Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

WEAK AND STRONG CONVERGENCE THEOREMS FOR AN ASYMPTOTICALLY k-STRICT PSEUDO-CONTRACTION AND A MIXED EQUILIBRIUM PROBLEM

  • Yao, Yong-Hong (DEPARTMENT OF MATHEMATICS TIANJIN POLYTECHNIC UNIVERSITY) ;
  • Zhou, Haiyun (DEPARTMENT OF MATHEMATICS SHIJIAZHUANG MECHANICAL ENGINEERING COLLEGE) ;
  • Liou, Yeong-Cheng (DEPARTMENT OF INFORMATION MANAGEMENT CHENG SHIU UNIVERSIT)
  • Published : 2009.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

We introduce two iterative algorithms for finding a common element of the set of fixed points of an asymptotically k-strict pseudo-contraction and the set of solutions of a mixed equilibrium problem in a Hilbert space. We obtain some weak and strong convergence theorems by using the proposed iterative algorithms. Our results extend and improve the corresponding results of Tada and Takahashi [16] and Kim and Xu [8, 9].

Keywords

References

  1. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123–145
  2. O. Chadli, I. V. Konnov, and J. C. Yao, Descent methods for equilibrium problems in a Banach space, Comput. Math. Appl. 48 (2004), no. 3-4, 609–616
  3. O. Chadli, S. Schaible, and J. C. Yao, Regularized equilibrium problems with application to noncoercive hemivariational inequalities, J. Optim. Theory Appl. 121 (2004), no. 3, 571–596 https://doi.org/10.1023/B:JOTA.0000037604.96151.26
  4. O. Chadli, N. C. Wong, and J. C. Yao, Equilibrium problems with applications to eigenvalue problems, J. Optim. Theory Appl. 117 (2003), no. 2, 245–266 https://doi.org/10.1023/A:1023627606067
  5. P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), no. 1, 117–136
  6. X. P. Ding, Y. C. Lin, and J. C. Yao, Predictor-corrector algorithms for solving generalized mixed implicit quasi-equilibrium problems, Appl. Math. Mech. (English Ed.) 27 (2006), no. 9, 1157–1164 https://doi.org/10.1007/s10483-006-0901-1
  7. S. D. Flam and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming 78 (1997), no. 1, Ser. A, 29–41 https://doi.org/10.1007/BF02614504
  8. T. H. Kim and H. K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006), no. 5, 1140–1152 https://doi.org/10.1016/j.na.2005.05.059
  9. T. H. Kim and H. K. Xu, Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions, Nonlinear Analysis: Theory, Methods & Applications 68 (2008), no. 9, 2828–2836 https://doi.org/10.1016/j.na.2007.02.029
  10. I. V. Konnov, S. Schaible, and J. C. Yao, Combined relaxation method for mixed equilibrium problems, J. Optim. Theory Appl. 126 (2005), no. 2, 309–322 https://doi.org/10.1007/s10957-005-4716-0
  11. C. Matinez-Yanes and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), no. 11, 2400–2411 https://doi.org/10.1016/j.na.2005.08.018
  12. M. A. Noor, Fundamentals of equilibrium problems, Math. Inequal. Appl. 9 (2006), no. 3, 529–566
  13. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597
  14. S. Plubtieng and R. Punpaeng , A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007), no. 1, 455–469 https://doi.org/10.1016/j.jmaa.2007.02.044
  15. A. Tada and W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, Nonlinear analysis and convex analysis, 609–617, Yokohama Publ., Yokohama, 2007
  16. A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl. 133 (2007), no. 3, 359–370 https://doi.org/10.1007/s10957-007-9187-z
  17. S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), no. 1, 506–515 https://doi.org/10.1016/j.jmaa.2006.08.036
  18. Y. Yao, Y. C. Liou, and J. C. Yao, Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, Fixed Point Theory Appl. 2007, Art. ID 64363, 12 pp https://doi.org/10.1155/2007/64363
  19. Y. Yao, Y. C. Liou, and J. C. Yao, A new hybrid iterative algorithm for fixed point problems, variational inequality problems and mixed equilibrium problems, Fixed Point Theory and Applications 2008 (2008), Article ID 417089, 15 pages https://doi.org/10.1155/2008/417089
  20. Y. Yao, M. A. Noor, and Y. C. Liou, On iterative methods for equilibrium problems, Nonlinear Anal. 70 (2009), no. 1, 497–509 https://doi.org/10.1016/j.na.2007.12.021
  21. L. C. Zeng and J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems , J. Comput. Appl. Math. 214 (2008), no. 1, 186–201 https://doi.org/10.1016/j.cam.2007.02.022
  22. L. C. Zeng, S. Y. Wu, and J. C. Yao, Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems, Taiwanese J. Math. 10 (2006), no. 6, 1497–1514

Cited by

  1. An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems vol.2009, pp.1, 2009, https://doi.org/10.1155/2009/632819
  2. Strong and Weak Convergence Criteria of Composite Iterative Algorithms for Systems of Generalized Equilibria vol.2014, 2014, https://doi.org/10.1155/2014/513678
  3. On Bounded Strictly Positive Operators of Closed Range and Some Applications to Asymptotic Hyperstability of Dynamic Systems vol.2013, 2013, https://doi.org/10.1155/2013/639576
  4. Hybrid Extragradient Method with Regularization for Convex Minimization, Generalized Mixed Equilibrium, Variational Inequality and Fixed Point Problems vol.2014, 2014, https://doi.org/10.1155/2014/436069
  5. Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces vol.247, 2013, https://doi.org/10.1016/j.cam.2013.01.004