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Generalized Convergence for Multi-Step Schemes under Weak Conditions

Author

Listed:
  • Ramandeep Behl

    (Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Ioannis K. Argyros

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

  • Hashim Alshehri

    (Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Samundra Regmi

    (Department of Mathematics, University of Houston, Houston, TX 77205, USA)

Abstract

We have developed a local convergence analysis for a general scheme of high-order convergence, aiming to solve equations in Banach spaces. A priori estimates are developed based on the error distances. This way, we know in advance the number of iterations required to reach a predetermined error tolerance. Moreover, a radius of convergence is determined, allowing for a selection of initial points assuring the convergence of the scheme. Furthermore, a neighborhood that contains only one solution to the equation is specified. Notably, we present the generalized convergence of these schemes under weak conditions. Our findings are based on generalized continuity requirements and contain a new semi-local convergence analysis (with a majorizing sequence) not seen in earlier studies based on Taylor series and derivatives which are not present in the scheme. We conclude with a good collection of numerical results derived from applied science problems.

Suggested Citation

  • Ramandeep Behl & Ioannis K. Argyros & Hashim Alshehri & Samundra Regmi, 2024. "Generalized Convergence for Multi-Step Schemes under Weak Conditions," Mathematics, MDPI, vol. 12(2), pages 1-15, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:220-:d:1315834
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    References listed on IDEAS

    as
    1. Huda J. Saeed & Ali Hasan Ali & Rayene Menzer & Ana Danca Poțclean & Himani Arora, 2023. "New Family of Multi-Step Iterative Methods Based on Homotopy Perturbation Technique for Solving Nonlinear Equations," Mathematics, MDPI, vol. 11(12), pages 1-13, June.
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