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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 KE is Q-nonexpansive if ρ(s·x-s·y)ρ(x-y) for all sS, x,yK and ρQ. 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 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 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 AP(S), the space of almost periodic functions on S, has a LIM but WAP(S) 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:
47H20Semigroups of nonlinear operators
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
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