About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 202095, 8 pages
http://dx.doi.org/10.1155/2013/202095
Research Article

Strong Convergence Theorems for Semigroups of Asymptotically Nonexpansive Mappings in Banach Spaces

1Department of Mathematics, Banaras Hindu University, Varanasi 221005, India
2Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
3Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan
4Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 4 February 2013; Accepted 4 April 2013

Academic Editor: Qamrul Hasan Ansari

Copyright © 2013 D. R. Sahu et al. 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. K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. T. D. Benavides and P. L. Ramírez, “Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 129, no. 12, pp. 3549–3557, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967. View at MathSciNet
  4. S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. L. C. Ceng, A. R. Khan, Q. H. Ansari, and J. C. Yao, “Viscosity approximation methods for strongly positive and monotone operators,” Fixed Point Theory, vol. 10, no. 1, pp. 35–71, 2009. View at Zentralblatt MATH · View at MathSciNet
  6. L. C. Ceng, H. K. Xu, and J. C. Yao, “The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1402–1412, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. L. C. Ceng, N. C. Wong, and J. C. Yao, “Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption,” Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2312–2322, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. 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, pp. 1349–1360, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. 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. 2–4, pp. 245–256, 2007. View at Zentralblatt MATH · View at MathSciNet
  10. T. C. Lim and H. K. Xu, “Fixed point theorems for asymptotically nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 22, no. 11, pp. 1345–1355, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. D. R. Sahu, R. P. Agarwal, and D. O'Regan, “Structure of the fixed point set of asymptotically nonexpansive mappings in Banach spaces with weak uniformly normal structure,” Journal of Applied Analysis, vol. 17, no. 1, pp. 51–68, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  12. D. R. Sahu, Z. Liu, and S. M. Kang, “Iterative approaches to common fixed points of asymptotically nonexpansive mappings,” The Rocky Mountain Journal of Mathematics, vol. 39, no. 1, pp. 281–304, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. Schu, “Approximation of fixed points of asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 112, no. 1, pp. 143–151, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. T. Suzuki, “On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133–2136, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. H. K. Xu, “A strong convergence theorem for contraction semigroups in Banach spaces,” Bulletin of the Australian Mathematical Society, vol. 72, no. 3, pp. 371–379, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. H. Zegeye and N. Shahzad, “Strong convergence theorems for continuous semigroups of asymptotically nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 30, no. 7-8, pp. 833–848, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, vol. 6 of Topological Fixed Point Theory and Its Applications, Springer, New York, NY, USA, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  19. F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,” Mathematische Zeitschrift, vol. 100, pp. 201–225, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, 1984. View at MathSciNet
  21. H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. C. H. Morales and J. S. Jung, “Convergence of paths for pseudo-contractive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 128, no. 11, pp. 3411–3419, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  23. C. H. Morales, “Strong convergence of path for continuous pseudo-contractive mappings,” Proceedings of the American Mathematical Society, vol. 135, no. 9, pp. 2831–2838, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. B. E. Rhoades, “Some theorems on weakly contractive maps,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 4, pp. 2683–2693, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  25. Y. I. Alber, C. E. Chidume, and H. Zegeye, “Approximating fixed points of total asymptotically nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2006, Article ID 10673, 20 pages, 2006. View at Zentralblatt MATH · View at MathSciNet