DOI QR코드

DOI QR Code

APPROXIMATION OF CUBIC MAPPINGS WITH n-VARIABLES IN β-NORMED LEFT BANACH MODULE ON BANACH ALGEBRAS

  • Gordji, Majid Eshaghi ;
  • Khodaei, Hamid ;
  • Najati, Abbas
  • Received : 2010.04.03
  • Published : 2011.09.30

Abstract

Let M = {1, 2, ${\ldots}$, n} and let V = {$I{\subseteq}M:1{\in}I$}. Denote M\I by $I^c$ for $I{\in}V$. The goal of this paper is to investigate the solution and the stability using the alternative fixed point of generalized cubic functional equation $ \sum\limits_{I{\in}V}f(\sum\limits_{i{\in}I}a_ix_i-\sum\limits_{i{\in}I^c}a_ix_i)=2{^{n-2}{a_1}}\sum\limits_{i=2}^na_i^2[f(x_1+x_i)+f(x_1-x_i)]+2{^{n-1}{a_1}(a^2_1-\sum\limits_{i=2}^2a^2_i)f(x_1)$ in ${\beta}$-Banach modules on Banach algebras, where $a_1,{\ldots},a_n{\in}\mathbb{Z}{\backslash}\{0\}$ with $a_1{\neq}={\pm}1$ and $a_n=1$.

Keywords

cubic functional equation;generalized Hyers-Ulam stability;Banach module

References

  1. V. K. Balachandran, Topological Algebras, Narosa Publishing House, New Delhi, Madras, Bombay, Calcutta, London, 1999.
  2. 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
  3. L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Iteration theory (ECIT '02), 43-52, Grazer Math. Ber., 346, Karl-Franzens- Univ. Graz, Graz, 2004.
  4. P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore, London, 2002.
  5. M. Eshaghi Gordji and H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Anal. 71 (2009), no. 11, 5629-5643. https://doi.org/10.1016/j.na.2009.04.052
  6. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434. https://doi.org/10.1155/S016117129100056X
  7. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately ad- ditive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436. https://doi.org/10.1006/jmaa.1994.1211
  8. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  9. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhaser, Basel, 1998.
  10. K. W. Jun and H. M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), no. 2, 267-278.
  11. K. W. Jun and H. M. Kim, Ulam stability problem for a mixed type of cubic and additive functional equa- tion, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 2, 271-285.
  12. K. W. Jun and H. M. Kim, On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces, J. Math. Anal. Appl. 332 (2007), no. 2, 1335-1350. https://doi.org/10.1016/j.jmaa.2006.11.024
  13. K. W. Jun, H. M. Kim, and I. S. Chang, On the Hyers-Ulam stability of an Euler- Lagrange type cubic functional equation, J. Comput. Anal. Appl. 7 (2005), no. 1, 21-33.
  14. K. Jun and S. Lee, On the generalized Hyers-Ulam stability of a cubic functional equa- tion, J. Chungcheong Math. Soc. 19 (2006), no. 2, 189-196.
  15. H. Khodaei and Th. M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. 1 (2010), 22-41.
  16. B. Margolis and J. B. Diaz, 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
  17. A. Najati, Hyers-Ulam-Rassias stability of a cubic functional equation, Bull. Korean Math. Soc. 44 (2007), no. 4, 825-840. https://doi.org/10.4134/BKMS.2007.44.4.825
  18. A. Najati and F. Moradlou, Stability of an Euler-Lagrange type cubic functional equa- tion, Turkish J. Math. 33 (2009), no. 1, 65-73.
  19. A. Najati and F. Moradlou, Stability of a mixed additive, quadratic and cubic functional equation in quasi- Banach spaces, Aust. J. Math. Anal. Appl. 5 (2008), no. 2, Article 10, 21 pp.
  20. A. Najati and C. Park, On the stability of a cubic functional equation, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 12, 1953-1964. https://doi.org/10.1007/s10114-008-6560-2
  21. A. Najati and G. Zamani Eskandani, Stability of a mixed additive and cubic functional equation in quasi-Banach spaces, J. Math. Anal. Appl. 342 (2008), no.2, 1318-1331. https://doi.org/10.1016/j.jmaa.2007.12.039
  22. V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91-96.
  23. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  24. Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), no. 2, 352-378. https://doi.org/10.1006/jmaa.2000.6788
  25. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130. https://doi.org/10.1023/A:1006499223572
  26. S. M. Ulam, Problems in Modern Mathematics, Chapter VI, science Editions., Wiley, New York, 1964.

Cited by

  1. Behavior of Bi-Cubic Functions in Lipschitz Spaces vol.39, pp.6, 2018, https://doi.org/10.1134/S1995080218060136