Some relationships between sufficient conditions for the fixed point property (Q2871105)

Let \(X\) be a Banach space without the Schur property. Let \(N_X\) denote the set of all weakly null sequences in the unit sphere of \(X\) and \(M_X\) denote the set weakly null sequences \(\{x_n\}\) in the closed unit ball of \(X\) such that \(\limsup_n\limsup_m\|x_n-x_m\|\leq1\). \textit{S. Prus} and \textit{M. Szczepanik} [J. Math. Anal. Appl. 307, No. 1, 255--273 (2005; Zbl 1079.46007)] introduced the following sufficient condition for the weak fixed point property: \(X\) satisfies the PSz condition if there exists an \(\varepsilon\in(0,1)\) such that, for every norm one element \(x\in X\), it follows that \(\max\{d(x),1-b(x)\}>\varepsilon\), where \(d(x)=\inf\{\limsup_m\|x+y_m\|-\|x\|:\{y_m\}\in N_X\}\) and \(b(x)=\sup\{\liminf_m\|x+y_m\|-\|x\|:\{y_m\}\in M_X\}\). The authors prove that, if \(X\) is an \(E\)-convex Banach space, then there exists an \(\varepsilon\in(0,1)\) such that \(b(x)<1-\varepsilon\) for all norm one elements \(x\in X\); and hence \(X\) satisfies the PSz condition. Moreover, the authors also discuss the independence between various sufficient conditions for the weak fixed point property for nonexpansive mappings.