The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space
The equation 𝐿𝑢=𝑓, where 𝐿=𝐴+𝐵, with 𝐴 being a K-positive definite operator and 𝐵 being a linear operator, is solved in a Banach space. Our scheme provides a generalization to the so-called method of moments studied in a Hilbert space by Petryshyn (1962), as well as Lax and Milgram (1954). Furtherm...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2010/376852 |
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| author | S. J. Aneke |
| author_facet | S. J. Aneke |
| author_sort | S. J. Aneke |
| collection | DOAJ |
| description | The equation
𝐿𝑢=𝑓, where 𝐿=𝐴+𝐵, with 𝐴 being a K-positive definite operator and 𝐵 being a linear operator, is solved in a Banach space. Our scheme provides a generalization to the so-called method of moments studied in a Hilbert space by Petryshyn (1962), as well as Lax and Milgram (1954). Furthermore,
an application of the inverse function theorem provides simultaneously a general solution to this
equation in some neighborhood of a point 𝑥𝑜, where 𝐿 is Fréchet differentiable and an iterative scheme which converges strongly to the unique solution of this equation. |
| format | Article |
| id | doaj-art-8c9cf7819a684b93ba776a4698bcfeca |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-8c9cf7819a684b93ba776a4698bcfeca2025-02-03T05:59:41ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252010-01-01201010.1155/2010/376852376852The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach SpaceS. J. Aneke0Department of Mathematics, University of Nigeria, Nsukka, NigeriaThe equation 𝐿𝑢=𝑓, where 𝐿=𝐴+𝐵, with 𝐴 being a K-positive definite operator and 𝐵 being a linear operator, is solved in a Banach space. Our scheme provides a generalization to the so-called method of moments studied in a Hilbert space by Petryshyn (1962), as well as Lax and Milgram (1954). Furthermore, an application of the inverse function theorem provides simultaneously a general solution to this equation in some neighborhood of a point 𝑥𝑜, where 𝐿 is Fréchet differentiable and an iterative scheme which converges strongly to the unique solution of this equation.http://dx.doi.org/10.1155/2010/376852 |
| spellingShingle | S. J. Aneke The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space International Journal of Mathematics and Mathematical Sciences |
| title | The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space |
| title_full | The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space |
| title_fullStr | The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space |
| title_full_unstemmed | The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space |
| title_short | The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space |
| title_sort | solution by iteration of a composed k positive definite operator equation in a banach space |
| url | http://dx.doi.org/10.1155/2010/376852 |
| work_keys_str_mv | AT sjaneke thesolutionbyiterationofacomposedkpositivedefiniteoperatorequationinabanachspace AT sjaneke solutionbyiterationofacomposedkpositivedefiniteoperatorequationinabanachspace |