Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE "TERRA INCOGNITA" FOR THE NEWTON-KANTROVICH METHOD WITH APPLICATIONS

  • Argyros, Ioannis Konstantinos (Department of Mathematics Sciences Cameron University) ;
  • Cho, Yeol Je (Department of Mathematics Education and the RINS Gyeongsang National University) ;
  • George, Santhosh (Department of Mathematical and Computational Sciences National Institute of Technology Karnataka)
  • Received : 2013.02.20
  • Accepted : 2013.08.14
  • Published : 2014.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]-[13], in some interesting cases, provided that the Fr$\acute{e}$chet-derivative of the operator involved is p-H$\ddot{o}$lder continuous (p${\in}$(0, 1]). Numerical examples involving two boundary value problems are also provided.

Keywords

References

  1. J. Appell, E. De Pascale, J. V. Lysenko, and P. P. Zabrejko, New results on Newton-Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997), no. 1-2, 1-17. https://doi.org/10.1080/01630569708816744
  2. I. K. Argyros, The theory and application of abstract polynomial equations, St. Lucie/CRC/Lewis Publ. Mathematics series, Boca Raton, Florida, 1998.
  3. I. K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004), no. 2, 374-397. https://doi.org/10.1016/j.jmaa.2004.04.008
  4. I. K. Argyros, Concerning the "terra incognita" between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005), no. 1, 179-194. https://doi.org/10.1016/j.na.2005.02.113
  5. I. K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer, New York, 2008.
  6. I. K. Argyros, Y. J. Cho, and S. Hilout, Numerical Methods for Equations and Its Applications, CRC Press, Taylor and Francis, New York, 2012.
  7. F. Cianciaruso, A further journey in the "terra incognita" of the Newton-Kantorovich method, Nonlinear Funct. Anal. Appl., to appear.
  8. F. Cianciaruso and E. De Pascale, Newton-Kantorovich approximations when the derivative is Holderian: old and new results, Numer. Funct. Anal. Optim. 24 (2003), no. 7-8, 713-723. https://doi.org/10.1081/NFA-120026367
  9. E. De Pascale and P. P. Zabrejko, Convergence of the Newton-Kantorovich method under Vertgeim conditions: a new improvement, Z. Anal. Anwendvugen 17 (1998), no. 2, 271-280. https://doi.org/10.4171/ZAA/821
  10. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
  11. J. V. Lysenko, Conditions for the convergence of the Newton-Kantorovich method for nonlinear equations with Holder linearizations, Dokl. Akad. Nauk Belarusi 38 (1994), no. 3, 20-24, 122-123.
  12. B. A. Vertgeim, On conditions for the applicability of Newton's method, (Russian) Dokl. Akad. N., SSSR 110 (1956), 719-722.
  13. B. A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957), 166-169 (in Russian); English transl.: Amer. Math. Soc. Transl. 16 (1960), 378-382.

Cited by

  1. On iterative computation of fixed points and optimization vol.2015, pp.1, 2015, https://doi.org/10.1186/s13663-015-0372-8
  2. LOCAL CONVERGENCE FOR SOME THIRD-ORDER ITERATIVE METHODS UNDER WEAK CONDITIONS vol.53, pp.4, 2016, https://doi.org/10.4134/JKMS.j150244