A Fixed Point Approach to the Stability of Quadratic Equations in Quasi Normed Spaces

Mirmostafaee, Alireza Kamel

  • Received : 2008.10.12
  • Accepted : 2008.12.11
  • Published : 2009.12.31


We use the fixed alternative theorem to establish Hyers-Ulam-Rassias stability of the quadratic functional equation where functions map a linear space into a complete quasi p-normed space. Moreover, we will show that the continuity behavior of an approximately quadratic mapping, which is controlled by a suitable continuous function, implies the continuity of a unique quadratic function, which is a good approximation to the mapping. We also give a few applications of our results in some special cases.


quasi p-norm;quadratic functional equation;fixed point alternative;Hyers-Ulam-Rassias stability


  1. T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo, 18(1942), 588-594.
  2. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2(1950) 64-66.
  3. Y. Benyamini and J. Lindenstrauss, Geomertic nonlineat functional analysis volume 1, Coll. Pub. ( Amer. Math. Soc.) Volume 48, (2000).
  4. L. Cadariu, Fixed points in generalized metric space and the stability of a quartic functional equation, Bul. Stiint. Univ. Politeh. Timis. Ser. Mat. Fiz., 50(64)(2)(2005), 25-34.
  5. L. Cadariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara Ser. Mat.-Inform., 41(1)(2003), 25-48.
  6. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27(1984), 76-86.
  7. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62(1992), 59-64.
  8. S. Czerwik, The stability of the quadratic functional equation, in: Stability of mappings of Hyers-Ulam type, 81-91, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1994.
  9. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002.
  10. J. B. Diaz and B. Margolis, A fixed point theorem of the alternative for the contractions on generalized complete metric space, Bull. Amer. Math. Soc., 74(1968), 305-309.
  11. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184(1994), 431-436.
  12. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A., 27(1941), 222-224.
  13. S. -M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
  14. S. -M. Jung, T. -S. Kim and K. -S. Lee, A fixed point approach to the stability of quadratic functional equation, Bull. Korean Math. Soc., 43(3)(2006), 531-541.
  15. S. -M. Jung and P. K. Sahoo, Hyers-Ulam stability of the quadratic equation of Pexider type, J. Korean Math. Soc., 38(3)(2001), 645-656.
  16. A. K. Mirmostafaee, Stability of quartic mappings in non-Archimedean normed spaces, to be appear in Kyungpook Mathematical Journal(KMJ).
  17. A. K. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 159(2008) 730-738.
  18. A. K. Mirmostafaee and M. S. Moslehian, Fuzzy approximately cubic mappings, Information Sciences, 78(19)(2008) 3791-3798.
  19. A. K. Mirmostafaee and M. S. Moslehian, Fuzzy almost quadratic functions Result. Math., 52(2008) 161-177.
  20. V. Radu, The fixed point alternative and stability of functional equations, Fixed Point Theory, 4(1), (2003), 91-96.
  21. J. M. Rassias, Alternative contraction principle and Ulam stability problem, Math. Sci. Res. J., 9(7) (2005), 190-199.
  22. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(1978), 297-300.
  23. Th. M. Rassias (ed.), Functional equations, inequalities and applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003.
  24. F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano, 53(1983) 113-129.
  25. S. M. Ulam, Problems in Modern Mathematics (Chapter VI, Some Questions in Analysis: x1, Stability), Science Editions, John Wiley & Sons, New York, 1964.