On an analogue of Zarantonello's inequality and a nonlinear mean ergodic theorem
In 1975, \textit{J. B. Baillon} [C. R. Acad. Sci. Paris 280, pp. 1511-1514 (1975; Zbl 0307.47006)] obtained the first mean ergodic theorem for contractions on closed convex subsets of a Hilbert space. Baillon's original proof used an inequality established by E. Zarantonello in 1971. To wit, let T be a contractive (or nonexpansive) self-map of a closed convex set C in a Hilbert space. Then for \(x_ i\in C,\lambda_ i\geq 0\) with \(\sum^{n}_{i=1}\lambda_ i=1\), \[ \| T(\sum^{n}_{i=1}\lambda_ ix_ i)-\sum^{n}_{i=1}\lambda_ iTx_ i\|^ 2\leq \sum_{1\leq j}\lambda_ i\lambda_ j\{\| x_ i-x_ j\|^ 2-\| Tx_ i-Tx_ j\|^ 2\}. \] The author obtains a new proof of the (weak) mean ergodic theorem for contractions on closed, bounded convex sets of a uniformly convex Banach space with a Fréchet differentiable norm. The proof is via a generalization of Zarantonello's inequality. The key lemma is as follows. Let T be a contraction on a closed bounded convex set C in a uniformly convex Banach space. Then there is a continuous increasing function \(\gamma\) from [0,\(\infty)\) to itself with \(\gamma (0)=0\) such that for all \(x,y\in C\) and \(c\in [0,1]\), \[ \gamma (\| cTx+(1-c)Ty-T(cx+(1-c)y)\|)\leq c(1-c)(\| x-y\| -\| Tx- Ty\|). \]
mean ergodic theorem for contractions on closed, bounded convex sets of a uniformly convex Banach space with a Fréchet differentiable norm
generalization of Zarantonello's inequality
scientific article; zbMATH DE number 4103811
scientific article