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.

In this paper we establish lower bounds for the weakly convergent sequence coefficient WCS(X) of a Banach space X, in terms of some well known moduli and coefficients. By mean of these bounds we identify several properties, of geometrical nature, which imply normal structure. We show that these properties are strictly more general than other previously known sufficient conditions for normal structure.

In this paper we exhibit some connections between the Dunkl-Williams constant and some other well-known constants and notions. We establish bounds for the Dunkl-Williams constant that explain and quantify a characterization of uniformly nonsquare Banach spaces in terms of the Dunkl-Williams constant given by M. Baronti and P.L. Papini. We also study the relationship between Dunkl-Williams constant, the fixed point property for nonexpansive mappings and normal structure.

No abstract

No abstract

We consider the modulus of u-convexity of a Banach space introduced by Ji Gao (1996) and we improve a sufficient condition for the fixed-point property (FPP) given by this author. We also give a sufficient condition for normal structure in terms of the modulus of u-convexity.Let X be a Banach space and let C be a nonempty subset of X. A mapping T : C → C is said to be nonexpansive wheneverfor all x, y ∈ C. A Banach space X has the weak fixed-point property (WFPP) (resp., fixed-point property (FPP)) if for each nonempty weakly compact convex (resp., bounded, closed, and convex) set C ⊂ X and each nonexpansive mappingIt is well known that the WFPP holds for Banach spaces with certain geometrical properties. Among such properties, weak normal structure is, maybe, the most widely studied (see [5, Chapter 3.2]). In order to give sufficient conditions for the WFPP or weak normal structure, different moduli of convexity of Banach spaces have been introduced by several authors (see [5, Chapter 4.5]).At the origin of these moduli is the classical modulus of convexity introduced by J.

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.

No abstract