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A FIXED POINT APPROACH TO THE STABILITY OF QUADRATIC FUNCTIONAL EQUATION

  • Jung, Soon-Mo (Mathematics Sectiion College of Science and Technology, Hong-Ik University) ;
  • Kim, Tae-Soo (Department of Mathematics, Chungbuk National University) ;
  • Lee, Ki-Suk (Department of Mathematics Education, Korea National University of Education)
  • Published : 2006.08.01

Abstract

[ $C\u{a}dariu$ ] and Radu applied the fixed point method to the investigation of Cauchy and Jensen functional equations. In this paper, we adopt the idea of $C\u{a}dariu$ and Radu to prove the Hyers-Ulam-Rassias stability of the quadratic functional equation for a large class of functions from a vector space into a complete ${\gamma}-normed$ space.

Keywords

Hyers-Ulam-Rassias stability;quadratic functional equation;fixed point method

References

  1. L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43-52
  2. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86 https://doi.org/10.1007/BF02192660
  3. 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
  4. J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309 https://doi.org/10.1090/S0002-9904-1968-11933-0
  5. G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1-2, 143-190 https://doi.org/10.1007/BF01831117
  6. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224
  7. D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153 https://doi.org/10.1007/BF01830975
  8. S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), no. 1, 126-137 https://doi.org/10.1006/jmaa.1998.5916
  9. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001
  10. V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91-96
  11. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300
  12. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130 https://doi.org/10.1023/A:1006449032100
  13. F. Skof, Proprieta locali e approssimazione di operatori , Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129 https://doi.org/10.1007/BF02924890
  14. S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publisher, New York, 1960
  15. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel/Boston, 1998
  16. S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations, Dynam. Sys tems Appl. 6 (1997), no. 4, 541-566
  17. L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation, JIPAM. J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Art. 4, 7 pp

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