Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2011 (2011), Article ID 824718, 14 pages
http://dx.doi.org/10.1155/2011/824718
Research Article

A Generalization of Suzuki's Lemma

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 4 February 2011; Accepted 27 April 2011

Academic Editor: D. Anderson

Copyright © 2011 B. Panyanak and A. Cuntavepanit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. A. Kirk, “Krasnoselskii's iteration process in hyperbolic space,” Numerical Functional Analysis and Optimization, vol. 4, no. 4, pp. 371–381, 1982. View at Publisher · View at Google Scholar
  2. W. A. Kirk, “Geodesic geometry and fixed point theory. II,” in International Conference on Fixed Point Theory and Applications, pp. 113–142, Yokohama Publ., 2004.
  3. M. A. Khamsi, “On asymptotically nonexpansive mappings in hyperconvex metric spaces,” Proceedings of the American Mathematical Society, vol. 132, no. 2, pp. 365–373, 2003. View at Publisher · View at Google Scholar
  4. K. Goebel and W. A. Kirk, “Iteration processes for nonexpansive mappings,” in Topological Methods in Nonlinear Functional Analysis, vol. 21 of Contemporary Mathematics, pp. 115–123, American Mathematical Society, Providence, RI, USA, 1983. View at Google Scholar
  5. T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, vol. 2005, no. 1, pp. 103–123, 2005. View at Publisher · View at Google Scholar
  6. Y. Song and H. Li, “Strong convergence of iterative sequences for nonexpansive mappings,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1500–1507, 2009. View at Publisher · View at Google Scholar
  7. M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, vol. 319 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1999.
  8. W. A. Kirk, “Fixed point theorems in CAT(0) spaces and -trees,” Fixed Point Theory and Applications, vol. 2004, no. 4, pp. 309–316, 2004. View at Publisher · View at Google Scholar
  9. K. S. Brown, Buildings, Springer, New York, NY, USA, 1989.
  10. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.
  11. W. A. Kirk, “Geodesic geometry and fixed point theory,” in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), vol. 64 of Colecc. Abierta, pp. 195–225, Universidad de Sevilla Secr. Publ., Seville, Spain, 2003. View at Google Scholar
  12. S. Dhompongsa, A. Kaewkhao, and B. Panyanak, “Lim's theorems for multivalued mappings in CAT(0) spaces,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 478–487, 2005. View at Publisher · View at Google Scholar
  13. P. Chaoha and A. Phon-on, “A note on fixed point sets in CAT(0) spaces,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 983–987, 2006. View at Publisher · View at Google Scholar
  14. L. Leustean, “A quadratic rate of asymptotic regularity for CAT(0)-spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 386–399, 2007. View at Publisher · View at Google Scholar
  15. S. Dhompongsa and B. Panyanak, “On Δ-convergence theorems in CAT(0) spaces,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2572–2579, 2008. View at Publisher · View at Google Scholar
  16. N. Shahzad and J. Markin, “Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1457–1464, 2008. View at Publisher · View at Google Scholar
  17. N. Shahzad, “Fixed point results for multimaps in CAT(0) spaces,” Topology and Its Applications, vol. 156, no. 5, pp. 997–1001, 2009. View at Publisher · View at Google Scholar
  18. R. Espínola and A. Fernández-León, “CAT(k)-spaces, weak convergence and fixed points,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 410–427, 2009. View at Publisher · View at Google Scholar
  19. N. Hussain and M. A. Khamsi, “On asymptotic pointwise contractions in metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4423–4429, 2009. View at Publisher · View at Google Scholar
  20. A. Razani and H. Salahifard, “Invariant approximation for CAT(0) spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 5, pp. 2421–2425, 2010. View at Publisher · View at Google Scholar
  21. S. Saejung, “Halpern's iteration in CAT(0) spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 471781, 13 pages, 2010. View at Publisher · View at Google Scholar
  22. A. R. Khan, M. A. Khamsi, and H. Fukhar-ud-din, “Strong convergence of a general iteration scheme in CAT(0) spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 783–791, 2011. View at Publisher · View at Google Scholar
  23. S. H. Khan and M. Abbas, “Strong and Δ-convergence of some iterative schemes in CAT(0) spaces,” Computers & Mathematics with Applications, vol. 61, no. 1, pp. 109–116, 2011. View at Publisher · View at Google Scholar
  24. A. Abkar and M. Eslamian, “Common fixed point results in CAT(0) spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp. 1835–1840, 2011. View at Publisher · View at Google Scholar
  25. M. Bestvina, “—trees in topology, geometry, and group theory,” in Handbook of Geometric Topology, pp. 55–91, North-Holland, Amsterdam, The Netherlands, 2002. View at Google Scholar
  26. C. Semple and M. Steel, Phylogenetics, vol. 24 of Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, Oxford, UK, 2003.
  27. R. Espínola and W. A. Kirk, “Fixed point theorems in —trees with applications to graph theory,” Topology and Its Applications, vol. 153, no. 7, pp. 1046–1055, 2006. View at Publisher · View at Google Scholar
  28. W. A. Kirk, “Some recent results in metric fixed point theory,” Journal of Fixed Point Theory and Applications, vol. 2, no. 2, pp. 195–207, 2007. View at Publisher · View at Google Scholar
  29. N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641–3645, 1997. View at Publisher · View at Google Scholar
  30. H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003. View at Publisher · View at Google Scholar