Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 615107, 17 pages
http://dx.doi.org/10.1155/2009/615107
Research Article

A New Iteration Process for Approximation of Common Fixed Points for Finite Families of Total Asymptotically Nonexpansive Mappings

1The Abdus Salam International Centre for Theoretical Physics, 34151 Trieste, Italy
2Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State, Nigeria

Received 10 July 2009; Accepted 26 October 2009

Academic Editor: Gelu Popescu

Copyright © 2009 C. E. Chidume and E. U. Ofoedu. 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 Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. R. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property,” Colloquium Mathematicum, vol. 65, no. 2, pp. 169–179, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. W. A. Kirk, “Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type,” Israel Journal of Mathematics, vol. 17, no. 4, pp. 339–346, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. G. E. Kim and T. H. Kim, “Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 42, no. 12, pp. 1565–1570, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. D. R. Sahu, “Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 46, no. 4, pp. 653–666, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. V. Berinde, Iterative Approximation of Fixed Points, vol. 1912 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2nd edition, 2007. View at MathSciNet
  7. R. E. Bruck Jr., “A common fixed point theorem for a commuting family of nonexpansive mappings,” Pacific Journal of Mathematics, vol. 53, no. 1, pp. 59–71, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. C. E. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009. View at MathSciNet
  9. S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974. View at Google Scholar · 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. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. O. Osilike and S. C. Aniagbosor, “Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings,” Mathematical and Computer Modelling, vol. 32, no. 10, pp. 1181–1191, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. G. B. Passty, “Construction of fixed points for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 84, no. 2, pp. 212–216, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. B. E. Rhoades, “Fixed point iterations for certain nonlinear mappings,” Journal of Mathematical Analysis and Applications, vol. 183, no. 1, pp. 118–120, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. N. Shahzad and A. Udomene, “Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2006, Article ID 18909, 10 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. Schu, “Iterative construction of fixed points of asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 158, no. 2, pp. 407–413, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. H. F. Senter and W. G. Dotson Jr., “Approximating fixed points of nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 44, pp. 375–380, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. C. Bose, “Weak convergence to the fixed point of an asymptotically nonexpansive map,” Proceedings of the American Mathematical Society, vol. 68, no. 3, pp. 305–308, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, no. 4, pp. 591–597, 1967. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. B. Xu and M. A. Noor, “Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 267, no. 2, pp. 444–453, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. K.-K. Tan and H. K. Xu, “A nonlinear ergodic theorem for asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 45, no. 1, pp. 25–36, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. K.-K. Tan and H. K. Xu, “The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 114, no. 2, pp. 399–404, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. K.-K. Tan and H. K. Xu, “Fixed point iteration processes for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 122, no. 3, pp. 733–739, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. S.-S. Chang, Y. J. Cho, and H. Zhou, “Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings,” Journal of the Korean Mathematical Society, vol. 38, no. 6, pp. 1245–1260, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. Górnicki, “Nonlinear ergodic theorems for asymptotically nonexpansive mappings in Banach spaces satisfying Opial's condition,” Journal of Mathematical Analysis and Applications, vol. 161, no. 2, pp. 440–446, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. C. E. Chidume, E. U. Ofoedu, and H. Zegeye, “Strong and weak convergence theorems for asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 280, no. 2, pp. 364–374, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Ya. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Y. J. Cho, G. T. Guo, and H. Y. Zhou, “Approximating fixed ponits of asymptotically quasi-nonexpansive mappings by the iteraive sequences with errors,” in Proceedings of the International Conference on Dynamical Systems and Applications, pp. 262–272, Antalya, Turkey, July 2004.
  31. M. K. Ghosh and L. Debnath, “Convergence of Ishikawa iterates of quasi-nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 207, no. 1, pp. 96–103, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. Q. Liu, “Iterative sequences for asymptotically quasi-nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 1–7, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. Z.-H. Sun, “Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 351–358, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. L. P. Belluce and W. A. Kirk, “Fixed-point theorems for families of contraction mappings,” Pacific Journal of Mathematics, vol. 18, no. 2, pp. 213–217, 1966. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, no. 4, pp. 1041–1044, 1965. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. R. De Marr, “Common fixed points for commuting contraction mappings,” Pacific Journal of Mathematics, vol. 13, no. 4, pp. 1139–1141, 1963. View at Google Scholar · View at MathSciNet
  37. T.-C. Lim, “A fixed point theorem for families on nonexpansive mappings,” Pacific Journal of Mathematics, vol. 53, no. 2, pp. 487–493, 1974. View at Google Scholar · View at MathSciNet
  38. H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 202, no. 1, pp. 150–159, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. S.-S. Chang, K. K. Tan, H. W. J. Lee, and C. K. Chan, “On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 313, no. 1, pp. 273–283, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. C. E. Chidume, H. Zegeye, and N. Shahzad, “Convergence theorems for a common fixed point of a finite family of nonself nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2005, no. 2, pp. 233–241, 2005. View at Publisher · View at Google Scholar · View at Scopus
  41. J. G. O'Hara, P. Pillay, and H.-K. Xu, “Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 54, no. 8, pp. 1417–1426, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. J. S. Jung, “Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 509–520, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. J. S. Jung, Y. J. Cho, and R. P. Agarwal, “Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2005, no. 2, pp. 125–135, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. 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 · View at Zentralblatt MATH · View at MathSciNet
  45. T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. H. Zhou, L. Wei, and Y. J. Cho, “Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces,” Applied Mathematics and Computation, vol. 173, no. 1, pp. 196–212, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. C. E. Chidume and E. U. Ofoedu, “Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 128–141, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. E. Zeidler, Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems, Springer, New York, NY, USA, 1986. View at MathSciNet
  49. S.-S. Chang, H. W. Joseph Lee, and C. K. Chan, “On Reich's strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 11, pp. 2364–2374, 2007. View at Publisher · View at Google Scholar · View at Scopus
  50. C. E. Chidume, J. Li, and A. Udomene, “Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 133, no. 2, p. 473–480, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  51. C. E. Chidume and B. Ali, “Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 960–973, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  52. 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, pp. 1295–1301, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  53. K. Nammanee, M. A. Noor, and S. Suantai, “Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 314, no. 1, pp. 320–334, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  54. T. Shimizu and W. Takahashi, “Strong convergence theorem for asymptotically nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 2, pp. 265–272, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  55. N. Shioji and W. Takahashi, “A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces,” Archiv der Mathematik, vol. 72, no. 5, pp. 354–359, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  56. N. Shioji and W. Takahashi, “Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces,” Journal of Approximation Theory, vol. 97, no. 1, pp. 53–64, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  57. S. Suantai, “Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 506–517, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  58. K.-K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet