Nonexpansive Mappings in Banach Spaces
Torrey M. Gallagher, Víctor Pérez-García, Łukasz Piasecki · 2025
This chapter comprises a brief survey of classical results pertaining to the fixed point properties of nonexpansive mappings defined on closed, bounded, convex subsets of Banach spaces. We present enough results to provided background and context for the remainder of the text. In the first theme of results, we show that all such mappings must admit approximate fixed point sequences, present examples of fixed point free nonexpansive mappings, theorems of Browder, Göhde, and Kirk which establish the fixed point property for nonexpansive mappings in uniformly convex spaces, Browder&s;s Demiclosedness Principle, asymptotic regularity results due to Opial, Browder and Petryshyn, Ishikawa, and Reich which establish convergence, weak convergence, and almost convergence properties for iterates. The next themes involve the asymptotic center technique of Edelstein and minimal invariant sets of nonexpansive mappings in spaces having normal structure (Kirk&s;s Theorem). The final themes involve Goebel–Kuczumow sets, characterizations of all separable L 1 -preduals whose duals have the weak ∧* fixed point property for nonexpansive mappings, and the fixed point property for uniformly lipschitzian mappings (in particular, theorem of Goebel and Kirk, and strong version of Lifshitz&s;s Theorem) in terms of some geometric moduli of metric and Banach spaces.