• Title, Summary, Keyword: quadratic mapping

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ON THE QUADRATIC MAPPING IN GENERALIZED QUASI-BANACH SPACES

  • Park, Choonkil;Jun, Kil-Woung;Lu, Gang
    • Journal of the Chungcheong Mathematical Society
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    • v.19 no.3
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    • pp.263-274
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    • 2006
  • In this paper, we prove the Hyers-Ulam-Rassias stability of the quadratic mapping in generalized quasi-Banach spaces, and of the quadratic mapping in generalized p-Banach spaces.

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LINEAR MAPPINGS, QUADRATIC MAPPINGS AND CUBIC MAPPINGS IN NORMED SPACES

  • Park, Chun-Gil;Wee, Hee-Jung
    • The Pure and Applied Mathematics
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    • v.10 no.3
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    • pp.185-192
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    • 2003
  • It is shown that every almost linear mapping $h{\;}:{\;}X{\;}{\rightarrow}{\;}Y$ of a complex normed space X to a complex normed space Y is a linen. mapping when h(rx) = rh(x) (r > 0,$r\;{\neq}\;1$$x{\;}{\in}{\;}X$, that every almost quadratic mapping $h{\;}:{\;}X{\;}{\rightarrow}{\;}Y$ of a complex normed space X to a complex normed space Y is a quadratic mapping when $h(rx){\;}={\;}r^2h(x){\;}(r{\;}>{\;}0,r\;{\neq}\;1)$ holds for all $x{\;}{\in}{\;}X$, and that every almost cubic mapping $h{\;}:{\;}X{\;}{\rightarrow}{\;}Y$ of a complex normed space X to a complex normed space Y is a cubic mapping when $h(rx){\;}={\;}r^3h(x){\;}(r{\;}>{\;}0,r\;{\neq}\;1)$ holds for all $x{\;}{\in}{\;}X$.

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A Fixed Point Approach to the Stability of Quadratic Equations in Quasi Normed Spaces

  • Mirmostafaee, Alireza Kamel
    • Kyungpook Mathematical Journal
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    • v.49 no.4
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    • pp.691-700
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    • 2009
  • We use the fixed alternative theorem to establish Hyers-Ulam-Rassias stability of the quadratic functional equation where functions map a linear space into a complete quasi p-normed space. Moreover, we will show that the continuity behavior of an approximately quadratic mapping, which is controlled by a suitable continuous function, implies the continuity of a unique quadratic function, which is a good approximation to the mapping. We also give a few applications of our results in some special cases.

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 AN n-DIMENSIONAL QUADRATIC EQUATION

  • Jun, Kil-Woung;Lee, Sang-Baek
    • Journal of the Chungcheong Mathematical Society
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    • v.20 no.1
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    • pp.23-29
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    • 2007
  • Let X and Y be vector spaces. In this paper we prove that a mapping $f:X{\rightarrow}Y$ satisfies the following functional equation $${\large}\sum_{1{\leq}k<l{\leq}n}\;(f(x_k+x_l)+f(x_k-x_l))-2(n-1){\large}\sum_{i=1}^{n}f(x_i)=0$$ if and only if the mapping f is quadratic. In addition we investigate the generalized Hyers-Ulam-Rassias stability problem for the functional equation.

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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.

JENSEN TYPE QUADRATIC-QUADRATIC MAPPING IN BANACH SPACES

  • Park, Choon-Kil;Hong, Seong-Ki;Kim, Myoung-Jung
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.4
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    • pp.703-709
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    • 2006
  • Let X, Y be vector spaces. It is shown that if an even mapping $f:X{\rightarrow}Y$ satisfies f(0) = 0 and $$(0.1)\;f(\frac {x+y} 2+z)+f(\frac {x+y} 2-z)+f(\frac {x-y} 2+z)+f(\frac {x-y} 2-z)=f(x)+f(y)+4f(z)$$ for all x, y, z ${\in}$X, then the mapping $f:X{\rightarrow}Y$ is quadratic. Furthermore, we prove the Cauchy-Rassias stability of the functional equation (0.1) in Banach spaces.

FUZZY STABILITY OF A CUBIC-QUADRATIC FUNCTIONAL EQUATION: A FIXED POINT APPROACH

  • Park, Choonkil;Lee, Sang Hoon;Lee, Sang Hyup
    • Korean Journal of Mathematics
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    • v.17 no.3
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    • pp.315-330
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    • 2009
  • Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following cubic-quadratic functional equation $$(0.1)\;\frac{1}{2}(f(2x+y)+f(2x-y)-f(-2x-y)-f(y- 2x))\\{\hspace{35}}=2f(x+y)+2f(x-y)+4f(x)-8f(-x)-2f(y)-2f(-y)$$ in fuzzy Banach spaces.

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