The fixed point property for some generalized nonexpansive mappings in a nonreflexive Banach space (Q2871106)

A Banach space \(X\) has the fixed point property (fpp) for nonexpansive mappings if for every bounded closed convex subset \(C\) of \(X\) and for every nonexpansive mapping \(T:C\to C\), that is, \(\|Tx-Ty\|\leq\|x-y\|\) for all \(x,y\in C\), there exists a point \(p\in C\) such that \(p=Tp\). Many reflexive Banach spaces have the fpp for nonexpansive mappings, for example, Hilbert or uniformly convex Banach spaces. It was for a long time an open problem if every Banach space with the fpp for nonexpansive mappings is reflexive. In 2008, \textit{P.-K. Lin} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 8, 2303--2308 (2008; Zbl 1151.46006)] solved this in the negative by giving an equivalent renorming \(|\cdot|\) of \(\ell_1\) for which \((\ell_1,|\cdot|)\) has the fpp for nonexpansive mappings. The authors of this paper prove that \(\ell_1\) can be equivalently renormed to have the fpp for a certain wider class of mappings. This class was given by \textit{E. Llorens Fuster} and \textit{E. Moreno Gálvez} [Abstr. Appl. Anal. 2011, Article ID 435686, 15 p. (2011; Zbl 1215.47042)].