Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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SOLUTION AND STABILITY OF AN n-VARIABLE ADDITIVE FUNCTIONAL EQUATION

  • Received : 2020.04.14
  • Accepted : 2020.09.15
  • Published : 2020.09.30
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Abstract

In this paper, we investigate the general solution and the Hyers-Ulam stability of n-variable additive functional equation of the form $${\Im}\(\sum\limits_{i=1}^{n}(-1)^{i+1}x_i\)=\sum\limits_{i=1}^{n}(-1)^{i+1}{\Im}(x_i)$$, where n is a positive integer with n ≥ 2, in Banach spaces by using the direct method.

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References

  1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989.
  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. E. Baktash, Y. Cho, M. Jalili, R. Saadati and S. M. Vaezpour, On the stability of cubic mappingsand quadratic mappings in random normed spaces, J. Inequal. Appl. 2008 (2008), Article ID 902187.
  4. I. Chang, E. Lee and H. Kim, On the Hyers-Ulam-Rassias stability of a quadratic functional equations, Math. Inequal. Appl. 6 (2003), 87-95.
  5. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  6. 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
  7. D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34, Progress in Nonlinear Differential Equations and Their Applications, Birkhauser, Boston, 1998.
  8. M. Eshaghi Gordji and H. Khodaie, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Anal. 71 (2009), 5629-5643. https://doi.org/10.1016/j.na.2009.04.052
  9. K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 267-278.
  10. C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, Bull. Sci. Math. 132 (2008), 87-96. https://doi.org/10.1016/j.bulsci.2006.07.004
  11. C. Park and J. Cui, Generalized stability of C*-ternary quadratic mappings, Abs. Appl. Anal. 2007 (2007), Article ID 23282.
  12. C. Park and A. Najati, Homomorphisms and derivations in C*-algebras, Abs. Appl. Anal. 2007 (2007), Article ID 80630.
  13. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  14. Th. M. Rassias, On the stability of the functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264-284. https://doi.org/10.1006/jmaa.2000.7046
  15. Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan, London, 2003.
  16. S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1964.