Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 378750, 19 pages
http://dx.doi.org/10.1155/2013/378750
Research Article

The Mann-Type Extragradient Iterative Algorithms with Regularization for Solving Variational Inequality Problems, Split Feasibility, and Fixed Point Problems

1Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
3Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 3 December 2012; Accepted 31 December 2012

Academic Editor: Jen-Chih Yao

Copyright © 2013 Lu-Chuan Ceng 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. A. Bnouhachem, M. Aslam Noor, and Z. Hao, “Some new extragradient iterative methods for variational inequalities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 3, pp. 1321–1329, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  2. L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Viscosity approximation methods for generalized equilibrium problems and fixed point problems,” Journal of Global Optimization, vol. 43, no. 4, pp. 487–502, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  3. L.-C. Ceng and S. Huang, “Modified extragradient methods for strict pseudo-contractions and monotone mappings,” Taiwanese Journal of Mathematics, vol. 13, no. 4, pp. 1197–1211, 2009. View at Google Scholar · View at MathSciNet
  4. L.-C. Ceng, C.-Y. Wang, and J.-C. Yao, “Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,” Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 375–390, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  5. L.-C. Ceng and J.-C. Yao, “An extragradient-like approximation method for variational inequality problems and fixed point problems,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 205–215, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  6. L.-C. Ceng and J.-C. Yao, “Relaxed viscosity approximation methods for fixed point problems and variational inequality problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3299–3309, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  7. Y. Yao, Y.-C. Liou, and S. M. Kang, “Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3472–3480, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  8. L.-C. Zeng and J.-C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293–1303, 2006. View at Google Scholar · View at MathSciNet
  9. N. Nadezhkina and W. Takahashi, “Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings,” SIAM Journal on Optimization, vol. 16, no. 4, pp. 1230–1241, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  10. 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 MathSciNet
  11. 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 · View at MathSciNet
  12. 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 MathSciNet
  13. R. U. Verma, “On a new system of nonlinear variational inequalities and associated iterative algorithms,” Mathematical Sciences Research Hot-Line, vol. 3, no. 8, pp. 65–68, 1999. View at Google Scholar · View at MathSciNet
  14. L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Finding common solutions of a variational inequality, a general system of variational inequalities, and a fixed-point problem via a hybrid extragradient method,” Fixed Point Theory and Applications, vol. 2011, Article ID 626159, 22 pages, 2011. View at Google Scholar · View at MathSciNet
  15. Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2–4, pp. 221–239, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  16. C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  17. Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “A unified approach for inversion problems in intensity-modulated radiation therapy,” Physics in Medicine & Biology, vol. 51, no. 10, pp. 2353–2365, 2006. View at Publisher · View at Google Scholar
  18. Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071–2084, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Y. Censor, A. Motova, and A. Segal, “Perturbed projections and subgradient projections for the multiple-sets split feasibility problem,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1244–1256, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  20. H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, Article ID 105018, 17 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  21. C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  22. B. Qu and N. Xiu, “A note on the CQ algorithm for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1655–1665, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  23. H.-K. Xu, “A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problem,” Inverse Problems, vol. 22, no. 6, pp. 2021–2034, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  24. Q. Yang, “The relaxed CQ algorithm solving the split feasibility problem,” Inverse Problems, vol. 20, no. 4, pp. 1261–1266, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  25. J. Zhao and Q. Yang, “Several solution methods for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1791–1799, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  26. M. I. Sezan and H. Stark, “Applications of convex projection theory to image recovery in tomography and related areas,” in Image Recovery Theory and Applications, H. Stark, Ed., pp. 415–462, Academic Press, Orlando, Fla, USA, 1987. View at Google Scholar
  27. B. Eicke, “Iteration methods for convexly constrained ill-posed problems in Hilbert space,” Numerical Functional Analysis and Optimization, vol. 13, no. 5-6, pp. 413–429, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  28. L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” American Journal of Mathematics, vol. 73, pp. 615–624, 1951. View at Google Scholar · View at MathSciNet
  29. L. C. Potter and K. S. Arun, “A dual approach to linear inverse problems with convex constraints,” SIAM Journal on Control and Optimization, vol. 31, no. 4, pp. 1080–1092, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  30. P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Modeling & Simulation, vol. 4, no. 4, pp. 1168–1200, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  31. L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “An extragradient method for solving split feasibility and fixed point problems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 633–642, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  32. N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 128, no. 1, pp. 191–201, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  33. D. P. Bertsekas and E. M. Gafni, “Projection methods for variational inequalities with application to the traffic assignment problem,” Mathematical Programming Study, vol. 17, pp. 139–159, 1982. View at Google Scholar · View at MathSciNet
  34. D. Han and H. K. Lo, “Solving non-additive traffic assignment problems: a descent method for co-coercive variational inequalities,” European Journal of Operational Research, vol. 159, no. 3, pp. 529–544, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  35. P. L. Combettes, “Solving monotone inclusions via compositions of nonexpansive averaged operators,” Optimization, vol. 53, no. 5-6, pp. 475–504, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  36. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  37. G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  38. M. O. Osilike, S. C. Aniagbosor, and B. G. Akuchu, “Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces,” Panamerican Mathematical Journal, vol. 12, no. 2, pp. 77–88, 2002. View at Google Scholar · View at MathSciNet
  39. 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 MathSciNet
  40. R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970. View at Google Scholar · View at MathSciNet
  41. 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 Google Scholar · View at MathSciNet