• Honam Mathematical Journal
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• v.34 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2165-2182
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• 2017

Let G be a group and $\mathbb{C}$ the field of complex numbers. Suppose ${\sigma}:G{\rightarrow}G$ is an endomorphism satisfying ${{\sigma}}({{\sigma}}(x))=x$ for all x in G. In this paper, we first determine the central solution, f : G or $G{\times}G{\rightarrow}\mathbb{C}$, of the functional equation $f(xy)+f({\sigma}(y)x)=2f(x)+2f(y)$ for all $x,y{\in}G$, which is a variant of the quadratic functional equation. Using the central solution of this functional equation, we determine the general solution of the functional equation f(pr, qs) + f(sp, rq) = 2f(p, q) + 2f(r, s) for all $p,q,r,s{\in}G$, which is a variant of the equation f(pr, qs) + f(ps, qr) = 2f(p, q) + 2f(r, s) studied by Chung, Kannappan, Ng and Sahoo in [3] (see also [16]). Finally, we determine the solutions of this equation on the free groups generated by one element, the cyclic groups of order m, the symmetric groups of order m, and the dihedral groups of order 2m for $m{\geq}2$.

• Communications of the Korean Mathematical Society
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• v.16 no.2
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• pp.211-223
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• 2001

In this paper, we obtain the modified Hyers-Ulam-Rassias stability for the family of the functional equation f(x o y) = H(f(x)(sup)1/t, f(y)(sup)1/t)(x,y) $\in$S), where H is a s homogeneous function of degree t and o is a square-symmetric operation on the set S.

• Journal of applied mathematics & informatics
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• v.42 no.1
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• pp.85-92
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• 2024

In this work we give the necessary and sufficient conditions for f-biharmonic curves in three-dimensional generalized symmetric spaces.

• Journal of applied mathematics & informatics
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• v.8 no.2
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• pp.569-580
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• 2001

In this paper, we obtain some results on the Hyers-Ulam stability for the family of the functional equation f(xоy) = $H(f(x)^{1/t},f(y)^{1/t}$) (x, $y{\in}S$), where H is a homogeneous function of degree t and о is a square-symmetric operation on the set S.

• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.65-73
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• 2003

In this paper we seek a positive, radially symmetric and energy minimizing solution of an m-Laplacian equation, -div$($\mid${\nabla}u$\mid$^{m-2}$\mid${\nabla}u)\;=\;h(u)$. In the variational sense, the solutions are the critical points of the associated functional called the energy, $J(v)\;=\;\frac{1}{m}\;\int_{R^N}\;$\mid${\nabla}v$\mid$^m\;-\;\int_{R^N}\;H(v)dx,\;where\;H(v)\;=\;{\int_0}^v\;h(t)dt$. A positive, radially symmetric critical point of J can be obtained by solving the constrained minimization problem; minimize{$\int_{R^N}$\mid${\nabla}u$\mid$^mdx$\mid$\;\int_{R^N}\;H(u)d;=\;1$}. Moreover, the solution minimizes J(v).

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.157-164
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• 2021

Recently we investigated a type of Hyers-Ulam stability of the Schrödinger equation with the symmetric parabolic wall potential that efficiently describes the quantum harmonic oscillations. In this paper we study a type of Hyers-Ulam stability of the Schrödinger equation when the potential barrier is a quartic wall in the solid crystal models.

• The Pure and Applied Mathematics
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• v.28 no.1
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• pp.1-13
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• 2021

In this note we use a generalization of coincidence point(a property which was defined by [1] in symmetric spaces) to prove common fixed point theorem on S-metric spaces for weakly compatible maps. Also the results are used to achieve the solution of an integral equation and the bounded solution of a functional equation in dynamic programming.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.239-251
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• 2019

In this paper we introduce a new formulation of symmetric homogeneous bivariate means that depends on the variation of a given continuous strictly increasing function on (0, ${\infty}$). It turns out that this class of means includes a lot of known bivariate means among them the arithmetic mean, the harmonic mean, the geometric mean, the logarithmic mean as well as the first and second Seiffert means. Using this new formulation we introduce a lot of new bivariate means and derive some mean-inequalities.

• Bulletin of the Society of Naval Architects of Korea
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• v.25 no.3
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• pp.3-12
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• 1988

A natural or an artificial harbor can exhibit frequency(or period) dependent water surface oscillations when excited by incident waves. Such oscillations in harbors can cause significant damage to moored ships and adjacent structures. This can also induce undesirable current in harbors. Many previous investigators have studied various aspects of harbor resonance problem. In the percent paper, both a localizes finite element method(LFEM) which is based on the functional constructed by Chen & Mei(1974) and Bai & Yeung(1974) and an integral equation method which was used by Lee(1969) are applied to harbor resonance problem. The present method(LFEM) shows computationally more efficient than the integral equation method. Our test results shows good agreement compared with other results. This enhanced computational efficiency is due to the fact that the present method gives a banded symmetric coefficients matrix and requires much less computational time in the calculation of the influence coefficients matrix than the integral equation method involved with Green's function. To test the present numerical scheme, two models are treated here. The present method(LFEM) can be extended to a fully three dimensional harbor problem with the similar computational advantage.