Strong Convergence of Mann’s Iteration Process in Banach Spaces
Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/6/954 |
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| author | Hong-Kun Xu Najla Altwaijry Souhail Chebbi |
| author_facet | Hong-Kun Xu Najla Altwaijry Souhail Chebbi |
| author_sort | Hong-Kun Xu |
| collection | DOAJ |
| description | Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants. |
| first_indexed | 2024-03-10T19:14:39Z |
| format | Article |
| id | doaj.art-08a8f96618c84905ad5626ac715e2308 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T19:14:39Z |
| publishDate | 2020-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-08a8f96618c84905ad5626ac715e23082023-11-20T03:30:28ZengMDPI AGMathematics2227-73902020-06-018695410.3390/math8060954Strong Convergence of Mann’s Iteration Process in Banach SpacesHong-Kun Xu0Najla Altwaijry1Souhail Chebbi2School of Science, Hangzhou Dianzi University, Hangzhou 310018, ChinaDepartment of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaMann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants.https://www.mdpi.com/2227-7390/8/6/954nonexpansive mappingmann iterationstrong convergenceduality mapbanach space |
| spellingShingle | Hong-Kun Xu Najla Altwaijry Souhail Chebbi Strong Convergence of Mann’s Iteration Process in Banach Spaces Mathematics nonexpansive mapping mann iteration strong convergence duality map banach space |
| title | Strong Convergence of Mann’s Iteration Process in Banach Spaces |
| title_full | Strong Convergence of Mann’s Iteration Process in Banach Spaces |
| title_fullStr | Strong Convergence of Mann’s Iteration Process in Banach Spaces |
| title_full_unstemmed | Strong Convergence of Mann’s Iteration Process in Banach Spaces |
| title_short | Strong Convergence of Mann’s Iteration Process in Banach Spaces |
| title_sort | strong convergence of mann s iteration process in banach spaces |
| topic | nonexpansive mapping mann iteration strong convergence duality map banach space |
| url | https://www.mdpi.com/2227-7390/8/6/954 |
| work_keys_str_mv | AT hongkunxu strongconvergenceofmannsiterationprocessinbanachspaces AT najlaaltwaijry strongconvergenceofmannsiterationprocessinbanachspaces AT souhailchebbi strongconvergenceofmannsiterationprocessinbanachspaces |