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The \(\tau\)-fixed point property for multivalued nonexpansive mappings and its permanence under renorming. (English) Zbl 1124.47032

Fetter Nathansky, Helga (ed.) et al., Proceedings of the 7th international conference on fixed-point theory and its applications, Guanajuato, Mexico, July 17–23, 2005. Yokohama: Yokohama Publishers (ISBN 4-946552-25-1/pbk). 73-85 (2006).
The authors study set-valued (with compact convex values) nonexpansive selfmaps of a closed bounded subset \(C\) of a Banach space \(X\) endowed with an additional linear topology \(\tau\) satisfying some special properties. The paper announces several interesting results on the existence of fixed points for such maps under assumptions concerning separability, \(\tau\)-compactness and some other hypotheses involving geometrical properties of the set \(C\). These properties involve, for instance, the so-called (DL)-condition with respect to the topology \(\tau\) which, in turn, implies the normal structure of \(C\) with respect to \(\tau\). Moreover, the authors discuss the relevance and the properties of the modulus of noncompact convexity and the Opial modulus of \(X\) with respect to \(\tau\). The permanence and behavior of these properties is studied under renorming, yielding results concerning the permanence of the fixed point property for a studied class of mappings.
For the entire collection see [Zbl 1097.47001].

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
47H04 Set-valued operators
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