Advertisement

Stability of general A-cubic functional equations in modular spaces

  • G. Zamani Eskandani
  • John Michael Rassias
Original Paper
  • 99 Downloads

Abstract

In this paper, by using fixed point theory, we investigate the generalized Hyers–Ulam stability of an \(\alpha \)-cubic functional equation in modular spaces.

Keywords

Fixed point Modular space Generalized Hyers–Ulam stability 

Mathematics Subject Classification

Primary 39B52 Secondary 39B72 47H09 

Notes

Acknowledgements

The first author was supported by University of Tabriz.

References

  1. 1.
    Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bouikhalene, B., Eloqrachi, E.: Hyers-Ulam stability of spherical functions. Georgian Math. J. 23(2), 181–189 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Czerwik, S.: Functional equations and inequalities in several variables. World Scientific, New Jersey, London, Singapore, Hong Kong (2002)CrossRefMATHGoogle Scholar
  4. 4.
    Eskandani, G.Z., Rassias, J.M.: Approximation of general \(\alpha \)-cubic functional equations in 2-Banach spaces. Ukr. Math. J. 10, 1430–1436 (2017)Google Scholar
  5. 5.
    Eskandani, G.Z., Rassias, J.M., Gavruta, P.: Generalized Hyers-Ulam stability for a general cubic functional equation in quasi–normed spaces, Asian-Eur. J. Math. 4(03), 413–425 (2011)Google Scholar
  6. 6.
    Găvruta, P.: On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings. J. Math. Anal. Appl. 261, 543–553 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Găvruta, P.: An answer to question of John M. Rassias concerning the stability of Cauchy equation, Advanced in Equation and Inequality, Edited by John M. Rassias, Hadronic Press Mathematics Series pp 67–71 (1999)Google Scholar
  8. 8.
    Găvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. 27, 222–224 (1941)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hyers, D.H., Isac, G., Rassias, Th.M: Stability of functional equations in several variables. Birkhäuser, Basel (1998)Google Scholar
  11. 11.
    Jung, S.M.: Hyers-Ulam-Rassias stability of functional equations in mathematical analysis. Hadronic Press, Palm Harbor (2001)MATHGoogle Scholar
  12. 12.
    Nakano, H.: Modulared semi-ordered linear spaces. Maruzen, Tokyo, Japan (1950)MATHGoogle Scholar
  13. 13.
    Khamsi, M. A.: Quasicontraction Mapping in modular spaces without \(\Delta _{2}\)-condition, Fixed Point Theory and Applications Volume, Artical ID 916187, 6 pages (2008)Google Scholar
  14. 14.
    Koshi, S., Shimogaki, T.: On F-norms of quasi-modular spaces. J. Fac. Sci. Hokkaido Univ. Ser. I 15(3), 202–218 (1961)MathSciNetMATHGoogle Scholar
  15. 15.
    Park, C.: Homomorphisms between Poisson \(JC^{*}\)-algebras. Bull. Braz. Math. Soc. 36, 79–97 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Rassias, J.M.: On approximation of approximately linear mappings by linear mappings, J Funct Anal. 46(1), 126—130 (1982)Google Scholar
  17. 17.
    Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. Bull. des Sci. Math. 108(4), 445–446 (1984)MathSciNetMATHGoogle Scholar
  18. 18.
    Rassias, J.M.: Solution of a problem of Ulam. J. Approx. Theory 57(3), 268–273 (1989)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Rassias, J.M.: Solution of a stability problem of Ulam. Discussiones Mathematicae 12, 95–103 (1992)MathSciNetMATHGoogle Scholar
  20. 20.
    Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)Google Scholar
  21. 21.
    Rassias, Th.M.: On a modified Hyers-Ulam sequence. J. Math. Anal. Appl. 158, 106–113 (1991)Google Scholar
  22. 22.
    Sadeghi, G.: A fixed point approach to stability of functional equations in modular spaces. Bull. Malaysian Math. Sci. Soc. 37, 333–344 (2014)Google Scholar
  23. 23.
    Ulam, S.M.: A collection of the mathematical problems, Interscience Publ. New York, 431–436 (1960)Google Scholar
  24. 24.
    Wongkum, K., Chaipunya, P., Kumam, P.: On the Generalized Ulam–Hyers–Rassias Stability of Quadratic Mappings in Modular Spaces without \(\Delta _{2}\)-Conditions, Journal of Function Spaces, Volume, Article ID 461719, 7 pages (2014)Google Scholar
  25. 25.
    Yamamuro, S.: On conjugate spaces of Nakano spaces. Trans. Amer. Math. Soc. 90, 291–311 (1959)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.Faculty of Mathematical ScienceUniversity of TabrizTabrizIran
  2. 2.Pedagogical DepartmentNational and Capodistrian University of AthensAghia ParaskeviGreece

Personalised recommendations