• Korean Journal of Mathematics
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• v.19 no.2
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• pp.149-161
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• 2011

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in Banach algebras and of derivations on Banach algebras for the following Jensen type functional equation: $$f({\frac{x+y}{2}})+f({\frac{x-y}{2}})=f(x)$$.

• Kyungpook Mathematical Journal
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• v.60 no.2
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• pp.415-421
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• 2020

In the proof of Theorem 3 on p.253 in [4], both right and left distributivity are assumed simultaneously which makes the proof invalid. We give a corrected proof for this theorem by introducing an extension of Lemma 2.2 in [2].

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.573-586
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• 2018

Let R be a noncommutative prime ring of characteristic different from 2, Q be its maximal right ring of quotients and C be its extended centroid. Suppose that $f(x_1,{\ldots},x_n)$ be a noncentral multilinear polynomial over $C,b{\in}Q,F$ a b-generalized derivation of R and d is a nonzero derivation of R such that d([F(f(r)), f(r)]) = 0 for all $r=(r_1,{\ldots},r_n){\in}R^n$. Then one of the following holds: (1) there exists ${\lambda}{\in}C$ such that $F(x)={\lambda}x$ for all $x{\in}R$; (2) there exist ${\lambda}{\in}C$ and $p{\in}Q$ such that $F(x)={\lambda}x+px+xp$ for all $x{\in}R$ with $f(x_1,{\ldots},x_n)^2$ is central valued in R.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.3
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• pp.149-153
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• 2023

The aim of this note is to derive two known combinatorial identities via a hypergeometric series approach using Saalschiitz's classical summation theorem.

• Communications of the Korean Mathematical Society
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• v.31 no.2
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• pp.229-235
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• 2016

In this paper we study the regularity of inside (or outside) (${\theta},{\phi}$)-derivations in BCI-algebras X and prove that let $d_{({\theta},{\phi})}:X{\rightarrow}X$ be an inside (${\theta},{\phi}$)-derivation of X. If there exists a ${\alpha}{\in}X$ such that $d_{({\theta},{\phi})}(x){\ast}{\theta}(a)=0$, then $d_{({\theta},{\phi})}$ is regular for all $x{\in}X$. It is also shown that if X is a BCK-algebra, then every inside (or outside) (${\theta},{\phi}$)-derivation of X is regular. Furthermore the concepts of ${\theta}$-ideal, ${\phi}$-ideal and invariant inside (or outside) (${\theta},{\phi}$)-derivations of X are introduced and their related properties are investigated. Finally we obtain the following result: If $d_{({\theta},{\phi})}:X{\rightarrow}X$ is an outside (${\theta},{\phi}$)-derivation of X, then $d_{({\theta},{\phi})}$ is regular if and only if every ${\theta}$-ideal of X is $d_{({\theta},{\phi})}$-invariant.

• Journal of the Korean Mathematical Society
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• v.41 no.3
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• pp.461-477
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• 2004

It is shown that every almost linear mapping h : A\longrightarrowB of a unital $C^{*}$ -algebra A to a unital $C^{*}$ -algebra B is a homomorphism under some condition on multiplication, and that every almost linear continuous mapping h : A\longrightarrowB of a unital $C^{*}$ -algebra A of real rank zero to a unital $C^{*}$ -algebra B is a homomorphism under some condition on multiplication. Furthermore, we are going to prove the generalized Hyers-Ulam-Rassias stability of *-homomorphisms between unital $C^{*}$ -algebras, and of C-linear *-derivations on unital $C^{*}$ -algebras./ -algebras.

• Communications of the Korean Mathematical Society
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• v.24 no.1
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• pp.5-16
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• 2009

In this paper, we obtain the general solution and the generalized Hyers-Ulam-Rassias stability for a following functional equation $$\sum\limits_{i=1}^mf(x_i+\frac{1}{m}\sum\limits_{{i=1\atop j{\neq}i}\.}^mx_j)+f(\frac{1}{m}\sum\limits_{i=1}^mx_i)=2f(\sum\limits_{i=1}^mx_i)$$ for a fixed positive integer m with $m\;{\geq}\;2$. This is applied to investigate derivations and their stability on proper Lie $CQ^*$-algebras. The concept of Hyers-Ulam-Rassias stability originated from the Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), 297-300.

• Journal of the Korean Mathematical Society
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• v.50 no.3
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• pp.591-605
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• 2013

The underlying field is of characteristic $p$ > 2. In this paper, we first use the method of computing the homogeneous derivations to determine the first cohomology of the so-called odd contact Lie algebra with coefficients in the even part of the generalized Witt Lie superalgebra. In particular, we give a generating set for the Lie algebra under consideration. Finally, as an application, the derivation algebra and outer derivation algebra of the Lie algebra are completely determined.

• The Pure and Applied Mathematics
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• v.13 no.4 s.34
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• pp.255-259
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• 2006

In 1994, Lavoie et al. have obtained twenty tree interesting results closely related to the classical Dixon's theorem on the sum of a $_3F_2$ by making a systematic use of some known relations among contiguous functions. We aim at showing that these results can be derived by using the same technique developed by Bailey with the help of Gauss's summation theorem and generalized Kummer's theorem obtained by Lavoie et al..

• Journal of the Korean Mathematical Society
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• v.52 no.1
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• pp.191-207
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• 2015

Let $\mathcal{R}$ be a prime ring of characteristic different from 2, $\mathcal{Q}_r$ be its right Martindale quotient ring and $\mathcal{C}$ be its extended centroid. Suppose that $\mathcal{G}$ is a nonzero generalized skew derivation of $\mathcal{R}$, ${\alpha}$ is the associated automorphism of $\mathcal{G}$, f($x_1$, ${\cdots}$, $x_n$) is a non-central multilinear polynomial over $\mathcal{C}$ with n non-commuting variables and $$\mathcal{S}=\{f(r_1,{\cdots},r_n)\left|r_1,{\cdots},r_n{\in}\mathcal{R}\}$$. If $\mathcal{G}$ acts as a Jordan homomorphism on $\mathcal{S}$, then either $\mathcal{G}(x)=x$ for all $x{\in}\mathcal{R}$, or $\mathcal{G}={\alpha}$.