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REFINED HYERS{ULAM STABILITY FOR JENSEN TYPE MAPPINGS
John Michael Rassias,Ju ri Lee,Hark Mahn Kim 충청수학회 2009 충청수학회지 Vol.22 No.1
In 1940 S.M. Ulam proposed the famous Ulam stability problem. In 1941 D.H. Hyers solved the well-known Ulam stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. In this paper we improve results for Jensen type mappings and establish new theorems about the Ulam stability of additive and alternative Jensen type mappings.
JOHN MICHAEL RASSIAS,HEMEN DUTTA,NARASIMMAN PASUPATHI 장전수학회 2019 Proceedings of the Jangjeon mathematical society Vol.22 No.2
The aim of this article is to study the Hyers-Ulam-Rassias stability and generalized Hyers-Ulam-Rassias stability in non-Archimedean Intuitionistic fuzzy normed spaces. The paper introduces a new A- quartic functional equation and obtain solution for the same functional equation. Further, stability problem is investigated for the newly intro- duced A-quartic functional equation in non-Archimedean intuitionistic fuzzy normed spaces.
A MEASURE ZERO STABILITY OF A FUNCTIONAL EQUATION ASSOCIATED WITH INNER PRODUCT SPACE
정재영,John Michael Rassias 대한수학회 2017 대한수학회지 Vol.54 No.2
Let $X, Y$ be real normed vector spaces. We exhibit all the solutions $f:X\to Y$ 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 $\Omega\cap\{(x, y)\in X^2:\|x\|+\|y\|\ge d\}$, 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 $f:\mathbb R\to Y$.
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 $f:\mathbb{R}{\rightarrow}Y$.
Orthogonal Stability of an Euler-Lagrange-Jensen (a, b)-cubic Functional Equation
NARASIMMAN PASUPATHI,JOHN MICHAEL RASSIAS,이정례,심은화 한국수학교육학회 2022 純粹 및 應用數學 Vol.29 No.2
In this paper, we introduce a new generalized (a,b)-cubic Euler-Lagrange-Jensen functional equation and obtain its general solution. Furthermore, we prove the Hyers-Ulam stability of the new generalized (a, b)-cubic Euler-Lagrange-Jensen functional equation in orthogonality normed spaces.
Dongwen Zhang,JOHN MICHAEL RASSIAS,Yongjin Li 강원경기수학회 2022 한국수학논문집 Vol.30 No.4
By established a Banach space with the Hausdorff distance, we introduce the alternative fixed-point theorem to explore the existence and uniqueness of a fixed subset of Y and investigate the stability of set-valued Euler-Lagrange functional equations in this space. Some properties of the Hausdorff distance are furthermore explored by a short and simple way.
ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION
CHUNG, JAEYOUNG,RASSIAS, JOHN MICHAEL Cambridge University Press 2016 Bulletin of the Australian Mathematical Society Vol.93 No.2
<P>Let $G$ be a commutative group, $Y$ a real Banach space and $f:G\rightarrow Y$. We prove the Ulam-Hyers stability theorem for the cyclic functional equation $$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$ for all $x,y\in {\rm\Omega}$, where $H$ is a finite cyclic subgroup of $\text{Aut}(G)$ and ${\rm\Omega}\subset G\times G$ satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation $$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$ for all $(z,{\it\zeta})\in {\rm\Omega}$, where $f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$ and ${\rm\Omega}\subset \mathbb{C}^{2}$ has four-dimensional Lebesgue measure $0$.</P>
THE GENERAL SOLUTION AND APPROXIMATIONS OF A DECIC TYPE FUNCTIONAL EQUATION IN VARIOUS NORMED SPACES
Arunkumar, Mohan,Bodaghi, Abasalt,Rassias, John Michael,Sathya, Elumalai Chungcheong Mathematical Society 2016 충청수학회지 Vol.29 No.2
In the current work, we define and find the general solution of the decic functional equation g(x + 5y) - 10g(x + 4y) + 45g(x + 3y) - 120g(x + 2y) + 210g(x + y) - 252g(x) + 210g(x - y) - 120g(x - 2y) + 45g(x - 3y) - 10g(x - 4y) + g(x - 5y) = 10!g(y) where 10! = 3628800. We also investigate and establish the generalized Ulam-Hyers stability of this functional equation in Banach spaces, generalized 2-normed spaces and random normed spaces by using direct and fixed point methods.