MR2645846 (2011f:46031) Day, Jerry B.; Lennard, Chris A characterization of the minimal invariant sets of Alspach's mapping. Nonlinear Anal. 73 (2010), no. 1, 221–227. (Reviewer: Diana Caponetti)

Weakly compact, convex subsets in a Banach spaceneed not have the fixed point property for nonexpansive mappings, asshown by D.E. Alspach in [Proc. Amer. Math. Soc. 82 (1981), no. 3,423–424; MR0612733 (82j:47070)], where the example of a weaklycompact, convex subset $C$ of $L_1[0,1]$ and of a nonexpansive selfmapping $T$ on $C$ fixed point free is provided. Then, by Zorn'slemma, there exist weakly compact, convex, $T$-invariant fixedpoint free subsets of the set $C$ which are minimal with respectto these properties. But these minimal invariant sets have not beenexplicitly characterized.In the paper under review the authors give an explicit formula forthe $n$th power $T^n$ of the Alspach's mapping $T$ and they provethat the sequence $(T^nf)$ converges weakly to $\|f\|_1\chi_{[0,1]}$, for all $f$ in $C$. As a result using [K. Goebel,Concise course on fixed point theorems, Yokohama Publ., Yokohama,2002; MR1996163 (2004e:47088)] they obtain a description of theminimal invariant sets of the Alspach's mapping $T$. They prove thatfor all $\alpha \in (0,1)$, Alspach's mapping $T$ is fixed pointfree on $C_\alpha:= \{f \in C : \|f\|_1 = \alpha \}$, and $\{D_\infty(\alpha \chi_{[0,1]}) : 0< \alpha <1 \}$ is the collectionof all fixed point free minimal invariant subsets of $C$ for $T$,where $D_0(\alpha \chi_{[0,1]}):= \{ \alpha \chi_{[0,1]}\}$,$D_{n+1}(\alpha \chi_{[0,1]}):= \mbox{conv} \{ D_n(\alpha\chi_{[0,1]})\cup T( D_n(\alpha \chi_{[0,1]}))\}$ inductively, and $D_\infty(\alpha \chi_{[0,1]}):= \overline{ \cup_{n=0}^\inftyD_n(\alpha \chi_{[0,1]})}$. The authors also give an alternativemethod to characterize the minimal invariant sets of the Alspach'smapping $T$ which does not require the formula for $T^n$.