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Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. (English) Zbl 1120.46006

Whether reflexive (or even superreflexive) Banach spaces have the fixed point property for nonexpansive mappings is a long-standing open question in metric fixed point theory. In this interesting and significant article, the authors use the modulus of nearly uniform smoothness and a geometric coefficient introduced by T. Domínguez-Benavides in [Houston J. Math. 22, 835–849 (1996; Zbl 0873.46012)] to prove a more general theorem, namely that uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. Since uniformly nonsquare Banach spaces are superreflexive, the authors’ results provide a partial answer to the open questions above.

MSC:

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

Citations:

Zbl 0873.46012
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Full Text: DOI

References:

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