On fixed point theorems of nonexpansive mappings in product spaces

Abstract

Abstract. We prove some fixed point theorems for nonexpansive self-and non-self-mappings in product spaces; in particular, we provide a constructive proof of a result of Kirk and Martinez and a partial answer to a question of Khamsi. Our proofs are elementary in the sense that we do not use any universal (or ultra) nets.

Products of uniformly noncreasy spaces

“…A certain extension of this theorem for spaces with the so-called property (P ) (see [18,19]) is also shown. We note that our results go beyond the conditions which guarantee normal structure and give new examples of Banach spaces without (weak) normal structure but with the (weak) fixed point property.…”

“…Recall [18] (see also [19]) that a Banach space X has property (P ) if lim inf x n − x < diam (x n ) whenever x n x and x n is nonconstant. It is not difficult to see that this condition is weaker than Bynum's condition WCS (X) > 1.…”

The common fixed point set of commuting nonexpansive mappings in Cartesian products of separable spaces

“…Now, we recall the fixed point theorem for nonexpansive mappings in product spaces. W. A. Kirk [9] used a retraction approach based on a method due to Bruck [4] to prove the following theorems (the analogous results for the product of convex weakly compact sets was obtained by the first author [10], see also [5], [6], [7], [8], [11], [12], [13] and [14]). …”

Geometric and Fixed Point Properties in Products of Normed Spaces

“…Another interesting unsolved problem is whether WFPP is preserved in the p-direct sum of two normed spaces that have WFPP for any 1 ≤ p ≤ ∞. Several conditions under which this happens have been identified (see, for example, [12,17,20]). We improve some existing results in the study of preservation of WFPP in products of normed spaces.…”