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On the Convergence of Harmonic Mean Newton Method Under \(\omega \) Continuity Condition in Banach Spaces

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Abstract

We present the local and semi-local convergence analysis of the cubically convergent harmonic mean Newton method (HNM) for obtaining solutions of Banach space valued nonlinear operator equations. The significance of this work is that the convergence study needs only the condition that the first-order Fréchet derivative obeys \(\omega \) continuity. Moreover, we avoid using higher order derivatives, which do not occur in this scheme. The dynamical properties of the scheme are also explored using basins of attraction technique for various complex polynomials. Finally, the convergence radii for benchmark numerical problems are computed applying our analytical results.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

  2. Department of Mathematics, Kalinga Institute of Industrial Technology, Bhubaneswar, Odisha, 751024, India

    Debasis Sharma

  3. Department of Computer Science, University of Oklahoma, Norman, OK, 73071, USA

    Christopher I. Argyros

  4. Department of Mathematics, Fakir Mohan University, Odisha, 756020, India

    Sanjaya Kumar Parhi

  5. Department of Mathematics, IIIT Bhubaneswar, Odisha, 751003, India

    Shanta Kumari Sunanda

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  1. Ioannis K. Argyros
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  2. Debasis Sharma
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  3. Christopher I. Argyros
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  4. Sanjaya Kumar Parhi
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  5. Shanta Kumari Sunanda
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Correspondence to Debasis Sharma.

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Argyros, I.K., Sharma, D., Argyros, C.I. et al. On the Convergence of Harmonic Mean Newton Method Under \(\omega \) Continuity Condition in Banach Spaces. Int. J. Appl. Comput. Math 7, 219 (2021). https://doi.org/10.1007/s40819-021-01159-3

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