Advertisement

Related Results on Stability of Functional Inequalities and Equations

  • Yeol Je Cho
  • Themistocles M. Rassias
  • Reza Saadati
Part of the Springer Optimization and Its Applications book series (SOIA, volume 86)

Abstract

In this chapter, we prove some stability results for certain functional inequalities and functional equations in latticetic random φ-normed spaces, r-divisible groups and homogeneous probabilistic (random) modular spaces.

References

  1. 2.
    J. Aczél, Lectures on Functional Equations and Their Applications (Academic Press, New York, 1966) zbMATHGoogle Scholar
  2. 4.
    J. Aczél, J. Dhombres, Functional Equations in Several Variables (Cambridge University Press, Cambridge, 1989) zbMATHCrossRefGoogle Scholar
  3. 27.
    L. Cădariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346, 43–52 (2004) zbMATHGoogle Scholar
  4. 39.
    P.W. Cholewa, The stability problem for a generalized Cauchy type functional equation. Rev. Roum. Math. Pures Appl. 29, 457–460 (1984) MathSciNetzbMATHGoogle Scholar
  5. 40.
    J.K. Chung, P.K. Sahoo, On the general solution of a quartic functional equation. Bull. Korean Math. Soc. 40, 565–576 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 45.
    S. Czerwik, On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 62, 59–64 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 53.
    A. Ebadian, A. Najati, M. Eshaghi Gordji, On approximate additive–quartic and quadratic–cubic functional equations in two variables on Abelian groups. Results Math. (2010). doi: 10.1007/s00025-010-0018-4 MathSciNetGoogle Scholar
  8. 56.
    M. Eshaghi Gordji, Stability of a functional equation deriving from quartic and additive functions. Bull. Korean Math. Soc. 47, 491–502 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 60.
    M. Eshaghi Gordji, M.B. Ghaemi, S.K. Gharetapeh, S. Shams, A. Ebadian, On the stability of J -derivations. J. Geom. Phys. 60, 454–459 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 64.
    M. Eshaghi Gordji, S.K. Gharetapeh, C. Park, S. Zolfaghri, Stability of an additive–cubic–quartic functional equation. Adv. Differ. Equ. 2009, 395693 (2009), 20 pp. Google Scholar
  11. 65.
    M. Eshaghi Gordji, S.K. Gharetapeh, J.M. Rassias, S. Zolfaghari, Solution and stability of a mixed type additive, quadratic and cubic functional equation. Adv. Differ. Equ. 2009, 826130 (2009), 17 pp. Google Scholar
  12. 68.
    M. Eshaghi Gordji, H. Khodaei, On the generalized Hyers–Ulam–Rassias stability of quadratic functional equations. Abstr. Appl. Anal. 2009, 923476 (2009), 11 pp. MathSciNetGoogle Scholar
  13. 86.
    W. Fechner, Stability of a functional inequalities associated with the Jordan–von Neumann functional equation. Aequ. Math. 71, 149–161 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 88.
    G.L. Forti, An existence and stability theorem for a class of functional equations. Stochastica 4, 22–30 (1980) Google Scholar
  15. 92.
    M.B. Ghaemi, M. Eshaghi Gordji, H. Majani, Approximately quintic and sextic mappings on the probabilistic normed spaces. Preprint Google Scholar
  16. 93.
    A. Gilányi, Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequ. Math. 62, 303–309 (2001) zbMATHCrossRefGoogle Scholar
  17. 94.
    A. Gilányi, On a problem by K. Nikodem. Math. Inequal. Appl. 5, 707–710 (2002) MathSciNetzbMATHGoogle Scholar
  18. 103.
    O. Hadžić, E. Pap, V. Radu, Generalized contraction mapping principles in probabilistic metric spaces. Acta Math. Hung. 101, 131–148 (2003) zbMATHCrossRefGoogle Scholar
  19. 116.
    K.W. Jun, H.M. Kim, The generalized Hyers–Ulam–Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274, 867–878 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 117.
    K.W. Jun, H.M. Kim, I.S. Chang, On the Hyers–Ulam stability of an Euler–Lagrange type cubic functional equation. J. Comput. Anal. Appl. 7, 21–33 (2005) MathSciNetzbMATHGoogle Scholar
  21. 125.
    S.M. Jung, Z.H. Lee, A fixed point approach to the stability of quadratic functional equation with involution. Fixed Point Theory Appl. 2008, 732086 (2008), 11 pp. MathSciNetGoogle Scholar
  22. 127.
    S.M. Jung, S. Min, A fixed point approach to the stability of the functional equation f(x+y)=F[f(x),f(y)]. Fixed Point Theory Appl. 2009, 912046 (2009), 8 pp. MathSciNetCrossRefGoogle Scholar
  23. 132.
    P. Kannappan, Quadratic functional equation and inner product spaces. Results Math. 27, 368–372 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 151.
    S. Lee, S. Im, I. Hwang, Quartic functional equations. J. Math. Anal. Appl. 307, 387–394 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 180.
    A. Najati, Hyers–Ulam–Rassias stability of a cubic functional equation. Bull. Korean Math. Soc. 44, 825–840 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 193.
    C. Park, Fixed points in functional inequalities. J. Inequal. Appl. 2008, 298050 (2008), pp. 8 CrossRefGoogle Scholar
  27. 195.
    C. Park, Y. Cho, M. Han, Functional inequalities associated with Jordan–von Neumann type additive functional equations. J. Inequal. Appl. 2007, 41820 (2007) MathSciNetGoogle Scholar
  28. 232.
    J. Rätz, On inequalities associated with the Jordan–von Neumann functional equation. Aequ. Math. 66, 191–200 (2003) zbMATHCrossRefGoogle Scholar
  29. 244.
    F. Skof, Proprietà locali e approssimazione di operatori. Rend. Semin. Mat. Fis. Milano 53, 113–129 (1983) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Yeol Je Cho
    • 1
  • Themistocles M. Rassias
    • 2
  • Reza Saadati
    • 3
  1. 1.College of Education, Department of Mathematics EducationGyeongsang National UniversityChinjuRepublic of South Korea
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece
  3. 3.Department of MathematicsIran University of Science and TechnologyBehshahrIran

Personalised recommendations