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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 $\left(E;Q\right)$. Then an action of $S$ on a subset $K\subset E$ is $Q$-nonexpansive if $\rho \left(s·x-s·y\right)\le \rho \left(x-y\right)$ 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 $\left(E;Q\right)$ 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 $\text{WAP}\left(S\right)$, 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 A. T.–M. 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 $\text{AP}\left(S\right)$, the space of almost periodic functions on $S$, has a LIM but $\text{WAP}\left(S\right)$ does not have a LIM, answering Problem 1 of A. T.–M. Lau [in: Fixed Point Theory Appl., Proc. Semin. Halifax 1975, 121–129 (1976; Zbl 0385.47037)].

##### MSC:
 47H20 Semigroups of nonlinear operators 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties