• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.253-267
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• 2003

In this paper we deal With the quadratic functional equation (equation omitted) deriving from an inequality of T. Popoviciu for convex functions. We solve this functional equation by proving that its solutions we the polynomials of degree at most two. Likewise, we investigate its stability in the spirit of Hyers, Ulam, and Rassias.

• Honam Mathematical Journal
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• v.26 no.1
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• pp.61-75
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• 2004

It is shown that every almost linear mapping $h:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ of a unital Poisson Banach algebra ${\mathcal{A}}$ to a unital Poisson Banach algebra ${\mathcal{B}}$ is a Poisson algebra homomorphism when h(xy) = h(x)h(y) holds for all $x,y{\in}\;{\mathcal{A}}$, and that every almost linear almost multiplicative mapping $h:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ is a Poisson algebra homomorphism when h(qx) = qh(x) for all $x\;{\in}\;{\mathcal{A}}$. Here the number q is in the functional equation given in the almost linear almost multiplicative mapping. We prove that every almost Poisson bracket $B:{\mathcal{A}}\;{\times}\;{\mathcal{A}}\;{\rightarrow}\;{\mathcal{A}}$ on a Banach algebra ${\mathcal{A}}$ is a Poisson bracket when B(qx, z) = B(x, qz) = qB(x, z) for all $x,z{\in}\;{\mathcal{A}}$. Here the number q is in the functional equation given in the almost Poisson bracket.

• Journal of the Korean Mathematical Society
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• v.43 no.2
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• pp.323-356
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• 2006

Let X and Y be vector spaces. It is shown that a mapping $f\;:\;X{\rightarrow}Y$ satisfies the functional equation $$mn_{mn-2}C_{k-2}f(\frac {x_1+...+x_{mn}} {mn})$$ $(\ddagger)\;+mn_{mn-2}C_{k-1}\;\sum\limits_{i=1}^n\;f(\frac {x_{mi-m+1}+...+x_{mi}} {m}) =k\;{\sum\limits_{1{\leq}i_1<... if and only if the mapping $f : X{\rightarrow}Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation $(\ddagger)$ in Banach modules over a unital $C^*-algebra$. Let A and B be unital $C^*-algebra$ or Lie $JC^*-algebra$. As an application, we show that every almost homomorphism h : $A{\rightarrow}B$ of A into B is a homomorphism when $h(2^d{\mu}y) = h(2^d{\mu})h(y)\;or\;h(2^d{\mu}\;o\;y)=h(2^d{\mu})\;o\;h(y)$ for all unitaries ${\mu}{\in}A,\;all\;y{\in}A$, and d = 0,1,2,..., and that every almost linear almost multiplicative mapping $h:\;A{\rightarrow}B$ is a homomorphism when h(2x)=2h(x) for all $x{\in}A$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*-algebras$ or in Lie $JC^*-algebras$, and of Lie $JC^*-algebra$ derivations in Lie $JC^*-algebras$.

• 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.