Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings

Abstract

It is shown that if the modulus X of nearly uniform smoothness of a reflexive Banach space satisfies X (0) < 1, then every bounded closed convex subset of X has the fixed point property for nonexpansive mappings. In particular, uniformly nonsquare Banach spaces have this property since they are properly included in this class of spaces. This answers a long-standing question in the theory.

The Locally Uniform Nonsquare in Generalized Cesàro Sequence Spaces

“…This result is related to the result of the paper by J. Falset et al 2006 by giving the examples and the motivation to find the geometric properties that are weaker than uniformly nonsquare but still possess the fixed point property in any Banach spaces. …”

A product space with the fixed point property

“…(c) Property M (X) >1 (see [12]). In particular this condition covers the uniformly nonsquare Banach spaces (see [13,14]). Other reflexive Banach spaces with M (X) >1 are those satisfying R(X) <2 (see [15]).…”

Non-squareness properties of Orlicz–Lorentz sequence spaces

“…Recently J. García-Falset, E. Llorens-Fuster and E.M Mazcuñan-Navarro have shown that uniformly non-square Banach spaces have the fixed point property (see [19]). Therefore, it was natural and interesting to look for criteria of non-squareness properties in various well-known classes of Banach spaces.…”