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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {ARUC}(S)$$\end{document} and LMC(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {LMC}(S)$$\end{document} and then study the existence of Q-nonexpansive retractions in the pointwise closure of S using a new method.