The closed, convex hull of every ai \(c_{0}\)-summing basic sequence fails the FPP for affine nonexpansive mappings
In [Proc.\ Am.\ Math.\ Soc.\ 132, No.\ 6, 1659--1666 (2004; Zbl 1057.47064)], \textit{P.\ N.\ Dowling}, \textit{C.\ J.\ Lennard} and the reviewer showed that, if \(C\) is a nonempty, closed, bounded, convex subset of \(c_0\) that is not weakly compact, there exists a nonexpansive mapping of \(C\) into itself that fails to have a fixed point. Whether the fixed point free nonexpansive mapping can also be taken to be affine is an open question, and it is this question that the authors of the article under review investigate. The authors identify several non-weakly compact, closed, bounded, convex subsets of \(c_0\) which admit affine, nonexpansive, fixed point free self-mappings. For example, if, for some \(L>0\), \((x_n)\) is an \(L\)-scaled asymptotically isometric \(c_0\)-summing basis, in the sense that there exists \(0\leq \varepsilon_n\to 0\) such that \[ L \sup_{n\geq 1} \frac{1}{1+\varepsilon_n} \left| \sum_{j=n}^\infty t_j\right| \leq \left\| \sum_{j=1}^\infty t_j x_j\right\| \leq L \sup_{n\geq 1} (1+\varepsilon_n) \left| \sum_{j=n}^\infty t_j\right|\,, \] the authors construct an affine, nonexpansive mapping of \(\overline{\text{co}} \{ x_n:n\in \mathbb{N}\}\) into itself without a fixed point. The authors also prove that, if \((e_n)\) is the canonical vector basis in \(c_0\) and \((b_n)\) is a sequence with \(0<\inf_{n\in\mathbb{N}} b_n\leq \sup_{n\in\mathbb{N}} b_n<\infty\), then there exists an affine, nonexpansive self-mapping of the set \(\{\,\sum_{n=1}^\infty t_n b_n e_n: 1=t_1\geq t_2\geq \cdots \geq t_n\downarrow 0\}\) that fails to have a fixed point. Moreover, in both of these results, the authors note that the affine, fixed point free mappings can be constructed to be contractive.
asymptotically isometric \(c_0\)-summing basis
scientific article; zbMATH DE number 5909848