• Title/Summary/Keyword: sine functional equation

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ON THE SUPERSTABILITY OF THE PEXIDER TYPE SINE FUNCTIONAL EQUATION

  • Kim, Gwang Hui
    • Journal of the Chungcheong Mathematical Society
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    • v.25 no.1
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    • pp.1-18
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    • 2012
  • The aim of this paper is to investigate the superstability of the pexider type sine(hyperbolic sine) functional equation $f(\frac{x+y}{2})^{2}-f(\frac{x+{\sigma}y}{2})^{2}={\lambda}g(x)h(y),\;{\lambda}:\;constant$ which is bounded by the unknown functions ${\varphi}(x)$ or ${\varphi}(y)$. As a consequence, we have generalized the stability results for the sine functional equation by P. M. Cholewa, R. Badora, R. Ger, and G. H. Kim.

ON THE SUPERSTABILITY FOR THE p-POWER-RADICAL SINE FUNCTIONAL EQUATION

  • Gwang Hui Kim
    • Nonlinear Functional Analysis and Applications
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    • v.28 no.3
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    • pp.801-812
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    • 2023
  • In this paper, we investigate the superstability for the p-power-radical sine functional equation $$f\(\sqrt[p]{\frac{x^p+y^p}{2}}\)^2-f\(\sqrt[p]{\frac{x^p-y^p}{2}}\)^2=f(x)f(y)$$ from an approximation of the p-power-radical functional equation: $$f(\sqrt[p]{x^p+y^p})-f(\sqrt[p]{x^p-y^p})={\lambda}g(x)h(y),$$ where p is an odd positive integer and f, g, h are complex valued functions. Furthermore, the obtained results are extended to Banach algebras.

ON THE STABILITY OF PEXIDER TYPE TRIGONOMETRIC FUNCTIONAL EQUATIONS

  • Kim, Gwang Hui
    • Korean Journal of Mathematics
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    • v.16 no.3
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    • pp.369-378
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    • 2008
  • The aim of this paper is to study the stability problem for the pexider type trigonometric functional equation f(x + y) − f(x−y) = 2g(x)h(y), which is related to the d'Alembert, the Wilson, the sine, and the mixed trigonometric functional equations.

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THE STABILITY OF THE GENERALIZED SINE FUNCTIONAL EQUATIONS III

  • Kim, Gwang Hui
    • Journal of the Chungcheong Mathematical Society
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    • v.20 no.4
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    • pp.465-476
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    • 2007
  • The aim of this paper is to investigate the stability problem bounded by function for the generalized sine functional equations as follow: $f(x)g(y)=f(\frac{x+y}{2})^2-f(\frac{x+{\sigma}y}{2})^2\\g(x)g(y)=f(\frac{x+y}{2})^2-f(\frac{x+{\sigma}y}{2})^2$. As a consequence, we have generalized the superstability of the sine type functional equations.

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TRANSFERRED SUPERSTABILITY OF THE p-RADICAL SINE FUNCTIONAL EQUATION

  • Kim, Gwang Hui;Roh, Jaiok
    • Journal of the Chungcheong Mathematical Society
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    • v.35 no.4
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    • pp.315-327
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    • 2022
  • In this paper, we investigate the transferred superstability for the p-radical sine functional equation $$f\(\sqrt[p]{\frac{x^p+y^p}{2}}\)^2-f\(\sqrt[p]{\frac{x^p-y^p}{2}}\)^2=f(x)f(y)$$ from the p-radical functional equations: $$f({\sqrt[p]{x^p+y^p}})+f({\sqrt[p]{x^p-y^p}})={\lambda}g(x)g(y),\;\\f({\sqrt[p]{x^p+y^p}})+f({\sqrt[p]{x^p-y^p}})={\lambda}g(x)h(y),$$ where p is an odd positive integer, λ is a positive real number, and f is a complex valued function. Furthermore, the results are extended to Banach algebras. Therefore, the obtained result will be forced to the pre-results(p=1) for this type's equations, and will serve as a sample to apply it to the extension of the other known equations.

ON THE SUPERSTABILITY OF THE p-RADICAL SINE TYPE FUNCTIONAL EQUATIONS

  • Kim, Gwang Hui
    • The Pure and Applied Mathematics
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    • v.28 no.4
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    • pp.387-398
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    • 2021
  • In this paper, we will find solutions and investigate the superstability bounded by constant for the p-radical functional equations as follows: $f\(\sqrt[p]{\frac{x^p+y^p}{2}}\)^2-f\(\sqrt[p]{\frac{x^p-y^p}{2}}\)^2=\;\{(i)\;f(x)f(y),\\(ii)\;g(x)f(y),\\(iii)\;f(x)g(y),\\(iv)\;g(x)g(y).$ with respect to the sine functional equation, where p is an odd positive integer and f is a complex valued function. Furthermore, the results are extended to Banach algebra.

ON THE SUPERSTABILITY OF SOME PEXIDER TYPE FUNCTIONAL EQUATION II

  • Kim, Gwang-Hui
    • The Pure and Applied Mathematics
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    • v.17 no.4
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    • pp.397-411
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    • 2010
  • In this paper, we will investigate the superstability for the sine functional equation from the following Pexider type functional equation: $f(x+y)-g(x-y)={\lambda}{\cdot}h(x)k(y)$ ${\lambda}$: constant, which can be considered an exponential type functional equation, the mixed functional equation of the trigonometric function, the mixed functional equation of the hyperbolic function, and the Jensen type equation.