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Local convergence of a parameter based iteration with Hölder continuous derivative in Banach spaces

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Abstract

The local convergence analysis of a parameter based iteration with Hölder continuous first derivative is studied for finding solutions of nonlinear equations in Banach spaces. It generalizes the local convergence analysis under Lipschitz continuous first derivative. The main contribution is to show the applicability to those problems for which Lipschitz condition fails without using higher order derivatives. An existence-uniqueness theorem along with the derivation of error bounds for the solution is established. Different numerical examples including nonlinear Hammerstein equation are solved. The radii of balls of convergence for them are obtained. Substantial improvements of these radii are found in comparison to some other existing methods under similar conditions for all examples considered.

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Acknowledgments

The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Sukhjit Singh, D. K. Gupta & Rakesh P. Badoni

  2. Instituto de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, Spain

    E. Martínez

  3. Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain

    José L. Hueso

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  1. Sukhjit Singh
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  2. D. K. Gupta
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  3. Rakesh P. Badoni
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  4. E. Martínez
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  5. José L. Hueso
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Correspondence to Sukhjit Singh.

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Singh, S., Gupta, D.K., Badoni, R.P. et al. Local convergence of a parameter based iteration with Hölder continuous derivative in Banach spaces. Calcolo 54, 527–539 (2017). https://doi.org/10.1007/s10092-016-0197-9

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  • DOI: https://doi.org/10.1007/s10092-016-0197-9

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