## Uniform convexity, hyperbolic geometry, and nonexpansive mappings.(English)Zbl 0537.46001

The theory of nonexpansive mappings (distances between images of points don’t exceed distances between these points) has flourished during the last twenty years having many connections with (non-linear) differential equations and with the geometry of Banach spaces. The theory of holomorphic mappings of domains in complex Banach spaces is quite young and abounds by unsolved problems. However, the related hyperbolic geometry is very old (150 years) and roots in famous works of Beltrami, Bolyai, Klein, Lobachewsky, Poincaré. According to the known Schwarz lemma any holomorphic mapping of the unit disc D in $${\mathbb{C}}$$ into D is a nonexpansive mapping with respect to the metric $$\rho_ 0(z,w)=| z- w| /| 1-z\bar w|.$$ More deeper links between the two theories is connected with the concept of uniform convexity. Requiring a modest background in functional analysis and complex analysis, this book describes the above connections in the multidimensional case and develops interesting results partly unpublished before.
The book is divided into three chapters: 1) Banach spaces, 2) Hyperbolic geometry and 3) Spherical geometry.
Chapter 1 deals with fixed points of nonexpansive (uniformly Lipschitzian) mappings of (uniformly) convex sets, nearest point projections, asymptotic centers of sequences in uniformly convex Banach spaces, nonexpansive retractions. There are two applications: to periodic solutions of evolution equations with a maximal monotone coercive operator and to the nonlinear mean ergodic theorem in a Hilbert space.
Chapter 2 is dedicated mainly (sections 11-32) to holomorphic mappings of the open unit ball B of a complex Hilbert space of dimension $$\geq 2$$ into B. The first 8 sections contain various materials concerning the concepts of Chapter 1 for the simplest (model) case of the unit disc D in $${\mathbb{C}}$$ with the Poincaré metric $$\rho$$ : balls in $$(D,\rho)$$, $$\rho$$-convexity, uniform convexity in $$(D,\rho)$$. In particular the convexity of the family of $$\rho$$-nonexpansive self-mappings of D is proved. Then, holomorphic mappings in Banach spaces and the Carathéodory-Reiffen-Finsler (pseudo)metric are introduced. With the aid of Cartan’s uniqueness theorem automorphisms of B are studied (even for Banach spaces) and the fixed point theorem of Earle and Hamilton (1979) (each holomorphic mapping of a bounded domain in a Banach space into itself has a unique fixed point) is given. Then, in sections 19-32 almost all abstract concepts of Chapter 1 are realized for B.
Theorem 25.4 says: if a norm continuous mapping of $$\bar B$$ into $$\bar B$$ is $$\rho$$-nonexpansive on B, then it has a fixed point in $$\bar B$$. Here $$\rho$$ is the hyperbolic metric on B defined in section 15. The small chapter 3 contains a number of analogous results (uniform convexity, asymptotic centers, fixed points) for the unit sphere $$\partial B$$ in a real Hilbert space.
Reviewer: V.Isakov

### MSC:

 46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory 46G20 Infinite-dimensional holomorphy 46B20 Geometry and structure of normed linear spaces 34C25 Periodic solutions to ordinary differential equations 51M10 Hyperbolic and elliptic geometries (general) and generalizations