Right Bregman nonexpansive operators in Banach spaces

Abstract

We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.

A Modified Iterative Algorithm for Split Feasibility Problems of Right Bregman Strongly Quasi-Nonexpansive Mappings in Banach Spaces with Applications

“…(2) T is called right Bregman strongly quasi-nonexpansive (shortly, R-BSQNE) (see [23,24]) with respect to a nonempty…”

Split common fixed point problem for Bregman demigeneralized mappings in Banach spaces with applications

“…We denote bỹ F(T ) the set of strong asymptotic fixed points of T . A map T : C → C is called quasi-Bregman relatively nonexpansive [16] if F(T ) = ∅,F(T ) = F(T ) and D f (T x, p) D f (x, p) for all x ∈ C and p ∈ F(T ). T is said to be quasi-Bregman strictly pseudocontractive [26] if there exists a constant λ ∈ [0, 1) and…”

Convergence Theorems for Right Bregman Strongly Nonexpansive Mappings in Reflexive Banach Spaces

“…or, equivalently, The class of right Bregman firmly nonexpansive mappings associated with the Bregman distance induced by a convex function was introduced and studied by Martin-Marques et al [30]. Examples of right Bregman firmly nonexpansive mappings are given in [30].…”

“…or, equivalently, The class of right Bregman firmly nonexpansive mappings associated with the Bregman distance induced by a convex function was introduced and studied by Martin-Marques et al [30]. Examples of right Bregman firmly nonexpansive mappings are given in [30]. If is a nonempty and closed subset of int(dom ), where is a Legendre and Fréchet differentiable function, and : → int(dom ) is a right Bregman strongly nonexpansive mapping, it is proved that ( ) is closed (see [30]).…”

“…Examples of right Bregman firmly nonexpansive mappings are given in [30]. If is a nonempty and closed subset of int(dom ), where is a Legendre and Fréchet differentiable function, and : → int(dom ) is a right Bregman strongly nonexpansive mapping, it is proved that ( ) is closed (see [30]). In addition, they have shown that this class of mappings is closed under composition and convex combination and proved weak convergence of the Picard iterative method to a fixed point of a mapping under suitable conditions (see [31]).…”