TY - GEN

T1 - 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)

AU - Caponetti, Diana

PY - 2010

Y1 - 2010

N2 - In the paper under review the authors mainlyinvestigate the existence of a fixed point for nonexpansivemappings in the general setting of strictly $L(\tau)$ Banachspaces. They consider a linear topology $\tau$ on a Banach space$(X, \|\cdot \|)$, weaker than the norm topology, then the Banachspace $X$ is a strictly $L(\tau)$ space if there exists acontinuous function $\delta:[0, \infty[ \times [0, \infty[ \to [0,\infty[$ such that $\delta(\cdot,s)$ and $\delta(r, \cdot)$ arestrictly 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 byT. 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 $x \in C$, $\|Tx - x_0\| \le \|x-x_0\|$. Thenone of the main tools of the paper is the characterization of thefixed point existence for mappings $T:C \to C$ admitting a centerby means of a compactness condition, concerning proximinal subsetsof $C$. They establish the connection, for strictly $L(\tau)$spaces, between mappings admitting a center and nonexpansivemappings. Precisely, if $C$ is a nonempty closed, bounded andconvex subset of a strictly $L(\tau)$ space $X$ such that$\overline{C}^\tau$ is $\tau$-sequentially compact, then $T: C \toC$ nonexpansive implies that $T$ admits a center in$\overline{C}^\tau$. Hence theresults on mappings admitting a center are used to obtain newfixed point theorems for nonexpansive mappings, which encompass anumber of earlier results. In particular their results imply thatthere is a class, more general than the class of weak star compactconvex subsets, of subsets of $l_1$ which have the fixed pointproperty for nonexpansive mappings. They also obtain fixed pointresults for multivalued nonexpansive mappings and asymptoticallynonexpansive mappings.

AB - In the paper under review the authors mainlyinvestigate the existence of a fixed point for nonexpansivemappings in the general setting of strictly $L(\tau)$ Banachspaces. They consider a linear topology $\tau$ on a Banach space$(X, \|\cdot \|)$, weaker than the norm topology, then the Banachspace $X$ is a strictly $L(\tau)$ space if there exists acontinuous function $\delta:[0, \infty[ \times [0, \infty[ \to [0,\infty[$ such that $\delta(\cdot,s)$ and $\delta(r, \cdot)$ arestrictly 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 byT. 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 $x \in C$, $\|Tx - x_0\| \le \|x-x_0\|$. Thenone of the main tools of the paper is the characterization of thefixed point existence for mappings $T:C \to C$ admitting a centerby means of a compactness condition, concerning proximinal subsetsof $C$. They establish the connection, for strictly $L(\tau)$spaces, between mappings admitting a center and nonexpansivemappings. Precisely, if $C$ is a nonempty closed, bounded andconvex subset of a strictly $L(\tau)$ space $X$ such that$\overline{C}^\tau$ is $\tau$-sequentially compact, then $T: C \toC$ nonexpansive implies that $T$ admits a center in$\overline{C}^\tau$. Hence theresults on mappings admitting a center are used to obtain newfixed point theorems for nonexpansive mappings, which encompass anumber of earlier results. In particular their results imply thatthere is a class, more general than the class of weak star compactconvex subsets, of subsets of $l_1$ which have the fixed pointproperty for nonexpansive mappings. They also obtain fixed pointresults for multivalued nonexpansive mappings and asymptoticallynonexpansive mappings.

KW - Fixed point

KW - Fixed point

UR - http://hdl.handle.net/10447/55818

M3 - Other contribution

ER -