On \(M\)-type structures and the fixed point property (Q2716454)

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 \(\varlimsup_\alpha\|ru+ sx_\alpha\|\leq \varlimsup_\alpha\|v+ x_\alpha\|\).NEWLINENEWLINENEWLINEA 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.NEWLINENEWLINENEWLINEProperty \(M^*(r,s)\), 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.