Abstract
The Opial property of Hilbert spaces and some other special Banach spaces is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally, nonspreading mappings. Unfortunately, not every Banach space shares the Opial property. However, every Banach space has a similar Bregman-Opial property for Bregman distances. In this paper, using Bregman distances, we introduce the classes of Bregman nonspreading mappings and investigate the Mann and Ishikawa iterations for these mappings. We establish weak and strong convergence theorems for Bregman nonspreading mappings.
Highlights
Let E be a Banach space with norm ‖ ⋅ ‖ and dual space E∗
Let us give an example of a Bregman nonspreading mapping with nonempty fixed point set, which is not quasinonexpansive
If C is a nonempty, closed, and convex subset of a reflexive Banach space E and g : E → R is a strongly coercive Bregman function, for each x in E, there exists a unique x0 in C such that
Summary
Let E be a (real) Banach space with norm ‖ ⋅ ‖ and dual space E∗. For any x in E, we denote the value of x∗ in E∗ at x by ⟨x, x∗⟩. A Banach space E is said to satisfy the Opial property [3] if for any weakly convergent sequence {xn}n∈N in E with weak limit x we have lim sup n→∞. Not every Banach space satisfies the Opial property; see, for example, [4, 5]. The Bregman distance [6] (see [7, 8]) corresponding to g is the function Dg : E×E → R defined by It follows from the strict convexity of g that Dg(x, y) ≥ 0 for all x, y in E. Let us give an example of a Bregman nonspreading mapping with nonempty fixed point set, which is not quasinonexpansive. Our results improve and generalize some known results in the current literature; see, for example, [11]
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