# ON A SYMMETRIC FUNCTIONAL EQUATION

• Accepted : 2012.07.01
• Published : 2012.09.25

#### Abstract

We find a general solution $f:G{\rightarrow}G$ of the symmetric functional equation $$x+f(y+f(x))=y+f(x+f(y)),\;f(0)=0$$ where G is a 2-divisible abelian group. We also prove that there exists no measurable solution $f:\mathbb{R}{\rightarrow}\mathbb{R}$ of the equation. We also find the continuous solutions $f:\mathbb{C}{\rightarrow}\mathbb{C}$ of the equation.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

#### References

1. J. Aczel, J. Dhombres, Functional equations in several variables, Cambridge University Press, New York-Sydney, 1989.
2. Nicole Brillouet-Belluot, On a symmetric functional equation in two variables, Aequationes Math. 68(2004), 10-20. https://doi.org/10.1007/s00010-004-2732-z
3. M. E. Kuczma, On the mutual noncompatiablity of homogeneous analytic nonpower means, Aequationes Math. 45(1993), 300-321. https://doi.org/10.1007/BF01855888