• Korean Journal of Mathematics
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• 제17권3호
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• pp.315-330
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• 2009

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following cubic-quadratic functional equation $$(0.1)\;\frac{1}{2}(f(2x+y)+f(2x-y)-f(-2x-y)-f(y- 2x))\\{\hspace{35}}=2f(x+y)+2f(x-y)+4f(x)-8f(-x)-2f(y)-2f(-y)$$ in fuzzy Banach spaces.

• 호남수학학술지
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• 제29권2호
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• pp.193-204
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• 2007

In this paper, we investigate some results concerning the stability of the following quadratic type functional equation: f(x + y) + f(x - y) + f(y + z) + f(y - z) + f(z + x) + f(z - x) = 4f(x) + 4f(y) + 4f(z).

• Journal of applied mathematics & informatics
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• 제18권1_2호
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• pp.563-572
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• 2005

In this paper, we obtain the general solution and the generalized Hyers-Ulam stability of the additive-quadratic functional equation f(x + y, z + w) + f(x + y, z - w) = 2f(x, z)+2f(x, w)+2f(y, z)+2f(y, w).

• 충청수학회지
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• 제30권4호
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• pp.467-476
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• 2017

In this paper, we obtain the stability of the additive-quadratic functional equation f(x+y, z+w)+f(x+y, z-w) = 2f(x, z)+2f(x, w)+2f(y, z)+2f(y, w) and the bi-Jensen functional equation $$4f(\frac{x+y}{2},\;\frac{z+w}{2})=f(x,\;z)+f(x,\;w)+f(y,\;z)+f(y,\;w)$$ in 2-Banach spaces.

• 대한수학회논문집
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• 제25권3호
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• pp.405-417
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• 2010

We prove the generalized Hyers-Ulam stability of the generalized quadratic functional equation $$f(rx\;+\;sy)\;=\;r^2f(x)\;+\;s^2f(y)\;+\;\frac{rs}{2}[f(x\;+\;y)\;-\;f(x\;-\;y)]$$ in fuzzy Banach spaces, where r, s are non-zero rational numbers with $r^2\;+\;s^2\;{\neq}\;1$.

• 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 $m{\geq}2$.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1421-1433
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• 2011

In this paper, using a fixed point approach, the generalized Hyeres-Ulam stability of the following quadratic functional equation $f(x+y+z)+f(x+{\sigma}(y))+f(y+{\sigma}(z))+f(x+{\sigma}(z))=3(f(x)+f(y)+f(z))$ will be studied, where f is a function from abelian group G into a quasi-${\beta}$-normed space and ${\sigma}$ is an involution on the group G. Next, we consider its pexiderized equation of the form $f(x+y+z)+f(x+{\sigma}(y))+f(y+{\sigma}(z))+f(x+{\sigma}(z))=g(x)+g(y)+g(z)$ and its generalized Hyeres-Ulam stability.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제18권3호
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• pp.269-277
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• 2014

A numerical scheme is proposed to control the BBMB (Benjamin-Bona-Mahony-Burgers) equation, and the scheme consists of three steps. Firstly, BBMB equation is converted to a finite set of nonlinear ordinary differential equations by the quadratic B-spline finite element method in spatial. Secondly, the controller is designed based on the linear quadratic regulator (LQR) theory; Finally, the system of the closed loop compensator obtained on the basis of the previous two steps is solved by the backward Euler method. The controlled numerical solutions are obtained for various values of parameters and different initial conditions. Numerical simulations show that the scheme is efficient and feasible.

• 한국수학사학회지
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• 제30권3호
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• pp.137-162
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• 2017

The Chinese algebraic method, the tian yuan shu, was developed during Song period (960-1279), of which Li Ye's works contain the earliest testimony. Two 18th century editors commentated on his works: the editor of the Siku quanshu and Li Rui, the latter responding to the former. Korean scholar Nam Byeong-gil added another response in 1855. Differences can be found in the way these commentators considered mathematical objects and procedures. The conflicting nature of these commentaries shows that the same object, the quadratic equation, can beget different interpretations, either a procedure or an assertion of equality. Textual elements in this paper help modern readers reconstruct different authors' understandings and reconsider the evolution of the definition of the object we now call 'equation'.

• 충청수학회지
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• 제25권2호
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• pp.171-182
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• 2012

In this paper, for any fixed integer $n\;>\;m\;{\geq}\;1$, we investigate the generalized Hyers-Ulam stability of the following quadratic functional equation $f(nx+my)\;+\;f(nx-my)\;=\;mn[f(x+y)\;+\;f(x-y)]\;+\;2(n-m)[nf(x)\;-\;mf(y)]$ in ${p}$-Banach spaces, where $0\;<\;p\;{\leq}\;1$. And we prove the same stability of the above functional equation in 2-Banach spaces.