(KK)-Properties, Normal Structure and Fixed Points of Nonexpansive Mappings in Orlicz Sequence Spaces

Abstract

In this paper we investigate Orlicz sequence spaces with regard to certain geometric properties that have proved to be important in fixed point theory. In particular, we shall consider various Kadec-Klee type properties, and weak and weak* normal structure. It turns out that many of these properties, though generally distinct, coincide in Orlicz sequence spaces and that all of them are intimately related to the so-called Δ2-condition. Some of our results extend to vector-valued Orlicz sequence spaces. For example, we prove a rather powerful theorem on the preservation of weak normal structure under the formation of substitution spaces. There is also a fixed point theorem: the Orlicz sequence space hM has the fixed point property if the complementary Orlicz function M* satisfies theΔ2-condition. Another one of our results implies that, under this assumption on M*, hM has weak normal structure if and only if M also satisfies the Δ2-condition.