Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces
Qingxia Li, Lili Su, Qian Wei · Journal of Mathematical Study · 2016
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 (Γ) (for any set Γ) 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 (Γ) has a fixed point.