On the modulus of u‐convexity of Ji Gao

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.

On the modulus of U‐convexity

“…This paper is organized as follows: in Section 2 we prove some inequalities concerning the modulus of U-convexity, introduced by Gao, and other constants. By these inequalities, we immediately obtain some results proved by Gao [4] and Mazcuñán-Navarro [10]. Finally, in Section 3, we prove that if a Banach space X is super-reflexive, then the moduli of U-convexity of the ultrapower X of X and of X itself coincide.…”

“…In particular, if u X (1) > 0 and X has the worth property, then X has normal structure [10,Corollary 8]. In the next section, we will see that this conclusion still holds regardless of whether or not X has the worth property.…”

“…It was proved in [3] that u X (•) is continuous on [0,2). Hence we restate [10,Theorem 5] the following.…”

“…Mazcuñán-Navarro [10] proved a relationship between two of the above notions. Namely, if there exists δ > 0 such that u X (1 − δ) > 0, then R(X) < 2 [10,Theorem 5].…”