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

Published online by Cambridge University Press:  20 November 2018

D. van Dulst  and
V. de Valk
D. van Dulst
Affiliation:
University of Amsterdam, Amsterdam, Netherlands
V. de Valk
Affiliation:
University of Amsterdam, Amsterdam, Netherlands
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 3 , 01 June 1986 , pp. 728 - 750
Copyright
Copyright © Canadian Mathematical Society 1986

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