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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Bibliographic Details
Published in:Mathematics (Basel) 2020-06, Vol.8 (6), p.1022
Main Authors: Naraghirad, Eskandar, Shi, Luoyi, Wong, Ngai-Ching
Format: Article
Language:English
Subjects:
Bregman absolute fixed point
Bregman generalized hybrid map/sequence
Bregman–Opial property
convergence theorem
fixed point theorem
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math8061022