Some generalized nonexpansive mappings and weak normal structure (Q6545465)

A Banach space \(X\) is said to have weak normal structure if each weakly compact convex set \(C\subset X\) containing more than one point has a nondiametral point \(x\), i.e., \(\sup \{\left\Vert x-y\right\Vert :y\in C\}< \mathrm{diam}\, C\). In the present paper, by clever modifications of \textit{V.~L. Klee jun.}'s construction in [Trans. Am. Math. Soc. 78, 30--45 (1955; Zbl 0064.10505)], weak normal structure is characterized in terms of the fixed point property for various classes of continuous generalized nonexpansive mappings introduced by \textit{R.~E. Jaggi} [Contemp. Math. 21, 147--149 (1983; Zbl 0539.47037)], \textit{A.~Nicolae} [Fixed Point Theory Appl. 2010, Article ID 458265 (2010; Zbl 1202.47061)], \textit{E.~Llorens-Fuster} [Bull. Aust. Math. Soc. 93, No.~3, 497--503 (2016; Zbl 1385.47022)], and \textit{E.~Llorens-Fuster} and \textit{E.~Moreno-Gálvez} [Abstr. Appl. Anal. 2011, Article ID 435686, 15~p. (2011; Zbl 1215.47042)]. In particular, it answers an open question in [\textit{H.~Fetter} and \textit{E.~Llorens-Fuster}, J. Nonlinear Convex Anal. 18, No.~10, 1771--1779 (2017; Zbl 1505.47056)].