Minimal displacement of points under holomorphic mappings and fixed point properties for unions of convex sets

Abstract

Let D be an open convex bounded subset of a complex Banach space ( X , ‖ ⋅ ‖ ) (X,\left \| \cdot \right \|) , and let C be the union of a finite number of closed convex sets lying strictly inside D. Using the Kuratowski measure of noncompactness with respect to the Kobayashi distance in D, we first show that if f : D → D f:D \to D is a holomorphic mapping which leaves C invariant, and if the Lefschetz number λ ( f | C ) ≠ 0 \lambda ({f_{|C}}) \ne 0 , then inf { ‖ x − f ( x ) ‖ : x ∈ C } = 0 \inf \{ \left \| {x - f(x)} \right \|:x \in C\} = 0 . We then deduce several new fixed point theorems for holomorphic and nonexpansive mappings.