Browse > Article
http://dx.doi.org/10.7858/eamj.2012.28.1.081

AN IMPLICIT ITERATES FOR NON-LIPSCHITZIAN ASYMPTOTICALLY QUASI-NONEXPANSIVE TYPE MAPPINGS IN CAT(0) SPACES  

Saluja, G.S. (Department of Mathematics and Information Technology, Govt. Nagarjuna P.G. College of Science)
Publication Information
East Asian mathematical journal / v.28, no.1, 2012 , pp. 81-92 More about this Journal
Abstract
The purpose of this paper is to establish strong convergence of an implicit iteration process to a common fixed point for a finite family of asymptotically quasi-nonexpansive type mappings in CAT(0) spaces. Our results improve and extend the corresponding results of Fukhar-ud-din et al. [15] and some others from the current literature.
Keywords
Asymptotically quasi-nonexpansive type mapping; implicit iteration process; common fixed point; strong convergence; CAT(0) space;
Citations & Related Records
연도 인용수 순위
  • Reference
1 W. A. Kirk, Some recent results in metric fixed point theory, J. Fixed Point Theory Appl. 2 (2007), no. 2, 195-207.   DOI
2 W. Laowang and B. Panyanak, Strong and ${\Delta}$ convergence theorems for multivalued mappings in CAT(0) spaces, J. Inequal. Appl. Vol. 2009, Article ID 730132, 16 pages.
3 L. Leustean, A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl. 325 (2007), no. 1, 386-399.   DOI   ScienceOn
4 S. Saejung, Halpern's iteration in CAT(0) spaces, Fixed Point Theory Appl., Vol. 2010, Article ID 471781, 13 pages.
5 C. Semple and M. Steel, Phylogenetics, Vol. 24 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, UK, 2003.
6 N. Shahzad, Fixed point results for multimaps in CAT(0) spaces, Topology Appl. 156 (2009), no. 5, 997-1001.   DOI   ScienceOn
7 Z. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 286 (2003), no. 1, 351-358.   DOI   ScienceOn
8 K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301-308.   DOI   ScienceOn
9 H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. Vol. 22 (2001), no. 5-6, 767-773.   DOI   ScienceOn
10 N. Hussain and M. A. Khamsi, On asymptotic pointwise contractions in metric spaces, Nonlinear Anal.: TMA, 71 (2009), no. 10, 4423-4429.   DOI   ScienceOn
11 S. Imnang and S. Suantai, Common fixed points of multi-step Noor iterations with errors for a finite family of generalized asymptotically quasi-nonexpansive mappings, Abstr. Appl. Anal. Vol. 2009, Article ID 728510, 14 pages.
12 M. A. Khamsi and W. A. Kirk, An introduction to metric spaces and fixed point theory, Pure Appl. Math, Wiley-Interscience, New York, NY, USA, 2001.
13 A. R. Khan, M. A. Khamsi and H. Fukhar-ud-din, Strong convergence of a general iteration scheme in CAT(0) spaces, Nonlinear Anal.: TMA, 74 (2011), no. 3, 783-791.   DOI   ScienceOn
14 A. R. Khan and M. A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in a convex metric spaces and applications, Comput. Math. Appl. 59 (2010), no. 8, 2990-2995.   DOI   ScienceOn
15 A. R. Khan, A. A. Domlo and H. Fukhar-ud-din, Common fixed points of Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 341 (2008), no. 1, 1-11.   DOI   ScienceOn
16 W. A. Kirk, Fixed point theory in CAT(0) spaces and $^{(R)}$-trees, Fixed Point Appl. 2004, no. 4, 309-316.
17 S. H. Khan and M. Abbas, Strong and ${\Delta}$-convergence of some iterative schemes in CAT(0) spaces, Comput. Math. Appl. 61 (2011), no. 1, 109-116.   DOI   ScienceOn
18 W. A. Kirk, Geodesic geometry and fixed point theory, in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Vol. 64 of Coleccion Abierta, 195-225, University of Seville Secretary of Publications, Seville, Spain, 2003.
19 W. A. Kirk, Geodesic geometry and fixed point theory II, in International Conference on Fixed point Theory and Applications, 113-142, Yokohama Publishers, Yokohama, Japan, 2004.
20 S. Dhompongsa and B. Panyanak, On ${\Delta}$-convergence theorem in CAT(0) spaces, Comput. Math. Appl. 56 (2008), no. 10, 2572-2579.   DOI   ScienceOn
21 R. Espinola and A. Fernandez-Leon, CAT(k)-spaces, weak convergence and fixed point, J. Math. Anal. Appl. 353 (2009), no. 1, 410-427.   DOI   ScienceOn
22 R. Espinola and W. A. Kirk, Fixed point theorems in $^{(R)}$-trees with applications to graph theory, Topology Appl. 153 (2007), no. 7, 1046-1055.
23 H. Fukhar-ud-din and A. R. Khan, Convergence of implicit iterates with errors for mappings with unbounded domain in Banach spaces, Int. J. Math. Math. Sci. 10 (2005), 1643-1653.
24 H. Fukhar-ud-din and A. R. Khan, Approximating common fixed points of asymptotically nonexpansive mappings in uniformly convex Banach spaces, Comput. Math. Appl. 53 (2007), no. 9, 1349-1360.   DOI   ScienceOn
25 H. Fukhar-ud-din, A. R. Khan, D. O'Regan and R. P. Agarwal, An implicit iteration scheme with errors for a finite family of uniformly continuous mappings, Funct. Diff. Equns. 14 (2007), no. 2-4, 245-256.
26 W. Guo and Y. J. Cho, On the strong convergence of the implicit iterative process with errors for a finite family of asymptotically nonexpansive mappings, Appl. Math. Lett. 21 (2008), no. 10, 1046-1052.   DOI   ScienceOn
27 H. Fukhar-ud-din and S. H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl. 328 (2007), no. 2, 821-829.   DOI   ScienceOn
28 H. Fukhar-ud-din, A. A. Domlo and A. R. Khan, Strong convergence of an implicit algorithm in CAT(0) spaces, Fixed Point Theory and Applications, (2011), Article ID 173621, 11 pages.
29 K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Vol. 83 of Monograph and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, NY, USA, 1984.
30 A. Akbar and M. Eslamian, Common fixed point results in CAT(0) spaces, Nonlinear Anal.: Theory, Method and Applications, Vol. 74 (2011), no. 5, 1835-1840.   DOI   ScienceOn
31 F. Bruhat and J. Tits, "Groups reductifs sur un corps local", Institut des Hautes Etudes Scienti ques. Publications Mathematiques, Vol. 41 (1972), 5-251.   DOI
32 M. Bestvina, $^{(R)}$-trees in topology, geometry, and graph theory, in Handbook of geometric topology, 55-91, North-Holland, Amsterdam, The Netherlands, 2002.
33 M. R. Bridson and A. Haeiger, Metric spaces of non-positive curvature, Vol. 319 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1999.
34 K. S. Brown, Buildings, Springer, New York, NY, USA, 1989.
35 P. Chaoha and A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl. 320 (2006), no. 2, 983-987.   DOI   ScienceOn
36 S. Dhompongsa, A. Kaewkho and B. Panyanak, Lim's theorems for multivalued mappings in CAT(0) spaces, J. Math. Anal. Appl. 312 (2005), no. 2, 478-487.   DOI   ScienceOn