Search results for "Minimal invariant set"

MR2645846 (2011f:46031) Day, Jerry B.; Lennard, Chris A characterization of the minimal invariant sets of Alspach's mapping. Nonlinear Anal. 73 (2010…

2011

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 …