Existence of fixed points and retractions for asymptotically nonexpansive semigroups in locally convex spaces

In this paper, we will be concerned with a problem raised by Lau (Fixed Point Theory Appl, 2010) for the existence of common fixed points for a right reversible semitopological semigroup S acting asymptotically nonexpansive on a nonempty compact convex set in the framework of locally convex spaces. We give answers to this problem under the left amenability of \(\text {ARUC}(S)\) and \(\text {LMC}(S)\) and then study the existence of Q-nonexpansive retractions in the pointwise closure of S using a new method.