Volume 49, Issue 1
Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces

J. Math. Study, 49 (2016), pp. 33-41.

Published online: 2016-03

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• Abstract

In this paper, we first show that for every mapping f from a metric space Ω to itself which is continuous off a countable subset of Ω, there exists a nonempty closed separable subspace S ⊂ Ω so that f|S is again a self mapping on S. Therefore, both the fixed point property and the weak fixed point property of a nonempty closed convex set in a Banach space are separably determined. We then prove that every separable subspace of c0(Γ) (for any set Γ) is again lying in c0. Making use of these results, we finally presents a simple proof of the famous result: Every non-expansive self-mapping defined on a nonempty weakly compact convex set of c0(Γ) has a fixed point.

• Keywords

Non-expansive mapping weakly compact convex set fixed point Banach space

52B10 65D18 68U05 68U07

1220950678@qq.com (Lili Su)

1173806177@qq.com (Qian Wei)

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@Article{JMS-49-33, author = {Li , Qingxia and Su , Lili and Wei , Qian }, title = {Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces}, journal = {Journal of Mathematical Study}, year = {2016}, volume = {49}, number = {1}, pages = {33--41}, abstract = {

In this paper, we first show that for every mapping f from a metric space Ω to itself which is continuous off a countable subset of Ω, there exists a nonempty closed separable subspace S ⊂ Ω so that f|S is again a self mapping on S. Therefore, both the fixed point property and the weak fixed point property of a nonempty closed convex set in a Banach space are separably determined. We then prove that every separable subspace of c0(Γ) (for any set Γ) is again lying in c0. Making use of these results, we finally presents a simple proof of the famous result: Every non-expansive self-mapping defined on a nonempty weakly compact convex set of c0(Γ) has a fixed point.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v49n1.16.04}, url = {http://global-sci.org/intro/article_detail/jms/986.html} }
TY - JOUR T1 - Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces AU - Li , Qingxia AU - Su , Lili AU - Wei , Qian JO - Journal of Mathematical Study VL - 1 SP - 33 EP - 41 PY - 2016 DA - 2016/03 SN - 49 DO - http://dor.org/10.4208/jms.v49n1.16.04 UR - https://global-sci.org/intro/article_detail/jms/986.html KW - Non-expansive mapping KW - weakly compact convex set KW - fixed point KW - Banach space AB -

In this paper, we first show that for every mapping f from a metric space Ω to itself which is continuous off a countable subset of Ω, there exists a nonempty closed separable subspace S ⊂ Ω so that f|S is again a self mapping on S. Therefore, both the fixed point property and the weak fixed point property of a nonempty closed convex set in a Banach space are separably determined. We then prove that every separable subspace of c0(Γ) (for any set Γ) is again lying in c0. Making use of these results, we finally presents a simple proof of the famous result: Every non-expansive self-mapping defined on a nonempty weakly compact convex set of c0(Γ) has a fixed point.

Qingxia Li , Lili Su & Qian Wei . (2020). Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces. Journal of Mathematical Study. 49 (1). 33-41. doi:10.4208/jms.v49n1.16.04
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