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d-ISOMETRIC LINEAR MAPPINGS IN LINEAR d-NORMED BANACH MODULES
박춘길,Themistocles M. Rassias 대한수학회 2008 대한수학회지 Vol.45 No.1
We prove the Hyers–Ulam stability of linear d-isometries in linear d-normed Banach modules over a unital C*-algebra and of linear isometries in Banach modules over a unital C*-algebra. The main purpose of this paper is to investigate d-isometric C*-algebra isomorphisms between linear d-normed C*-algebras and isometric C*-algebra isomorphisms between C*-algebras, and d-isometric Poisson C*-algebra isomorphisms between linear d-normed Poisson C*-algebras and isometric Poisson C*-algebra isomorphisms between Poisson C*-algebras. We moreover prove the Hyers–Ulam stability of their d-isometric homomorphisms and of their isometric homomorphisms. We prove the Hyers–Ulam stability of linear d-isometries in linear d-normed Banach modules over a unital C*-algebra and of linear isometries in Banach modules over a unital C*-algebra. The main purpose of this paper is to investigate d-isometric C*-algebra isomorphisms between linear d-normed C*-algebras and isometric C*-algebra isomorphisms between C*-algebras, and d-isometric Poisson C*-algebra isomorphisms between linear d-normed Poisson C*-algebras and isometric Poisson C*-algebra isomorphisms between Poisson C*-algebras. We moreover prove the Hyers–Ulam stability of their d-isometric homomorphisms and of their isometric homomorphisms.
Moradlou, Fridoun,Rassias, Themistocles M. Korean Mathematical Society 2013 대한수학회보 Vol.50 No.6
In this paper, we investigate the generalized HyersUlam-Rassias stability of the following additive functional equation $$2\sum_{j=1}^{n}f(\frac{x_j}{2}+\sum_{i=1,i{\neq}j}^{n}\;x_i)+\sum_{j=1}^{n}f(x_j)=2nf(\sum_{j=1}^{n}x_j)$$, in quasi-${\beta}$-normed spaces.
Fridoun Moradlou,Themistocles M. Rassias 대한수학회 2013 대한수학회보 Vol.50 No.6
In this paper, we investigate the generalized HyersUlam– Rassias stability of the following additive functional equation [수식] in quasi-β-normed spaces.
On a Generalized Trif's Mapping in Banach Modules over a$C^*$-Algebra
박춘길,Themistocles M. Rassias 대한수학회 2006 대한수학회지 Vol.43 No.2
Let $X$ and $Y$ be vector spaces. It is shown that a mapping $f : X \rightarrow Y$ satisfies the functional equation $$ \aligned &\ mn \ {_{mn-2}}C_{k-2}f\bigg(\frac{x_1+\cdots + x_{mn}}{mn}\bigg) \\& + m \ {_{mn-2}}C_{k-1}\ \sum_{i=1}^n f\bigg(\frac{x_{mi-m+1} + \cdots + x_{mi}}{m}\bigg) \\ =&\ k \ \sum_{1\le i_1<\cdots <i_k\le mn}f\bigg(\frac{x_{i_1}+\cdots + x_{i_k}}{k}\bigg) \endaligned\tag\ddag $$ if and only if the mapping $f : X \rightarrow Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation {\rm $(\ddag)$} in Banach modules over a unital $C^*$-algebra. Let $\Cal A$ and $\Cal B$ be unital $C^*$-algebras or Lie $JC^*$-algebras. As an application, we show that every almost homomorphism $h : \Cal A \rightarrow \Cal B$ of $\Cal A$ into $\Cal B$ is a homomorphism when $h(2^d u y) = h(2^d u) h(y)$ or $h(2^d u \circ y) = h(2^d u)\circ h(y)$ for all unitaries $u \in \Cal A$, all $y \in \Cal A$, and $d = 0, 1, 2, \ldots$, and that every almost linear almost multiplicative mapping $h : \Cal A \rightarrow \Cal B$ is a homomorphism when $h(2 x) = 2 h(x)$ for all $x \in \Cal A$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*$-algebras or in Lie $JC^*$-algebras, and of Lie $JC^*$-algebra derivations in Lie $JC^*$-algebras.
On a composite functional equation related to the Golab-Schinzel equation
Madjid Eshaghi Gordji,Themistocles M. Rassias,Mohamed Tial,Driss Zeglami 대한수학회 2016 대한수학회보 Vol.53 No.2
Let $X$ be a vector space over a field $K$ of real or complex numbers and $ k\in \mathbb{N}$. We prove the superstability of the following generalized Golab--Schinzel type equation \begin{equation*} f(x_{1}+\sum_{i=2}^{p}x_{i}f(x_{1})^{k} f(x_{2})^{k}\cdots f(x_{i-1})^{k})=\prod \limits_{i=1}^{p}f(x_{i}),\ x_{1},x_{2},\ldots,x_{p}\in X, \end{equation*} where $f:X\rightarrow K$ is an unknown function which is hemicontinuous at the origin.
INEQUALITIES FOR COORDINATED HARMONIC PREINVEX FUNCTIONS
MUHAMMAD ASLAM NOOR,Themistocles M. Rassias,Khalida Inayat Noor,SABAH IFTIKHAR 장전수학회 2017 Proceedings of the Jangjeon mathematical society Vol.20 No.4
In this paper, we introduce and investigate the co-ordinated harmonic preinvex functions. Some new Hermite-Hadamard inequalities for co-ordinated harmonic preinvex functions are derived. These new results can be viewed as significant refinement and improvement of the known results. Some special cases are discussed as applications of main results. The ideas and techniques of this paper may stimulate further research in this area.
ON A COMPOSITE FUNCTIONAL EQUATION RELATED TO THE GOLAB-SCHINZEL EQUATION
Gordji, Madjid Eshaghi,Rassias, Themistocles M.,Tial, Mohamed,Zeglami, Driss Korean Mathematical Society 2016 대한수학회보 Vol.53 No.2
Let X be a vector space over a field K of real or complex numbers and $k{\in}{\mathbb{N}}$. We prove the superstability of the following generalized Golab-Schinzel type equation $f(x_1+{\limits\sum_{i=2}^p}x_if(x_1)^kf(x_2)^k{\cdots}f(x_{i-1})^k)={\limits\prod_{i=1}^pf(x_i),x_1,x_2,{\cdots},x_p{\in}X$, where $f:X{\rightarrow}K$ is an unknown function which is hemicontinuous at the origin.
d-ISOMETRIC LINEAR MAPPINGS IN LINEAR d-NORMED BANACH MODULES
Park, Choon-Kil,Rassias, Themistocles M. Korean Mathematical Society 2008 대한수학회지 Vol.45 No.1
We prove the Hyers-Ulam stability of linear d-isometries in linear d-normed Banach modules over a unital $C^*-algebra$ and of linear isometries in Banach modules over a unital $C^*-algebra$. The main purpose of this paper is to investigate d-isometric $C^*-algebra$ isomor-phisms between linear d-normed $C^*-algebras$ and isometric $C^*-algebra$ isomorphisms between $C^*-algebras$, and d-isometric Poisson $C^*-algebra$ isomorphisms between linear d-normed Poisson $C^*-algebras$ and isometric Poisson $C^*-algebra$ isomorphisms between Poisson $C^*-algebras$. We moreover prove the Hyers-Ulam stability of their d-isometric homomorphisms and of their isometric homomorphisms.
ON DISTANCE-PRESERVING MAPPINGS
Jung, Soon-Mo,M.Rassias, Themistocles Korean Mathematical Society 2004 대한수학회지 Vol.41 No.4
We generalize a theorem of W. Benz by proving the following result: Let $H_{\theta}$ be a half space of a real Hilbert space with dimension $\geq$ 3 and let Y be a real normed space which is strictly convex. If a distance $\rho$ > 0 is contractive and another distance N$\rho$ (N $\geq$ 2) is extensive by a mapping f : $H_{\theta}$ \longrightarrow Y, then the restriction f│$_{\theta}$ $H_{+}$$\rho$/2// is an isometry, where $H_{\theta}$+$\rho$/2/ is also a half space which is a proper subset of $H_{\theta}$. Applying the above result, we also generalize a classical theorem of Beckman and Quarles.
Hyers–Ulam stability of a generalized Apollonius type quadratic mapping
Park, Chun-Gil,Rassias, Themistocles M. Elsevier 2006 Journal of mathematical analysis and applications Vol.322 No.1
<P><B>Abstract</B></P><P>Let X,Y be linear spaces. It is shown that if a mapping Q:X→Y satisfies the following functional equation:<ce:label>(0.1)</ce:label>Q((∑i=1n<SUB>zi</SUB>)−(∑i=1n<SUB>xi</SUB>))+Q((∑i=1n<SUB>zi</SUB>)−(∑i=1n<SUB>yi</SUB>))=12Q((∑i=1n<SUB>xi</SUB>)−(∑i=1n<SUB>yi</SUB>))+2Q((∑i=1n<SUB>zi</SUB>)−(∑i=1n<SUB>xi</SUB>)+(∑i=1n<SUB>yi</SUB>)2) then the mapping Q:X→Y is quadratic. We moreover prove the Hyers–Ulam stability of the functional equation (0.1) in Banach spaces.</P>