Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERAL SOLUTION AND ULAM-HYERS STABILITY OF VIGINTI FUNCTIONAL EQUATIONS IN MULTI-BANACH SPACES

  • Received : 2017.12.30
  • Accepted : 2018.01.31
  • Published : 2018.05.15
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce the general form of a viginti functional equation. Then, we find the general solution and study the generalized Ulam-Hyers stability of such functional equation in multi-Banach spaces by using fixed point technique. Also, we indicate an example for non-stability case regarding to this new functional equation.

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, J. M. Rassias, and E. Sathiya, The general solution and approximations of a decic type functional equation in various normed spaces, J. Chungcheong Math. Soc. 29 (2016), 287-328. https://doi.org/10.14403/jcms.2016.29.2.287
  3. A. Bodaghi, Stability of a mixed type additive and quartic function equation, Filomat, 28 (2014), no. 8, 1629-1640. https://doi.org/10.2298/FIL1408629B
  4. A. Bodaghi, Stability of a quartic functional equation, Scientific World Journal, 2014, Article ID 752146, 9pages.
  5. 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
  6. A. Bodaghi, Approximate mixed type additive and quartic functional equation, Bol. Soc. Paran. Mat. 35 (2017), 43-56. https://doi.org/10.5269/bspm.v35i1.29014
  7. A. Bodaghi, D. Kang, and J. M. Rassias, The mixed cubic-quartic functional equation, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), LXIII (2017), 215-227.
  8. A. Bodaghi and S. O. Kim, Ulam's type stability of a functional equation deriving from quadratic and additive functions, J. Math. Ineq. 9 (2015), 73-84.
  9. A. Bodaghi, S. M. Moosavi, and H. Rahimi, The generalized cubic functional equation and the stability of cubic Jordan ∗-derivations, Ann. Univ. Ferrara, 59 (2013), 235-50. https://doi.org/10.1007/s11565-013-0185-9
  10. A. Bodaghi and P. Narasimman, Stability of the general form of quadratic-quartic functional equations in non-Archimedean L-fuzzy normed spaces, Tbilisi Math. J. 11 (2018), no. 1, 15-29.
  11. 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 (2016), 729-743. https://doi.org/10.4134/CKMS.c150147
  12. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
  13. H. G. Dales and M. Sal Moslehian, Stability of mappings on multi-normed spaces, Glas. Math. J. 49 (2007), 321-332. https://doi.org/10.1017/S0017089507003552
  14. 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
  15. Z. Gajda, On stability of additive mappings, Int. J. Math. Sci. 14 (1991), 431-434. https://doi.org/10.1155/S016117129100056X
  16. 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
  17. M. E. Gordji, tability of a functional equation deriving from quartic and additive functions, Bull. Korean Math. Soc. 47 (2010), 491-502. https://doi.org/10.4134/BKMS.2010.47.3.491
  18. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  19. K. W. Jun and H. M. Kim, The generalized Hyers-Ulam-Russias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867-878. https://doi.org/10.1016/S0022-247X(02)00415-8
  20. D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567-572. https://doi.org/10.1016/j.jmaa.2008.01.100
  21. P. Narasimman and A. Bodaghi, Solution and stability of a mixed type functional equation, Filomat, 31 (2017), 1229-1239. https://doi.org/10.2298/FIL1705229N
  22. J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Matematicki Series III, 34 (1999) 243-252.
  23. J. M. Rassias, Solution of the Ulam stability problem for cubic mappings, Glas. Mat. Ser. III, 36 (2001), no. 56, 63-72.
  24. J. M. Rassias, M. Arunkumar, E. Sathya, and T. Namachivayam, Various generalized Ulam-Hyers Stabilities of a nonic functional equations, Tbilisi Math. J. 9 (2016), 159-196.
  25. J. M. Rassias and M. Eslamian, Fixed points and stability of nonic functional equation in quasi-${\beta}$-normed spaces, Cont. Anal. Appl. Math. 3 (2015), 293-309.
  26. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  27. K. Ravi, J. M. Rassias, and B. V. Senthil Kumar, Ulam-Hyers stability of undecic functional equation in quasi-beta normed spaces fixed point method, Tbilisi Math. Sci. 9 (2016), 83-103.
  28. K. Ravi, J. M. Rassias, S. Pinelas, and S. Suresh, General solution and stability of quattuordecic functional equation in quasi beta-normed spaces, Adv. Pure Math. 6 (2016), 921-941. https://doi.org/10.4236/apm.2016.612070
  29. F. Skof, Propriet localie approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  30. M. Turinici, Sequentially iterative processes and applications to Volterra func- tional equations, Annales Univ. Mariae-Curie Sklodowska, (Sect A), 32 (1978), 127-134.
  31. S. M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.
  32. T. Z. Xu and J. M. Rassias, Approximate septic and octic mappings in quasi-${\beta}$- normed spaces, J. Comput. Anal. Appl. 15 (2013), no. 6, 1110-1119.
  33. T. Z. Xu, J. M. Rassias, and W. X. Xu, A generalized mixed quadratic-quartic functional equation, Bulletin Malay. Math. Sci. Soc. 35 (2012), 633-649.
  34. T. Z. Xu, J. M. Rassias, M. J. Rassias, and W. X. Xu, A fixed point approach to the stability of quintic and sextic functional equations in quasi-${\beta}$-normed spaces, J. Inequal. Appl. 2010, Article ID 423231, 23pages, doi:10.1155/2010/423231.