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 ARUC(S) and LMC(S) and then study the existence of Q-nonexpansive retractions in the pointwise closure of S using a new method.