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

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 ℓ1[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 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.