Convergence of the Newton–Kurchatov Method Under Weak Conditions

  • S. M. Shakhno
  • H. P. Yarmola

We study the semilocal convergence of the combined Newton–Kurchatov method to a locally unique solution of the nonlinear equation under weak conditions imposed on the derivatives and first-order divided differences. The radius of the ball of convergence is established and the rate of convergence of the method is estimated. As a special case of these conditions, we consider the classical Lipschitz conditions.


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  1. 1.
    V. A. Kurchatov, “A method of linear interpolation for the solution of functional equations,” Dokl. Akad. Nauk SSSR, 198, No. 3, 524–526 (1971).MathSciNetGoogle Scholar
  2. 2.
    J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York (1970).zbMATHGoogle Scholar
  3. 3.
    S. M. Shakhno and G. P. Yarmola, “A two-point method for the solution of nonlinear equations with nondifferentiable operator,” Mat. Studii, 36, No. 2, 213–220 (2011).MathSciNetzbMATHGoogle Scholar
  4. 4.
    S. M. Shakhno and G. P. Yarmola, “Convergence of the Newton–Kurchatov method under the classical Lipschitz conditions,” Zh. Obchysl. Prykl. Mat., No. 1 (121), 89–97 (2016).Google Scholar
  5. 5.
    S. M. Shakhno, “On the difference method with quadratic convergence for the solution of nonlinear operator equations,” Mat. Studii, 26, No. 1, 105–110 (2006).MathSciNetzbMATHGoogle Scholar
  6. 6.
    S. Shakhno, “The method of chords under the generalized Lipschitz conditions for the first-order divided differences,” Mat. Visn. NTSh., 4, 296–303 (2007).zbMATHGoogle Scholar
  7. 7.
    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, No. 2, 374–397 (2004).MathSciNetCrossRefGoogle Scholar
  8. 8.
    I. K. Argyros and S. Hilout, “Newton–Kantorovich approximations under weak continuity conditions,” J. Appl. Math. Comput., 37, Nos. 1-2, 361–375 (2011).MathSciNetCrossRefGoogle Scholar
  9. 9.
    I. K. Argyros and H. Ren, “On the convergence of a Newton-like method under weak conditions,” Comm. Korean Math. Soc., 26, No. 4, 575–584 (2011).MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. Cătinaş, “On some iterative methods for solving nonlinear equations,” Rev. d’Anal. Numér. Théor. de l’Appr., 23, No. 1, 47–53 (1994).MathSciNetzbMATHGoogle Scholar
  11. 11.
    H. Ren and I. K. Argyros, “A new semilocal convergence theorem for a fast iterative method with nondifferentiable operators,” J. Appl. Math. Comput., 34, Nos. 1-2, 39–46 (2010).MathSciNetCrossRefGoogle Scholar
  12. 12.
    S. M. Shakhno, “On a Kurchatov’s method of linear interpolation for solving nonlinear equations,” Proc. Appl. Math. Mech., 4, No. 1, 650–651 (2004). Scholar
  13. 13.
    S. M. Shakhno, “Combined Newton–Kurchatov method under the generalized Lipschitz conditions for the derivatives and divided differences,” Zh. Obchysl. Prykl. Mat., No. 2 (119), 78–89 (2015).Google Scholar
  14. 14.
    P. P. Zabrejko and D. F. Nguen, “The majorant method in the theory of Newton–Kantorovich approximations and the Pták error estimates,” Numer. Funct. Anal. Optim., 9, Nos. 5-6, 671–684 (1987).MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • S. M. Shakhno
    • 1
  • H. P. Yarmola
    • 1
  1. 1.I. Franko Lviv National UniversityLvivUkraine

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