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Zbl 1235.39024
Kenary, Hassan Azadi; Cho, Yeol Je
(English)
[J] Comput. Math. Appl. 61, No. 9, 2704-2724 (2011). ISSN 0898-1221

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.
[Jens Schwaiger (Graz)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains

Keywords: generalized Hyers-Ulam stability; random normed space; non-archimedean normed space; fixed point method; Jensen type functional equation

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