Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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STABILITY OF ZEROS OF POWER SERIES EQUATIONS

  • Wang, Zhihua (School of Science Hubei University of Technology) ;
  • Dong, Xiuming (School of Science Hubei University of Technology) ;
  • Rassias, Themistocles M. (Department of Mathematics National Technical University of Athens) ;
  • Jung, Soon-Mo (Mathematics Section College of Science and Technology Hongik University)
  • Received : 2012.09.13
  • Published : 2014.01.31
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Abstract

We prove that if ${\mid}a_1{\mid}$ is large and ${\mid}a_0{\mid}$ is small enough, then every approximate zero of power series equation ${\sum}^{\infty}_{n=0}a_nx^n$=0 can be approximated by a true zero within a good error bound. Further, we obtain Hyers-Ulam stability of zeros of the polynomial equation of degree n, $a_nz^n$ + $a_{n-1}z^{n-1}$ + ${\cdots}$ + $a_1z$ + $a_0$ = 0 for a given integer n > 1.

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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. M. Bidkham, H. A. Soleiman Mezerji, and M. Eshaghi Gordji, Hyers-Ulam stability of polynomial equations, Abstr. Appl. Anal. 2010 (2010), Article ID 754120, 7 pages.
  3. M. Bidkham, H. A. Soleiman Mezerji, and M. Eshaghi Gordji, Hyers-Ulam stability of power series equations, Abstr. Appl. Anal. 2011 (2011), Article ID 194948, 6 pages.
  4. G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1-2, 143-190. https://doi.org/10.1007/BF01831117
  5. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 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. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  7. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several variables, Birkhauser, Basel, 1998.
  8. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications Vol. 48, Springer, New York, 2011.
  9. S.-M. Jung, Hyers-Ulam stability of zeros of polynomial, Appl. Math. Lett. 24 (2011), no. 8, 1322-1325. https://doi.org/10.1016/j.aml.2011.03.002
  10. Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, 2009.
  11. Y. Li and L. Hua, Hyers-Ulam stability of a polynomial equation, Banach J. Math. Anal. 3 (2009), no. 2, 86-90. https://doi.org/10.15352/bjma/1261086712
  12. 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
  13. 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
  14. Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic, Dordrecht, 2003.
  15. E. Schechter, Handbook of Analysis and its Foundations, Academic Press, New York, 1997.
  16. S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1964.