Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

STABILITY AND HYPERSTABILITY OF MULTI-ADDITIVE-CUBIC MAPPINGS IN INTUITIONISTIC FUZZY NORMED SPACES

  • Ramzanpour, Elahe (Department of Mathematics, South Tehran Branch, Islamic Azad University) ;
  • Bodaghi, Abasalt (Department of Mathematics, Garmsar Branch, Islamic Azad University) ;
  • Gilani, Alireza (Department of Mathematics, South Tehran Branch, Islamic Azad University)
  • Received : 2019.09.25
  • Accepted : 2020.02.18
  • Published : 2020.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

In the current paper, the intuitionistic fuzzy normed space version of Hyers-Ulam stability for multi-additive, multi-cubic and multi-additive-cubic mappings by using a fixed point method are studied. Moreover, a few corollaries corresponding to some known stability and hyperstability outcomes in intuitionistic fuzzy normed space are presented.

Keywords

References

  1. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan. 2 (1950), 64.66. https://doi.org/10.2969/jmsj/00210064
  2. 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 No. 4 (2017), 915-932. https://doi.org/10.4134/CKMS.c160272
  3. A. Bodaghi, Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations, J. Intel. Fuzzy Syst. 30 (2016), 2309-2317. https://doi.org/10.3233/IFS-152001
  4. A. Bodaghi, C. Park and O. T. Mewomo, Multiquartic functional equations, Adv. Diff. Equa. 2019, 2019:312, https://doi.org/10.1186/s13662-019-2255-5
  5. A. Bodaghi and B. Shojaee, On an equation characterizing multi-cubic mappings and its stability and hyperstability, Fixed Point Theory, to appear, arXiv:1907.09378v2
  6. J. Brzdek and K. Cieplinski, Hyperstability and superstability, Abstr. Appl. Anal. 2013, Article ID 401756, 13 pp.
  7. K. Cieplinski, On the generalized Hyers-Ulam stability of multi-quadratic mappings, Comput. Math. Appl. 62 (2011), 3418-3426. https://doi.org/10.1016/j.camwa.2011.08.057
  8. K. Cieplinski, Generalized stability of multi-additive mappings, Appl. Math. Lett. 23 (2010), 1291-1294. https://doi.org/10.1016/j.aml.2010.06.015
  9. L. Cadariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara, Ser. Mat. Inform. 41 (2003), 25-48.
  10. L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: A fixed point approach, Grazer Math. Ber. 346 (2004), 43-52.
  11. J. B. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
  12. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  13. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  14. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality, Birkhauser Verlag, Basel, 2009.
  15. A. Nejati, A. Bodaghi and A. Gharibkhajeh, Characterization, stability and hyperstability of multi-additive-cubic mappings, unpublished manuscript.
  16. J. H. Park, Intuitionistic fuzzy metric spaces, Chaos. Sol. Frac. 22 (2004), 1039-1046. https://doi.org/10.1016/j.chaos.2004.02.051
  17. C. Park and A. Bodaghi, Two multi-cubic functional equations and some results on the stability in modular spaces, J. Inequ. Appl. (2020) 2020:6, https://doi.org/10.1186/s13660-019-2274-5
  18. J. M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA. 46, (1982) 126-130. https://doi.org/10.1016/0022-1236(82)90048-9
  19. Th. M. Rassias, On the stability of the linear mapping in Banach Space, Proc. Amer. Math. Soc. 72 (2),(1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  20. R. Saadati, A note on "Some results on the IF-normed spaces", Chaos. Sol. Frac. 41 (2009), 206-213. https://doi.org/10.1016/j.chaos.2007.11.027
  21. R. Saadati, Y. J. Cho and J. Vahidi, The stability of the quartic functional equation in various spaces, Comp. Math. Appl. 60 (2010), 1994-2002. https://doi.org/10.1016/j.camwa.2010.07.034
  22. R. Saadati and C. Park, Non-archimedean L-fuzzy normed spaces and stability of functional equations, Comp. Math. Appl. 60 (2010), 2488-2496. https://doi.org/10.1016/j.camwa.2010.08.055
  23. R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos. Sol. Frac. 27 (2006), 331-344. https://doi.org/10.1016/j.chaos.2005.03.019
  24. R. Saadati, S. Sedghi and N. Shobe, Modified intuitionistic fuzzy metric spaces and some fixed point theorems, Chaos. Sol. Frac., 38 (2008), 36-47. https://doi.org/10.1016/j.chaos.2006.11.008
  25. S. Salimi and A. Bodaghi, A fixed point application for the stability and hyper-stability of multi-Jensen-quadratic mappings, J. Fixed Point Theory Appl. 2020 22:9, https://doi.org/10.1007/s11784-019-0738-3
  26. S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Ed., Wiley, New York, 1940.
  27. T. Z. Xu, M. J. Rassias and W. X. Xu, Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces, J. Math. Physics. 093508, 19 pages, 51 (2010).
  28. T. Z. Xu, M. J. Rassias, W. X. Xu and J. M. Rassias, A fixed point approach to the intuitionistic fuzzy stability of quintic and sextic functional equations, Iran. J. Fuzzy Sys. 9, No 5 (2012), 21-40.
  29. L. A. Zadeh, Fuzzy sets, Inform. Control. 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
  30. X. Zhao, X. Yang and C.-T. Pang, Solution and stability of the multiquadratic functional equation, Abstr. Appl. Anal. (2013) Art. ID 415053, 8 pp.