An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. (English) Zbl 1345.54055

Summary: In this article, we propose and analyze an implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Results concerning \(\Delta\)-convergence as well as strong convergence of the proposed algorithm are proved. Our results are refinement and generalization of several recent results in CAT(0) spaces and uniformly convex Banach spaces.


54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
Full Text: DOI


[1] Kohlenbach, U, Some logical metatheorems with applications in functional analysis, Trans Am Math Soc, 357, 89-128, (2005) · Zbl 1079.03046
[2] Goebel K, Kirk WA: Iteration processes for nonexpansive mappings Topological Methods in Nonlinear Functional Analysis. In Contemp Math Am Math Soc AMS, Providence, RI Edited by: Singh SP, Thomeier S, Watson B. 1983, 21:115-123. · Zbl 0525.47040
[3] Reich, S; Shafrir, I, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal, 15, 537-558, (1990) · Zbl 0728.47043
[4] Goebel, K; Reich, S, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, (1984) · Zbl 0537.46001
[5] Takahashi, W, Iterative methods for approximation of fixed points and thier applications, J Oper Res Soc Jpn, 43, 87-108, (2000) · Zbl 1004.65069
[6] Xu, HK; Ori, RG, An implicit iteration process for nonexpansive mappings, Num Funct Anal Optim, 22, 767-773, (2001) · Zbl 0999.47043
[7] Fukhar-ud-din, H; Khan, AR, Convergence of implicit iterates with errors for mappings with unbounded domain in Banach spaces, Int J Math Math Sci, 10, 1643-1653, (2005) · Zbl 1096.47059
[8] Liu, JA, Some convergence theorems of implicit iterative process for nonexpansive mappings in Banach spaces, Math Commun, 7, 113-118, (2002) · Zbl 1025.47044
[9] Plubtieng, S; Ungchittrakool, K; Wangkeeree, R, Implicit iteration of two finite families for nonexpansive mappings in Banach spaces, Numer Funct Anal Optim, 28, 737-749, (2007) · Zbl 1122.47054
[10] Sun, ZH; He, C; Ni, YQ, Strong convergence of an implicit iteration process for nonexpansive mappings in Banach space, Nonlinear Funct Anal Appl, 8, 595-602, (2003) · Zbl 1074.47031
[11] Khan, AR; Khamsi, MA; Fukhar-ud-din, H, Strong convergence of a general iteration scheme in CAT(0)-spaces, Nonlinear Anal, 74, 783-791, (2011) · Zbl 1202.47076
[12] Takahashi, W, A convexity in metric spaces and nonexpansive mappings, Kodai Math Sem Rep, 22, 142-149, (1970) · Zbl 0268.54048
[13] Shimizu, T; Takahashi, W, Fixed points of multivalued mappings in certain convex metric spaces, Topol Methods Nonlinear Anal, 8, 197-203, (1996) · Zbl 0902.47049
[14] Leustean, L, A quadratic rate of asymptotic regularity for CAT(0)-spaces, J Math Anal Appl, 325, 386-399, (2007) · Zbl 1103.03057
[15] Leustean L: Nonexpansive iterations in uniformly convex{\bfW}-hyperbolic spaces. In Contemp Math Am Math Soc AMS Edited by: Leizarowitz A, Mordukhovich BS, Shafrir I, Zaslavski A. 2010, 513:193-209. Nonlinear Analysis and Optimization I: Nonlinear Analysis
[16] Lim, TC, Remarks on some fixed point theorems, Proc Am Math Soc, 60, 179-182, (1976) · Zbl 0346.47046
[17] Dhompongsa, S; Panyanak, B, On δ-convergence theorems in CAT(0)-spaces, Comp Math Appl, 56, 2572-2579, (2008) · Zbl 1165.65351
[18] Mann, WR, Mean value methods in iteration, Proc Am Math Soc, 4, 506-510, (1953) · Zbl 0050.11603
[19] Ishikawa, S, Fixed points by a new iteration method, Proc Am Math Soc, 44, 147-150, (1974) · Zbl 0286.47036
[20] Chidume, CE; Mutangadura, SA, An example on the Mann iteration method for lipschits pseudo-contarctions, Proc Am Math Soc, 129, 2359-2363, (2001) · Zbl 0972.47062
[21] Kirk WA: Geodesic geometry and FIxed point theory Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003). Univ Sevilla Secr Publ, Seville; 2003:195-225.
[22] Laowang, W; Panyanak, B, Approximating fixed points of nonexpansive nonself mappings in CAT(0) spaces, No. 2010, (2010) · Zbl 1188.54021
[23] Schu, J, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull Aust Math Soc, 43, 153-159, (1991) · Zbl 0709.47051
[24] Bose, SC; Laskar, SK, Fixed point theorems for certain class of mappings, J Math Phys Sci, 19, 503-509, (1985) · Zbl 0613.47048
[25] Khan, SH; Fukhar-ud-din, H, Weak and strong convergence of a scheme for two nonexpansive mappings, Nonlinear Anal, 8, 1295-1301, (2005) · Zbl 1086.47050
[26] Bauschke HH, Combettes PL: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer-Verlag, New York; 2011. · Zbl 1218.47001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.