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

Abstract

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 Sα,n = {x ∈ S H : x, n ≥ α}. We show that the distance between the set of contractions T : Sα,n → Sα,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.