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Fixed point properties of semigroups of non-expansive mappings. (English) Zbl 1149.47046
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 $\rho(s\cdot x- s\cdot y)\le \rho(x- y)$ 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 {\it 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 {\it A. T.--M.\thinspace Lau} [in: Fixed Point Theory Appl., Proc. Semin. Halifax 1975, 121--129 (1976; Zbl 0385.47037)].

47H20Semigroups of nonlinear operators
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
Full Text: DOI
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