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Mathematics
Volume 13
Issue 10
10.3390/math13101590
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

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

by
Ioannis K. Argyros
1,†,
Sania Qureshi
2,3,4,*,†,
Amanullah Soomro 
2,†,
Muath Awadalla
5,*,†,
Ausif Padder
6,† and
Michael I. Argyros 
7,†
1
Department of Computing and Mathematics Sciences, Cameron University, Lawton, OK 73505, USA
2
Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan
3
Department of Mathematics, Near East University TRNC, 99138 Mersin, Turkey
4
Research Center of Applied Mathematics, Khazar University, Baku 1009, Azerbaijan
5
Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Al Ahsa, Saudi Arabia
6
Symbiosis Institute of Technology, Hyderabad Campus, Symbiosis International (Deemed University), Pune 412115, India
7
College of Computing and Engineering , Nova Southeastern University, Fort Lauderdale, FL 33328, USA
*
Authors to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Mathematics 2025, 13(10), 1590; https://doi.org/10.3390/math13101590
Submission received: 10 April 2025 / Revised: 2 May 2025 / Accepted: 9 May 2025 / Published: 12 May 2025
(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis: 2nd Edition)
Versions Notes

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 semilocal convergence; majorizing sequence; banach space; Fréchet derivative; bounded linear operator

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MDPI and ACS Style

Argyros, I.K.; Qureshi, S.; Soomro , A.; Awadalla, M.; Padder, A.; Argyros , M.I. Abstract Convergence Analysis for a New Nonlinear Ninth-Order Iterative Scheme. Mathematics 2025, 13, 1590. https://doi.org/10.3390/math13101590

AMA Style

Argyros IK, Qureshi S, Soomro  A, Awadalla M, Padder A, Argyros  MI. Abstract Convergence Analysis for a New Nonlinear Ninth-Order Iterative Scheme. Mathematics. 2025; 13(10):1590. https://doi.org/10.3390/math13101590

Chicago/Turabian Style

Argyros, Ioannis K., Sania Qureshi, Amanullah Soomro , Muath Awadalla, Ausif Padder, and Michael I. Argyros . 2025. "Abstract Convergence Analysis for a New Nonlinear Ninth-Order Iterative Scheme" Mathematics 13, no. 10: 1590. https://doi.org/10.3390/math13101590

APA Style

Argyros, I. K., Qureshi, S., Soomro , A., Awadalla, M., Padder, A., & Argyros , M. I. (2025). Abstract Convergence Analysis for a New Nonlinear Ninth-Order Iterative Scheme. Mathematics, 13(10), 1590. https://doi.org/10.3390/math13101590

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