The fixed point property and normal structure for some B-convex Banach spaces

Abstract

We give a sufficient condition for normal structure more general than the well known ɛ0(X) < 1. Moreover we obtain sufficient conditions for the fixed point property for some B-convex Banach spaces.

$\ell_1$ spreading models and FPP in Banach spaces with monotone Schauder basis

“…, x k+1 in B X with min x i − x j : i = j ≥ ε. In [19] García Falset, Llorens Fuster and Mazcuñán Navarro introduced the geometric coefficient εk 0 (X) given by εk 0 (X) = sup ε ∈ [0, s k (X)) : δk (ε) = 0 where δk : [0, s k (X)) → [0, 1] is defined as follows:…”

“…So, general super-reflexive spaces are B-convex. Further aspects of B-convexity including geometric characterizations can be found in [11,22,23,26]; see also [18,19] for its usefulness in metric fixed point problems. In line with Lin's result [33], for example, García Falset [18] proved that every Banach space satisfying the weak-BSP and having a strongly bimonotone basis has the weak-FPP.…”

Three-dimensional convexity and the fixed point property for nonexpansive mappings

“…It is known that Banach spaces with ε 0 (X) < 1 have the FPP since they have indeed normal structure [3]. In [8,3] the authors identify properties which, added to ε 0 (X) < 2, imply the FPP.…”

“…In [8,3] the authors identify properties which, added to ε 0 (X) < 2, imply the FPP. But, as far as we know, there is no result which guarantees the FPP for Banach spaces X with ε 0 (X) < r for some r ∈ (1, 2].…”