Castillo Santos, Francisco EduardoDowling, Patrick N.Fetter Nathansky, Helga AndreaJapón Pineda, María de los ÁngelesLennard, Christopher J.Sims, BraileyTurett, Barry2018-11-142018-11-142018-08-01Castillo Santos, F.E., Dowling, P.N., Fetter Nathansky, H.A., Japón Pineda, M.d.l.Á., Lennard, C.J., Sims, B. y Turett, B. (2018). Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets. Journal of Functional Analysis, 275 (3), 559-576.0022-1236https://hdl.handle.net/11441/80137In this paper we define the concept of a near-infinity concentrated norm on a Banach space X with a boundedly complete Schauder basis. When k · k is such a norm, we prove that (X, k · k) 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 [P.K. Lin, There is an equivalent norm on l1 that has the fixed point property, Nonlinear Anal. 68 (8) (2008), 2303-2308] and the norm νp(·) (with p = (pn) and limn pn = 1) introduced in [P.N. Dowling, W.B. Johnson, C.J. Lennard and B. Turett, The optimality of James’s distortion theorems, Proc. Amer. Math. Soc. 124 (1) (1997), 167-174] are examples of near-infinity concentrated norms. When νp(·) is equivalent to the l1-norm, it was an open problem as to whether (l1, νp(·)) had the FPP. We prove that the norm νp(·) always generates a nonreflexive Banach space X = R ⊕p1(R ⊕p2(R ⊕p3. . . )) satisfying the FPP, regardless of whether νp(·) is equivalent to the l1-norm. We also obtain some stability results.application/pdfengAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Fixed point propertyNonexpansive mappingsRenorming theoryinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccesshttps://doi.org/10.1016/j.jfa.2018.04.007