Convergence Criteria of a Three-Step Scheme under the Generalized Lipschitz Condition in Banach Spaces

Akanksha Saxena, Jai Prakash Jaiswal (), Kamal Raj Pardasani and Ioannis K. Argyros ()
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Akanksha Saxena: Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal 462003, India
Jai Prakash Jaiswal: Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur 495009, India
Kamal Raj Pardasani: Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal 462003, India
Ioannis K. Argyros: Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

Mathematics, 2022, vol. 10, issue 21, 1-22

Abstract: In the given study, we investigate the three-step NTS’s ball convergence for solving nonlinear operator equations with a convergence order of five in a Banach setting. A nonlinear operator’s first-order derivative is assumed to meet the generalized Lipschitz condition, also known as the κ -average condition. Furthermore, several theorems on the convergence of the same method in Banach spaces are developed with the conditions that the derivative of the operators must satisfy the radius or center-Lipschitz condition with a weak κ -average and that κ is a positive integrable but not necessarily non-decreasing function. This novel approach allows for a more precise convergence analysis even without the requirement for new circumstances. As a result, we broaden the applicability of iterative approaches. The theoretical results are supported further by illuminating examples. The convergence theorem investigates the location of the solution ϵ * and the existence of it. In the end, we achieve weaker sufficient convergence criteria and more specific knowledge on the position of the ϵ * than previous efforts requiring the same computational effort. We obtain the convergence theorems as well as some novel results by applying the results to some specific functions for κ ( u ). Numerical tests are carried out to corroborate the hypotheses established in this work.

Keywords: nonlinear problem; convergence radius; ball convergence; banach space; Lipschitz condition; ?-average (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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