## Abstract and Applied Analysis

### The $\tau$-fixed point property for nonexpansive mappings

#### Abstract

Let $X$ be a Banach space and $\tau$ a topology on $X$. We say that $X$ has the $\tau$-fixed point property ($\tau$-FPP) if every nonexpansive mapping $T$ defined from a bounded convex $\tau$-sequentially compact subset $C$ of $X$ into $C$ has a fixed point. When $\tau$ satisfies certain regularity conditions, we show that normal structure assures the $\tau$-FPP and Goebel-Karlovitz′s Lemma still holds. We use this results to study two geometrical properties which imply the $\tau$-FPP: the $\tau$-GGLD and $M(\tau)$ properties. We show several examples of spaces and topologies where these results can be applied, specially the topology of convergence locally in measure in Lebesgue spaces. In the second part we study the preservence of the $\tau$-FPP under isomorphisms. In order to do that we study some geometric constants for a Banach space $X$ such that the $\tau$-FPP is shared by any isomorphic Banach space $Y$ satisfying that the Banach-Mazur distance between $X$ and $Y$ is less than some of these constants.

#### Article information

Source
Abstr. Appl. Anal., Volume 3, Number 3-4 (1998), 343-362.

Dates
First available in Project Euclid: 8 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1049832730

Digital Object Identifier
doi:10.1155/S1085337598000591

Mathematical Reviews number (MathSciNet)
MR1749415

Zentralblatt MATH identifier
0971.47035

#### Citation

Benavides, Tomás Domínguez; Falset, jesús García; Japón Pineda, Maria A. The $\tau$-fixed point property for nonexpansive mappings. Abstr. Appl. Anal. 3 (1998), no. 3-4, 343--362. doi:10.1155/S1085337598000591. https://projecteuclid.org/euclid.aaa/1049832730