• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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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.

• Korean Journal of Mathematics
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• v.23 no.3
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• pp.393-399
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• 2015

W. Park [J. Math. Anal. Appl. 376 (2011) 193-202] proved the Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces. But there are serious problems in the control functions given in all theorems of the paper. In this paper, we correct the statements of these results and prove the corrected theorems. Moreover, we prove the superstability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces under the original given conditions.

• Communications of the Korean Mathematical Society
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• v.23 no.3
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• pp.371-376
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• 2008

In this paper, it is shown that if f satisfies the following functional inequality (0.1) $${\parallel}\sum\limits_{i,j=1}^3\;f{(xi,yj)}{\parallel}{\leq}{\parallel}f(x_1+x_2+x_3,\;y_1+y_2+y_3){\parallel}$$ then f is a bi-additive mapping. We moreover prove that if f satisfies the following functional inequality (0.2) $${\parallel}2\sum\limits_{j=1}^3\;f{(x_j,\;z)}+2\sum\limits_{j=1}^3\;f{(x_j,\;w)-f(\sum\limits_{j=1}^3\;xj,\;z-w)}{\parallel}{\leq}f(\sum\limits_{j=1}^3\;xj,\;z+w){\parallel}$$ then f is an additive-quadratic mapping.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.1
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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.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.2
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• pp.195-205
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• 2019

We consider additive mappings similar to derivations on Banach ${\ast}$-algebras and we will first study the conditions for such additive mappings on Banach ${\ast}$-algebras. Then we prove some theorems concerning approximate linear mappings of derivation-type on Banach ${\ast}$-algebras. As an application, approximate linear mappings of derivation-type on $C^{\ast}$-algebra are characterized.

• Kyungpook Mathematical Journal
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• v.60 no.1
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• pp.133-144
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• 2020

We will prove the generalized Hyers-Ulam stability of cubic-quadratic-additive type functional equations and general cubic functional equations whose solutions are cubic-quadratic-additive mappings and general cubic mappings, respectively.

• Journal of Applied and Pure Mathematics
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• v.4 no.1_2
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• pp.1-7
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• 2022

A mapping T : R → R (not necessarily additive) is called multiplicative left 𝛼-centralizer if T(xy) = T(x)𝛼(y) for all x, y ∈ R. A mapping F : R → R (not necessarily additive) is called multiplicative (generalized)(𝛼, 𝛽)-derivation if there exists a map (neither necessarily additive nor derivation) f : R → R such that F(xy) = F(x)𝛼(y) + 𝛽(x)f(y) for all x, y ∈ R, where 𝛼 and 𝛽 are automorphisms on R. The main purpose of this paper is to study some algebraic identities with multiplicative (generalized) (𝛼, 𝛽)-derivations and multiplicative left 𝛼-centralizer on the left ideal of a prime ring R.

• Communications for Statistical Applications and Methods
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• v.30 no.3
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• pp.259-272
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• 2023

In this paper, we suggest a new method for the prediction of sharp changes in particulate matter (PM₁₀) using quantile mapping. To predict the current PM₁₀ density in Seoul, we consider PM₁₀ and precipitation in Baengnyeong and Ganghwa monitoring stations observed a few hours before. For the PM₁₀ distribution estimation, we use the extreme value mixture model, which is a combination of conventional probability distributions and the generalized Pareto distribution. Furthermore, we also consider a quantile generalized additive model (QGAM) for the relationship modeling between precipitation and PM₁₀. To prove the validity of our proposed model, we conducted a simulation study and showed that the proposed method gives lower mean absolute differences. Real data analysis shows that the proposed method could give a more accurate prediction when there are sharp changes in PM₁₀ in Seoul.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.219-231
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• 2009

For a mapping satisfying the inequality ${\parallel}{\lambda}f(x)+2{\lambda}f(y)+2f({\lambda}z){\parallel}{\leq}{\parallel}2f({\lambda}({\frac{2}{x}}+y+z)){\parallel}+{\phi}(x,y,z)$, we will study the stability problem of this mapping.