• Bulletin of the Korean Mathematical Society
• /
• v.47 no.1
• /
• pp.195-209
• /
• 2010

In this paper, we prove the generalized Hyers-Ulam stability of bi-homomorphisms in $C^*$-ternary algebras and of bi-derivations on $C^*$-ternary algebras for the following bi-additive functional equation f(x + y, z - w) + f(x - y, z + w) = 2f(x, z) - 2f(y, w). This is applied to investigate bi-isomorphisms between $C^*$-ternary algebras.

• Korean Journal of Mathematics
• /
• v.20 no.2
• /
• pp.213-223
• /
• 2012

In this paper, we will prove the superstability of the following generalized Pexider type exponential equation $${f(x+y)}^m=g(x)h(y)$$, where $f,g,h\;:\;G{\rightarrow}\mathbb{R}$ are unknown mappings and $m$ is a fixed positive integer. Here G is an Abelian group (G, +), and $\mathbb{R}$ the set of real numbers. Also we will extend the obtained results to the Banach algebra. The obtained results are generalizations of P. G$\check{a}$vruta's result in 1994 and G. H. Kim's results in 2011.

• Communications of the Korean Mathematical Society
• /
• v.31 no.1
• /
• pp.101-113
• /
• 2016

In this article, we considered the stability of the following (${\alpha}$, ${\beta}$, ${\gamma}$)-derivation $${\alpha}D[x,y]={\beta}[D(x),y]+{\gamma}[x,D(y)]$$ and homomorphisms associated to the quadratic type functional equation $$f(kx+y)+f(kx+{\sigma}(y))=2kg(x)+2g(y),\;x,y{\in}A$$, where ${\sigma}$ is an involution of the Lie $C^*$-algebra A and k is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

• Journal of the Chungcheong Mathematical Society
• /
• v.31 no.1
• /
• pp.29-42
• /
• 2018

In this paper, we investigate the stability of two functional equations f(ax+by + cz) - abf(x + y) - bcf(y + z) - acf(x + z) + bcf(y) - a(a - b - c)f(x) - b(b - a)f(-y) - c(c - a - b)f(z) = 0, f(ax+by + cz) + abf(x - y) + bcf(y - z) + acf(x - z) - a(a + b + c)f(x) - b(a + b + c)f(y) - c(a + b + c)f(z) = 0 by applying the direct method in the sense of Hyers and Ulam.

• Korean Journal of Mathematics
• /
• v.18 no.3
• /
• pp.311-322
• /
• 2010

In this paper, we prove stabilities of the reciprocal difference functional equation $$r(\frac{x+y+z}{3})-r(x+y+z)=\frac{2r(x)r(y)r(z)}{r(x)r(y)+r(y)r(z)+r(z)r(x)}$$ and the reciprocal adjoint functional equation $$r(\frac{x+y+z}{3})+r(x+y+z)=\frac{4r(x)r(y)r(z)}{r(x)r(y)+r(y)r(z)+r(z)r(x)}$$ with three variables. Stabilities of the reciprocal difference functional equation and the reciprocal adjoint functional equation in two variables was proved by K. Ravi, J. M. Rassias and B. V. Senthil Kumar. We extend their results to three variables in similar types.

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

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.1
• /
• pp.45-60
• /
• 2009

In this paper, we investigate homomorphisms in proper $JCQ^*$-triples and derivations on proper $JCQ^*$-triples associated to the following Pexiderized functional equation $$f(x+y+z)=f_0(x)+f_1(y)+f_2(z)$$. This is applied to investigate homomorphisms and derivations in proper $JCQ^*$-triples.

• Journal of the Chungcheong Mathematical Society
• /
• v.32 no.3
• /
• pp.337-347
• /
• 2019

In this paper, we investigate the generalized Hyers-Ulam stability of the quadratic and quartic type functional equations $$f(kx+y)+f(kx-y)-k^2f(x+y)-k^2f(x-y)-2f(kx)\\{\hfill{67}}+2k^2f(x)+2(k^2-1)f(y)=0,\\f(x+5y)-5f(x+4y)+10f(x+3y)-10f(x+2y)+5f(x+y)\\{\hfill{67}}-f(-x)=0,\\f(kx+y)+f(kx-y)-k^2f(x+y)-k^2f(x-y)\\{\hfill{67}}-{\frac{k^2(k^2-1)}{6}}[f(2x)-4f(x)]+2(k^2-1)f(y)=0$$ by using the fixed point theory in the sense of L. $C{\breve{a}}dariu$ and V. Radu.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.5
• /
• pp.1063-1078
• /
• 2011

Let M = {1, 2, ${\ldots}$, n} and let V = {$I{\subseteq}M:1{\in}I$}. Denote M\I by $I^c$ for $I{\in}V$. The goal of this paper is to investigate the solution and the stability using the alternative fixed point of generalized cubic functional equation $ \sum\limits_{I{\in}V}f(\sum\limits_{i{\in}I}a_ix_i-\sum\limits_{i{\in}I^c}a_ix_i)=2{^{n-2}{a_1}}\sum\limits_{i=2}^na_i^2[f(x_1+x_i)+f(x_1-x_i)]+2{^{n-1}{a_1}(a^2_1-\sum\limits_{i=2}^2a^2_i)f(x_1)$ in ${\beta}$-Banach modules on Banach algebras, where $a_1,{\ldots},a_n{\in}\mathbb{Z}{\backslash}\{0\}$ with $a_1{\neq}={\pm}1$ and $a_n=1$.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.4
• /
• pp.1157-1169
• /
• 2016

We find the distributional solutions of the Wilson's functional equations $$u{\circ}T+u{\circ}T^{\sigma}-2u{\otimes}v=0,\\u{\circ}T+u{\circ}T^{\sigma}-2v{\otimes}u=0,$$ where $u,v{\in}{\mathcal{D}}^{\prime}({\mathbb{R}}^n)$, the space of Schwartz distributions, T(x, y) = x + y, $T^{\sigma}(x,y)=x+{\sigma}y$, $x,y{\in}{\mathbb{R}}^n$, ${\sigma}$ an involution, and ${\circ}$, ${\otimes}$ are pullback and tensor product of distributions, respectively. As a consequence, we solve the $Erd{\ddot{o}}s$' problem for the Wilson's functional equations in the class of locally integrable functions. We also consider the Ulam-Hyers stability of the classical Wilson's functional equations $$f(x+y)+f(x+{\sigma}y)=2f(x)g(y),\\f(x+y)+f(x+{\sigma}y)=2g(x)f(y)$$ in the class of Lebesgue measurable functions.