# Superstability of Pexiderized functional equations arising from distance measures

Volume 9, Issue 2, pp 413--423
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### Authors

Gwang Hui Kim - Department of Applied Mathematics, Kangnam University, Yongin, Gyeonggi, 446-702, Korea. Young Whan Lee - Department of Computer Hacking and Information Security, College of Natural Science, Daejeon University, Daejeon, 300-716, Republic of Korea.

### Abstract

In this paper, we obtain the superstability of the functional equation $f(pr; qs) + g(ps; qr) = \theta(pq; rs)h(p; q)k(r; s)$ for all $p; q; r; s \in G$, where $G$ is an Abelian group, $f; g; h; k$ are functionals on $G^2$, and $\theta$ is a cocycle on $G^2$. This functional equation is a generalized form of the functional equation $f(pr; qs)+f(ps; qr) = f(p; q) f(r; s)$, which arises in the characterization of symmetrically compositive sum-form distance measures and the information measures, and also they can be represented as products of some multiplicative functions and the exponential functional equations. As corollaries, we obtain the superstability of the many functional equations (combination of three variables functions, for example: $f(pr; qs) + g(ps; qr) = \theta(pq; rs)h(p; q)g(r; s))$.

### Share and Cite

##### ISRP Style

Gwang Hui Kim, Young Whan Lee, Superstability of Pexiderized functional equations arising from distance measures, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 2, 413--423

##### AMA Style

Kim Gwang Hui, Lee Young Whan, Superstability of Pexiderized functional equations arising from distance measures. J. Nonlinear Sci. Appl. (2016); 9(2):413--423

##### Chicago/Turabian Style

Kim, Gwang Hui, Lee, Young Whan. "Superstability of Pexiderized functional equations arising from distance measures." Journal of Nonlinear Sciences and Applications, 9, no. 2 (2016): 413--423

### Keywords

• Distance measure
• superstability
• multiplicative function
• stability of functional equation.

•  39B82
•  46S40

### References

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