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On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions

Author

Listed:
  • Akanksha Saxena

    (Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal 462003, M.P., India)

  • Ioannis K. Argyros

    (Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA)

  • Jai P. Jaiswal

    (Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur 495009, C.G., India)

  • Christopher Argyros

    (Department of Computer Science, University of Oklahoma, Norman, OK 73071, USA)

  • Kamal R. Pardasani

    (Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal 462003, M.P., India)

Abstract

The motive of this paper is to discuss the local convergence of a two-step Newton-type method of convergence rate three for solving nonlinear equations in Banach spaces. It is assumed that the first order derivative of nonlinear operator satisfies the generalized Lipschitz i.e., L -average condition. Also, some results on convergence of the same method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L -average particularly it is assumed that L is positive integrable function but not necessarily non-decreasing. Our new idea gives a tighter convergence analysis without new conditions. The proposed technique is useful in expanding the applicability of iterative methods. Useful examples justify the theoretical conclusions.

Suggested Citation

  • Akanksha Saxena & Ioannis K. Argyros & Jai P. Jaiswal & Christopher Argyros & Kamal R. Pardasani, 2021. "On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions," Mathematics, MDPI, vol. 9(6), pages 1-20, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:6:p:669-:d:521382
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    Cited by:

    1. Akanksha Saxena & Jai Prakash Jaiswal & Kamal Raj Pardasani & Ioannis K. Argyros, 2022. "Convergence Criteria of a Three-Step Scheme under the Generalized Lipschitz Condition in Banach Spaces," Mathematics, MDPI, vol. 10(21), pages 1-22, October.

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