Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces (Journal of Mathematical Study, 2018)
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 nonemptyclosed 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 closedconvex set in a Banach space are separably determined. We then prove that everyseparable subspace of $c_0(Γ)$ (for any set $Γ$) is again lying in $c_0.$ Making use of theseresults, we finally presents a simple proof of the famous result: Every non-expansiveself-mapping defined on a nonempty weakly compact convex set of $c_0(Γ)$ has a fixedpoint.
Keywords: Non-expansive mapping; weakly compact convex set; fixed point; Banach space.