• 한국수학교육학회지시리즈B:순수및응용수학
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• 제15권2호
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• pp.169-178
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• 2008

In this paper we generalize the superstability of the exponential functional equation proved by J. Baker et al. [2], that is, we solve an exponential type functional equation $$f(x+y)\;=\;a^{xy}f(x)f(y)$$ and obtain the superstability of this equation. Also we generalize the stability of the exponential type equation in the spirt of R. Ger[4] of the following setting $$|{\frac{f(x\;+\;y)}{{a^{xy}f(x)f(y)}}}\;-\;1|\;{\leq}\;{\delta}.$$

• 호남수학학술지
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• 제27권1호
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• pp.51-68
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• 2005

In this paper we solve a generalized Popoviciu functional equation and investigate the stability of this equation.

• 대한수학회보
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• 제54권6호
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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 $m{\geq}2$.

• 충청수학회지
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• 제20권4호
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• pp.449-455
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• 2007

We obtain a discrete version of a fundamental result by Yoshizawa related to the existence of almost periodic solutions of functional differential equations with finite delay.

• 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.

• 대한수학회지
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• 제59권2호
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• pp.279-298
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• 2022

In this paper, the existence and uniqueness for the global solution of neutral stochastic functional differential equation is investigated under the locally Lipschitz condition and the contractive condition. The implicit iterative methodology and the Lyapunov-Razumikhin theorem are used. The stability analysis for such equations is also applied. One numerical example is provided to illustrate the effectiveness of the theoretical results obtained.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.517-522
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• 2006

In this paper, we obtain the general solution and the generalized Hyers-Ulam stability of the multi-additive functional equation $f(x1+x2,y1+y2,z1+z2)={\Sigma}_{1{\le}i,j,k{\le}2}\;f(x1,yj,zk)$.

• 호남수학학술지
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• 제30권2호
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• pp.219-225
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• 2008

In this paper, we obtain the general solution and the stability of the 2-dimensional vector variable quadratic functional equation f( x + y, z - w) + f(x - y, z + w) = 2f(x, z ) + 2f(y, ${\omega}$). The quadratic form f( x, y) = $ax^2$ + $by^2$ without cross product terms is a solution of the above functional equation.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제17권4호
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• pp.363-372
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• 2010

In this paper we obtain the Hyers-Ulam stability of functional equations $f(x+y)=f(x)+f(y)+In\;{\alpha}^{2xy-1}$ and $f(x+y)=f(x)+f(y)+In\;{\beta(x,y)^{-1}$ which is related to the exponential and beta functions.

• 대한수학회보
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• 제43권3호
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• pp.499-507
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• 2006

In this paper, we obtain the general solution and the stability of the hi-Jensen functional equation $$4f(\frac {x+y} 2,\;\frac {z+w} 2)=f(x,\;z)+f(x,\;w)+f(y,\;z)+f(y,\;w)$$.