• 대한수학회논문집
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• 제31권4호
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• pp.819-825
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• 2016

In this paper, we establish approximation of bi-quadratic functional equations in Lipschitz spaces.

• 대한수학회논문집
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• 제31권1호
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• pp.101-113
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• 2016

In this article, we considered the stability of the following (${\alpha}$, ${\beta}$, ${\gamma}$)-derivation $${\alpha}D[x,y]={\beta}[D(x),y]+{\gamma}[x,D(y)]$$ and homomorphisms associated to the quadratic type functional equation $$f(kx+y)+f(kx+{\sigma}(y))=2kg(x)+2g(y),\;x,y{\in}A$$, where ${\sigma}$ is an involution of the Lie $C^*$-algebra A and k is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제27권1호
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• pp.25-33
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• 2020

In this paper, we consider the following quadratic (ρ₁, ρ₂)-functional equation (0, 1) $$N(2f({\frac{x+y}{2}})+2f({\frac{x-y}{2}})-f(x)-f(y)-{\rho}_1(f(x+y)+f(x-y)-2f(x)-2f(y))-{\rho}_2(4f({\frac{x+y}{2}})+f(x-y)-f(x)-f(y)),t){\geq}{\frac{t}{t+{\varphi}(x,y)}}$$, where ρ₂ are fixed nonzero real numbers with ρ₂ ≠ 1 and 2ρ₁ + 2ρ₂≠ 1, in fuzzy normed spaces. Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic (ρ₁, ρ₂)-functional equation (0.1) in fuzzy Banach spaces.

• 충청수학회지
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• 제29권1호
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• pp.83-93
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• 2016

In this paper, we introduce a generalized quadratic functional equation with several variables and then investigate its generalized Hyers-Ulam stability in normed spaces.

• 한국콘텐츠학회논문지
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• 제3권3호
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• pp.103-109
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• 2003

함수 방정식은 함수 자체의 정확한 형태를 가정하지 않고 단순히 기본적인 성질만을 언급하는 한정적이지 않은 방정식을 통하여 일반적인 관점의 수학적 형상화를 공식화하는데 매우 중요한 구실을 하기 때문에 실험적인 학문에서 유용하다. 본 논문에서는 범함수 방정식 문제 중 하나인 일반화된 2차 범함수 방정식(equation omitted) 의 해를 구하고 이 방정식의 안정성을 증명한다.

• 대한수학회보
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• 제38권2호
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• pp.325-336
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• 2001

In this paper, we investigate the generalized Hyers-Ulam-Rassias stability of a quadratic function equation f(x+y+z)+f(x-y)+f(y-z)+f(z-x)=3f(x)+3f(y)+3f(z) and prove the Hyers-Ulam-Rassias stability of the equation on bounded domain.

• Kyungpook Mathematical Journal
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• 제49권4호
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• pp.691-700
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• 2009

We use the fixed alternative theorem to establish Hyers-Ulam-Rassias stability of the quadratic functional equation where functions map a linear space into a complete quasi p-normed space. Moreover, we will show that the continuity behavior of an approximately quadratic mapping, which is controlled by a suitable continuous function, implies the continuity of a unique quadratic function, which is a good approximation to the mapping. We also give a few applications of our results in some special cases.

• 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 $m{\geq}2$.

• 대한수학회지
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• 제38권6호
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• pp.1179-1190
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• 2001

Quadratic invariant curve is one of the simplest nonlinear invariant curves and was considered by C. T. Ng and the author in order to study the one-dimensional nonlinear dynamics displayed by a second order delay differential equation with piecewise constant argument. In this paper a functional equation derived from the problem of invariant curves is discussed. Using a different method from what C. T. Ng and the author once used, we define solutions piecewisely and give results in the remaining difficult case left in C. T. Ng and the authors work. A problem of analytic extension given in their work is also answered negatively.

• 충청수학회지
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• 제17권1호
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• pp.57-75
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• 2004

The aim of this paper is to solve the general solution of a quadratic functional equation f(x + 2y) + 2f(x - y) = f(x - 2y) + 2f(x + y) in the class of functions between real vector spaces and to obtain the generalized Hyers-Ulam stability problem for the equation.