PROJECTION-ITERATION REALIZATION OF A NEWTON-LIKE METHOD FOR SOLVING NONLINEAR OPERATOR EQUATIONS
We consider the problem of existence and location of a solution of a nonlinear operator equation with a Fr´echet differentiable operator in a Banach space and present the convergence results for a projection-iteration method based on a Newton-like method under the Cauchy’s conditions, which generali...
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| Format: | Article |
| Language: | English |
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Oles Honchar Dnipro National University
2019-03-01
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| Series: | Journal of Optimization, Differential Equations and Their Applications |
| Subjects: | |
| Online Access: | https://model-dnu.dp.ua/index.php/SM/article/view/134 |
| _version_ | 1818965678827241472 |
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| author | Liudmyla L. Hart |
| author_facet | Liudmyla L. Hart |
| author_sort | Liudmyla L. Hart |
| collection | DOAJ |
| description | We consider the problem of existence and location of a solution of a nonlinear
operator equation with a Fr´echet differentiable operator in a Banach space and present the convergence results for a projection-iteration method based on a Newton-like method under the Cauchy’s conditions, which generalize the results for the projection-iteration realization of the Newton-Kantorovich method. The proposed method unlike the traditional interpretation is based on the idea of whatever approximation of the original equation by a sequence of approximate operator equations defined on subspaces of the basic space with the subsequent application of the Newton-like method to their approximate solution. We prove the convergence theorem, obtain the error estimate and discuss the advantages of the proposed approach and some of its modifications. |
| first_indexed | 2024-12-20T13:20:50Z |
| format | Article |
| id | doaj.art-4b6e0b3b9364491c9e093259cd1056ac |
| institution | Directory Open Access Journal |
| issn | 2617-0108 2663-6824 |
| language | English |
| last_indexed | 2024-12-20T13:20:50Z |
| publishDate | 2019-03-01 |
| publisher | Oles Honchar Dnipro National University |
| record_format | Article |
| series | Journal of Optimization, Differential Equations and Their Applications |
| spelling | doaj.art-4b6e0b3b9364491c9e093259cd1056ac2022-12-21T19:39:24ZengOles Honchar Dnipro National UniversityJournal of Optimization, Differential Equations and Their Applications2617-01082663-68242019-03-01271586810.15421/141903126PROJECTION-ITERATION REALIZATION OF A NEWTON-LIKE METHOD FOR SOLVING NONLINEAR OPERATOR EQUATIONSLiudmyla L. Hart0Department of Computational Mathematics and Mathematical Cybernetics, Oles Honchar Dnipro National UniversityWe consider the problem of existence and location of a solution of a nonlinear operator equation with a Fr´echet differentiable operator in a Banach space and present the convergence results for a projection-iteration method based on a Newton-like method under the Cauchy’s conditions, which generalize the results for the projection-iteration realization of the Newton-Kantorovich method. The proposed method unlike the traditional interpretation is based on the idea of whatever approximation of the original equation by a sequence of approximate operator equations defined on subspaces of the basic space with the subsequent application of the Newton-like method to their approximate solution. We prove the convergence theorem, obtain the error estimate and discuss the advantages of the proposed approach and some of its modifications.https://model-dnu.dp.ua/index.php/SM/article/view/134nonlinear equationFr´echet differentiable operatorNewton-like methodprojection-iteration methodapproximationconvergenceerror estimate |
| spellingShingle | Liudmyla L. Hart PROJECTION-ITERATION REALIZATION OF A NEWTON-LIKE METHOD FOR SOLVING NONLINEAR OPERATOR EQUATIONS Journal of Optimization, Differential Equations and Their Applications nonlinear equation Fr´echet differentiable operator Newton-like method projection-iteration method approximation convergence error estimate |
| title | PROJECTION-ITERATION REALIZATION OF A NEWTON-LIKE METHOD FOR SOLVING NONLINEAR OPERATOR EQUATIONS |
| title_full | PROJECTION-ITERATION REALIZATION OF A NEWTON-LIKE METHOD FOR SOLVING NONLINEAR OPERATOR EQUATIONS |
| title_fullStr | PROJECTION-ITERATION REALIZATION OF A NEWTON-LIKE METHOD FOR SOLVING NONLINEAR OPERATOR EQUATIONS |
| title_full_unstemmed | PROJECTION-ITERATION REALIZATION OF A NEWTON-LIKE METHOD FOR SOLVING NONLINEAR OPERATOR EQUATIONS |
| title_short | PROJECTION-ITERATION REALIZATION OF A NEWTON-LIKE METHOD FOR SOLVING NONLINEAR OPERATOR EQUATIONS |
| title_sort | projection iteration realization of a newton like method for solving nonlinear operator equations |
| topic | nonlinear equation Fr´echet differentiable operator Newton-like method projection-iteration method approximation convergence error estimate |
| url | https://model-dnu.dp.ua/index.php/SM/article/view/134 |
| work_keys_str_mv | AT liudmylalhart projectioniterationrealizationofanewtonlikemethodforsolvingnonlinearoperatorequations |