• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2165-2182
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• 2017

Let G be a group and $\mathbb{C}$ the field of complex numbers. Suppose ${\sigma}:G{\rightarrow}G$ 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 $G{\times}G{\rightarrow}\mathbb{C}$, of the functional equation $f(xy)+f({\sigma}(y)x)=2f(x)+2f(y)$ for all $x,y{\in}G$, 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) = 2f(p, q) + 2f(r, s) for all $p,q,r,s{\in}G$, which is a variant of the equation f(pr, qs) + f(ps, qr) = 2f(p, q) + 2f(r, s) studied by Chung, Kannappan, Ng and Sahoo in [3] (see also [16]). 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 $m{\geq}2$.

• Korean Journal of Mathematics
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• v.21 no.2
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• pp.161-170
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• 2013

Bae and W. Park [3] proved the Hyers-Ulam stability of bi-homomorphisms and bi-derivations in $C^*$-ternary algebras. It is easy to show that the definitions of bi-homomorphisms and bi-derivations, given in [3], are meaningless. So we correct the definitions of bi-homomorphisms and bi-derivations. Under the conditions in the main theorems, we can show that the related mappings must be zero. In this paper, we correct the statements and the proofs of the results, and prove the corrected theorems.