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Journal of Applied Mathematics
Volume 2012, Article ID 720192, 14 pages
http://dx.doi.org/10.1155/2012/720192
Research Article

Implicit and Explicit Iterations with Meir-Keeler-Type Contraction for a Finite Family of Nonexpansive Semigroups in Banach Spaces

1School of Control and Computer Engineering, North China Electric Power University, Baoding 071003, China
2School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
3Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 31 December 2011; Accepted 28 January 2012

Academic Editor: Rudong Chen

Copyright © 2012 Jiancai Huang 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. N. Shioji and W. Takahashi, “Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces,” Nonlinear Analysis, vol. 34, no. 1, pp. 87–99, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. T. Suzuki, “On strong convergence to common fixed poitns of nonexpansive semigroups in HIlbert spaces,” Proceedings of the American Mathematical Society, vol. 131, pp. 2133–2136, 2002. View at Google Scholar
  3. 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 Scopus
  4. Y. Song and S. Xu, “Strong convergence theorems for nonexpansive semigroup in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 152–161, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  5. R. D. Chen and H. M. He, “Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space,” Applied Mathematics Letters, vol. 20, no. 7, pp. 751–757, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. R. D. Chen, H. M. He, and M. A. Noor, “Modified mann iterations for nonexpansive semigroups in Banach space,” Acta Mathematica Sinica, English Series, vol. 26, no. 1, pp. 193–202, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. I. K. Argyros, Y. J. Cho, and X. Qin, “On the implicit iterative process for strictly pseudo-contractive mappings in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 208–216, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. S. S. Chang, Y. J. Cho, H. W. Joseph Lee, and C. K. Chan, “Strong convergence theorems for Lipschitzian demi-contraction semigroups in Banach spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 583423, 10 pages, 2011. View at Publisher · View at Google Scholar
  9. Y. J. Cho, L. B. Ćirić, and S. H. Wang, “Convergence theorems for nonexpansive semigroups in CAT (0) spaces,” Nonlinear Analysis, vol. 74, no. 17, pp. 6050–6059, 2011. View at Publisher · View at Google Scholar
  10. Y. J. Cho, S. M. Kang, and X. Qin, “Strong convergence of an implicit iterative process for an infinite family of strict pseudocontractions,” Bulletin of the Korean Mathematical Society, vol. 47, no. 6, pp. 1259–1268, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. W. Guo and Y. J. Cho, “On the strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings,” Applied Mathematics Letters, vol. 21, no. 10, pp. 1046–1052, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. H. He, S. Liu, and Y. J. Cho, “An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4128–4139, 2011. View at Publisher · View at Google Scholar
  13. X. Qin, Y. J. Cho, and M. Shang, “Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 542–550, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. J. P. Gossez and E. L. Dozo, “Some geometric properties related to the fixed point theoryfor nonexpansive mappings,” Pacific Journal of Mathematics, vol. 40, pp. 565–573, 1972. View at Google Scholar
  15. 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 Scopus
  16. J. Schu, “Approximation of fixed points of asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 112, pp. 143–151, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. T. C. Lim, “On characterizations of Meir-Keeler contractive maps,” Nonlinear Analysis, vol. 46, no. 1, pp. 113–120, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. A. Petrusel and J. C. Yao, “Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings,” Nonlinear Analysis, vol. 69, no. 4, pp. 1100–1111, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. T. Suzuki, “Moudafi's viscosity approximations with Meir-Keeler contractions,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 342–352, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. R. E. Bruck, “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,” Transactions of the American Mathematical Society, vol. 179, pp. 251–262, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. 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 · View at Zentralblatt MATH · View at Scopus