Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

STABILITY OF A QUADRATIC TYPE FUNCTIONAL EQUATION

  • Received : 2007.01.26
  • Accepted : 2007.04.23
  • Published : 2007.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate some results concerning the stability of the following quadratic type functional equation: f(x + y) + f(x - y) + f(y + z) + f(y - z) + f(z + x) + f(z - x) = 4f(x) + 4f(y) + 4f(z).

Keywords

References

  1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, 1989.
  2. J. 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
  3. I.-S. Chang and Y.-S. Jung, Stability of a functional equation deriving from cubic and quadratic functions, J. Math. Anal. Appl. 283(2) (2003), 491-500. https://doi.org/10.1016/S0022-247X(03)00276-2
  4. I.-S. Chang, K.-W. Jun and Y.-S. Jung, The modified Hyers-Ulam-Rassias sta­bility of a cubic type functional equation, Math. Ineq. Appl. 8(4) (2005), 675-683.
  5. I.-S. Chang, E. H. Lee and H.-M. Kim, On Hyers-Ulam-Rassias stability of a quadratic functional equation, Math. Ineq. Appl. 6(1) (2003), 87-95.
  6. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  7. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
  8. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approxi­mately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  9. P. Gavruta, On the Hyers- Ulam-Rassias stability of the quadractic mappings, Non­linear Funct. Anal. Appl., 9(3) (2004), 415-428.
  10. R. Ger, Superstability is not natural, Rocznik Naukowo-dydaktyczny WSP w Krakowie, Prace Math., 159 (1993), 109-123.
  11. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  12. D. H. Hyers, G. Isac and Th. M. Rassias, "Stability of Functional Equations in Several Variables", Birkhauser, Basel, 1998.
  13. K.-W. Jun and H.-M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274(2) (2002), 867-878. https://doi.org/10.1016/S0022-247X(02)00415-8
  14. K.-W. Jun, H.-M. Kim and I.-S. Chang, On the Hyers-Ulam stability of an Euler-Lagrange type functional equation, J. Comp. Anal. Appl., 7(1) (2005), 21-33.
  15. S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl., 222 (1998), 126-137. https://doi.org/10.1006/jmaa.1998.5916
  16. S.-M. Jung, On the stability of gamma functional equation, Results. Math., 33 (1998), 306-309. https://doi.org/10.1007/BF03322090
  17. Y.-S. Jung and I.-S. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl., 306(2) (2005), 752-760. https://doi.org/10.1016/j.jmaa.2004.10.017
  18. Y.-S. Jung and K.-H. Park, On the stability of the functional equation f(x + y + xy) = f(x) + f(y) + xf(y)+yf(x), J. Math. Anal. Appl, 274(2) (2002), 659-666. https://doi.org/10.1016/S0022-247X(02)00328-1
  19. Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368-372. https://doi.org/10.1007/BF03322841
  20. B. Margolis and J. B. Diaz, A fixed point theorem of the alternative for contrac­tions on a generalized complete metric space, Bull. Amer. Math. Soc., 126, 74 (1968), 305-309.
  21. V. Radu, The fixed point alternative and the stability of functional equations, Seminar on Fixed Point Theory Cluj-Napoca, (to appear in vol. IV on 2003).
  22. 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
  23. Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), 264-284. https://doi.org/10.1006/jmaa.2000.7046
  24. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl., 62 (2000), 23-130. https://doi.org/10.1023/A:1006499223572
  25. Th. M. Rassias (Ed.), "Functional Equations and inequalities", Kluwer Academic, Dordrecht/ Boston/ London, 2000.
  26. Th. M. Rassias and J. Tabor, What is left of Hyers-Ulam stability?, Journal of Natural Geometry, 1 (1992), 65-69.
  27. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  28. T. Trif, On the stability of a general gamma-type functional equation, Pulb. Math. Debrecen, 60/1-2 (2002), 47-61.
  29. S. M. Ulam, Problems in Modern Mathematics, (1960) Chap. VI, Science ed, Wiley, New York.