Let be a Banach space, be a nonempty bounded closed convex subset of , be a semitopological semigroup. Let be a fixed family of continuous seminorms on a separated locally convex space which determines the topology of . We denote the space . Then an action of on a subset is -nonexpansive if for all , and . Consider the following fixed point property:
(F) Whenewer acts on a weakly compact convex subset of a separated locally convex space and the action is weakly separately continuous, weakly quasi-equicontinuous and -nonexpansive, then contains a common fixed point for .
In the present paper, the authors prove the following Theorem. Let S be a separable semitopological semigroup. Then , the space of continuous weakly almost periodic functions on , has LIM (left invariant mean) if and only if 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 such that , the space of almost periodic functions on , has a LIM but 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)].