# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Convergence of Krasnoselskii-Mann iterations of nonexpansive operators. (English) Zbl 0977.47046
This article deals with nonexpansive operators in hyperbolic metric spaces. A metric space $(X,\rho)$ is called hyperbolic if (a) $X$ contains a family $M$ of metric lines such that for each pair of $x,y\in X$, $x\ne y$ there is a unique metric line in $M$ which passes through $x$ and $y$; metric line, by definition, is the image of a metric embedding $c:\bbfR\to X$ with the property $\rho(c(s),c(t))= |s- t|$ $(s,t\in\bbfR)$, and (b) $\rho({1\over 2} x\oplus{1\over 2} y,{1\over 2} w\oplus{1\over 2} z)\le{1\over 2}(\rho(x, w)+ \rho(y,z))$ $(x,y,z,w\in X)$ ($(1- t)x\oplus ty$ is defined as a point $z$ for which $\rho(x,z)= t\rho(x,y)$ and $\rho(z,y)= (1- t)\rho(x,y)$; such a point exists due to (a)). The main results of the article are three theorems in which it is proved that a generic nonexpansive operator $A$ on a closed and convex (but not necessarily bounded) subset of a hyperbolic space has a unique fixed point which attracts the Krasnosel’skii-Mann iterations of $A$.

##### MSC:
 47H09 Mappings defined by “shrinking” properties 47J25 Iterative procedures (nonlinear operator equations)
Full Text:
##### References:
 [1] Bauschke, H. H.; Borwein, J. M.: On projection algorithms for solving convex feasibility problems. SIAM review 38, 367-426 (1996) · Zbl 0865.47039 [2] Bauschke, H. H.; Borwein, J. M.; Lewis, A. S.: The method of cyclic projections for closed convex sets in Hilbert space. Recent developments in optimization theory and nonlinear analysis, contemporary mathematics 204, 1-38 (1997) · Zbl 0874.47029 [3] Borwein, J.; Reich, S.; Shafrir, I.: Krasnoselskii-Mann iterations in normed spaces. Canad. math. Bull. 35, 21-28 (1992) · Zbl 0712.47050 [4] Censor, Y.; Reich, S.: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 37, 323-339 (1996) · Zbl 0883.47063 [5] Cohen, J. E.: Ergodic theorems in demography. Bull. amer. Math. soc. 1, 275-295 (1979) · Zbl 0401.60065 [6] Dye, J.; Reich, S.: Random products of nonexpansive mappings. Optimization and nonlinear analysis, Pitman research notes in mathematics series 244, 106-118 (1992) · Zbl 0815.47067 [7] Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. Journal of mathematical analysis and applications 67, 274-276 (1979) · Zbl 0423.47026 [8] Deblasi, F. S.; Myjak, J.: Sur la convergence des approximations successives pour LES contractions non lineaires dans un espace de Banach. CR acad. Sc. Paris 283, 185-187 (1976) · Zbl 0332.47028 [9] Deblasi, F. S.; Myjak, J.: Generic flows generated by continuous vector fields in Banach spaces. Adv. in math. 50, 266-280 (1983) · Zbl 0601.34033 [10] Myjak, J.: Orlicz type category theorems for functional and differential equations. Dissertationes math. 206, 1-81 (1983) [11] Zaslavski, A. J.: Optimal programs on infinite horizon, 1 and 2. SIAM journal on control and optimization 33, 1643-1686 (1995) · Zbl 0847.49022 [12] Zaslavski, A. J.: Dynamic properties of optimal solutions of variational problems. Nonlinear analysis: theory, methods and applications 27, 895-932 (1996) · Zbl 0860.49003 [13] Reich, S.: The alternating algorithm of von Neumann in the Hilbert ball. Dynamic systems and appl. 2, 21-26 (1993) · Zbl 0768.41032 [14] Reich, S.; Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear analysis: theory, methods and applications 15, 537-558 (1990) · Zbl 0728.47043 [15] Kelley, J. L.: General topology. (1955) · Zbl 0066.16604