2023/07/24 by Cleon S. Barroso, Barroso, C. S., Verônica M. Ferreira +1 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2307.12958
openalex publication_date 2023/07/24 · openalex created_date 2023/07/26 · openalex updated_date 2026/07/28
In this paper we consider the class of Lipschitz maps on the unit ball BX of a Banach space X, and the question we deal with is whether for any λ>1 there exists a λ-Lipschitz fixed-point free mapping T\colon BX→ BX with d(T,BX)=0. We also consider its Hölder version. New related results are obtained. We show that if X has a spreading Schauder basis then such mappings can always be built, answering a question posed by the first author in \citeBar. In the general case, using a recent approach of R. Medina \citeM concerning Hölder retractions of (rn)-flat closed convex sets, we show that for any decreasing null sequence (rn)⊂ ℝ and α∈ (0,1), there exists a fixed-point free mapping T on BX so that ‖Tnx - Tn y‖≤ rn(‖ x - y‖α+1) for all x, y∈ BX and n∈ℕ.