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 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 v_p(•) (with p = (p_n) and lim_n p_n = 1) introduced in [3] are examples of near-infinity concentrated norms. When v_p(•) is equivalent to the ℓ₁-norm, it was an open problem as to whether (ℓ₁, v_p(•)) had the FPP. We prove that the norm v_p(•) always generates a nonreflexive Banach space X= ℝ ⊕_p₁ (ℝ ⊕_p₂ (ℝ ⊕_p₃...)) satisfying the FPP, regardless of whether v_p(•) is equivalent to the ℓ₁-norm. We also obtain some stability results.