- Title
- Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets
- Creator
- Castillo-Sántos, F. E.; Dowling, P. N.; Fetter, H.; Japón, M.; Lennard, C. J.; Sims, B.; Turett, B.
- Relation
- Journal of Functional Analysis Vol. 275, Issue 3, p. 559-576
- Publisher Link
- http://dx.doi.org/10.1016/j.jfa.2018.04.007
- Publisher
- Elsevier
- Resource Type
- journal article
- Date
- 2018
- Description
- In this paper we define the concept of a near-infinity concentrated norm on a Banach space χ with a boundedly complete Schauder basis. When ‖•‖ is such a norm, we prove that (X,‖•‖) has the fixed point property (FPP); that is, every nonexpansive self-mapping defined on a closed, bounded, convex subset has a fixed point. In particular, P.K. Lin's norm in ℓ₁[14] and the norm vp(•) (with p = (pn) and limn pn = 1) introduced in [3] are examples of near-infinity concentrated norms. When vp(•) is equivalent to the ℓ₁-norm, it was an open problem as to whether (ℓ₁, vp(•)) had the FPP. We prove that the norm vp(•) always generates a nonreflexive Banach space X= ℝ ⊕p₁ (ℝ ⊕p₂ (ℝ ⊕p₃...)) satisfying the FPP, regardless of whether vp(•) is equivalent to the ℓ₁-norm. We also obtain some stability results.
- Subject
- fixed point property; nonexpansive mappings; renorming theory
- Identifier
- http://hdl.handle.net/1959.13/1399460
- Identifier
- uon:34604
- Identifier
- ISSN:0022-1236
- Rights
- © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/.
- Language
- eng
- Full Text
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