On the stability of the quadratic functional equation on bounded domains

  • Soon-Mo Jung
  • Byungbae Kim


A result of Skof and Terracini will be generalized; More precisely, we will prove that if a functionf : [-t, t]nE satisfies the inequality (1) for some δ > 0 and for allx, y ∈ [-t, t]n withx + y, x - y ∈ [-t, t]n, then there exists a quadratic functionq: ℝnE such that ∥f(x) -q(x)∥ < (2912n2 + 1872n + 334)δ for anyx ∈ [-t, t] n .

Key words and phrases

Hyers-Ulam stability quadratic equation 


  1. [1]
    P. W. Cholewa, Remarks on the stability of functional equations.Aequ. Math. 27 (1984), 76–86.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    S. Czerwik, On the Stability of the Quadratic Mapping in Normed Spaces.Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    G. L. Forti, Hyers-Ulam stability of functional equations in several variables.Aequ. Math. 50(1995), 143–190.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.J. Math. Anal. Appl. 184 (1994), 431–436.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Z. Gajda, On stability of additive mappings.Internat. J. Math. Math. Sci. 14 (1991), 431–434.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    D. H. Hyers, On the stability of the linear functional equation.Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.CrossRefMathSciNetGoogle Scholar
  7. [7]
    D. H. Hyers, G. Isac, andTh. M. Rassias, On the asymptoticity aspect of Hyers-Ulam stability of mappings.Proc. Amer. Math. Soc. 126 (1998), 425–430.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    _,Stability of Functional Equations in Several Variables. Birkhäuser, 1998.Google Scholar
  9. [9]
    D. H. Hyers andTh. M. Rassias, Approximate homomorphisms.Aequ. Math. 44 (1992), 125–153.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    S.-M. Jung, On the Hyers-Ulam-Rassias stability of approximately additive mappings.J. Math. Anal. Appl. 204 (1996), 221–226.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    —, Hyers-Ulam-Rassias stability of functional equations.Dynamic Systems and Appl. 6(1997), 541–566.MATHMathSciNetGoogle Scholar
  12. [12]
    —, On the Hyers-Ulam stability of the functional equations that have the quadratic property.J. Math. Anal. Appl. 222 (1998), 126–137.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    P. Kannappan, Quadratic functional equation and inner product spaces.Results Math. 27 (1995), 368–372.MathSciNetGoogle Scholar
  14. [14]
    Z. Kominek, On a local stability of the Jensen functional equation.Demonstratio Math. 22 (1989), 499–507.MATHMathSciNetGoogle Scholar
  15. [15]
    Th. M. Rassias, On the stability of the linear mapping in Banach spaces.Proc. Amer. Math. Soc. 72 (1978), 297–300.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    —, The stability of mappings and related topics. In ‘Report on the 27th ISFE,’,Aequ. Math. 39 (1990), pp. 292–293.Google Scholar
  17. [17]
    —, On a modified Hyers-Ulam sequence.J. Math. Anal. Appl. 158 (1991), 106–113.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Th. M. Rassias andP. Šemrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability.Proc. Amer. Math. Soc. 114 (1992), 989–993.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    —, On the Hyers-Ulam stability of linear mappings.J. Math. Anal. Appl. 173 (1993), 325–338.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    F. Skof, Proprieta’ locali e approssimazione di operatori.Rend. Sem. Mat. Fis. Milano 53(1983), 113–129.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    F. Skof andS. Terracini, Sulla stabilità dell’equazione funzionale quadratica su un dominio ristretto.Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 121 (1987), 153–167.MATHGoogle Scholar
  22. [22]
    S. M. ULAM,Problems in modern mathematics. Ch. VI, Wiley, 1964.Google Scholar

Copyright information

© Mathematische Seminar 1999

Authors and Affiliations

  1. 1.Mathematics Section/Physics SectionCollege of Science & Technology, Hong-Ik UniversityChochiwonSouth Korea

Personalised recommendations