A Study of at Least Sixth Convergence Order Methods Without or with Memory and Divided Differences for Equations Under Generalized Continuity

Ioannis K. Argyros, Ramandeep Behl, Sattam Alharbi () and Abdulaziz Mutlaq Alotaibi
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Ioannis K. Argyros: Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
Ramandeep Behl: Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Sattam Alharbi: Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
Abdulaziz Mutlaq Alotaibi: Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia

Mathematics, 2025, vol. 13, issue 5, 1-23

Abstract: Multistep methods typically use Taylor series to attain their convergence order, which necessitates the existence of derivatives not naturally present in the iterative functions. Other issues are the absence of a priori error estimates, information about the radius of convergence or the uniqueness of the solution. These restrictions impose constraints on the use of such methods, especially since these methods may converge. Consequently, local convergence analysis emerges as a more effective approach, as it relies on criteria involving only the operators of the methods. This expands the applicability of such methods, including in non-Euclidean space scenarios. Furthermore, this work uses majorizing sequences to address the more challenging semi-local convergence analysis, which was not explored in earlier research. We adopted generalized continuity constraints to control the derivatives and obtain sharper error estimates. The sufficient convergence criteria are demonstrated through examples.

Keywords: Banach space; ball convergence; generalized continuity; multistep method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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