On the weak-approximate fixed point property
In [Proc.\ Am.\ Math.\ Soc.\ 93, 633--639 (1985; Zbl 0566.47039)], the second author and \textit{Y.\,Sternfeld} proved that, if \(C\) is a noncompact, closed, convex subset of a Banach space, then there exists a Lipschitz map \(f:C\to C\) without an approximate fixed point sequence, i.e., without a sequence \((x_n)\) in \(C\) such that \((x_n-f(x_n))\) converges to \(0\) in norm. It is possible, however, that there exists a sequence \((x_n)\) in \(C\) such that \((x_n-f(x_n))\) converges to \(0\) in the weak topology. A Banach space \(X\) has the \textit{weak approximate fixed point property} if, for every nonempty, closed, bounded, convex subset \(C\) of \(X\) and every continuous map \(f:C\to C\), there exists a sequence \((x_n)\) in \(C\) such that \((x_n-f(x_n))\) converges weakly to \(0\). In [Discrete Contin.\ Dyn.\ Syst.\ 25, No.\,2, 467--479 (2009; Zbl 1178.47034)], the first author proved that every reflexive space has the weak approximate fixed point property. In the article under review, the authors generalize this by showing that every Asplund space has the weak approximate fixed point property. The authors ask if every Banach space containing no isomorphic copy of \(\ell_1\) has the weak approximate fixed point property.
scientific article; zbMATH DE number 5673266
scientific article