Abstract
We study the fixed point property (FPP) in the Banach space c0 with the equivalent norm ∥·∥D. The space c0 with this norm has the weak fixed point property. We prove that every infinite‐dimensional subspace of (c0, ∥·∥D) contains a complemented asymptotically isometric copy of c0, and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of (c0, ∥·∥D) which are not ω‐compact and do not contain asymptotically isometric c0—summing basis sequences. Then we define a family of sequences which are asymptotically isometric to different bases equivalent to the summing basis in the space (c0, ∥·∥D), and we give some of its properties. We also prove that the dual space of (c0, ∥·∥D) over the reals is the Bynum space l1∞ and that every infinite‐dimensional subspace of l1∞ does not have the fixed point property.
Highlights
We start with some notations and terminologies
First we prove that every infinite-dimensional subspace Y of c0, · D has a complemented asymptotically isometric copy of c0 and by a result proved by Dowling et al in 4, Y does not have the FPP
We prove that if Θ1 / ± Θ2, the aiΘ1bc0D and aiΘ2bc0D sequences are different in the sense that there exists a nonempty, closed, convex, and bounded subset of c0, · D, which is not ω-compact, contains an aiΘ1bc0D sequence, and does not contain aiΘ2bc0D sequences
Summary
We start with some notations and terminologies. Let K be a nonempty, convex, closed and bounded subset of a Banach space X, ·. To study the FPP in the space c0, · D using aisbc[0] sequences, we would expect that nonempty, convex, closed and bounded subsets K of c0, · D , which are not ω-compact, contain an aisbc[0] sequence. This fact is true for some ω-compact sets in c0, · D , since the space c0 embeds isometrically in c0, · D. If K contains an aisbc0D sequence with this {εn}, there exists a nonempty, convex and closed subset C of K and T : C → C affine, nonexpansive, and fixed-point-free. If tnζnΘ : tn ≥ 0, tn 1 , 2.25 the set C contains neither aisbc0D sequences nor aisbc[0] sequences with the norm · D
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