• Title, Summary, Keyword: Cauchy functional equation

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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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THE STABILITY OF A GENERALIZED CAUCHY FUNCTIONAL EQUATION

  • LEE, EUN HWI;CHOI, YOUNG HO;NA, YOUNG YOON
    • Honam Mathematical Journal
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    • v.22 no.1
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    • pp.37-46
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    • 2000
  • We prove the stability of a generalized Cauchy functional equation of the form ; $$f(a_1x+a_2y)=b_1f(x)+b_2f(y)+w.$$ That is, we obtain a partial answer for the open problem which was posed by the Th.M Rassias and J. Tabor on the stability for a generalized functional equation.

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GENERALIZED STABILITIES OF CAUCHY'S GAMMA-BETA FUNCTIONAL EQUATION

  • Lee, Eun-Hwi;Han, Soon-Yi
    • Honam Mathematical Journal
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    • v.30 no.3
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    • pp.567-579
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    • 2008
  • We obtain generalized super stability of Cauchy's gamma-beta functional equation B(x, y) f(x + y) = f(x)f(y), where B(x, y) is the beta function and also generalize the stability in the sense of R. Ger of this equation in the following setting: ${\mid}{\frac{B(x,y)f(x+y)}{f(x)f(y)}}-1{\mid}$ < H(x,y), where H(x,y) is a homogeneous function of dgree p(0 ${\leq}$ p < 1).

STABILITY OF THE CAUCHY FUNCTIONAL EQUATION IN BANACH ALGEBRAS

  • Lee, Jung Rye;Park, Choonkil
    • Korean Journal of Mathematics
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    • v.17 no.1
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    • pp.91-102
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    • 2009
  • Using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in Banach algebras and of derivations on Banach algebras for the 3-variable Cauchy functional equation.

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SUPERSTABILITY OF A GENERALIZED EXPONENTIAL FUNCTIONAL EQUATION OF PEXIDER TYPE

  • Lee, Young-Whan
    • Communications of the Korean Mathematical Society
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    • v.23 no.3
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    • pp.357-369
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    • 2008
  • We obtain the superstability of a generalized exponential functional equation f(x+y)=E(x,y)g(x)f(y) and investigate the stability in the sense of R. Ger [4] of this equation in the following setting: $$|\frac{f(x+y)}{(E(x,y)g(x)f(y)}-1|{\leq}{\varphi}(x,y)$$ where E(x, y) is a pseudo exponential function. From these results, we have superstabilities of exponential functional equation and Cauchy's gamma-beta functional equation.

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.