Iterative methods for monotone nonexpansive mappings in uniformly convex spaces
We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1n∑i=0n−1Ti(x)$ \frac{1}{n}\...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2021-03-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2020-0170 |
| _version_ | 1819120293647482880 |
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| author | Shukla Rahul Wiśnicki Andrzej |
| author_facet | Shukla Rahul Wiśnicki Andrzej |
| author_sort | Shukla Rahul |
| collection | DOAJ |
| description | We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1n∑i=0n−1Ti(x)$ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard’s iteration {Tn(x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well. |
| first_indexed | 2024-12-22T06:18:22Z |
| format | Article |
| id | doaj.art-5020926b1755482e8b2568d35f75cf1b |
| institution | Directory Open Access Journal |
| issn | 2191-9496 2191-950X |
| language | English |
| last_indexed | 2024-12-22T06:18:22Z |
| publishDate | 2021-03-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-5020926b1755482e8b2568d35f75cf1b2022-12-21T18:36:01ZengDe GruyterAdvances in Nonlinear Analysis2191-94962191-950X2021-03-011011061107010.1515/anona-2020-0170Iterative methods for monotone nonexpansive mappings in uniformly convex spacesShukla Rahul0Wiśnicki Andrzej1Department of Mathematics & Applied Mathematics, University of Johannesburg, Kingsway Campus, Auckland Park 2006, South AfricaDepartment of Mathematics, Pedagogical University of Krakow, PL-30-084Cracow, PolandWe show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1n∑i=0n−1Ti(x)$ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard’s iteration {Tn(x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.https://doi.org/10.1515/anona-2020-0170monotone mappingnonexpansive mappingfixed pointergodic theorempicard iterationordered banach space47j2646b2047h0754f05 |
| spellingShingle | Shukla Rahul Wiśnicki Andrzej Iterative methods for monotone nonexpansive mappings in uniformly convex spaces Advances in Nonlinear Analysis monotone mapping nonexpansive mapping fixed point ergodic theorem picard iteration ordered banach space 47j26 46b20 47h07 54f05 |
| title | Iterative methods for monotone nonexpansive mappings in uniformly convex spaces |
| title_full | Iterative methods for monotone nonexpansive mappings in uniformly convex spaces |
| title_fullStr | Iterative methods for monotone nonexpansive mappings in uniformly convex spaces |
| title_full_unstemmed | Iterative methods for monotone nonexpansive mappings in uniformly convex spaces |
| title_short | Iterative methods for monotone nonexpansive mappings in uniformly convex spaces |
| title_sort | iterative methods for monotone nonexpansive mappings in uniformly convex spaces |
| topic | monotone mapping nonexpansive mapping fixed point ergodic theorem picard iteration ordered banach space 47j26 46b20 47h07 54f05 |
| url | https://doi.org/10.1515/anona-2020-0170 |
| work_keys_str_mv | AT shuklarahul iterativemethodsformonotonenonexpansivemappingsinuniformlyconvexspaces AT wisnickiandrzej iterativemethodsformonotonenonexpansivemappingsinuniformlyconvexspaces |