The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces
The Opial property of Hilbert spaces is essential in many fixed point theorems of non-expansive maps. While the Opial property does not hold in every Banach space, the Bregman–Opial property does. This suggests to study fixed point theorems for various Bregman non-expansive like maps in the general...
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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/2227-7390/8/6/1022 |
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| author | Eskandar Naraghirad Luoyi Shi Ngai-Ching Wong |
| author_facet | Eskandar Naraghirad Luoyi Shi Ngai-Ching Wong |
| author_sort | Eskandar Naraghirad |
| collection | DOAJ |
| description | The Opial property of Hilbert spaces is essential in many fixed point theorems of non-expansive maps. While the Opial property does not hold in every Banach space, the Bregman–Opial property does. This suggests to study fixed point theorems for various Bregman non-expansive like maps in the general Banach space setting. In this paper, after introducing the notion of Bregman generalized hybrid sequences in a reflexive Banach space, we prove (with using the Bregman–Opial property instead of the Opial property) convergence theorems for such sequences. We also provide new fixed point theorems for Bregman generalized hybrid maps defined on an arbitrary but not necessarily convex subset of a reflexive Banach space. We end this paper with a brief discussion of the existence of Bregman absolute fixed points of such maps. |
| first_indexed | 2024-03-10T18:58:22Z |
| format | Article |
| id | doaj.art-89827e1c75244523a72e483ccf8b5638 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T18:58:22Z |
| publishDate | 2020-06-01 |
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| series | Mathematics |
| spelling | doaj.art-89827e1c75244523a72e483ccf8b56382023-11-20T04:37:12ZengMDPI AGMathematics2227-73902020-06-0186102210.3390/math8061022The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach SpacesEskandar Naraghirad0Luoyi Shi1Ngai-Ching Wong2Department of Mathematics, Yasouj University, Yasouj 75918, IranSchool of Mathematical Sciences, Tiangong University, Tianjin 300387, ChinaDepartment of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, TaiwanThe Opial property of Hilbert spaces is essential in many fixed point theorems of non-expansive maps. While the Opial property does not hold in every Banach space, the Bregman–Opial property does. This suggests to study fixed point theorems for various Bregman non-expansive like maps in the general Banach space setting. In this paper, after introducing the notion of Bregman generalized hybrid sequences in a reflexive Banach space, we prove (with using the Bregman–Opial property instead of the Opial property) convergence theorems for such sequences. We also provide new fixed point theorems for Bregman generalized hybrid maps defined on an arbitrary but not necessarily convex subset of a reflexive Banach space. We end this paper with a brief discussion of the existence of Bregman absolute fixed points of such maps.https://www.mdpi.com/2227-7390/8/6/1022Bregman–Opial propertyBregman generalized hybrid map/sequenceBregman absolute fixed pointconvergence theoremfixed point theorem |
| spellingShingle | Eskandar Naraghirad Luoyi Shi Ngai-Ching Wong The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces Mathematics Bregman–Opial property Bregman generalized hybrid map/sequence Bregman absolute fixed point convergence theorem fixed point theorem |
| title | The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces |
| title_full | The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces |
| title_fullStr | The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces |
| title_full_unstemmed | The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces |
| title_short | The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces |
| title_sort | bregman opial property and bregman generalized hybrid maps of reflexive banach spaces |
| topic | Bregman–Opial property Bregman generalized hybrid map/sequence Bregman absolute fixed point convergence theorem fixed point theorem |
| url | https://www.mdpi.com/2227-7390/8/6/1022 |
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