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Convergence of a Two-Step Iterative Method for Nondifferentiable Operators in Banach Spaces

Author

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  • Abhimanyu Kumar
  • Dharmendra K. Gupta
  • Eulalia Martínez
  • Sukhjit Singh

Abstract

The semilocal and local convergence analyses of a two-step iterative method for nonlinear nondifferentiable operators are described in Banach spaces. The recurrence relations are derived under weaker conditions on the operator. For semilocal convergence, the domain of the parameters is obtained to ensure guaranteed convergence under suitable initial approximations. The applicability of local convergence is extended as the differentiability condition on the involved operator is avoided. The region of accessibility and a way to enlarge the convergence domain are provided. Theorems are given for the existence-uniqueness balls enclosing the unique solution. Finally, some numerical examples including nonlinear Hammerstein type integral equations are worked out to validate the theoretical results.

Suggested Citation

  • Abhimanyu Kumar & Dharmendra K. Gupta & Eulalia Martínez & Sukhjit Singh, 2018. "Convergence of a Two-Step Iterative Method for Nondifferentiable Operators in Banach Spaces," Complexity, Hindawi, vol. 2018, pages 1-11, May.
    • Handle: RePEc:hin:complx:7352780
    DOI: 10.1155/2018/7352780
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    References listed on IDEAS

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    1. Behl, Ramandeep & Cordero, Alicia & Motsa, S.S. & Torregrosa, Juan R., 2015. "On developing fourth-order optimal families of methods for multiple roots and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 520-532.
    2. Ezquerro, J.A. & Hernández-Verón, M.A. & Velasco, A.I., 2015. "An analysis of the semilocal convergence for secant-like methods," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 883-892.
    3. Hernández-Verón, M.A. & Rubio, M.J., 2016. "On the ball of convergence of secant-like methods for non-differentiable operators," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 506-512.
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