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Mathematics
Volume 12
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10.3390/math12132145
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Article

Multistep Iterative Methods for Solving Equations in Banach Space

by
Ramandeep Behl
1,*,
Ioannis K. Argyros
2,
Sattam Alharbi
3,
Hashim Alshehri
1 and
Michael Argyros
4
1
Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2
Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
3
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
4
Computing Sciences and Mathematics, Franklin University, 201 S Grant Ave., Columbus, OH 43215, USA
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(13), 2145; https://doi.org/10.3390/math12132145
Submission received: 6 June 2024 / Revised: 2 July 2024 / Accepted: 5 July 2024 / Published: 8 July 2024
(This article belongs to the Section E: Applied Mathematics)
Versions Notes

Abstract

The novelty of this article lies in the fact that we extend the use of a multistep method for developing a sequence whose limit solves a Banach space-valued equation. We suggest the error estimates, local convergence, and semi-local convergence, a radius of convergence, the uniqueness of the required solution that can be computed under ω-continuity, and conditions on the first derivative, which is on the method. But, earlier studies used high-order derivatives, even though those derivatives do not appear in the body structure of the proposed method. In addition to this, they did not propose computable estimates and semi-local convergence. We checked the applicability of our study to three real-life problems for semi-local convergence and two problems chosen for local convergence. Based on the obtained results, we conclude that our approach improves its applicability and makes it suitable for challenges in applied science.
Keywords: multistep method; ball convergence; w-continuity; Banach space multistep method; ball convergence; w-continuity; Banach space

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MDPI and ACS Style

Behl, R.; Argyros, I.K.; Alharbi, S.; Alshehri, H.; Argyros, M. Multistep Iterative Methods for Solving Equations in Banach Space. Mathematics 2024, 12, 2145. https://doi.org/10.3390/math12132145

AMA Style

Behl R, Argyros IK, Alharbi S, Alshehri H, Argyros M. Multistep Iterative Methods for Solving Equations in Banach Space. Mathematics. 2024; 12(13):2145. https://doi.org/10.3390/math12132145

Chicago/Turabian Style

Behl, Ramandeep, Ioannis K. Argyros, Sattam Alharbi, Hashim Alshehri, and Michael Argyros. 2024. "Multistep Iterative Methods for Solving Equations in Banach Space" Mathematics 12, no. 13: 2145. https://doi.org/10.3390/math12132145

APA Style

Behl, R., Argyros, I. K., Alharbi, S., Alshehri, H., & Argyros, M. (2024). Multistep Iterative Methods for Solving Equations in Banach Space. Mathematics, 12(13), 2145. https://doi.org/10.3390/math12132145

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