Remarks on asymptotically non-expansive mappings in Hilbert space (Q1612545)

Let \(H\) be a real Hilbert space. In the present paper, the author proves the following ergodic theorem: Let \(D\) be a nonemppty subset of \(H\) and \(T:D\to D\) a mapping of asymptotically nonexpansive type (i.e., for each \(x\in D\), \(\lim_{n\to\infty} \sup_{y\in D}(|T^nx-T^n y|-|x-y |)\leq 0\). Then for each \(x\in D\), the sequence \(S_nx= {1\over n}\sum^{n-1}_{i=0} T^ix\) converges weakly in \(H\) if and only if \(\liminf_{n\to\infty} |S_nx|<+\infty\). In this case, \(\{T^nx\}\) is bounded, the weak limit of \(\{S_nx\}\) coincides with the asymptotic center \(c\) of \(\{T^n x\}\), and for every \(y\in D\) the following equality holds: \(\lim_{n\to \infty} |T^ny-c|=\inf_{n\geq 0}|T^ny-c|\). If, moreover, \(T^N\) is nonexpansive for some integer \(N\geq 1\), then the sequence \(\{|T^{kN}y-c |\}_{k\geq 0}\) is nonincreasing. Next, the author applies these results to give constructive proofs of fixed point theorems for such self-mappings of \(D\), extending Goebel-Schoenberg's theorem, see \textit{K. Goebel} and \textit{R. Schöenberg} [Bull. Aust. Math. Soc. 17, 463-466 (1977; Zbl 0365.47031)]. The main fixed point theorem is the following: Theorem 3.1. Let \(T\) be an asymptotically nonexpansive type self-mapping of a nonempty subset \(D\) of \(H\). Assume that \(T^N\) is nonexpansive for some \(N\geq 1\). Then \(T\) has a fixed point in \(D\) if and only if \(\{T^nx\}\) is bounded for some (hence for all) \(x\in D\), and for any \(y\in clco\{T^nx: n\geq 0\}\) there is a unique \(p\in D\) such that \(|y-p|= \inf_{z\in D}|y-z|\).