Korean J. Math.  Vol 24, No 2 (2016)  pp.297-305
DOI: https://doi.org/10.11568/kjm.2016.24.2.297

### A fixed point approach to the stability of the functional equation related to distance measures

Hwan-Yong Shin, Gwang Hui Kim

#### Abstract

In this paper, by using fixed point theorem, we obtain the stability of the following functional equations
\begin{align*}
f(pr,qs)+g(ps,qr)&=\theta(p,q, r,s)f(p,q)h(r,s) \\
f(pr,qs)+g(ps,qr)&=\theta(p,q, r,s)g(p,q)h(r,s),
\end{align*}
where $G$ is a commutative semigroup, $\theta : G^{4} \rightarrow \mathbb{R}_{k}$ a function and $f,g,h$ are functionals on $G^{2}$.

#### Keywords

distance measure, superstability, stability of functional equation, fixed point theorem

39B82, 39B52.

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