Some fixed point results in \(\ell^1\) and \(c_0\)
\textit{R. C. James} [Ann. of Math., II. Ser. 80, 542-550 (1964; Zbl 0132.08902)] proved the necessary and sufficient conditions in order that a Banach space contains an isomorphic copy of \(\ell^1\) or \(c_0\), respectively. This result is used by the authors to construct some Lipschitzian maps defined on a closed, bounded and convex subset into itself without fixed point property. The authors introduce the idea that a Banach space contains an ``asymptotically isometric'' copy of \(\ell^1\) or \(c_0\), respectively and they proved some necessary and sufficient conditions related to this problem. Let us recall that a mapping \(T: C\to C\) is said to be asymptotically nonexpansive if \[ \|T^nx- T^ny\|\leq k_n\|x- y\|\quad\text{for all }x,y\in C\quad\text{and }n\in\mathbb{N}, \] where a sequence of real numbers \(k_n\) converges to one. The following result proved. Theorem. If a Banach space \(X\) contains an isomorphic copy of \(c_0\), then there exists a closed, bounded, convex subset \(C\) of \(X\) and an asymptotically nonexpansive mapping \(T: C\to C\) without a fixed point. The authors pointed out that it is unknown whether the corresponding result for \(\ell^1\) holds.
scientific article; zbMATH DE number 1419537