10.30495/maca.2021.680135

Generalized Ulam-Hyers stability of an alternate additive-quadratic-quartic functional equation in fuzzy Banach spaces

  1. Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens, 4 Agamemnonos Street, Aghia Paraskevi, Athens 15342, Greece
  2. Department of Mathematics, Shanmuga Industries Arts and Science College, Tiruvannamalai-606603, Tamil Nadu, India
  3. Department of Mathematics, Government Arts College, Tiruvannamalai-606603, Tamil Nadu, India

Published in Issue 2021-01-01

How to Cite

Generalized Ulam-Hyers stability of an alternate additive-quadratic-quartic functional equation in fuzzy Banach spaces. (2021). Mathematical Analysis and Its Contemporary Applications, 3(1). https://doi.org/10.30495/maca.2021.680135

PDF views: 21

Abstract

In this paper, we obtain and establish the generalized Ulam-Hyers stability of an additive-quadratic-quartic functional equation in fuzzy Banach spaces.

Keywords

  • Additive functional equations,
  • Quadratic functional equations,
  • Quartic functional equations,
  • Mixed type functional equations,
  • Ulam-Hyers-Rassias stability,
  • Fuzzy Banach Space

References

  1. J. Aczel and J. Dhombres, Functional equations in several variables, Cambridge Univ. Press, 1989.
  2. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.
  3. M. Arunkumar, Three dimensional quartic functional equation in fuzzy normed spaces, Far East J. Appl. Math., 41(2) (2010), 88-94.
  4. M. Arunkumar, A. Bodaghi, T. Namachivayam and E. Sathya, A new type of the additive functional equations on intuitionistic fuzzy normed spaces, Commun. Korean Math. Soc., 32(4) (2017), 915-932.
  5. M. Arunkumar, S. H. Latha and C. D. Shaymala Mary, Functional equation originating from arithmetic Mean of consecutive terms of an arithmetic Progression are stable in Banach space: Direct and fixed point methods, JP J. Math. Sci., 3(1) (2012), 27-43.
  6. M. Arunkumar and S. H. Latha, Additive-Quartic functional equations are stable in random normed space, Jamal Acad. Research J. Interdis., (2015), 31-38.
  7. K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87-96.
  8. T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11(3) (2003), 687-705.
  9. T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst., 151 (2005), 513-547.
  10. B. Belaid and El. Elhoucien, Ulam-Gavruta-Rassias stability of the Pexider functional equation, Inter. J. Math. Stat., 7 (2007), 27-39.
  11. A. Bodaghi, Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations, J. Intell. Fuzzy Syst., 30 (2016), 2309-2317.
  12. A. Bodaghi, Stability of a quartic functional equation, The Sci. World J., 2014, Art. ID 752146, 9 pages, doi:10.1155/2014/752146.
  13. A. Bodaghi, I. A. Alias and M. H. Ghahramani, Approximately cubic functional equations and cubic multipliers, J. Ineq. Appl., 2011 (2011), 53.
  14. A. Bodaghi and S. O. Kim, Approximation on the quadratic reciprocal functional equation, J. Func. Spac., 2014, Article ID 532463, 5 pages. http://dx.doi.org/10.1155/2014/532463
  15. A. Bodaghi, C. Park and J. M. Rassias, Fundamental stabilities of the nonic functional equation in intuitionistic fuzzy normed spaces, Commun. Korean Math. Soc., 31(4) (2016), 729-743.
  16. S. Czerwik, Functional equations and inequalities in several variables, World Scientific, River Edge, NJ, 2002.
  17. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.
  18. M. Eshaghi Gordji and A. Bodaghi, On the Hyers-Ulam-Rasias stability problem for quadratic functional equations, East. J. Approx. 16(2) (2010), 123-130.
  19. M. E. Gordji, A. Bodaghi and C. Park, A fixed point approach to the stability of double Jordan centralizers and Jordan multipliers on Banach algebras, U. Politeh. Buch. Ser. A., 73 (2011), 65-73.
  20. M. Eshaghi Gordji, M. B. Savadkouhi and C. Park, Quadratic-Quartic Functional Equations in RN-Spaces, J. Inequal. Appl., Volume 2009, Article ID 868423, 14 pages doi:10.1155/2009/868423.
  21. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., U.S.A., 27 (1941) 222-224.
  22. D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkhauser, Basel, 1998.
  23. Pl. Kannappan, Functional equations and inequalities with applications, Springer Monographs in Mathematics, 2009.
  24. L. Maligranda, A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functionsa question of priority, Aequ. Math., 75 (2008), 289-296.
  25. A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Set. Systems, 159(6) (2008), 720-729.
  26. A. K. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst., 159(6) (2008), 730-738.
  27. A. K. Mirmostafaee, M. S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci., 178(19) (2008), 3791-3798.
  28. A. K. Mirmostafaee and M. S. Moslehian, Fuzzy almost quadratic functions, Results Math., 52(1-2) (2008), 161-177.
  29. A. Najati, M. B. Moghimi, On the stability of a quadratic and additive functional equation, J. Math. Anal. Appl., 337 (2008), 399-415.
  30. A. Niazi Motlagh, The generalized Hyers-Ulam stability of derivations in non-Archimedean Banach algebras, Math. Anal. Cont. Appl., 2 (2020), 17-22.
  31. J. M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal., 46, (1982) 126-130.
  32. J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Math., 34(2) (1999), 243-252.
  33. J. M. Rassias, Solution of the Ulam stability problem for cubic mappings, Glasnik Math., Ser. III., 36(1) (2001), 63-72.
  34. J. M. Rassias, K. Ravi, M. Arunkumar and B. V. Senthil Kumar, Ulam Stability of Mixed type Cubic and Additive functional equation, Functional Ulam Notions (F.U.N) Nova Science Publishers, 2010, Chapter 13, 149-175.
  35. J. M. Rassias, M. Arunkumar and E. Sathya, Non-stabilities of mixed type Euler-Lagrange k-cubic-quartic functional equation in various normed spaces, Math. Anal. Cont. Appl., 1 (2019), 1-43.
  36. J. M. Rassias, M. Arunkumar, E.Sathya and N. Mahesh Kumar, Generalized Ulam-Hyers stability of a (AQQ): additive-quadratic-quartic functional equation, Malaya J. Mat., 5(1) (2017), 122-142.
  37. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.
  38. K. Ravi, M. Arunkumar and J. M. Rassias, On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Sci., 3(A08) (2008), 36-47.
  39. S. M. Ulam, Problems in modern mathematics, Science Editions, Wiley, NewYork, 1964.
  40. A. Zivari-Kazempour, Stability of cosine type functional equations on module extension Banach algebras, Math. Anal. Cont. Appl., 1 (2020), 44-49.