Dirichlet summability and strong nonlinear ergodic theorems in Hilbert spaces (Q1811775)

Let \(C\) be a nonempty closed convex subset of a real Hilbert space \(H\). A selfmap \(T\) of \(C\) is called asymptotically nonexpansive with Lipschitz constants \(\{\alpha_n\}\) if \[ \|T^nx - T^ny\|\leq (1 + \alpha_n)\|x - y\| \] for each \(n \geq 0\) and all \(x, y \in C\), where \(\alpha_n \geq 0\) and \(\lim \alpha_n = 0\). Let \((D, \mu)\) denote a Dirichlet method of summability, where the \(\{\mu_n\}\) satisfy certain conditions. Define \[ \begin{alignedat}{2} a_{\mu}(T; x) =& \lim \sup (\log\|\sum_{k = 0}^nT^kx\|)/\mu_n\quad&&\text{if }\lim \sup \|\sum_{l = 0}^mT^kx\|> 0, \text{ and}\\ &-\infty&&\text{if }\lim\sup \|\sum_{l = 0}^mT^kx\|= 0.\end{alignedat} \] The first result of the present paper is that, if \(T\) is any nonlinear selfmap of \(C\) and if \(\displaystyle\sum_{n = 0}^{\infty}e^{- \mu_0s}T^nx\) converges for any \(s > 0\), then \(a_{\mu}(T,; x) \leq 0\). Further, if \(a_{\mu}(T,; x) < \infty\) for some \(x\) in \(C\), then \(\sum_{n = 0}^{\infty}e^{- \mu_0s}T^nx\) converges for any \(s > 0\) with \(s > \max\{ 0, a_{\mu}(T,; x)\}\). If now \(T\) is asymptotically nonexpansive and, for each \(m, \langle T^jx, T^{j+m}x\rangle\) converges as \(j \to \infty\), uniformly for \(m \geq 0\), then the Dirichlet means of \(\{T^n\}\) converge strongly as \(s \to 0+\) to the asymptotic center of \(\{T^n\}\). The paper contains other theorems including the strong convergence of the Wittman type scheme \(x_{n+1} = \beta_nx + (1 - \beta_n)D_{s_n}^{(\mu)}[T]x_n,\) for \(T \in \) Nonext\((C)\), to the metric projection \(Px \) onto \(F(T)\).