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Semilocal and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition

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  • Singh, Sukhjit
  • Gupta, Dharmendra Kumar
  • Martínez, E.
  • Hueso, José L.

Abstract

The semilocal and local convergence in Banach spaces is described for a fifth order iteration for the solutions of nonlinear equations when the Fréchet derivative satisfies the Hölder condition. The Hölder condition generalizes the Lipschtiz condition. The importance of our work lies in the fact that many examples are available which fail to satisfy the Lipschtiz condition but satisfy the Hölder condition. The existence and uniqueness theorem is established with error bounds for the solution. The convergence analysis is finally worked out on different examples and convergence balls for each of them are obtained. These examples include nonlinear Hammerstein and Fredholm integral equations and a boundary value problem. It is found that the larger radius of convergence balls is obtained for all the examples in comparison to existing methods using stronger conditions.

Suggested Citation

  • Singh, Sukhjit & Gupta, Dharmendra Kumar & Martínez, E. & Hueso, José L., 2016. "Semilocal and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 266-277.
    • Handle: RePEc:eee:apmaco:v:276:y:2016:i:c:p:266-277
    DOI: 10.1016/j.amc.2015.11.062
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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. Hernández-Verón, M.A. & Yadav, Sonia & Martínez, Eulalia & Singh, Sukhjit, 2021. "Solving nonlinear integral equations with non-separable kernel via a high-order iterative process," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    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. Abhimanyu Kumar & D. K. Gupta & Shwetabh Srivastava, 2017. "Influence of the Center Condition on the Two-Step Secant Method," International Journal of Analysis, Hindawi, vol. 2017, pages 1-9, September.

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