y}2)+f(\frac{x-y}2)+f(\frac{y-x}2)=f(x)+f(y).$$ They give the general odd and the general even solution. They also consider stability questions for the equation in various settings but also divided into the cases of even or odd functions. It seems that the authors do not give the (most) general solution of the equation which---by the way---could be quite easily derived from the general even and odd solution. The stability results also seem to rest heavily on the additional assumptions on the function to be even or odd. It should be remarked that the definition of Cauchy sequences in RN-spaces given at the very beginning seems to be incorrect. Transformed to “classical” normed spaces the definition would mean that a sequence $(x_n)$ is Cauchy if $x_{n+p}-x_n$ tends to zero for $n$ to infinity for all (fixed) $p$. The sequence of the numbers $\ln(n)$ obviously satisfies the above condition, is unbounded, and thus not a Cauchy sequence.

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges
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