On the fixed point property in direct sums of Banach spaces with strictly monotone norms

Abstract

It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y has the asymptotic (P) property (which is weaker than the condition WCS(Y) > 1), then X ⊕ Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.

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

“…where (x, y) ∈ X × Y . The following lemma was proved in [28,Lemma 4]. Similar arguments can be found in [11,29].…”

“…Recently, two general fixed point theorems in direct sums were proved in [28]. In the present paper we are able to remove additional assumptions imposed on the space X in that paper.…”

Hyper-extensions in metric fixed point theory

“…Recently, a few general fixed point theorems in direct sums were proved in [31,32] (see also [24,30]). Although their proofs were formulated in standard terms, the original ideas came from nonstandard analysis.…”

“…In Section 4 we present a nonstandard approach to fixed-point problems in direct sums of Banach spaces. We show how to use the notion of intra-convex sets and a counterpart of Mazur's lemma to improve the results in [24]. The reader may compare this approach, closer to the original idea, with its classical translation in [31].…”

A product space with the fixed point property

“…In [23] it was shown that X 1 ⊕ 1 X 2 has the (WFPP) provided that X 1 is a Banach space with both, the weak Banach-Saks property and the (WFPP) while X 2 has asymptotic (P). Asymptotic (P) is a geometric property which implies normal structure, which in turn implies (ANS) and (WFPP), introduced by B. Sims and M.A.…”