Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets

Abstract

In this paper we define the concept of a near-infinity concentrated norm on a Banach space X 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 1 [14] and the norm νp(·) (with p = (pn) and limn pn = 1) introduced in [3] are examples of near-infinity concentrated norms. When νp(·) is equivalent to the 1 -norm, it was an open problem as to whether ( 1 , νp(·)) had the FPP. We prove that the norm νp(·) always generates a nonreflexive Banach space X = R ⊕p 1 (R ⊕p 2 (R ⊕p 3 . . . )) satisfying the FPP, regardless of whether νp(·) is equivalent to the 1 -norm. We also obtain some stability results.|t n | 1991 Mathematics Subject Classification. 46B03, 47H09, 47H10.

$\ell_1$ spreading models and FPP in Banach spaces with monotone Schauder basis

“…Since then, many authors inspired by Lin's approach have provided several other renorming fixed-point techniques (e.g. see [13] and its references). Conjectured by R. C. James in [26], the problem of whether or not B-convex spaces are reflexive remained an open problem for some time, until in 1974 he solved the conjecture himself in [28] by constructing an example of a non-reflexive B-convex space.…”