MR2524371 (2010g:47114) Domínguez Benavides, T.; García Falset, J.; Llorens-Fuster, E.; Lorenzo Ramírez, P. Fixed point properties and proximinality in Banach spaces. Nonlinear Anal. 71 (2009), no. 5-6, 1562–1571. (Reviewer: Diana Caponetti), 47H10 (46B20)

Abstract

In the paper under review the authors mainly investigate the existence of a fixed point for nonexpansive mappings in the general setting of strictly $L(\tau)$ Banach spaces. They consider a linear topology $\tau$ on a Banach space $(X, \|\cdot \|)$, weaker than the norm topology, then the Banach space $X$ is a strictly $L(\tau)$ space if there exists a continuous function $\delta:[0, \infty[ \times [0, \infty[ \to [0, \infty[$ such that $\delta(\cdot,s)$ and $\delta(r, \cdot)$ are strictly increasing; $\delta(0,s)=s$, for every $s \in [0, \infty[$; and $\phi_{(x_n)}(y)= \delta (\phi_{(x_n)}(0), \|y\|)$, for every $y \in X$ and for every bounded and $\tau$-null sequence $(x_n)$, where $\phi_{(x_n)}(y):= \limsup_{n \to \infty} \|x_n -y\|$. Strictly $L(\tau)$ spaces were considered in the paper by T. Dominguez Benavides, J. Garcia-Falset and M. A. Jap´on Pineda [Abstr. Appl. Anal. 3 (1998), no. 3-4, 343–362; MR1749415 (2001f:47095)]. Given a bounded closed convex subset $C$ of $X$, a point $x_0 \in X$ is said to be a center for a mapping $T: C \to X$ if, for each $x \in C$, $\|Tx - x_0\| \le \|x-x_0\|$. Then one of the main tools of the paper is the characterization of the fixed point existence for mappings $T:C \to C$ admitting a center by means of a compactness condition, concerning proximinal subsets of $C$. They establish the connection, for strictly $L(\tau)$ spaces, between mappings admitting a center and nonexpansive mappings. Precisely, if $C$ is a nonempty closed, bounded and convex subset of a strictly $L(\tau)$ space $X$ such that $\overline{C}^\tau$ is $\tau$-sequentially compact, then $T: C \to C$ nonexpansive implies that $T$ admits a center in $\overline{C}^\tau$. Hence the results on mappings admitting a center are used to obtain new fixed point theorems for nonexpansive mappings, which encompass a number of earlier results. In particular their results imply that there is a class, more general than the class of weak star compact convex subsets, of subsets of $l_1$ which have the fixed point property for nonexpansive mappings. They also obtain fixed point results for multivalued nonexpansive mappings and asymptotically nonexpansive mappings.