• Title, Summary, Keyword: additive mapping

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STABILITY OF ADDITIVE (n, 2)-MAPPINGS

  • Kang, Pyung-Lyun;Park, Chun-Gil
    • Journal of the Chungcheong Mathematical Society
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    • v.17 no.1
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    • pp.19-27
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    • 2004
  • We define an additive (n, 2)-mapping, and prove the stability of additive (n, 2)-mappings.

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STABILITY AND HYPERSTABILITY OF MULTI-ADDITIVE-CUBIC MAPPINGS IN INTUITIONISTIC FUZZY NORMED SPACES

  • Ramzanpour, Elahe;Bodaghi, Abasalt;Gilani, Alireza
    • Honam Mathematical Journal
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    • v.42 no.2
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    • pp.391-409
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    • 2020
  • In the current paper, the intuitionistic fuzzy normed space version of Hyers-Ulam stability for multi-additive, multi-cubic and multi-additive-cubic mappings by using a fixed point method are studied. Moreover, a few corollaries corresponding to some known stability and hyperstability outcomes in intuitionistic fuzzy normed space are presented.

ON 3-ADDITIVE MAPPINGS AND COMMUTATIVITY IN CERTAIN RINGS

  • Park, Kyoo-Hong;Jung, Yong-Soo
    • Communications of the Korean Mathematical Society
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    • v.22 no.1
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    • pp.41-51
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    • 2007
  • Let R be a ring with left identity e and suitably-restricted additive torsion, and Z(R) its center. Let H : $R{\times}R{\times}R{\rightarrow}R$ be a symmetric 3-additive mapping, and let h be the trace of H. In this paper we show that (i) if for each $x{\in}R$, $$n=<<\cdots,\;x>,\;\cdots,x>{\in}Z(R)$$ with $n\geq1$ fixed, then h is commuting on R. Moreover, h is of the form $$h(x)=\lambda_0x^3+\lambda_1(x)x^2+\lambda_2(x)x+\lambda_3(x)\;for\;all\;x{\in}R$$, where $\lambda_0\;{\in}\;Z(R)$, $\lambda_1\;:\;R{\rightarrow}R$ is an additive commuting mapping, $\lambda_2\;:\;R{\rightarrow}R$ is the commuting trace of a bi-additive mapping and the mapping $\lambda_3\;:\;R{\rightarrow}Z(R)$ is the trace of a symmetric 3-additive mapping; (ii) for each $x{\in}R$, either $n=0\;or\;<n,\;x^m>=0$ with $n\geq0,\;m\geq1$ fixed, then h = 0 on R, where denotes the product yx+xy and Z(R) is the center of R. We also present the conditions which implies commutativity in rings with identity as motivated by the above result.

A General Uniqueness Theorem concerning the Stability of AQCQ Type Functional Equations

  • Lee, Yang-Hi;Jung, Soon-Mo
    • Kyungpook Mathematical Journal
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    • v.58 no.2
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    • pp.291-305
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    • 2018
  • In this paper, we prove a general uniqueness theorem which is useful for proving the uniqueness of the relevant additive mapping, quadratic mapping, cubic mapping, quartic mapping, or the additive-quadratic-cubic-quartic mapping when we investigate the (generalized) Hyers-Ulam stability.

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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REMARKS ON THE STABILITY OF ADDITIVE FUNCTIONAL EQUATION

  • Jun, Kil-Woung;Kim, Hark-Mahn
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.4
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    • pp.679-687
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    • 2001
  • In this paper, using an idea from the direct method of Hyers, we give the conditions in order for a linear mapping near an approximately additive mapping to exist.

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APPROXIMATE ADDITIVE-QUADRATIC MAPPINGS AND BI-JENSEN MAPPINGS IN 2-BANACH SPACES

  • Park, Won-Gil
    • Journal of the Chungcheong Mathematical Society
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    • v.30 no.4
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    • pp.467-476
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    • 2017
  • In this paper, we obtain the stability of the additive-quadratic functional equation f(x+y, z+w)+f(x+y, z-w) = 2f(x, z)+2f(x, w)+2f(y, z)+2f(y, w) and the bi-Jensen functional equation $$4f(\frac{x+y}{2},\;\frac{z+w}{2})=f(x,\;z)+f(x,\;w)+f(y,\;z)+f(y,\;w)$$ in 2-Banach spaces.

HYERS{ULAM STABILITY OF FUNCTIONAL INEQUALITIES ASSOCIATED WITH CAUCHY MAPPINGS

  • Kim, Hark-Mahn;Oh, Jeong-Ha
    • Journal of the Chungcheong Mathematical Society
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    • v.20 no.4
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    • pp.503-514
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    • 2007
  • In this paper, we investigate the generalized Hyers-Ulam stability of the functional inequality $$||af(x)+bf(y)+cf(z)||{\leq}||f(ax+by+cz))||+{\phi}(x,y,z)$$ associated with Cauchy additive mappings. As a result, we obtain that if a mapping satisfies the functional inequality with perturbing term which satisfies certain conditions then there exists a Cauchy additive mapping near the mapping.

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APPROXIMATE ADDITIVE MAPPINGS IN 2-BANACH SPACES AND RELATED TOPICS: REVISITED

  • YUN, SUNGSIK
    • Korean Journal of Mathematics
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    • v.23 no.3
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    • pp.393-399
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    • 2015
  • W. Park [J. Math. Anal. Appl. 376 (2011) 193-202] proved the Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces. But there are serious problems in the control functions given in all theorems of the paper. In this paper, we correct the statements of these results and prove the corrected theorems. Moreover, we prove the superstability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces under the original given conditions.