• Title, Summary, Keyword: Gamma-beta 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 (α, β, γ)-DERIVATIONS ON LIE C*-ALGEBRA ASSOCIATED TO A PEXIDERIZED QUADRATIC TYPE FUNCTIONAL EQUATION

  • Eghbali, Nasrin;Hazrati, Somayeh
    • Communications of the Korean Mathematical Society
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    • v.31 no.1
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    • pp.101-113
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    • 2016
  • In this article, we considered the stability of the following (${\alpha}$, ${\beta}$, ${\gamma}$)-derivation $${\alpha}D[x,y]={\beta}[D(x),y]+{\gamma}[x,D(y)]$$ and homomorphisms associated to the quadratic type functional equation $$f(kx+y)+f(kx+{\sigma}(y))=2kg(x)+2g(y),\;x,y{\in}A$$, where ${\sigma}$ is an involution of the Lie $C^*$-algebra A and k is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

ON THE STABILITY OF FUNCTIONAL EQUATIONS IN n-VARIABLES AND ITS APPLICATIONS

  • KIM, GWANG-HUI
    • Communications of the Korean Mathematical Society
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    • v.20 no.2
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    • pp.321-338
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    • 2005
  • In this paper we investigate a generalization of the Hyers-Ulam-Rassias stability for a functional equation of the form $f(\varphi(X))\;=\;\phi(X)f(X)$, where X lie in n-variables. As a consequence, we obtain a stability result in the sense of Hyers, Ulam, Rassias, and Gavruta for many other equations such as the gamma, beta, Schroder, iterative, and G-function type's equations.

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 PEXIDERIZED EXPONENTIAL TYPE FUNCTIONS

  • Lee, Young-Whan
    • The Pure and Applied Mathematics
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    • v.19 no.2
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    • pp.193-198
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    • 2012
  • We show that every unbounded approximate Pexiderized exponential type function has the exponential type. That is, we obtain the superstability of the Pexiderized exponential type functional equation $$f(x+y)=e(x,y)g(x)h(y)$$. From this result, we have the superstability of the exponential functional equation $$f(x+y)=f(x)f(y)$$.

Development of Mathematical Model for the Hydrolysis Fish Oil (물고기 기름의 가수분해에 대한 수학적 모형개발)

  • Kim Won-Ho;Lee Yong-Hoon;Park Ji-Suk;Hur Byung-Ki
    • KSBB Journal
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    • v.20 no.2
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    • pp.106-111
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    • 2005
  • The functional relationship between the number of mole of an i-fatty acid (Si) included in fish oil and the hydrolysis time(t) was expressed as a mathematical model, $S_i=-{\alpha_i}1n(t)+\beta_i$. The average errors of calculated values on the basis of the measured values were distributed in the range of less than $5\%$ for all the 15 fatty aids composing of fish oil. The equation of hydrolysis rate of each fatty acid was deduced as $v_i={\gamma_i}exp(\frac{S_i}{\alpha_i})$ from the above-mentioned $S_i=-{\alpha_i}ln(t)+{\beta_i}$. Therefore the hydrolysis yields of fatty acids were analyzed using the equation of $S_i\;Vs.\;t.$. The 15 fatty acids were categorized into 4groups from the view point of hydrolysis yield. The hydrolysis yields of the first group, including C14:0, C16:0, C16:1, C18:0, C18:1 (n-7) and 1l8:1 (n-9), were higher than $70\%$ at 48 hr of hydrolysis. Those of the second group, C20:1, C22:1, C18:3, C20:4 and C20:5, were distributed from $40\%,\;to\;60\%$, and third group were around $30\%$. The final group containing only C22:6 was very hard to be hydrolyzed and the yield was less than $20\%$ at the same time.