Nonexpansive iterations in hyperbolic spaces. (English) Zbl 0728.47043

Reich, Simeon; Shafrir, Itai

Nonlinear Anal., Theory Methods Appl. 15, No. 6, 537-558 (1990).

The authors study the nature of hyperbolic spaces (which includes all normed linear spaces and Hadamard manifolds etc.). They give some important results about accretive operators (§ 3) and uniform convexity (§ 4). They show the relations between “locally uniformly Fréchet (Gâteaux) differentiable” and “(weakly) locally uniformly convex”, and the connection between these geometric properties and accretive operators. At last they show the convergence nature for explicit and implicit iterations under different conditions. These results are the summary of their works in several years, which give us many useful tools and methods for further studying, on hyperbolic space, accretive operators, nonexpansive iteration etc.

Reviewer: Ling Yongxiang (Beijing)

MSC:

47J25

Iterative procedures involving nonlinear operators

Keywords:

co-accretive; hyperbolic spaces; Hadamard manifolds; accretive operators; uniform convexity; locally uniformly Fréchet (Gâteaux) differentiable; (weakly) locally uniformly convex; convergence nature for explicit and implicit iterations; nonexpansive iteration

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