A variant of the quadratic functional equation on groups and an application

Bull. Korean Math. Soc. 2017 Vol. 54, No. 6, 2165-2182 https://doi.org/10.4134/BKMS.b160824 Published online August 4, 2017 Printed November 30, 2017

Heather Hunt Elfen, Thomas Riedel, Prasanna K. Sahoo Robert Morris University, University of Louisville, University of Louisville

Abstract : Let $G$ be a group and $\mathbb{C}$ the field of complex numbers. Suppose $\sigma : G \to G$ is an endomorphism satisfying $\sigma (\sigma (x)) = x$ for all $x$ in $G$. In this paper, we first determine the central solution, $f: G$ or $ G\times G \to \mathbb{C}$, of the functional equation \begin{align*} f(xy) + f(\sigma (y) x) = 2 f(x) + 2 f(y) \quad \text{for all } x, y \in G, \end{align*} which is a variant of the quadratic functional equation. Using the central solution of this functional equation, we determine the general solution of the functional equation $f(pr,qs)+f(sp,rq) = 2 f(p,q) + 2 f(r, s)$ for all $p, q, r, s \in G$, which is a variant of the equation $f(pr,qs)+f(ps,qr) = 2 f(p,q) + 2 f(r, s)$ studied by Chung, Kannappan, Ng and Sahoo in \cite{CKNS} (see also \cite{PKSPK}). Finally, we determine the solutions of this equation on the free groups generated by one element, the cyclic groups of order $m$, the symmetric groups of order $m$, and the dihedral groups of order $2m$ for $m \geq 2$.

Keywords : bi-homomorphism, central function, cyclic group, dihedral group, endomorphism, free group generated by one element, homomorphism, quadratic functional equation, symmetric bi-homomorphism, and symmetric group