The approximate fixed point property in product spaces

In this paper we generalize to unbounded   convex subsets CC of hyperbolic   spaces results obtained by W.A. Kirk and R. Espínola on approximate fixed points of nonexpansive mappings in product spaces (C×M)∞(C×M)∞, where MM is a metric space and CC 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(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 CC and classes of self-mappings of CC. The paper ends with an open problem.