Nonexpansive maps with surjective displacement

On the minimal displacement vector of compositions and convex combinations of nonexpansive mappings

Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. Five years ago, it was shown that if finitely many firmly nonexpansive mappings have or "almost have" fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and of convex combinations of firmly nonexpansive mappings was obtained in terms of the displacement vectors of the underlying operators. Using a new proof technique based on the Brezis-Haraux theorem and…

The magnitude of the minimal displacement vector for compositions and convex combinations of firmly nonexpansive mappings

Maximally monotone operators and firmly nonexpansive mappings play key roles in modern optimization and nonlinear analysis. Five years ago, it was shown that if finitely many firmly nonexpansive operators are all asymptotically regular (i.e., the have or "almost have" fixed points), then the same is true for compositions and convex combinations. In this paper, we derive bounds on the magnitude of the minimal displacement vectors of compositions and of convex combinations in terms of the displacement vectors of the underlying operators. Our results completely generalize earlier works. Moreover,…

关于极值非扩张映射

On extremal nonexpansive mappings

本文研究赋范空间非空有界闭凸子集上非扩张映射的极值性，证明满射等距映射在具有Radon-Nikodym性质的巴拿赫空间及C(K)空间中为极值，并表明Baire范畴意义下典型非扩张映射接近极值。

关于 $L$-$ω$-非扩张映射

On $L$-$ω$-nonexpansive maps

本文研究了 Banach 空间凸子集上的 $L$-$ω$-非扩张映射，分析了其 AFPP 和 FPP 性质，给出了映射为常数或满足 FPP 的模条件，并证明了包含 $\co$ 副本的空间在某些条件下不满足 FPP。

Applications of uniform asymptotic regularity to fixed point theorems

We show that there are no nontrivial surjective uniformly asymptotically regular mappings acting on a metric space and derive some consequences of this fact. In particular, we prove that a jointly continuous left amenable or left reversible semigroup generated by firmly nonexpansive mappings on a bounded $τ$-compact subset of a Banach space has a common fixed point, and give a qualitative complement to the Markov-Kakutani theorem.

Hölder-contractive mappings, nonlinear extension problem and fixed point free results

For a bounded closed convex set $K$, in this note, we study the FPP for $α$-Hölder nonexpansive maps, i.e. mappings $T\colon K\to K$ for which $\|T x -Ty\| \leq\| x - y\|^α$ for all $x, y\in K$, $α\in (0,1)$. First, we note that only finite-dimensional spaces have the Hölder-FPP. Moreover, the unit ball $B_X$ of any infinite-dimensional space fails the FPP for Hölder maps with $\mathrm{d}(T, B_X)>0$, where $\mathrm{d}(T, K)$ denotes the minimal displacement of $T$. We further show that reflexivity and weak sequential continuity are sufficient conditions to capture fixed points of Hölder-Lipsch…