References & Citations
Mathematics > Functional Analysis
Title: $\ell_1$ spreading models and the FPP for Cesàro mean nonexpansive maps
(Submitted on 26 Mar 2024 (v1), last revised 4 May 2024 (this version, v5))
Abstract: Let $K$ be a nonempty subset of a Banach space $X$. A mapping $T\colon K\to K$ is called $\mathfrak{cm}$-nonexpansive if for any sequence $(u_i)_{i=1}^\infty$ and $y$ in $K$, $\limsup_{i\to\infty} \sup_{A\subset\{1,\dots, n\}}\|\sum_{k\in A} \big(T u_{i+k} - Ty\big)\|\leq \limsup_{i\to\infty} \sup_{A\subset\{1,\dots, n\}}\|\sum_{k\in A} (u_{i+k} - y)\|$ for all $n\in\mathbb{N}$. As a subclass of the class of nonexpansive maps, its FPP is well-established in a wide variety of spaces. The main result of this paper is a fixed point result relating $\mathfrak{cm}$-nonexpansiveness, $\ell_1$ spreading models and Schauder bases with not-so-large basis constants. As a consequence, we deduce that Banach spaces with the weak Banach-Saks property have the fixed point property for $\mathfrak{cm}$-nonexpansive maps.
Submission history
From: Cleon Barroso S. [view email][v1] Tue, 26 Mar 2024 21:36:27 GMT (18kb)
[v2] Fri, 29 Mar 2024 13:25:24 GMT (0kb,I)
[v3] Wed, 3 Apr 2024 05:22:46 GMT (19kb)
[v4] Mon, 22 Apr 2024 01:40:32 GMT (19kb)
[v5] Sat, 4 May 2024 02:26:21 GMT (21kb)
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