Normal structure and fixed point property

Abstract

Introduction. The most classical sufficient condition for the fixed point property of non-expansive mappings FPP in Banach spaces is the normal structure (see [6] and [10] Moreover, the permanence of the FPP for the /]-product of two Banach spaces with this property is still only partially understood. In a recent paper [11] T. Kukzumov, S. Reich and M. Schmidt have given sufficient conditions for a product of two Banach Spaces A", and X 2 endowed with the /]-product norm to have FPP. They introduce the semi-Opial property, mainly with the technical role of guaranteeing the FPP for a space Ri®A"2 provided that the space X 2 has FPP. Thus, if the space A'j is uniformly convex in every direction UCED and the second space X 2 verifies the semi-Opial property, then the /,-product X A ®, X 2 have the FPP.Here we also show that if the space A^ has the generalized Gossez-Lami Dozo property (a known sufficient condition for weak normal structure [8]) and the second space X 2 verifies the semi-Opial property, then the /,-product X 1 <8> i X 2 has the FPP. The scope of this last result is different from that of Theorem (1) of [11]. For example, a non UCED James space J has the generalized Gossez-Lami Dozo property while the space c 0 has a UCED renorming without the GGLD property.

Uniform property (K) and its related properties

“…where H-H^ is the usual /<" norm. It is known [5] Since property (K) implies weak normal structure, while uniform property ( K ) is the uniform version of property ( K ) , one might conjecture that uniform property (K) would imply the weak uniform normal structure, the uniform version of weak normal structure. However, unfortunately, the answer is negative as the example below shows.…”