DOI QR코드

DOI QR Code

ON THE STABILITY OF A GENERAL QUADRATIC-CUBIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN NORMED SPACES

  • Lee, Yang-Hi (Department of Mathematics Education, Gongju National University of Education)
  • Received : 2018.12.19
  • Accepted : 2019.05.08
  • Published : 2019.05.31

Abstract

In this paper, we investigate the stability for the functional equation $f(x+ky)-kf(x+y)+kf(x-y)-f(x-ky)-(k^3-k)f(y)+(k^3-k)f(-y)=0$ in the sense of M. S. Moslehian and Th. M. Rassias.

Keywords

References

  1. J. Baker, A general functional equation and its stability, Proc. Natl. Acad. Sci. 133(6) (2005), 1657-1664.
  2. I.-S. Chang and Y.-S. Jung, Stability of a functional equation deriving from cubic and quadratic functions, J. Math. Anal. Appl. 283 (2003), 491-500. https://doi.org/10.1016/S0022-247X(03)00276-2
  3. Y.-J. Cho, M. Eshaghi Gordji, and S. Zolfaghari, Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces, J. Inequal. Appl. 2010 (2010), Art. ID 403101.
  4. S. Czerwik, On the stability of the quadratic mapping in the normed space, Abh. Math. Sem. Hamburg 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
  5. 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
  6. 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
  7. K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), no. 2, 867-878. https://doi.org/10.1016/S0022-247X(02)00415-8
  8. K.-W. Jun and Y.-H. Lee, On the stability of a cubic functional equation, J. Chungcheong Math. Soc. 21 (2008), No. 3, 377-384.
  9. S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137. https://doi.org/10.1006/jmaa.1998.5916
  10. C.-J. Lee and Y.-H. Lee, On the stability of a mixed type quadratic and cubic functional equation, J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 19 (2012), 383-396.
  11. Y.-H. Lee, On the Hyers-Ulam-Rassias stability of a quadratic and cubic functional equation, Int. J. Math. Anal. (Ruse) 12 (2018), 577-583. https://doi.org/10.12988/ijma.2018.81064
  12. Y.-H. Lee and S.-M. Jung, Fuzzy stability of the cubic and quadratic functional equation, Appl. Math. Sci. (Ruse) 10 (2016), 2671-2686.
  13. M. S. Moslehian and Th. M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math. 1 (2007), 325-334. https://doi.org/10.2298/AADM0702325M
  14. J. M. Rassias, Solution of the Ulam stability problem for cubic, Glasnik Matematicki Serija III, vol. 36(56) (2001), no. 1, 63-72.
  15. 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
  16. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  17. W. Towanlong and P. Nakmahachalasint, A mixed-type quadratic and cubic functional equation and its stability, Thai J. Math. 8(4) (2012), 61-71.
  18. S.M. Ulam, Problems in Modern Mathematics, Wiley, New York 1964.
  19. Z. Wang and W. X. Zhang, Fuzzy stability of quadratic-cubic functional equations, Acta Math. Sin. (Engl. Ser.) 27 (2011), 2191-2204. https://doi.org/10.1007/s10114-011-9250-4