The property WORTH\(^\ast\) and the weak fixed point property (Q2925031)

A Banach space \(X\) is said to have the weak fixed point property if every nonexpansive mapping \(T:C\rightarrow C\) acting on a weakly compact convex subset \(C\) of \(X\) has a fixed point. A Banach space \(X\) has the property WORTH\(^{\ast }\) if, for every weak\(^{\ast }\) null sequence \((x_{n}^{\ast })\) and every \(x^{\ast }\in X^{\ast }\), NEWLINE\[NEWLINE\limsup_{n}\left\| x_{n}^{\ast }-x^{\ast }\right\| =\limsup_{n}\left\| x_{n}^{\ast }+x^{\ast }\right\| . NEWLINE\]NEWLINE It is shown that if the Banach-Mazur distance \(d(X,Y)<\frac{\sqrt{33}-3}{2}\) and \(Y^{\ast }\) has the WORTH\(^{\ast }\) property, then \(X\) has the weak fixed point property. In particular, WORTH\(^{\ast }\) implies the weak fixed point property which is a slight extension of the recent result of \textit{H. Fetter} and \textit{B. G. De Buen} [Fixed Point Theory Appl. 2010, Article ID 342691, 7 p. (2010; Zbl 1217.46011)].