Cabello, Juan Carlos; Nieto, Eduardo On \(M\)-type structures and the fixed point property. (English) Zbl 0984.46009 Houston J. Math. 26, No. 3, 549-560 (2000). Given \(r,s\in (0,1]\) a Banach space \(X\) is said to have the property \(M(r,s)\) if whenever \(u,v\in X\) with \(\|u\|\leq\|v\|\) and \((x_\alpha)\) is a bounded weakly null net in \(X\) then \(\varlimsup_\alpha\|ru+ sx_\alpha\|\leq \varlimsup_\alpha\|v+ x_\alpha\|\).A Banach space is said to have weak fixed point property (w-fpp) if every weakly compact convex subset of \(X\) has the fixed point property. For \(r,s\in (0,1]\) with \(r+ s>1\), the authors show that if a Banach space has properly \(M(r,s)\), then \(X\) has the w-fpp.Property \(M^*(r,s)\), similar to \(M(r,s)\) is defined for \(X\) and it is investigated when \(X\) has the property w\(^*\)-fpp. Relations between \(M(r,s)\) and \(M^*(r,s)\) are found and example of \(X\) with \(M^*(r,s)\) are given. Reviewer: S.Ganguly (Kolkata) Cited in 1 ReviewCited in 2 Documents MSC: 46B20 Geometry and structure of normed linear spaces 47H10 Fixed-point theorems Keywords:weak normal structure; \(M\)-type structure; weak fixed point property; weakly compact convex subset PDFBibTeX XMLCite \textit{J. C. Cabello} and \textit{E. Nieto}, Houston J. Math. 26, No. 3, 549--560 (2000; Zbl 0984.46009)