Abstract Convergence Analysis for a New Nonlinear Ninth-Order Iterative Scheme

Ioannis K. Argyros, Sania Qureshi (), Amanullah Soomro, Muath Awadalla (), Ausif Padder and Michael I. Argyros
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Ioannis K. Argyros: Department of Computing and Mathematics Sciences, Cameron University, Lawton, OK 73505, USA
Sania Qureshi: Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan
Amanullah Soomro: Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan
Muath Awadalla: Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Al Ahsa, Saudi Arabia
Ausif Padder: Symbiosis Institute of Technology, Hyderabad Campus, Symbiosis International (Deemed University), Pune 412115, India
Michael I. Argyros: College of Computing and Engineering , Nova Southeastern University, Fort Lauderdale, FL 33328, USA

Mathematics, 2025, vol. 13, issue 10, 1-18

Abstract: This study presents a comprehensive analysis of the semilocal convergence properties of a high-order iterative scheme designed to solve nonlinear equations in Banach spaces. The investigation is carried out under the assumption that the first derivative of the associated nonlinear operator adheres to a generalized Lipschitz-type condition, which broadens the applicability of the convergence analysis. Furthermore, the research demonstrates that, under an additional mild assumption, the proposed scheme achieves a remarkable ninth-order rate of convergence. This high-order convergence result significantly contributes to the theoretical understanding of iterative schemes in infinite-dimensional settings. Beyond the theoretical implications, the results also have practical relevance, particularly in the context of solving complex systems of equations and integral equations that frequently arise in applied mathematics, physics, and engineering disciplines. Overall, the findings provide valuable insights into the behavior and efficiency of advanced iterative schemes in Banach space frameworks. The comparative analysis with existing schemes also demonstrates that the ninth-order iterative scheme achieves faster convergence in most cases, particularly for smaller radii.

Keywords: semilocal convergence; majorizing sequence; banach space; Fréchet derivative; bounded linear operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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