The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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A NEW GENERALIZED CUBIC FUNCTIONAL EQUATION AND ITS STABILITY PROBLEMS

  • Received : 2019.09.19
  • Accepted : 2021.01.14
  • Published : 2021.02.28
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Abstract

The purpose of this paper is to introduce a new type of a cubic functional equation and then investigate its stability problems in a convex modular space with a generalized ∆a-condition.

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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. J.A. Baker: The stability of certain functional equations. Proc. Amer. Math. Soc. 112 (1991), no. 3, 729-732 https://doi.org/10.1090/S0002-9939-1991-1052568-7
  3. L. Cadariu & V. Radu: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Appl. 2008 (2008), Art. ID 749392
  4. L. Cadariu & V. Radu: On the stability of the Cauchy functional equation: a fixed point approach. Iteration theory (ECIT '02), Grazer Math. Ber. 346 (2004), Karl-Franzens-Univ. Graz., 43-52
  5. I.S. Chang, K.W. Jun & Y.S. Jung: The modified Hyers-Ulam-Rassias stability of a cubic type functional equation. Math. Inequal. Appl. 8 (2005), no. 4, 675-683
  6. D.H. Hyers: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 27 (1941), 222-224 https://doi.org/10.1073/pnas.27.4.222
  7. G. Isac & Th.M. Rassias: Stability of ϕ-additive mappings: Applications to nonlinear analysis. Internat. J. Math. Math. Sci. 19 (1996), no. 2, 219-228 https://doi.org/10.1155/S0161171296000324
  8. K.W. Jun & H.M. Kim: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274 (2002), 867-878 https://doi.org/10.1016/S0022-247X(02)00415-8
  9. K.W. Jun & H.M. Kim: On the Hyers-Ulam-Rassias stability of a general cubic functional equation. Math. Inequal. Appl. 6 (2003), no. 2, 289-302
  10. H.K. Kim & H.Y. Shin: Refined stability of additive and quadratic functional equations in modular spaces. J. Inequal. Appl. 2017 (2017), DOI 10.1186/s13660-017-1422-z
  11. B. Margolis & J.B. Diaz: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 126 (1968), no. 74, 305-309 https://doi.org/10.1090/S0002-9904-1968-11933-0
  12. D. Mihet & V. Radu: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343 (2008), no. 1, 567-572 https://doi.org/10.1016/j.jmaa.2008.01.100
  13. H. Nakano: Modulared Semi-Ordered Linear Spaces. Maruzen (1950), Tokyo
  14. C. Park: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory and Appl. 2007 (2007), Art. ID 50175
  15. A. Najati: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. Turk. J. Math. 31 (2007), 395-408
  16. V. Radu: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4 (2003), no. 1, 91-96
  17. 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
  18. I.A. Rus: Principles and Appications of Fixed Point Theory. Ed. Dacia (1979), ClujNapoca (in Romanian)
  19. P.K. Sahoo: A generalized cubic functional equation. Acta Math. Sinica 21 (2005), no. 5, 1159-1166 https://doi.org/10.1007/s10114-005-0551-3
  20. S.M. Ulam: Problems in Morden Mathematics. Wiley, New York (1960)
  21. T.Z. Xu, J.M. Rassias & W.X. Xu: A generalized mixed quadratic-quartic functional equation. Bull. Malaysian Math. Scien. Soc. 35 (2012), no. 3, 633-649