A Note on Properties That Imply the Fixed Point Property

A Banach space X is said to satisfy the weak fixed point property (fpp) if every nonempty weakly compact convex subsetC, and every nonexpansivemapping T : C→ C (i.e., ‖Tx− Ty‖ ≤ ‖x− y‖ for every x, y ∈ C) has a fixed point, that is, there exists x ∈ C such that T(x) = x. Many properties have been shown to imply fpp. The most recent one is the uniform nonsquareness which is proved by Mazcuñán [20] solving a long stand open problem. Other well known properties include Opial property (Opial [21]), weak normal structure (Kirk [17]), property (M) (Garcı́a-Falset and Sims [12]), R(X) < 2 (Garcı́aFalset [10]), and UCED (Garkavi [13]). Connection between these properties were investigated in Dalby [3] and Xu et al. [27]. We aim to continue the study in this direction. In contrast to [3], we do not assume that all Banach spaces are separable.