Fixed point properties of semigroups of non-expansive mappings
Let \(E\) be a Banach space, \(K\) be a nonempty bounded closed convex subset of \(E\), \(S\) be a semitopological semigroup. Let \(Q\) be a fixed family of continuous seminorms on a separated locally convex space \(E\) which determines the topology of \(E\). We denote the space \((E;Q)\). Then an action of \(S\) on a subset \(K\subset E\) is \(Q\)-nonexpansive if \(\rho(s\cdot x- s\cdot y)\leq \rho(x- y)\) for all \(s\in S\), \(x,y\in K\) and \(\rho\in Q\). Consider the following fixed point property: (F) Whenewer \(S\) acts on a weakly compact convex subset \(K\) of a separated locally convex space \((E;Q)\) and the action is weakly separately continuous, weakly quasi-equicontinuous and \(Q\)-nonexpansive, then \(K\) contains a common fixed point for \(S\). In the present paper, the authors prove the following Theorem. Let S be a separable semitopological semigroup. Then \(\text{WAP}(S)\), the space of continuous weakly almost periodic functions on \(S\), has LIM (left invariant mean) if and only if \(S\) has the fixed point property (F). Next, the authors give an example of a semigroup which is not left reversible but has the fixed point property (F), answering a question raised be T. Mitchell (1984) [see \textit{A.\,T.--M.\thinspace Lau}, in: The analytic and topological theory of semigroups, Conf., Oberwolfach/Ger.\ 1989, 313--334 (1990; Zbl 0713.43002)], and an example of a semigroup \(S\) such that \(\text{AP}(S)\), the space of almost periodic functions on \(S\), has a LIM but \(\text{WAP}(S)\) does not have a LIM, answering Problem~1 of \textit{A.\,T.--M.\thinspace Lau} [in: Fixed Point Theory Appl., Proc.\ Semin.\ Halifax 1975, 121--129 (1976; Zbl 0385.47037)].
scientific article; zbMATH DE number 5276380
scientific article