The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces

Eskandar Naraghirad, Luoyi Shi and Ngai-Ching Wong
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Eskandar Naraghirad: Department of Mathematics, Yasouj University, Yasouj 75918, Iran
Luoyi Shi: School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
Ngai-Ching Wong: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Mathematics, 2020, vol. 8, issue 6, 1-13

Abstract: 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.

Keywords: Bregman–Opial property; Bregman generalized hybrid map/sequence; Bregman absolute fixed point; convergence theorem; fixed point theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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