• Bulletin of the Korean Mathematical Society
• /
• v.48 no.4
• /
• pp.673-688
• /
• 2011

Let $n{\geq}2$ be an even integer. We investigate that if an odd mapping f : X ${\rightarrow}$ Y satisfies the following equation $2_{n-2}C_{\frac{n}{2}-1}rf\(\sum\limits^n_{j=1}{\frac{x_j}{r}}\)\;+\;{\sum\limits_{i_k{\in}\{0,1\} \atop {{\sum}^n_{k=1}\;i_k={\frac{n}{2}}}}\;rf\(\sum\limits^n_{i=1}(-1)^{i_k}{\frac{x_i}{r}}\)=2_{n-2}C_{{\frac{n}{2}}-1}\sum\limits^n_{i=1}f(x_i),$ then f : X ${\rightarrow}$ Y is additive, where $r{\in}R$. We also prove the stability in normed group by using shadowing property and the Hyers-Ulam stability of the functional equation in Banach spaces and in Banach modules over unital C-algebras. As an application, we show that every almost linear bijection h : A ${\rightarrow}$ B of unital $C^*$-algebras A and B is a $C^*$-algebra isomorphism when $h(\frac{2^s}{r^s}uy)=h(\frac{2^s}{r^s}u)h(y)$ for all unitaries u ${\in}$ A, all y ${\in}$ A, and s = 0, 1, 2,....

• The Pure and Applied Mathematics
• /
• v.25 no.3
• /
• pp.219-227
• /
• 2018

In this paper, we solve the additive ${\rho}$-functional equations $$(0.1)\;f(x+y)+f(x-y)-2f(x)={\rho}\left(2f\left({\frac{x+y}{2}}\right)+f(x-y)-2f(x)\right)$$, where ${\rho}$ is a fixed non-Archimedean number with ${\mid}{\rho}{\mid}$ < 1, and $$(0.2)\;2f\left({\frac{x+y}{2}}\right)+f(x-y)-2f(x)={\rho}(f(x+y)+f(x-y)-2f(x))$$, where ${\rho}$ is a fixed non-Archimedean number with ${\mid}{\rho}{\mid}$ < |2|. Furthermore, we prove the Hyers-Ulam stability of the additive ${\rho}$-functional equations (0.1) and (0.2) in non-Archimedean Banach spaces.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.2
• /
• pp.309-315
• /
• 2002

Let I be an open interval and X a complex Banach space. Let$\varepsilon\geq0\;and\;\lambda$ a non-zero complex number with Re $\lambda\neq0$. If $\varphi$ is a strongly differentiable map from I to X with $\parallel\varphi^'(t)-\lambda\varphi(t)\parallel\leq\varepsilon\;for\;all\;t\in\;I$, then we show that the distance between $\varphi$ and the set of all solutions to the differential equation y'=$\lambda$y is at most $\varepsilon/$\mid$Re\lambda$\mid$$.

• Korean Journal of Mathematics
• /
• v.19 no.1
• /
• pp.17-24
• /
• 2011

Let X, Y be vector spaces. It is shown that if an even mapping $f:X{\rightarrow}Y$ satisfies f(0) = 0, and $$2(_{2d-2}C_{d-1}-_{2d-2}C_d)f\({\sum_{j=1}^{2d}}x_j\)+{\sum_{{\iota}(j)=0,1,{{\small\sum}_{j=1}^{2d}}{\iota}(j)=d}}\;f\({\sum_{j=1}^{2d}}(-1)^{{\iota}(j)}x_j\)=2(_{2d-1}C_d+_{2d-2}C_{d-1}-_{2d-2}C_d){\sum_{j=1}^{2d}}f(x_j)$$ for all $x_1$, ${\cdots}$, $x_{2d}{\in}X$, then the even mapping $f:X{\rightarrow}Y$ is quadratic. Furthermore, we prove the Hyers-Ulam stability of the above functional equation in Banach spaces.

• Korean Journal of Mathematics
• /
• v.20 no.1
• /
• pp.33-46
• /
• 2012

Using the direct method, we prove the Hyers-Ulam stability of the orthogonally additive-quartic functional equation (0.1) $f(2x+y)+f(2x-y)=4f(x+y)+4f(x-y)+10f(x)+14f(-x)-3f(y)-3f(-y)$ for all $x$, $y$ with $x{\perp}y$, in non-Archimedean Banach spaces. Here ${\perp}$ is the orthogonality in the sense of R$\ddot{a}$tz.

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

• Bulletin of the Korean Mathematical Society
• /
• v.41 no.3
• /
• pp.541-557
• /
• 2004

In this paper we prove the Hyers-Ulam-Rassias stability by considering the cases that the approximate remainder ${\varphi}$ is defined by (x * y) ＋ (x * $y^{-1}$) - 2 (x) - 2 (y) =<${\varphi}$(x,y), (x*y*z)＋ (x)＋ (y)＋ (z)－ (x*y)－ (y*z)－ (z*x)＝${\varphi}$(x, y, z), where (G,*) is a group, X is a real or complex Hausdorff topological vector space, and is a function from G into X.

• Journal of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.37-47
• /
• 2001

We present the general solution of the functional equation f(x$_1$y$_1$,x$_2$y$_2$) + f(x$_1$y$_1$(sup)-1,x$_2$) + f(x$_1$,x$_2$y$_2$(sup)-1) = f(x$_1$y$_1$(sup)-1,x$_2$y$_2$(sup)-1) + f(x$_1$y$_1$,x$_2$) + f(x$_1$,x$_2$y$_2$). Furthermore, we also prove the Hyers-Ulam stability of the above functional equation.

• Journal of applied mathematics & informatics
• /
• v.33 no.3_4
• /
• pp.435-445
• /
• 2015

In this paper, using the fixed point method, we investigate the generalized Hyers-Ulam stability of the system of quintic functional equation $f(x_1+x_2,y)+f(x_1-x_2,y)=2f(x_1,y)+2f(x_2,y)\;f(x,2_{y1}+y_2)+f(x,2_{y1}-y_2)=f(x,y_1-2_{y2})+f(x,y_1+y_2)\;-f(x,y_1-y_2)+15f(x,y_1)+6f(x,y_2)$ in non-Archimedean 2-Banach spaces.

• Journal of the Chungcheong Mathematical Society
• /
• v.29 no.2
• /
• pp.287-328
• /
• 2016

In the current work, we define and find the general solution of the decic functional equation g(x + 5y) - 10g(x + 4y) + 45g(x + 3y) - 120g(x + 2y) + 210g(x + y) - 252g(x) + 210g(x - y) - 120g(x - 2y) + 45g(x - 3y) - 10g(x - 4y) + g(x - 5y) = 10!g(y) where 10! = 3628800. We also investigate and establish the generalized Ulam-Hyers stability of this functional equation in Banach spaces, generalized 2-normed spaces and random normed spaces by using direct and fixed point methods.