The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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GENERALIZED HYERS-ULAM STABILITY OF A QUADRATIC-CUBIC FUNCTIONAL EQUATION IN MODULAR SPACES

  • Lee, Yang-Hi (Department of Mathematics Education, Gongju National University of Education)
  • Received : 2018.11.24
  • Accepted : 2018.12.22
  • Published : 2019.02.28
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Abstract

In this paper, I prove the stability problem for a quadratic-cubic functional equation $$f(x+ky)-k^2f(x+y)-k^2f(x-y)+f(x-ky)+f(kx)-{\frac{k^3-3k^2+4}{2}}f(x)+{\frac{k^3-k^2}{2}}f(-x)=0$$ in modular spaces by applying the direct method.

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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. Baker: A general functional equation and its stability. Proc. Natl. Acad. Sci. 133 (2005), no. 6, 1657-1664.
  3. I.-S. Chang & 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
  4. Y.-J. Cho, M. Eshaghi Gordji & S. Zolfaghari: Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces. J. Inequal. Appl. 2010 (2010), Art. ID 403101.
  5. 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
  6. P. Gavruta: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. and Appl. 184 (1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  7. 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
  8. S.-S. Jin & Y.-H. Lee: Generalized Hyers-Ulam stability of a 3-dimensional quadratic functional equation in modular spaces. Int. J. Math. Anal. (Ruse) 10 (2016), 953-963. https://doi.org/10.12988/ijma.2016.6461
  9. C.-J. Lee & 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.
  10. 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
  11. Y.-H. Lee & K.-W. Jun: On the stability of approximately additive mappings. Proc. Amer. Math. Soc. 128 (2000), 1361-1369. https://doi.org/10.1090/S0002-9939-99-05156-4
  12. Y.-H. Lee & S.-M. Jung: Fuzzy stability of the cubic and quadratic functional equation. Appl. Math. Sci. (Ruse) 10 (2016), 2671-2686.
  13. J. Musielak & W. Orlicz: On modular spaces. Studia Mathematica 18 (1959), 591-597.
  14. H. Nakano: Modular Semi-Ordered Spaces. Tokyo, Japan, 1959.
  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. G. Sadeghi: A fixed point approach to stability of functional equations in modular spaces. Bull. the Malays. Math. Sci. Soc. 37 (2014) 333-344.
  17. F. Skof: Proprieta locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  18. W. Towanlong & P. Nakmahachalasint: A mixed-type quadratic and cubic functional equation and its stability. Thai J. Math. 8 (2012), no. 4, 61-71.
  19. S.M. Ulam: Problems in Modern Mathematics. Wiley, New York, 1964.
  20. Z. Wang & 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
  21. K. Wongkum, P. Chaipunya & Poom Kumam: On the generalized Ulam-Hyers-Rassias stability of quadratic mappings in modular spaces without ${\Delta}_2$-conditions. J. Funct. Spaces (2015), Article ID 461719, 6 pages.