• Title, Summary, Keyword: Cauchy-Rassias stability

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CAUCHY-RASSIAS STABILITY OF DERIVATIONS ON QUASI-BANACH ALGEBRAS

  • An, Jong Su;Boo, Deok-Hoon;Park, Choonkil
    • Journal of the Chungcheong Mathematical Society
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    • v.20 no.2
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    • pp.173-182
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    • 2007
  • In this paper, we prove the Cauchy-Rassias stability of derivations on quasi-Banach algebras associated to the Cauchy functional equation and the Jensen functional equation. We use the Cauchy-Rassias inequality that was first introduced by Th. M. Rassias in the paper "On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300".

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A FIXED POINT APPROACH TO THE CAUCHY-RASSIAS STABILITY OF GENERAL JENSEN TYPE QUADRATIC-QUADRATIC MAPPINGS

  • Park, Choon-Kil;Gordji, M. Eshaghi;Khodaei, H.
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.5
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    • pp.987-996
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    • 2010
  • In this paper, we investigate the Cauchy-Rassias stability in Banach spaces and also the Cauchy-Rassias stability using the alternative fixed point for the functional equation: $$f(\frac{sx+ty}{2}+rz)+f(\frac{sx+ty}{2}-rz)+f(\frac{sx-ty}{2}+rz)+f(\frac{sx-ty}{2}-rz)=s^2f(x)+t^2f(y)+4r^2f(z)$$ for any fixed nonzero integers s, t, r with $r\;{\neq}\;{\pm}1$.

THE STABILITY OF LINEAR MAPPINGS IN BANACH MODULES ASSOCIATED WITH A GENERALIZED JENSEN MAPPING

  • Lee, Sung Jin
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.2
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    • pp.287-301
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    • 2011
  • Let X and Y be vector spaces. It is shown that a mapping $f\;:\;X{\rightarrow}Y$ satisfies the functional equation $$(\ddag)\hspace{50}dk\;f\left(\frac{\sum_{j=1}^{dk}x_j}{dk}\right)=\displaystyle\sum_{j=1}^{dk}f(x_j)$$ if and only if the mapping $f$ : X ${\rightarrow}$ Y is Cauchy additive, and prove the Cauchy-Rassias stability of the functional equation ($\ddag$) in Banach modules over a unital $C^{\ast}$-algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^{\ast}$-algebras. As an application, we show that every almost homomorphism $h\;:\;\mathcal{A}{\rightarrow}\mathcal{B}$ of $\mathcal{A}$ into $\mathcal{B}$ is a homomorphism when $h((k-1)^nuy)=h((k-1)^nu)h(y)$ for all unitaries $u{\in}\mathcal{A}$, all $y{\in}\mathcal{A}$, and $n$ = 0,1,2,$\cdots$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^{\ast}$-algebras.

LINEAR 𝜃-DERIVATIONS ON JB*-TRIPLES

  • Bak, Chunkil
    • Journal of the Chungcheong Mathematical Society
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    • v.19 no.1
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    • pp.27-36
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    • 2006
  • In [1], the concept of generalized (${\theta}$, ${\phi}$)-derivations on rings was introduced. We introduce the concept of linear ${\theta}$-derivations on $JB^*$-triples, and prove the Cauchy-Rassias stability of linear ${\theta}$-derivations on $JB^*$-triples.

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APPROXIMATELY ADDITIVE MAPPINGS OVER p-ADIC FIELDS

  • Park, Choonkil;Boo, Deok-Hoon;Rassias, Themistocles M.
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.1
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    • pp.1-14
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    • 2008
  • In this paper, we prove the Hyers-Ulam-Rassias stability of the Cauchy functional equation f(x+y) = f(x)+f(y) and of the Jensen functional equation $2f(\frac{x+y}{2})=f(x)+f(y)$ over the p-adic field ${\mathbb{Q}}_p$. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

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ON THE STABILITY OF A GENERALIZED ADDITIVE FUNCTIONAL EQUATION II

  • Lee, Jung-Rye;Lee, Tae-Keug;Shin, Dong-Yun
    • The Pure and Applied Mathematics
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    • v.14 no.2
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    • pp.111-125
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    • 2007
  • For an odd mapping, we study a generalized additive functional equation in Banach spaces and Banach modules over a $C^*-algebra$. And we obtain generalized solutions of a generalized additive functional equation and so generalize the Cauchy-Rassias stability.

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APPROXIMATELY ADDITIVE MAPPINGS IN NON-ARCHIMEDEAN NORMED SPACES

  • Mirmostafaee, Alireza Kamel
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.2
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    • pp.387-400
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    • 2009
  • We establish a new strategy to study the Hyers-Ulam-Rassias stability of the Cauchy and Jensen equations in non-Archimedean normed spaces. We will also show that under some restrictions, every function which satisfies certain inequalities can be approximated by an additive mapping in non-Archimedean normed spaces. Some applications of our results will be exhibited. In particular, we will see that some results about stability and additive mappings in real normed spaces are not valid in non-Archimedean normed spaces.