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진선숙,이양희 영남수학회 2020 East Asian mathematical journal Vol.36 No.1
In this paper, we investigate Hyers-Ulam-Rassias stability of an additive-quartic functional equation, of a quadratic-quartic functional equation, and of a cubic-quartic functional equation.
Jin, Sun-Sook,Lee, Yang-Hi The Youngnam Mathematical Society 2020 East Asian mathematical journal Vol.36 No.1
In this paper, we investigate Hyers-Ulam-Rassias stability of an additive-quartic functional equation, of a quadratic-quartic functional equation, and of a cubic-quartic functional equation.
Stability of a Quadratic Jensen Type Functional Equation
Lee, Sang-Han 한국전산응용수학회 2002 The Korean journal of computational & applied math Vol.9 No.1
In this paper we solve a quadratic Jensen type functional equation (equation omitted) and prove the stability of this equation.
FOR THE HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION
Lee, Eun-Hwi,Chang, Ick-Soon 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.15 No.1
In this paper, we obtain the general solution of a quadratic functional equation $b^2f(\frac{x+y+z}{b})+f(x-y)+f(x-z)=\;a^2[f(\frac{x-y-z}{a})+f(\frac{x+y}{a})+f(\frac{x+z}{a})]$ and prove the stability of this equation.
Yang-Hi Lee,정순모 경남대학교 수학교육과 2018 Nonlinear Functional Analysis and Applications Vol.23 No.2
We will investigate a fuzzy version of Hyers-Ulam stability for a class of functionalm equations of the form, which includes the quadratic-additive functional equations.
Stability of a Generalized Quadratic Type Functional Equation
Kim, Mi-Hye,Hwang, In-Sung The Korea Contents Association 2002 한국콘텐츠학회논문지 Vol.2 No.4
함수 방정식은 연구원들이 함수 자체의 정확한 형태를 가정하지 않고 단순히 기본적인 함수의 성질만을 언급하는 한정적이지 않은 방정식을 통하여 일반적인 관점의 수학적 형상화를 공식화하는데 매우 중요한 구실을 하기 때문에 실험적인 학문에서 유용하다. 그러한 많은 함수 방정식 가운데에서 이 논문은 다소 일반화된 2차 함수 방정식을 선택해 해를 구하며 이 방정식의 안정성을 증명한다. a$^2$f((x+y/a))+b$^2$f((x-y/b)) = 2f(x)+2f(y) Functional equations are useful in the experimental science because they play very important role for researchers to formulate mathematical models in general terms, through some not very restrictive equations that only stipulate basic properties of functions showing in these equations, without postulating the exact forms of such functions. Of lots of such functional equations, in this paper we adopt and solve some generalized quadratic functional equation a$^2$f((x+y/a))+b$^2$f((x-y/b)) = 2f(x)+2f(y)
HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION
Trif, Tiberiu Korean Mathematical Society 2003 대한수학회보 Vol.40 No.2
In this paper we deal With the quadratic functional equation (equation omitted) deriving from an inequality of T. Popoviciu for convex functions. We solve this functional equation by proving that its solutions we the polynomials of degree at most two. Likewise, we investigate its stability in the spirit of Hyers, Ulam, and Rassias.
A VARIANT OF THE QUADRATIC FUNCTIONAL EQUATION ON GROUPS AND AN APPLICATION
Elfen, Heather Hunt,Riedel, Thomas,Sahoo, Prasanna K. Korean Mathematical Society 2017 대한수학회보 Vol.54 No.6
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 $m{\geq}2$.
A variant of the quadratic functional equation on groups and an application
Heather Hunt Elfen,Thomas Riedel,Prasanna K. Sahoo 대한수학회 2017 대한수학회보 Vol.54 No.6
Let $G$ be a group and $\mathbb{C}$ the field of complex numbers. Suppose $\sigma : G \to G$ 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 \to \mathbb{C}$, of the functional equation \begin{align*} f(xy) + f(\sigma (y) x) = 2 f(x) + 2 f(y) \quad \text{for all } x, y \in G, \end{align*} 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 $f(pr,qs)+f(sp,rq) = 2 f(p,q) + 2 f(r, s)$ for all $p, q, r, s \in G$, which is a variant of the equation $f(pr,qs)+f(ps,qr) = 2 f(p,q) + 2 f(r, s)$ studied by Chung, Kannappan, Ng and Sahoo in \cite{CKNS} (see also \cite{PKSPK}). 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 $m \geq 2$.
A MEASURE ZERO STABILITY OF A FUNCTIONAL EQUATION ASSOCIATED WITH INNER PRODUCT SPACE
Chun, Jaeyoung,Rassias, John Michael Korean Mathematical Society 2017 대한수학회지 Vol.54 No.2
Let X, Y be real normed vector spaces. We exhibit all the solutions $f:X{\rightarrow}Y$ of the functional equation f(rx + sy) + rsf(x - y) = rf(x) + sf(y) for all $x,y{\in}X$, where r, s are nonzero real numbers satisfying r + s = 1. In particular, if Y is a Banach space, we investigate the Hyers-Ulam stability problem of the equation. We also investigate the Hyers-Ulam stability problem on a restricted domain of the following form ${\Omega}{\cap}\{(x,y){\in}X^2:{\parallel}x{\parallel}+{\parallel}y{\parallel}{\geq}d\}$, where ${\Omega}$ is a rotation of $H{\times}H{\subset}X^2$ and $H^c$ is of the first category. As a consequence, we obtain a measure zero Hyers-Ulam stability of the above equation when $f:\mathbb{R}{\rightarrow}Y$.