On the fixed points of nonexpansive mappings in direct sums of Banach spaces

Abstract

We show that if a Banach space X has the weak fixed point property for nonexpansive mappings and Y has the generalized Gossez-Lami Dozo property or is uniformly convex in every direction, then the direct sum X ⊕ Y with a strictly monotone norm has the weak fixed point property. The result is new even if Y is finite-dimensional.2010 Mathematics Subject Classification. 47H10, 46B20, 47H09.

The Fixed Point Property in Direct Sums and Modulus

“…, g (r−1) , g ) ψ and |g | ≥ |g | + ε 0 , which contradicts the strict monotonicity of • ψ . Thus, using Lemma 3.1, we can follow the arguments in [29,Lemma 3.3] and obtain the following result. L 3.2.…”

“…, 0) ∈ K such that lim n→∞ x (r) n = 0. The following construction (for r = 2) was proposed in [29]. Fix an integer k ≥ 1 and a sequence (ε n ) in (0, 1).…”

“…Nowadays, there exist many results concerning permanence properties of conditions which imply normal structure (see [8] for a survey). However, the problem is more difficult if at least one of the spaces lacks weak normal structure (see [29] and references therein) and it is quite well understood only for nonexpansive mappings defined on rectangles C 1 × C 2 (see [20,22]).…”

The super fixed point property for asymptotically nonexpansive mappings

“…It was proved in [19] that all uniformly noncreasy spaces are superreflexive and have SFPP. This yields Recently, a fixed point theorem in direct sums of two Banach spaces was proved in [23]. Assume that X has SFPP (for nonexpansive mappings) and Y is uniformly convex, uniformly smooth or finite dimensional.…”

A product space with the fixed point property

“…Smyth in 1995. Very recently in [24], A. Wiśnicki proved that the assumption of the Banach-Saks property could be dropped in the above result.…”