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 $c_0(\Gamma)$ (for any set $\Gamma$) is again lying in $c_0.$ 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 $c_0(\Gamma)$ 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 = {Qingxia and Li and and 7108 and Quanzhou Preschool Education College, Quanzhou 362000, Fujian, P.R. China and Qingxia Li and Lili and Su and 1220950678@qq.com and 7109 and School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, P.R. China and Lili Su and Qian and Wei and 1173806177@qq.com and 7110 and School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, P.R. China and Qian Wei}, 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 $c_0(\Gamma)$ (for any set $\Gamma$) is again lying in $c_0.$ 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 $c_0(\Gamma)$ 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://doi.org/10.4208/jms.v49n1.16.04 UR - https://global-sci.org/intro/article_detail/jms/986.html KW - Non-expansive mapping, weakly compact convex set, fixed point, 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 $c_0(\Gamma)$ (for any set $\Gamma$) is again lying in $c_0.$ 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 $c_0(\Gamma)$ 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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