COMPARISON BETWEEN SEVERAL CONDITIONS IMPLYING WEAK NORMAL STRUCTURE AND THE WEAK FIXED POINT PROPERTY

摘要

In 1996,T. Dominguez-Benavides associated to each Banach space X the family of coefficients {R(a, X): a ≥ 0}, and he proved the two following results: (I) X has weak normal structure whenever R(0, X) < 1. (II) X has the weak fixed point property for nonexpansive mappings whenever R(a, X) < 1 + a for some a>0. Later, in 2008, S. Saejung proved that F-convex Banach spaces have uniform normal structure. In this paper, we prove that any F-convex Banach space X verifies R(O,X) く 1. This shows that the result by Saejung can be derived from Also in 2008, P.N. Dowling, B. Randrianantoanina and B. Turret identified several sufficient conditions for the weak fixed point property for nonexpansive mappings, including E-convexity. In this paper we prove that if a Banach space X verifies any of these conditions, in particular if X is a E-convex, then R(a, X) < 1 + a for some a > 0. Consequently, the fixed point results by Dowling, Randrianantoanina and Turret can be obtained as a byproduct of (II).