• Korean Journal of Mathematics
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• v.19 no.1
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• pp.87-99
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• 2011

Using the Hyers-Ulam-Rassias stability method of functional equations, we investigate homomorphisms in proper $CQ^*$-algebras and proper Lie $CQ^*$-algebras, and derivations on proper $CQ^*$-algebras and proper Lie $CQ^*$-algebras associated with the following functional equation $$\frac{1}{k}f(kx+ky+kz)=f(x)+f(y)+f(z)$$ for a fixed positive integer $k$.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.2
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• pp.239-249
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• 2017

In this paper, we approximate a fuzzy almost cubic function by a cubic function in a fuzzy sense. Indeed, we investigate solutions of the following cubic functional equation $$3f(kx+y)+3f(kx-y)-kf(x+2y)-2kf(x-y)-3k(2k^2-1)f(x)+6kf(y)=0$$. and prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces.

• Kyungpook Mathematical Journal
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• v.58 no.2
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• pp.291-305
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• 2018

In this paper, we prove a general uniqueness theorem which is useful for proving the uniqueness of the relevant additive mapping, quadratic mapping, cubic mapping, quartic mapping, or the additive-quadratic-cubic-quartic mapping when we investigate the (generalized) Hyers-Ulam stability.

• Honam Mathematical Journal
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• v.42 no.3
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• pp.449-460
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• 2020

In this paper, we investigate the stability of the additive-cubic functional equations f(x+ky)+f(x-ky)-k² f(x+y)-k² f(x-y)+(k²-1)f(x) - (k²-1)f(-x) = 0, f(x+ky)-f(ky-x)-k² f(x+y)+k² f(y-x)+2(k²-1)f(x)= 0, f(kx+y)+f(kx-y)-kf(x+y)-kf(x-y)-2f(kx)+2kf(x)= 0 by using the fixed point theory in the sense of L. Cădariu and V. Radu.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.617-630
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• 2011

Let X, Y be vector spaces, and let r be 2 or 4. It is shown that if an odd mapping $f:X{\rightarrow}Y$ satisfies the functional equation $${\hspace{50}}rf(\frac{\sum_{j=1}^{d}\;x_j} {r})+\;{\sum\limits_{\iota(j)=0,1 \atop {\sum_{j=1}^{d}}\;{\iota}(j)=l}}\;rf(\frac{\sum_{j=1}^{d}{(-1)^{\iota(j)}x_j}}{r}) \\({\ddag}){\hspace{160}}=(_{d-1}C_l-_{d-1}C_{l-1}+1)\;{\sum\limits_{j=1}^{d}\;f(x_j)}$$ then the odd mapping $f:X{\rightarrow}Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation in Banach modules over a unital $C^*$-algebra. As an application, we show that every almost linear bijection $h:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ of a unital $C^*$-algebra ${\mathcal{A}}$ onto a unital $C^*$-algebra ${\mathcal{B}}$ is a $C^*$-algebra isomorphism when $h(2^nuy)=h(2^nu)h(y)$ for all unitaries $u{\in}{\mathcal{A}}$, all $y{\in}{\mathcal{A}}$, and $n=0,1,2,{\cdots}$.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.851-860
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• 2007

In this paper, we prove that a function satisfying the following inequality $${\parallel}f(x)+2f(y)+2f(z){\parallel}{\leq}{\parallel}2f(\frac{x}{2}+y+z){\parallel}+{\epsilon}({\parallel}x{\parallel}^r{\cdot}{\parallel}y{\parallel}^r{\cdot}{\parallel}z{\parallel}^r)$$ for all x, y, z ${\in}$ X and for $\epsilon{\geq}0$, is Cauchy additive. Moreover, we will investigate for the stability in Banach spaces.

• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.415-426
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• 2020

In this paper, we will prove some uniqueness theorems that can be applied to the generalized Hyers-Ulam stability of some additive-quadratic-cubic functional equation in complete modular spaces without Δ₂-conditions.

• The Pure and Applied Mathematics
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• v.15 no.4
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• pp.393-400
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• 2008

In this paper, we investigate the following additive functional inequality (0.1) ||f(x)+f(y)+f(z)+f(w)||${\leq}$||f(x+y)+f(z+w)|| in normed modules over a $C^*$-algebra. This is applied to understand homomor-phisms in $C^*$-algebra. Moreover, we prove the generalized Hyers-Ulam stability of the functional inequality (0.2) ||f(x)+f(y)+f(z)f(w)||${\leq}$||f(x+y+z+w)||+${\theta}||x||^p||y||^p||z||^p||w||^p$ in real Banach spaces, where ${\theta}$, p are positive real numbers with $4p{\neq}1$.

• Bulletin of the Korean Mathematical Society
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• v.41 no.2
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• pp.347-357
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• 2004

In this note we will find out the general solution and investigate the generalized Hyers-Ulam-Rassias stability for the cubic functional equation f(3$\chi$+y) + f($3\chi$-y) = $3f(\chi$+ y) + $3f(\chi$-y) + ($48f(\chi)$ on abelian groups.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.4
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• pp.815-830
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• 2009

In this paper we establish the general solution of the following functional equation with mixed type of quadratic and additive mappings f(mx+y)+f(mx-y)+2f(x)=f(x+y)+f(x-y)+2f(mx), where $m{\geq}2$ is a positive integer, and then investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces.