Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

STABILITY OF A BETA-TYPE FUNCTIONAL EQUATION WITH A RESTRICTED DOMAIN

  • Lee, Young-Whan (Department of Computer and Information Security Daejeon University) ;
  • Choi, Byung-Mun (Department of Computer and Information Security Daejeon University)
  • Published : 2004.10.01
Download PDF
⟨ Previous Next ⟩

Abstract

We obtain the Hyers-Ulam-Rassias stability of a betatype functional equation $f(\varphi(x),\phi(y))$ = $ \psi(x,y)f(x,y)+ \lambda(x,y)$ with a restricted domain and the stability in the sense of R. Ger of the equation $f(\varphi(x),\phi(y))$ = $ \psi(x,y)f(x,y)$ with a restricted domain in the following settings: $g(\varphi(x),\phi(y))-\psi(x,y)g(s,y)-\lambda(x,y)$\mid$\leq\varepsilon(x,y)$ and $\frac{g(\varphi(x),\phi(y))}{\psi(x,y),g(x,y)}-1 $\mid$ \leq\epsilon(x,y)$.

Keywords

References

  1. J. Baker, J. Lawrence and F. Zorzitto, The stability of the equation f(x+y)=f(x)+f(y), Proc. Amer. Math. Soc. 74 (1979), 242-246. https://doi.org/10.1090/S0002-9939-1979-0524294-6
  2. C. Borelli, On Hyers-Ulam stability for a class of functional equations, Aequationes Math. 54 (1997), 74-86. https://doi.org/10.1007/BF02755447
  3. G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), 146-190. https://doi.org/10.1007/BF01831117
  4. R. Ger, Superstability is not natural, Roczik Naukowo-Dydaktyczny WSP W. Krakowie, Prace Mat. 159 (1993), 109-123.
  5. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  6. D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125-153. https://doi.org/10.1007/BF01830975
  7. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of functional equations in several variables, Birkhauser-Basel-Berlin (1998).
  8. K. W. Jun, G. H. Kim and Y. W. Lee, Stability of generalized gamma and beta functional equations, Aequationes Math. 60 (2000), 15-24. https://doi.org/10.1007/s000100050132
  9. S. M. Jung, On the general Hyers-Ulam stability of gamma functional equation, Bull. Korean Math. Soc. 34 (1997), no. 3, 437-446.
  10. S. M. Jung, On the stability of the gamma functional equation, Results Math. 33 (1998), 306-309. https://doi.org/10.1007/BF03322090
  11. G. H. Kim and Y. W. Lee, The stability of the beta functional equation, Babes-Bolyai Mathematica, XLV (1) (2000), 89-96.
  12. Y. W. Lee, On the stability of a quadratic Jensen type functional equation, J. Math. Anal. Appl. 270 (2002), 590-601. https://doi.org/10.1016/S0022-247X(02)00093-8
  13. Y. W. Lee, The stability of derivations on Banach algebras, Bull. Inst. Math. Acad. Sinica. 28 (2000), 113-116.
  14. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  15. T. Trif, On the stability of a gamma-type functional equation, to appear.
  16. S. M. Ulam, "Problems in Modern Mathematics" Chap. VI, Science editions, Wiley, New York (1964).