Journal of Mathematical Chemistry

, Volume 55, Issue 7, pp 1392–1406 | Cite as

Convergence of Newton’s method under Vertgeim conditions: new extensions using restricted convergence domains

  • I. K. Argyros
  • J. A. Ezquerro
  • M. A. Hernández-Verón
  • Á. A. Magreñán
Original Paper


We present new sufficient convergence conditions for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear equation in a Banach space. We use Hölder and center Hölder conditions, instead of just Hölder conditions, for the first derivative of the operator involved in combination with our new idea of restricted convergence domains. This way, we find a more precise location where the iterates lie, leading to at least as small Hölder constants as in earlier studies. The new convergence conditions are weaker, the error bounds are tighter and the information on the solution at least as precise as before. These advantages are obtained under the same computational cost. Numerical examples show that our results can be used to solve equations where older results cannot.


Newton’s method Recurrent functions Hölder continuity Semilocal convergence Integral equation Differential equation 

Mathematics Subject Classification

65H10 65G99 65J15 47H17 49M15 


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Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • I. K. Argyros
    • 1
  • J. A. Ezquerro
    • 2
  • M. A. Hernández-Verón
    • 2
  • Á. A. Magreñán
    • 3
  1. 1.Department of Mathematics SciencesCameron UniversityLawtonUSA
  2. 2.Department of Mathematics and ComputationUniversity of La RiojaLogroñoSpain
  3. 3.Escuela Superior de Ingeniería y TecnologíaInternational University of La RiojaLogroñoSpain

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