The approximate fixed point property in product spaces

In this paper we generalize to unbounded convex subsets C of hyperbolic spaces results obtained by W.A. Kirk and R. Esṕınola on approximate fixed points of nonexpansive mappings in product spaces (C×M)∞, where M is a metric space and C is a nonempty, convex, closed and bounded subset of a normed or a CAT(0)-space. We extend the results further, to families (Cu)u∈M of unbounded convex subsets of a hyperbolic space. The key ingredient in obtaining these generalizations is a uniform quantitative version of a theorem due to Borwein, Reich and Shafrir, obtained by the authors in a previous paper using techniques from mathematical logic. Inspired by that, we introduce in the last section the notion of uniform approximate fixed point property for sets C and classes of self-mappings of C. The paper ends with an open problem.