On properties of contractions and nonexpansive mappings on spherical caps in Hilbert spaces


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	On properties of contractions and nonexpansive mappings on spherical caps in Hilbert spaces


	Fixed Point Theory, Volume 18, No. 2, 2017, 471-480, June 1st, 2017 DOI: 10.24193/fpt-ro.2017.2.37 Authors: Krzysztof Bolibok and Mariusz Szczepanik Abstract: Let H be an at least two-dimensional real Hilbert space with the unit sphere S_H. For α ∈ [ -1,1] and n ∈ S_H we define an (α,n)-spherical cap by $S_{\alpha,n}=\left\{ x\in S_{H}:\left\langle x,n\right\rangle \geq \alpha \right\}$ . We show that the distance between the set of contractions $T:S_{\alpha,n}\rightarrow S_{\alpha ,n}$ and the identity mapping is positive iff α <0. We also study the fixed point property and the minimal displacement problem in this setting for nonexpansive mappings. Key Words and Phrases: Contractions, nonexpansive mappings, fixed point property, almost fixed point property, minimal displacement. 2010 Mathematics Subject Classification: 47H09, 47H10. Published on-line: June 1st, 2017. Abstract pdf Fulltext pdf Back to volume's table of contents