We establish weak and strong convergence of Ishikawa type iterates of two pointwise asymptotic nonexpansive maps in a Hadamard space. For weak and strong convergence results, we drop “rate of convergence condition”, namely (C n (x)-1)＜ to answer in the affirma-tive to the open question posed by Tan and Xu [1] even in a general setup.
K. K. Tan and H. K. Xu, “Fixed Point Iteration Processes for Asymptotically Nonexpansive Mapping,” Proceedings of the American Mathematical Society, Vol. 122, No. 3, 1994, pp. 733-739. doi:10.1090/S0002-9939-1994-1203993-5
M. Bridson and A. Haefliger, “Metric Spaces of Non-Positive Curvature,” Springer-Verlag, Berlin, 1999.
S. Dhompongsa and B. Panyanak, “On Δ-Convergence Theorems in CAT(0) Spaces,” Computers & Mathematics with Applications, Vol. 56, No. 10, 2008, pp. 2572-2579. doi:10.1016/j.camwa.2008.05.036
F. Bruhat and J. Tits, “Groupes Réductifs Sur Un Corps Local. I. Données Radicielles Valuées,” Institut des Hautes études Scientifiques Publications Mathématiques, Vol. 41, No. 1, 1972, pp. 5-251. doi:10.1007/BF02715544
W. A. Kirk, “A Fixed Point Theorem in CAT(0) Spaces and R-Trees,” Fixed Point Theory and Applications, Vol. 4, 2004, pp. 309-316.
W. A. Kirk and H. K. Xu, “Asymptotic Pointwise Contractions,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 69, No. 12, 2008, pp. 4706-4712. doi:10.1016/j.na.2007.11.023
K. Goebel and W. A. Kirk, “A Fixed Point Theorem for Asymptotically Nonexpansive Mappings,” Proceedings of the American Mathematical Society, Vol. 35, 1972, pp. 171-174. doi:10.1090/S0002-9939-1972-0298500-3
N. Hussain and M. A. Khamsi, “On Asymptotic Pointwise Contractions in Metric Spaces,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 10, 2009, pp. 4423-4429. doi:10.1016/j.na.2009.02.126
C. W. Groetsch, “A Note on Segmenting Mann Iterates,” Journal of Mathematical Analysis and Applications, Vol. 40, No. 2, 1972, pp. 369-372. doi:10.1016/0022-247X(72)90056-X
W. R. Mann, “Mean Value Methods in Iteration,” Proceedings of the American Mathematical Society, Vol. 4, No. 3, 1953, pp. 506-510. doi:10.1090/S0002-9939-1953-0054846-3
S. Ishikawa, “Fixed Points by a New Iteration Method,” Proceedings of the American Mathematical Society, Vol. 44, No. 1, 1974, pp. 147-150. doi:10.1090/S0002-9939-1974-0336469-5
S. C. Bose, “Weak Convergence to the Fixed Point of an Asymptotically Nonexpansive,” Proceedings of the American Mathematical Society, Vol. 68, No. 3, 1978, pp. 305-308. doi:10.1090/S0002-9939-1978-0493543-4
R. Bruck, T. Kuczumow and S. Reich, “Convergence of Iterates of Asymptotically Nonexpansive Mappings in Banach Spaces with the Uniform Opial Property,” Collectanea Mathematica, Vol. 65, No. 2, 1993, pp. 169-179.
S. Chang, Y. J. Cho and H. Zhou, “Demiclosed Principle and Weak Convergence Problems for Asymptotically Nonexpansive Mappings,” Journal of the Korean Mathematical Society, Vol. 38, 2001, pp. 145-1260.
H. Fukhar-ud-din and S. H. Khan, “Convergence of Two-Step Iterative Scheme with Errors for Two Asymptotically Nonexpansive Mappings,” International Journal of Mathematics and Mathematical Sciences, Vol. 2004, No. 37, 2004, pp. 1965-1971. doi:10.1155/S0161171204308161
H. Fukhar-ud-din and S. H. Khan, “Convergence of Iterates with Errors of Asymptotically Quasi-Nonexpansive Mappings and Applications,” Journal of Mathematical Analysis and Applications, Vol. 328, No. 2, 2007, pp. 821-829. doi:10.1016/j.jmaa.2006.05.068
H. Fukhar-ud-din and A. R. Khan, “Approximating Common Fixed Points of Asymptotically Nonexpansive Maps in Uniformly Convex Banach Spaces,” Computers & Mathematics with Applications, Vol. 53, No. 9, 2007, pp. 1349-1360. doi:10.1016/j.camwa.2007.01.008
H. Fukhar-ud-din and A. R. Khan, “Convergence of Implicit Iterates with Errors for Mappings with Unbounded Domain in Banach Spaces,” International Journal of Mathematics and Mathematical Sciences, Vol. 10, 2005, pp. 1643-1653. doi:10.1155/IJMMS.2005.1643
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,” Functional Differential Equations, Vol. 14, No. 3-4, 2007, pp. 245-256.
S. H. Khan and H. Fukhar-ud-din, “Weak and Strong Convergence of a Scheme with Errors for Two Nonexpansive Mappings,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 61, No. 8, 2005, pp. 1295-1301. doi:10.1016/j.na.2005.01.081
A. R. Khan and N. Hussain, “Iterative Approximation of Fixed Points of Nonexpansive Maps,” Scientiae Mathematicae Japonicae, Vol. 54, No. 3, 2001, pp. 503-511.
S. H. Khan and W. Takahashi, “Approximating Common Fixed Points of Two Asymptotically Nonexpansive Mappings,” Scientiae Mathematicae Japonicae, Vol. 53, No. 1, 2001, pp. 143-148.
W. M. Kozlowski, “Fixed Point Iteration Processes for Asymptotic Pointwise Nonexpansive Mappings in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 377, No. 1, 2011, pp. 43-52. doi:10.1016/j.jmaa.2010.10.026
T. Laokul and B. Panyanak, “Approximating Fixed Points of Nonexpansive Mappings in CAT(0) Spaces,” International Journal of Mathematics and Mathematical Sciences, Vol. 3, No. 25-28, 2009, pp. 1305-1315.
W. Laowang and B. Panyanak, “Approximating Fixed Points of Nonexpansive Nonself Mappings in CAT(0) Spaces,” Fixed Point Theory and Applications, 2010, 11 pages, Article ID: 367274.
B. Nanjaras and B. Panyanak, “Demiclosed Principle for Asymptotically Nonexpansive Mapping in CAT(0) Spaces,” Fixed Point Theory and Applications, 2010, 14 pages, Article ID: 268780.
M. A. Noor and B. Xu, “Fixed Point Iterations for Asymptotically Nonexpansive Mappings in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 267, No. 2, 2002, pp. 444-453. doi:10.1006/jmaa.2001.7649
Z. Opial, “Weak Convergence of Successive Approximations for Nonexpansive Mappings,” Bulletin of the American Mathematical Society, Vol. 73, 1967, pp. 591-597. doi:10.1090/S0002-9904-1967-11761-0
G. B. Passty, “Construction of Fixed Points for Asymptotically Nonexpansive Mappings,” Proceedings of the American Mathematical Society, Vol. 84, 1982, pp. 212- 216. doi:10.1090/S0002-9939-1982-0637171-7
B. E. Rhoades, “Fixed Point Iterations for Certain Nonlinear Mappings,” Journal of Mathematical Analysis and Applications, Vol. 183, No. 1, 1994, pp. 118-120. doi:10.1006/jmaa.1994.1135
J. Schu, “Iterative Construction of Fixed Points of Asymptotically Nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 158, No. 2, 1991, pp. 407-413. doi:10.1016/0022-247X(91)90245-U
J. Schu, “Weak and Strong Convergence to Fixed Points of Asymptotically Nonexpansive Mappings,” Bulletin of the Australian Mathematical Society, Vol. 43, No. 1, 1991, pp. 153-159. doi:10.1017/S0004972700028884