Advertisement

Measure Zero Stability Problem for Drygas Functional Equation with Complex Involution

  • Ahmed Nuino
  • Muaadh Almahalebi
  • Ahmed Charifi
Chapter
First Online:

Abstract

In this chapter, we discuss the Hyers–Ulam stability theorem for the σ-Drygas functional equation
$$\displaystyle f(x+y)+f\big (x+\sigma (y)\big )=2f(x)+f(y)+f\big (\sigma (y)\big ) $$
for all \((x,y)\in \varOmega \subset \mathbb {C}^{2}\) for Lebesgue measure m(Ω) = 0, where \(f:\mathbb {C}\to Y\) and σ : X → X is an involution.
This is a preview of subscription content, log in to check access.

References

  1. 1.
    M. Ait Sibaha, B. Bouikhalene, E. Elqorachi, Hyers-Ulam-Rassias stability of the K-quadratic functional equation. J. Inequal. Pure Appl. Math. 8(3) (2007)Google Scholar
  2. 2.
    C. Alsina, J.L. Garcia-Roig, On a conditional Cauchy equation on rhombuses, in Functional Analysis, Approximation Theory and Numerical Analysis, ed. by J.M. Rassias (World Scientific, Singapore, 1994)zbMATHGoogle Scholar
  3. 3.
    L.M. Arriola, W.A. Beyer, Stability of the Cauchy functional equation over p-adic fields. Real Anal. Exch. 31(1), 125–132 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Bahyrycz, J. Brzdȩk, On solutions of the d’Alembert equation on a restricted domain. Aequationes Math. 85, 169–183 (2013)Google Scholar
  5. 5.
    B. Batko, Stability of an alternative functional equation. J. Math. Anal. Appl. 339, 303–311 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    B. Batko, On approximation of approximate solutions of Dhombres’ equation. J. Math. Anal. Appl. 340, 424–432 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Brzdȩk, On the quotient stability of a family of functional equations. Nonlinear Anal. 71, 4396–4404 (2009)Google Scholar
  8. 8.
    J. Brzdȩk, On a method of proving the Hyers-Ulam stability of functional equations on restricted domains. Aust. J. Math. Anal. Appl. 6, 1–10 (2009)Google Scholar
  9. 9.
    J. Brzdȩk, J. Sikorska, A conditional exponential functional equation and its stability. Nonlinear Anal. 72, 2929–2934 (2010)Google Scholar
  10. 10.
    J. Chung, Stability of functional equations on restricted domains in a group and their asymptotic behaviors. Comput. Math. Appl. 60, 2653–2665 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J. Chung, Stability of a conditional Cauchy equation on a set of measure zero. Aequationes Math. (2013). http://dx.doi.org/10.1007/s00010-013-0235-5
  12. 12.
    J. Chung, J.M. Rassias, Quadratic functional equations in a set of Lebesgue measure zero. J. Math. Anal. Appl. 419(2), 1065–1075 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type (Hadronic Press, Inc., Palm Harbor, 2003)zbMATHGoogle Scholar
  14. 14.
    D.Z̆. Djoković, A representation theorem for (X 1 − 1)(X 2 − 1)…(X n − 1) and its applications. Ann. Polon. Math. 22(2), 189–198 (1969)Google Scholar
  15. 15.
    H. Drygas, Quasi-inner products and their applications, in Advances in Multivariate Statistical Analysis, ed. by A.K. Gupta (Reidel Publishing Company, Boston, 1987), pp. 13–30CrossRefGoogle Scholar
  16. 16.
    B.R. Ebanks, P.L. Kannappan, P.K. Sahoo, A common generalization of functional equations characterizing normed and quasi-inner-product spaces. Can. Math. Bull. 35, 321–327 (1992)MathSciNetCrossRefGoogle Scholar
  17. 17.
    V.A. Faĭziev, P.K. Sahoo, On Drygas functional equation on groups. Int. J. Appl. Math. Stat. 7, 59–69 (2007)Google Scholar
  18. 18.
    M. Fochi, An alternative functional equation on restricted domain. Aequationes Math. 70, 201–212 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    G.L. Forti, J. Sikorska, Variations on the Drygas equations and its stability. Nonlinear Anal. 74, 343–350 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    R. Ger, J. Sikorska, On the Cauchy equation on spheres. Ann. Math. Sil. 11, 89–99 (1997)MathSciNetzbMATHGoogle Scholar
  21. 21.
    D.H. Hyers, On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)MathSciNetCrossRefGoogle Scholar
  22. 22.
    S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl. 222, 126–137 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis (Springer, New York, 2011)CrossRefGoogle Scholar
  24. 24.
    S.-M. Jung, P.K. Sahoo, Stability of functional equation of Drygas. Aequationes Math. 64, 263–273 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    M. Kuczma, Functional equations on restricted domains. Aequationes Math. 18, 1–34 (1978)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Y.-H. Lee, Hyers-Ulam-Rassias stability of a quadratic-additive type functional equation on a restricted domain. Int. J. Math. Anal. 7(55), 2745–2752 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    J.C. Oxtoby, Measure and Category (Springer, New York, 1980)CrossRefGoogle Scholar
  28. 28.
    J.M. Rassias, On the Ulam stability of mixed type mappings on restricted domains. J. Math. Anal. Appl. 281, 747–762 (2002)MathSciNetCrossRefGoogle Scholar
  29. 29.
    J.M. Rassias, M.J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains. J. Math. Anal. Appl. 281, 516–524 (2003)MathSciNetCrossRefGoogle Scholar
  30. 30.
    J. Sikorska, On two conditional Pexider functional equations and their stabilities. Nonlinear Anal. 70, 2673–2684 (2009)MathSciNetCrossRefGoogle Scholar
  31. 31.
    J. Sikorska, On a direct method for proving the Hyers-Ulam stability of functional equations. J. Math. Anal. Appl. 372, 99–109 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    H. Stetkær, Functional equations on abelian groups with involution. II. Aequationes Math. 55, 227–240 (1998)MathSciNetCrossRefGoogle Scholar
  33. 33.
    H. Stetkær, Functional equations involving means of functions on the complex plane. Aequationes Math. 55, 47–62 (1998)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Gy. Szabo, Some functional equations related to quadratic functions. Glasnik Math. 38, 107–118 (1983)Google Scholar
  35. 35.
    S.M. Ulam, A Collection of Mathematical Problems (Interscience Publication, New York, 1961). Problems in Modern Mathematics, Wiley, New York 1964Google Scholar
  36. 36.
    D. Yang, Remarks on the stability of Drygas equation and the Pexider-quadratic equation. Aequationes Math. 68, 108–116 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ahmed Nuino
    • 1
  • Muaadh Almahalebi
    • 1
  • Ahmed Charifi
    • 1
  1. 1.Department of Mathematics, Faculty of SciencesUniversity of Ibn TofailKenitraMorocco

Personalised recommendations

Cite chapter

Buy options