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Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces

Author

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  • Martínez, Eulalia
  • Singh, Sukhjit
  • Hueso, José L.
  • Gupta, Dharmendra K.

Abstract

In this work we introduce a new form of setting the general assumptions for the local convergence studies of iterative methods in Banach spaces that allows us to improve the convergence domains. Specifically a local convergence result for a family of higher order iterative methods for solving nonlinear equations in Banach spaces is established under the assumption that the Fréchet derivative satisfies the Lipschitz continuity condition. For some values of the parameter, these iterative methods are of fifth order. The importance of our work is that it avoids the usual practice of boundedness conditions of higher order derivatives which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained.

Suggested Citation

  • Martínez, Eulalia & Singh, Sukhjit & Hueso, José L. & Gupta, Dharmendra K., 2016. "Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 252-265.
    • Handle: RePEc:eee:apmaco:v:281:y:2016:i:c:p:252-265
    DOI: 10.1016/j.amc.2016.01.036
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    References listed on IDEAS

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    1. Cordero, A. & Ezquerro, J.A. & Hernández-Verón, M.A. & Torregrosa, J.R., 2015. "On the local convergence of a fifth-order iterative method in Banach spaces," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 396-403.
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    Cited by:

    1. Janak Raj Sharma & Harmandeep Singh & Ioannis K. Argyros, 2022. "A Unified Local-Semilocal Convergence Analysis of Efficient Higher Order Iterative Methods in Banach Spaces," Mathematics, MDPI, vol. 10(17), pages 1-16, September.
    2. Sukjith Singh & Eulalia Martínez & P. Maroju & Ramandeep Behl, 2020. "A study of the local convergence of a fifth order iterative method," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(2), pages 439-455, June.
    3. Ramandeep Behl & Ioannis K. Argyros & Ali Saleh Alshomrani, 2019. "High Convergence Order Iterative Procedures for Solving Equations Originating from Real Life Problems," Mathematics, MDPI, vol. 7(9), pages 1-12, September.
    4. Sukhjit Singh & Eulalia Martínez & Abhimanyu Kumar & D. K. Gupta, 2020. "Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations," Mathematics, MDPI, vol. 8(3), pages 1-11, March.

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