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Extending the Applicability of Highly Efficient Iterative Methods for Nonlinear Equations and Their Applications

Author

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  • Ramandeep Behl

    (Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Ioannis K. Argyros

    (Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA)

  • Fouad Othman Mallawi

    (Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Samaher Khalaf Alharbi

    (Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
    Department of Mathematics, College of Science and Arts, Qassim University, Ar Rass 51921, Saudi Arabia)

Abstract

Numerous three-step methods of high convergence order have been developed to produce sequences approximating solutions of equations usually defined on the Euclidean space with a finite dimension. The local convergence order is determined by Taylor expansions requiring the existence of derivatives that are not present on the methods. The more interesting semi-local convergence analysis for these methods has not been considered before. The semi-local is also provided based on generalized ω -continuity conditions on the derivative of the operator involved and the majorizing sequences, thus limiting their usage to only solving equations with operators that are many times differentiable. However, these methods may convergence to a solution of the equation even if these high-order derivatives do not exist. That is why a methodology is utilized on two sixth convergence order methods and in the more general setting of a Banach space. This time, the convergence depends only on the operators and the first derivative on the method. Therefore, by this methodology the applicability of the methods is in the extended area. Although this methodology is demonstrated on two competing and efficient methods, it can also be utilized for the same reasons on other methods involving inverses of operators that are linear. This is the motivation and novelty of the paper. The numerical applications further validate the theoretical results both in the local as well as the semi-local convergence case.

Suggested Citation

  • Ramandeep Behl & Ioannis K. Argyros & Fouad Othman Mallawi & Samaher Khalaf Alharbi, 2022. "Extending the Applicability of Highly Efficient Iterative Methods for Nonlinear Equations and Their Applications," Mathematics, MDPI, vol. 11(1), pages 1-18, December.
  • Handle: RePEc:gam:jmathe:v:11:y:2022:i:1:p:125-:d:1016683
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    References listed on IDEAS

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    1. Artidiello, Santiago & Cordero, Alicia & Torregrosa, Juan R. & Vassileva, Maria P., 2015. "Multidimensional generalization of iterative methods for solving nonlinear problems by means of weight-function procedure," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 1064-1071.
    2. Zhanlav, T. & Otgondorj, Kh., 2021. "Higher order Jarratt-like iterations for solving systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    3. Ioannis K. Argyros, 2021. "Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications," Mathematics, MDPI, vol. 9(16), pages 1-15, August.
    4. Ramandeep Behl & Ioannis K. Argyros & Jose Antonio Tenreiro Machado, 2020. "Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators," Mathematics, MDPI, vol. 8(5), pages 1-12, April.
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