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Abstract and Applied Analysis
Volume 2013, Article ID 174302, 7 pages
http://dx.doi.org/10.1155/2013/174302
Research Article

Strong Convergence for a Strongly Quasi-Nonexpansive Sequence in Hilbert Spaces

Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received 7 May 2013; Accepted 6 November 2013

Academic Editor: Simeon Reich

Copyright © 2013 Satit Saejung and Kanokwan Wongchan. 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. A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. P. E. Maingé, “The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 74–79, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. 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
  4. 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 MathSciNet
  5. B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. 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
  7. S. Reich, “Approximating fixed points of nonexpansive mappings,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 23–28, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. C. E. Chidume and C. O. Chidume, “Iterative approximation of fixed points of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 288–295, 2006. View at Publisher · View at Google Scholar
  9. T. Suzuki, “A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 135, no. 1, pp. 99–106, 2007. View at Google Scholar
  10. S. Saejung, “Halpern's iteration in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 10, pp. 3431–3439, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. E. Bruck and S. Reich, “Nonexpansive projections and resolvents of accretive operators in Banach spaces,” Houston Journal of Mathematics, vol. 3, no. 4, pp. 459–470, 1977. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. Reich, “A limit theorem for projections,” Linear and Multilinear Algebra, vol. 13, no. 3, pp. 281–290, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. K. Wongchan and S. Saejung, “On the strong convergence of viscosity approximation process for quasinonexpansive mappings in Hilbert spaces,” Abstract and Applied Analysis, vol. 2011, Article ID 385843, 9 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland, , Amsterdam, The Netherlands, 2001. View at Google Scholar
  15. M. Tian and X. Jin, “Strong convergent result for quasi-nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2011, article 88, 8 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Tian and X. Jin, “A general iterative method for quasi-nonexpansive mappings in Hilbert space,” Journal of Inequalities and Applications, vol. 2012, article 38, 8 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. 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. View at MathSciNet
  18. R. Kraikaew and S. Saejung, “On Maingé's approach for hierarchical optimization problems,” Journal of Optimization Theory and Applications, vol. 154, no. 1, pp. 71–87, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2350–2360, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. G. Marino and H. K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet