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Stability of an n-Dimensional Functional Equation in Banach Space and Fuzzy Normed Space

  • Sandra Pinelas
  • V. Govindan
  • K. Tamilvanan
Chapter

Abstract

In this paper, the authors investigate the general solution of a new additive functional equation
$$\displaystyle f\left (\sum ^{n}_{i=1}x_i\right )+\sum _{{j=1; i \neq j}}^{n}f\left (-x_j-x_i+\sum _{1\leq i < j < k \leq n}x_{k}\right ){=}\left (\frac {n^2-5n+6}{2}\right ) \sum ^{n}_{i=1}f\left (x_i\right )\\ $$
where n is a positive integer with \(\mathbb {N}-\{1,2,3,4 \}\) and discuss its generalized Hyers–Ulam stability in Banach spaces and stability in fuzzy normed spaces using two different methods.

References

  1. 1.
    T. Aoki, On the stability of the linear transformation in Banach space. Int. J. Math. Soc. 2(1–2), 64–66 (1950)MathSciNetCrossRefGoogle Scholar
  2. 2.
    T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11(3), 687–705 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    T. Bag, S.K. Samanta, Fuzzy bounded linear operators. Fuzzy Sets Syst. 151, 513–547 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Biswas, Fuzzy inner product space and fuzzy norm functions. Inf. Sci. 53, 185–190 (1991)CrossRefGoogle Scholar
  5. 5.
    I.S. Chang, H.M. Kim, On the Hyers-Ulam stability of quadratic functional equations. J. Inequal. Pure Appl. Math. 3, 33 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    S.C. Cheng, J.N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces. Bull. Calcuta Math. Soc. 86, 429–436 (1994)MathSciNetzbMATHGoogle Scholar
  7. 7.
    C. Felbin, Finite dimensional fuzzy normed space. Fuzzy Sets Syst. 48, 239–248 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    V. Govindan, K. Tamilvanan, Stability of functional equation in Banach space: using two different methods. Int. J. Math. Appl. 6(1-C), 527–536 (2018)Google Scholar
  10. 10.
    V. Govindan, S. Murthy, M. Saravanan, Solution and stability of New type of (aaq,bbq,caq,daq) mixed type functional equation in various normed spaces: using two different methods. Int. J. Math. Appl. 5(1-B), 187–211 (2017)Google Scholar
  11. 11.
    D.H. Hyers, On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27, 222–224 (1941)MathSciNetCrossRefGoogle Scholar
  12. 12.
    P. L. Kannappan, Quadratic functional equation and inner product spaces. Results Math. 27, 368–372 (1995)MathSciNetCrossRefGoogle Scholar
  13. 13.
    A.K. Katsaras, Fuzzy topological vector spaces II. Fuzzy Sets Syst. 12, 143–154 (1984)MathSciNetCrossRefGoogle Scholar
  14. 14.
    I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces. Kybernetica 11, 326–334 (1975)MathSciNetzbMATHGoogle Scholar
  15. 15.
    A.K. Mirmostafee, M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets Syst. 159, 720–729 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A.K. Mirmostafee, M.S. Moslehian, Fuzzy almost quadratic functions. Results Math. https://doi.org/10.1007/s00025-007-0278-9
  17. 17.
    S. Murthy, V. Govindhan, General solution and generalized HU (Hyers-Ulam) stability of new dimension cubic functional equation in Fuzzy ternary Banach algebras: using two different methods. Int. J. Pure Appl. Math. 113( 6), 394–403 (2017)Google Scholar
  18. 18.
    S. Pinelas, V. Govindan, K. Tamilvanan, Stability of a quartic functional equation. J. Fixed Point Theory Appl. 20(148), 10pp. (2018). https://doi.org/10.007/s11784-018-0629-z
  19. 19.
    S. Pinelas, V. Govindan, K. Tamilvanan, Stability of non- additive functional equation. IOSR J. Math. 14( 2 - I), 60–78 (2018)Google Scholar
  20. 20.
    T.M. Rassias, On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)MathSciNetCrossRefGoogle Scholar
  21. 21.
    J.M. Rassias, On approximation of approximately linear mappings by linear mapping. J. Funct. Anal. 46(1), 126–130 (1982)MathSciNetCrossRefGoogle Scholar
  22. 22.
    J.M. Rassias, On approximation of approximately linear mappings by linear mappings. Bull. Sci. Math. 108(4), 445–446 (1984)MathSciNetzbMATHGoogle Scholar
  23. 23.
    K. Ravi, P. Narasimman, R. Kishore Kumar, Generalized Hyers-Ulam-Rassias stability and J. M. Rassias stability of a quadratic functional equation. Int. J. Math. Sci. Eng. Appl. 3(2), 79–94 (2009)Google Scholar
  24. 24.
    K. Ravi, R. Kodandan, P. Narasimman, Ulam stability of a quadratic functional equation. Int. J. Pure Appl. Math. 51(1), 87–101 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    B. Shieh, Infinite fuzzy relation equations with continuous t-norms. Inf. Sci. 178, 1961–1967 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    S.M. Ulam, A Collection of the Mathematical Problems (Interscience, New York, 1960).zbMATHGoogle Scholar
  27. 27.
    C. Wu, J. Fang, Fuzzy generalization of Klomogoroffs theorem. J. Harbin Inst. Technol. 1, 1–7 (1984)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sandra Pinelas
    • 1
  • V. Govindan
    • 2
  • K. Tamilvanan
    • 3
  1. 1.Departmento de Ciências Exatas e EngenhariaAcademia MilitarLisboaPortugal
  2. 2.Department of MathematicsSri Vidya Mandir Arts and Science CollegeUthangaraiIndia
  3. 3.Department of MathematicsGovernment Arts and Science College (for Men)KrishnagiriIndia

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