Weak convergence and nonlinear ergodic theorems for reversible semigroup of nonexpansive mappings (Q1072094)

Let S be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banach space E with a Fréchet differentiable norm and \({\mathcal S}=\{T_ a\); \(a\in S\}\) be a continuous representation of S as nonexpansive mappings of C into C such that the common fixed point set F(\({\mathcal S})\) of \({\mathcal S}\) in C is nonempty. We prove in this paper that if S is right reversible (i.e. S has finite intersection property for closed right ideals), then for each \(x\in C\), the closed convex set W(x)\(\cap F({\mathcal S})\) consists of at most one point, where \(W(x)=\cap \{K_ s(x)\); \(s\in S\}\), \(K_ s(x)\) is the closed convex hull of \(\{T_ tx\); \(t\geq s\}\) and \(t\geq s\) means \(t=s\) or \(t\in \overline{Ss}\). This result is applied to study the problem of weak convergence of the net \(\{T_ sx\); \(x\in S\}\), with S directed as above, to a common fixed point of \({\mathcal S}\). We also prove that if E is uniformly convex with a uniformly Fréchet differentiable norm, S is reversible and the space of bounded right uniformly continuous functions on S has a right invariant mean, then the intersection W(x)\(\cap F({\mathcal S})\) is nonempty for each \(x\in C\) if and only if there exists a nonexpansive retraction P of C onto F(\({\mathcal S})\) such that \(PT_ s=T_ sP=P\) for all \(s\in S\) and P(x) is in the closed convex hull of \(\{T_ s(x)\); \(s\in S\}\), \(x\in C\).