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We construct an equivalent renorming of ℓ 1 , which turns out to produce a degenerate ℓ 1-analog Lorentz-Marcinkiewicz space ℓ δ,1 , where the weight sequence δ = (δn) n∈N = (2, 1, 1, 1, • • •) is a decreasing positive sequence in ℓ ∞ \c0 , rather than in c0\ℓ 1 (the usual Lorentz situation). Then we obtain its isometrically isomorphic predual ℓ 0 δ,∞ and dual ℓ δ,∞ , corresponding degenerate c0-analog and ℓ ∞-analog Lorentz-Marcinkiewicz spaces, respectively. We prove that both spaces ℓ δ,1 and ℓ 0 δ,∞ enjoy the weak fixed point property (w-fpp) for nonexpansive mappings yet they fail to have the fixed point property (fpp) for nonexpansive mappings since they contain an asymptotically isometric copy of ℓ 1 and c0 , respectively. In fact, we prove for both spaces that there exist nonempty, closed, bounded, and convex subsets with invariant fixed point-free affine, nonexpansive mappings on them and so they fail to have fpp for affine nonexpansive mappings. Also, we show that any nonreflexive subspace of l 0 δ,∞ contains an isomorphic copy of c0 and so fails fpp for strongly asymptotically nonexpansive maps. Finally, we get a Goebel and Kuczumow analogy by proving that there exists an infinite dimensional subspace of ℓ δ,1 with fpp for affine nonexpansive mappings.

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P.K. Lin gave the first example of a non-reflexive Banach space (X, •) with the fixed point property (FPP) for nonexpansive mappings and showed this fact for (1 , • 1) with the equivalent norm • given by x = sup k∈N 8 k 1 + 8 k ∞ n=k |x n |, for all x = (x n) n∈N ∈ 1. We wonder (c 0 , • ∞) analogue of P.K. Lin's work and we give positive answer if functions are affine nonexpansive. In our work, for x = (ξ k) k ∈ c 0 , we define for every weakly null sequence (x n) n and for all x ∈ c 0 , then for every λ > 0, c 0 with the norm • ρ = ρ(•)+λ|||•||| has the FPP for affine • ρ-nonexpansive self-mappings.