An Extension on the Local Convergence for the Multi-Step Seventh Order Method with <i>ψ</i>-Continuity Condition in the Banach Spaces
The local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires a weak hypothesis where the Fréchet derivative of the nonlinear operator satisfies the <inline-formula><...
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MDPI AG
2022-11-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/12/713 |
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| author | Mohammad Taghi Darvishi R. H. Al-Obaidi Akanksha Saxena Jai Prakash Jaiswal Kamal Raj Pardasani |
| author_facet | Mohammad Taghi Darvishi R. H. Al-Obaidi Akanksha Saxena Jai Prakash Jaiswal Kamal Raj Pardasani |
| author_sort | Mohammad Taghi Darvishi |
| collection | DOAJ |
| description | The local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires a weak hypothesis where the Fréchet derivative of the nonlinear operator satisfies the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-continuity condition, which thereby extends the applicability of the method when both Lipschitz and Hölder conditions fail. The convergence in this study is considered under the hypotheses on the first-order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised here. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some concrete numerical examples showing the performance of the method over some existing schemes are presented in this article. |
| first_indexed | 2024-03-09T16:35:45Z |
| format | Article |
| id | doaj.art-f01e398688e842429693591fb230dc45 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T16:35:45Z |
| publishDate | 2022-11-01 |
| publisher | MDPI AG |
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| series | Fractal and Fractional |
| spelling | doaj.art-f01e398688e842429693591fb230dc452023-11-24T14:57:23ZengMDPI AGFractal and Fractional2504-31102022-11-0161271310.3390/fractalfract6120713An Extension on the Local Convergence for the Multi-Step Seventh Order Method with <i>ψ</i>-Continuity Condition in the Banach SpacesMohammad Taghi Darvishi0R. H. Al-Obaidi1Akanksha Saxena2Jai Prakash Jaiswal3Kamal Raj Pardasani4Department of Mathematics, Faculty of Science, Razi University, Kermanshah 67149, IranDepartment of Mathematics, Faculty of Science, Razi University, Kermanshah 67149, IranDepartment of Mathematics, Maulana Azad National Institute of Technology, Bhopal 462003, IndiaDepartment of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur 495009, IndiaDepartment of Mathematics, Maulana Azad National Institute of Technology, Bhopal 462003, IndiaThe local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires a weak hypothesis where the Fréchet derivative of the nonlinear operator satisfies the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-continuity condition, which thereby extends the applicability of the method when both Lipschitz and Hölder conditions fail. The convergence in this study is considered under the hypotheses on the first-order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised here. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some concrete numerical examples showing the performance of the method over some existing schemes are presented in this article.https://www.mdpi.com/2504-3110/6/12/713nonlinear equationBanach spacemulti-step method<i>ψ</i>-continuity conditionlocal convergence |
| spellingShingle | Mohammad Taghi Darvishi R. H. Al-Obaidi Akanksha Saxena Jai Prakash Jaiswal Kamal Raj Pardasani An Extension on the Local Convergence for the Multi-Step Seventh Order Method with <i>ψ</i>-Continuity Condition in the Banach Spaces Fractal and Fractional nonlinear equation Banach space multi-step method <i>ψ</i>-continuity condition local convergence |
| title | An Extension on the Local Convergence for the Multi-Step Seventh Order Method with <i>ψ</i>-Continuity Condition in the Banach Spaces |
| title_full | An Extension on the Local Convergence for the Multi-Step Seventh Order Method with <i>ψ</i>-Continuity Condition in the Banach Spaces |
| title_fullStr | An Extension on the Local Convergence for the Multi-Step Seventh Order Method with <i>ψ</i>-Continuity Condition in the Banach Spaces |
| title_full_unstemmed | An Extension on the Local Convergence for the Multi-Step Seventh Order Method with <i>ψ</i>-Continuity Condition in the Banach Spaces |
| title_short | An Extension on the Local Convergence for the Multi-Step Seventh Order Method with <i>ψ</i>-Continuity Condition in the Banach Spaces |
| title_sort | extension on the local convergence for the multi step seventh order method with i ψ i continuity condition in the banach spaces |
| topic | nonlinear equation Banach space multi-step method <i>ψ</i>-continuity condition local convergence |
| url | https://www.mdpi.com/2504-3110/6/12/713 |
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