Article

# On the approximate fixed point property in abstract spaces

Mathematische Zeitschrift (Impact Factor: 0.88). 01/2011; DOI:10.1007/s00209-011-0915-6
Source: arXiv

ABSTRACT Let $X$ be a Hausdorff topological vector space, $X^*$ its topological dual
and $Z$ a subset of $X^*$. In this paper, we establish some results concerning
the $\sigma(X,Z)$-approximate fixed point property for bounded, closed convex
subsets $C$ of $X$. Three major situations are studied. First when $Z$ is
separable in the strong topology. Second when $X$ is a metrizable locally
convex space and $Z=X^*$, and third when $X$ is not necessarily metrizable but
admits a metrizable locally convex topology compatible with the duality. Our
approach focuses on establishing the Fr\'echet-Urysohn property for certain
sets with regarding the $\sigma(X,Z)$-topology. The support tools include the
Brouwer's fixed point theorem and an analogous version of the classical
Rosenthal's $\ell_1$-theorem for $\ell_1$-sequences in metrizable case. The
results are novel and generalize previous work obtained by the authors in
Banach spaces.

0 0
·
0 Bookmarks
·
67 Views
• Source
##### Article: The approximate fixed point property in Hausdorff topological vector spaces and applications
[hide abstract]
ABSTRACT: We establish an approximate fixed point result for compact convex subsets of Hausdorff topological vector spaces where continuity is not a necessary condition.
11/2008;
• ##### Article: On the weak-approximate fixed point property
[hide abstract]
ABSTRACT: Let X be a Banach space and C a bounded, closed, convex subset of X. C is said to have the weak-approximate fixed point property if for any norm-continuous mapping f:C→C, there exists a sequence {xn} in C such that (xn−f(xn))n converges to 0 weakly. It is known that every infinite-dimensional Banach space with the Schur property does not have the weak-approximate fixed point property. In this article, we show that every Asplund space has the weak-approximate fixed point property. Applications to the asymptotic fixed point theory are given.
Journal of Mathematical Analysis and Applications - J MATH ANAL APPL. 01/2010; 365(1):171-175.
• Source
##### Article: Weak compactness and fixed point property for affine mappings
[hide abstract]
ABSTRACT: It is shown that a closed convex bounded subset of a Banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings. The class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly Lipschitzian with some constant M > 1 in the case of c 0 , the class of affine mappings which are uniformly Lipschitzian with some constant M > √ 6 in the case of quasi-reflexive James' space J and the class of nonexpansive affine mappings in the case of L-embedded spaces.
Journal of Functional Analysis - J FUNCT ANAL. 01/2004; 209(1).