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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 865810, 20 pages
http://dx.doi.org/10.1155/2012/865810
Research Article

On Variational Inclusion and Common Fixed Point Problems in q-Uniformly Smooth Banach Spaces

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 9 June 2012; Accepted 18 August 2012

Academic Editor: Alicia Cordero

Copyright © 2012 Yanlai Song 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. S. Reich, “Asymptotic behavior of contractions in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 44, pp. 57–70, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. Q.-L. Dong, S. He, and F. Su, “Strong convergence of an iterative algorithm for an infinite family of strict pseudo-contractions in Banach spaces,” Applied Mathematics and Computation, vol. 216, no. 3, pp. 959–969, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. H. Y. Zhou, “Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces,” Acta Mathematica Sinica, vol. 26, no. 4, pp. 743–758, 2010. View at Publisher · View at Google Scholar
  4. G. Cai and C. S. Hu, “Strong convergence theorems of a general iterative process for a finite family of λi-strict pseudo-contractions in q-uniformly smooth Banach spaces,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 149–160, 2010. View at Publisher · View at Google Scholar
  5. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, P. Noordhoff, Groningen, The Netherlands, 1976.
  6. S.-S. Zhang, J. H. W. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variational inclusion and fixed point problems,” Applied Mathematics and Mechanics, vol. 29, no. 5, pp. 571–581, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. J.-W. Peng, Y. Wang, D. S. Shyu, and J.-C. Yao, “Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems,” Journal of Inequalities and Applications, vol. 2008, Article ID 720371, 16 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. X. Qin, S. S. Chang, Y. J. Cho, and S. M. Kang, “Approximation of solutions to a system of variational inclusions in Banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 916806, 16 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. Y. Yao, M. Aslam Noor, K. Inayat Noor, Y.-C. Liou, and H. Yaqoob, “Modified extragradient methods for a system of variational inequalities in Banach spaces,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1211–1224, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Y. Hao, “On variational inclusion and common fixed point problems in Hilbert spaces with applications,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3000–3010, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. Y. Yao and J.-C. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551–1558, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. X. L. Qin, Y. J. Cho, J. I. Kang, and S. M. Kang, “Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 230, no. 1, pp. 121–127, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. G. M. Korpelevič, “An extragradient method for finding saddle points and for other problems,” Èkonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976. View at Google Scholar
  15. Y. L. Song and L. C. Zeng, “Strong convergence of a new general iterative method for variational inequality problems in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2012, article 46, 2012. View at Publisher · View at Google Scholar
  16. 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
  17. H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis, vol. 16, no. 12, pp. 1127–1138, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. Y. L. Song and C. S. Hu, “Strong convergence theorems of a new general iterative process with Meir-Keeler contractions for a countable family of strict pseudocontractions in q-uniformly smooth Banach spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 354202, 19 pages, 2010. View at Publisher · View at Google Scholar
  19. S.-Y. Matsushita and W. Takahashi, “Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions,” Nonlinear Analysis, vol. 68, no. 2, pp. 412–419, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. H. Zhang and Y. F. Su, “Strong convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces,” Nonlinear Analysis, vol. 70, no. 9, pp. 3236–3242, 2009. View at Publisher · View at Google Scholar
  21. P. Sunthrayuth and P. Kumam, “Iterative methods for variational inequality problems and fixed point problems of a countable family of strict pseudo-contractions in a q-uniformly smooth Banach space,” Fixed Point Theory and Applications, vol. 2012, article 65, 2012. View at Publisher · View at Google Scholar
  22. F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. Y. Shehu, “An iterative method for fixed point problems, variational inclusions and generalized equilibrium problems,” Mathematical and Computer Modelling, vol. 54, no. 5-6, pp. 1394–1404, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. W.-Y. Zeng, N.-J. Huang, and C.-W. Zhao, “Viscosity approximation methods for generalized mixed equilibrium problems and fixed points of a sequence of nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 714939, 15 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. 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
  26. A. Moudafi, “Weak convergence theorems for nonexpansive mappings and equilibrium problems,” Journal of Nonlinear and Convex Analysis, vol. 9, no. 1, pp. 37–43, 2008. View at Google Scholar · View at Zentralblatt MATH
  27. S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis, vol. 69, no. 3, pp. 1025–1033, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. Y. Shehu, “Iterative methods for family of strictly pseudocontractive mappings and system of generalized mixed equilibrium problems and variational inequality problems,” Fixed Point Theory and Applications, vol. 2011, Article ID 852789, 22 pages, 2011. View at Google Scholar · View at Zentralblatt MATH
  29. 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
  30. 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, vol. 67, no. 8, pp. 2350–2360, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. D. S. Mitrinović, Analytic Inequalities, Springer, New York, NY, USA, 1970.