• 호남수학학술지
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• 제26권1호
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• pp.45-59
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• 2004

In this paper we obtain the general solution the functional equation $a^2f(\frac{x-2y}{a})+f(x)+2f(y)=2a^2f(\frac{x-y}{a})+f(2y).$ The type of the solution of this equation is Q(x)+A(x)+C, where Q(x), A(x) and C are quadratic, additive and constant, respectively. Also we prove the stability of this equation in the spirit of Hyers, Ulam, Rassias and $G\check{a}vruta$.

• 호남수학학술지
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• 제34권3호
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• pp.375-379
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• 2012

We find a general solution $f:G{\rightarrow}G$ of the symmetric functional equation $$x+f(y+f(x))=y+f(x+f(y)),\;f(0)=0$$ where G is a 2-divisible abelian group. We also prove that there exists no measurable solution $f:\mathbb{R}{\rightarrow}\mathbb{R}$ of the equation. We also find the continuous solutions $f:\mathbb{C}{\rightarrow}\mathbb{C}$ of the equation.

• 대한수학회보
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• 제50권6호
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• pp.2061-2070
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• 2013

In this paper, we investigate the generalized HyersUlam-Rassias stability of the following additive functional equation $$2\sum_{j=1}^{n}f(\frac{x_j}{2}+\sum_{i=1,i{\neq}j}^{n}\;x_i)+\sum_{j=1}^{n}f(x_j)=2nf(\sum_{j=1}^{n}x_j)$$, in quasi-${\beta}$-normed spaces.

• 충청수학회지
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• 제21권1호
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• pp.1-14
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• 2008

In this paper, we prove the Hyers-Ulam-Rassias stability of the Cauchy functional equation f(x+y) = f(x)+f(y) and of the Jensen functional equation $2f(\frac{x+y}{2})=f(x)+f(y)$ over the p-adic field ${\mathbb{Q}}_p$. 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.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제22권4호
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• pp.343-357
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• 2015

Using the fixed point method, we prove the Hyers-Ulam stability of lin- ear mappings in Banach modules over a unital C*-algebra and in non-Archimedean Banach modules over a unital C*-algebra associated with the orthogonally Cauchy- Jensen additive functional equation.

• 충청수학회지
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• 제20권4호
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• pp.503-514
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• 2007

In this paper, we investigate the generalized Hyers-Ulam stability of the functional inequality $$||af(x)+bf(y)+cf(z)||{\leq}||f(ax+by+cz))||+{\phi}(x,y,z)$$ associated with Cauchy additive mappings. As a result, we obtain that if a mapping satisfies the functional inequality with perturbing term which satisfies certain conditions then there exists a Cauchy additive mapping near the mapping.

• Journal of applied mathematics & informatics
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• 제10권1_2호
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• pp.283-295
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• 2002

In this Paper we show the Solution of the following Jensen functional equation with three variables and prove the stability of this equations in the spirit of Hyers, Ulam, Rassias and Gavruta: (equation omitted).

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제23권3호
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• pp.265-285
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• 2016

In this paper, we introduce functional equations in G-normed spaces and we prove the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in complete G-normed spaces by using the fixed point method.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제16권3호
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• pp.297-306
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• 2009

In this paper, we obtain the general solution and the generalized HyersUlam stability theorem for an additive functional equation $af(x+y)+2f({\frac{x}{2}}+y)+2f(x+{\frac{y}{2})=(a+3)[f(x)+f(y)]$for any fixed integer a.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제25권3호
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• pp.219-227
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• 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.