Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE SUPERSTABILITY OF THE PEXIDER TYPE SINE FUNCTIONAL EQUATION

  • Published : 2012.02.15
⟨ Previous Next ⟩

Abstract

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.

Keywords

References

  1. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japen, 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064
  2. R. Badora, On the stability of cosine functional equation, Rocznik Naukowo- Dydak., Prace Mat. 15 (1998), 1-14.
  3. R. Badora and R. Ger, On some trigonometric functional inequalities, Func- tional Equations- Results and Advances, (2002), 3-15.
  4. J. A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1980), 411-416. https://doi.org/10.1090/S0002-9939-1980-0580995-3
  5. 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.
  6. D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385-397. https://doi.org/10.1215/S0012-7094-49-01639-7
  7. P. W. Cholewa, The stability of the sine equation, Proc. Amer. Math. Soc. 88 (1983), 631-634. https://doi.org/10.1090/S0002-9939-1983-0702289-8
  8. P. Gavruta, A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mapping, Journal of Mathematical Analysis and Applications 184 (1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  9. R. Ger, Superstability is not natural,Roczik Naukowo-Dydaktyczny WSP w Krakowie, Prace Mat. 159 (1993), 109-123.
  10. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  11. S. Kalra, J. C. Parnami, H. L. Vasudeva, A sine equation in Banach algebras, J. Indian Math. Soc. (N.S.) 50 (1988), 93-101.
  12. Pl. Kannappan, The functional equation f(xy) + $f(xy^{-1})$ = 2f(x)f(y) for groups, Proc. Amer. Math. Soc. 19 (1968), 69-74.
  13. Pl. Kannappan and G. H. Kim, On the stability of the generalized cosine functional equations, Ann. Acad. Paedagog. Crac. Stud. Math. 1 (2001), 49-58.
  14. G. H. Kim, The Stability of the d'Alembert and Jensen type functional equations, Jour. Math. Anal & Appl. 325 (2007), 237-248. https://doi.org/10.1016/j.jmaa.2006.01.062
  15. G. H. Kim, A Stability of the generalized sine functional equations, Jour. Math. Anal & Appl. 331 (2007), 886-894. https://doi.org/10.1016/j.jmaa.2006.09.037
  16. G. H. Kim, On the Stability of the generalized sine functional equations, Acta Math. Sin. (Engl. Ser.) 25 (2009), 965-972
  17. 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
  18. S. M. Ulam, "Problems in Modern Mathematics" Chap. VI, Science editions, Wiley, New York, 1964.