• Kyungpook Mathematical Journal
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• 제48권1호
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• pp.45-61
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• 2008

We investigate the generalized Hyers-Ulam-Rassias stability in p-Banach spaces for the following functional equation which is two types, that is, either cubic or quadratic: 2f(x+3y) + 6f(x-y) + 12f(2y) = 2f(x - 3y) + 6f(x + y) + 3f(4y). The concept of Hyers-Ulam-Rassias stability originated essentially with 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.

• 한국환경과학회지
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• 제28권11호
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• pp.1027-1032
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• 2019

This study aimed to analyze the prioritized area for the functional change of agricultural reservoir according to the rapid urbanization and social changes through the mapping method. Changwon-si, Gimhae-si, and Jinju-si in Gyeongnam province were selected as the study area, considering the results of land-use and cluster analysis. As the planning unit of management area, watershed was used and land coverage map from 1975 to 2015 were analyzed for changes of land use. The reduction ratio (%) of farmland was calculated for identifying the changes in 2013 compared to 1975. As a result, the reduction ratio was 11.9% for Changwon-si, 12.2% for Gimhae-si and 9.3% for Jinju-si, and the number of watershed having functionally reduced potential-area was relatively high in the proportion to the city size.

• Nonlinear Functional Analysis and Applications
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• 제26권4호
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• pp.749-780
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• 2021

In this paper, we give the notion of M(., .)-𝜂-proximal mapping for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. As an application, we introduce and investigate a new system of variational-like inclusions in Banach spaces. By means of M(., .)-𝜂-proximal mapping method, we give the existence of solution for the system of variational inclusions. Further, propose an iterative algorithm for finding the approximate solution of this class of variational inclusions. Furthermore, we discuss the convergence and stability analysis of the iterative algorithm. The results presented in this paper may be further expolited to solve some more important classes of problems in this direction.

• Korean Journal of Mathematics
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• 제31권3호
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• pp.363-372
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• 2023

In this paper, we consider two functional equations: (1) h(𝓕(x, y, z) + 2x + y + z) + h(xy + z) + yh(x) + yh(z) = h(𝓕(x, y, z) + 2x + y) + h(xy) + yh(x + z) + 2h(z), (2) h(𝓕(x, y, z) - y + z + 2e) + 2h(x + y) + h(xy + z) + yh(x) + yh(z) = h(𝓕(x, y, z) - y + 2e) + 2h(x + y + z) + h(xy) + yh(x + z), without any regularity assumption for all x, y, z in a unital algebra A, where 𝓕 : A³ → A is defined by 𝓕(x, y, z) := h(x + y + z) - h(x + y) - h(z) for all x, y, z ∈ A. Also, we find general solutions of these equations in unital algebras. Finally, we prove the Hyers-Ulam stability of (1) and (2) in unital Banach algebras.

• 생물정신의학
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• 제22권3호
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• pp.101-108
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• 2015

Recent breakthroughs in functional neuroimaging techniques have launched the quest of mapping the connections of the human brain, otherwise known as the human connectome. Imaging connectomics is an umbrella term that refers to the neuroimaging techniques used to generate these maps, which recently has enabled comprehensive brain mapping of network connectivity combined with graph theoretic methods. In this review, we present an overview of the key concepts in functional connectomics. Furthermore, we discuss articles that applied task-based and/or resting-state functional magnetic resonance imaging to examine network deficits in post-traumatic stress disorder (PTSD). These studies have provided important insights regarding the etiology of PTSD, as well as the overall organization of the brain network. Advances in functional connectomics are expected to provide insight into the pathophysiology and the development of biomarkers for diagnosis and treatment of PTSD.

• 충청수학회지
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• 제21권4호
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• pp.455-466
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• 2008

In, [7], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer $n{\geq}2$ $$n{\left\|{\frac{1}{n}}{\sum\limits_{i=1}^{n}}x_i{\left\|^2+{\sum\limits_{i=1}^{n}}\right\|}{x_i-{\frac{1}{n}}{\sum\limits_{j=1}^{n}x_j}}\right\|^2}={\sum\limits_{i=1}^{n}}{\parallel}x_i{\parallel}^2$$ holds for all $x_1,{\cdots},x_{n}{\in}V$. Let V,W be real vector spaces. It is shown that if a mapping $f:V{\rightarrow}W$ satisfies $$(0.1){\hspace{10}}nf{\left({\frac{1}{n}}{\sum\limits_{i=1}^{n}}x_i \right)}+{\sum\limits_{i=1}^{n}}f{\left({x_i-{\frac{1}{n}}{\sum\limits_{j=1}^{n}}x_i}\right)}\\{\hspace{140}}={\sum\limits_{i=1}^{n}}f(x_i)$$ for all $x_1$, ${\dots}$, $x_{n}{\in}V$ $$(0.2){\hspace{10}}2f\(\frac{x+y}{2}\)+f\(\frac{x-y}{2} \)+f\(\frac{y}{2}-x\)\\{\hspace{185}}=f(x)+f(y)$$ for all $x,y{\in}V$. Furthermore, we prove the generalized Hyers-Ulam stability of the functional equation (0.2) in real Banach spaces.

• 충청수학회지
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• 제24권4호
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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 $n=0,1,2,{\cdots}$.

• 한국정보전자통신기술학회논문지
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• 제12권3호
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• pp.257-264
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• 2019

최근 이더넷 신호를 융통성 있게 구성하도록 하는 FlexE는 OTN을 통해 투명하게 전달하기 위하여 OTN으로 수용하는 것을 ITU-T에서 권고하고 있다. FlexE신호를 OTN 페이로드로 맵핑할 때 발생할 수 있는 비트율 차이를 보상하기 위하여 유휴 코드워드를 제거하거나 삽입하는 방법을 사용한다. 그러나 이와 같은 방법을 구현하기 위해 필요한 세부적인 기능 블록은 아직 공개되고 있지 않다. 본 논문에서는 최근 ITU-T 요구사항을 기초로 하여 400G급을 기준으로 FlexE 신호의 OTN 맵핑을 위한 세부적인 기능 블록을 살펴보았다. 또한 세부 기능 블록을 바탕으로 유휴 코드워드의 제거 및 삽입되는 발생 특성을 얻기 위하여 수학적인 분석을 하였고, 이를 이용한 시뮬레이션을 결과를 보였다.

• 대한수학회보
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• 제38권3호
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• pp.435-441
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• 2001

We obtain a characterization of linear operators on vector spaces and homomorphisms on algebras applying the stability properties of functional equations.

• 충청수학회지
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• 제32권1호
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• pp.1-14
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• 2019

We prove the generalized Hyers-Ulam-Rassias stability problem of the quadratic functional equation with involution in the fuzzy quasi ${\beta}$-normed space by using the fixed point method.