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)

Abstract

Weakly compact, convex subsets in a Banach space need not have the fixed point property for nonexpansive mappings, as shown by D.E. Alspach in [Proc. Amer. Math. Soc. 82 (1981), no. 3, 423–424; MR0612733 (82j:47070)], where the example of a weakly compact, convex subset $C$ of $L_1[0,1]$ and of a nonexpansive self mapping $T$ on $C$ fixed point free is provided. Then, by Zorn's lemma, there exist weakly compact, convex, $T$-invariant fixed point free subsets of the set $C$ which are minimal with respect to these properties. But these minimal invariant sets have not been explicitly characterized. In the paper under review the authors give an explicit formula for the $n$th power $T^n$ of the Alspach's mapping $T$ and they prove that 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 the minimal invariant sets of the Alspach's mapping $T$. They prove that for all $\alpha \in (0,1)$, Alspach's mapping $T$ is fixed point free on $C_\alpha:= \{f \in C : \|f\|_1 = \alpha \}$, and $\{ D_\infty(\alpha \chi_{[0,1]}) : 0< \alpha <1 \}$ is the collection of 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}^\infty D_n(\alpha \chi_{[0,1]})}$. The authors also give an alternative method to characterize the minimal invariant sets of the Alspach's mapping $T$ which does not require the formula for $T^n$.