Taiwanese Journal of Mathematics

A NEW HYBRID-EXTRAGRADIENT METHOD FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS, FIXED POINT PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS

Jian-Wen Peng and Jen-Chih Yao

Full-text: Open access

Abstract

In this paper, we introduce a new iterative scheme based on the hybrid method and the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of a nonexpansive mapping and the set of the variational inequality for a monotone, Lipschitz-continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend and unify some well-known strong convergence theorems in the literature.

Article information

Source
Taiwanese J. Math., Volume 12, Number 6 (2008), 1401-1432.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405033

Digital Object Identifier
doi:10.11650/twjm/1500405033

Mathematical Reviews number (MathSciNet)
MR2444865

Zentralblatt MATH identifier
1185.47079

Subjects
Primary: 47H05: Monotone operators and generalizations 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 47H17

Keywords
generalized mixed equilibrium problem extragradient method hybrid method nonexpansive mapping monotone mapping variational inequality strong convergence fixed point

Citation

Peng, Jian-Wen; Yao, Jen-Chih. A NEW HYBRID-EXTRAGRADIENT METHOD FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS, FIXED POINT PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS. Taiwanese J. Math. 12 (2008), no. 6, 1401--1432. doi:10.11650/twjm/1500405033. https://projecteuclid.org/euclid.twjm/1500405033


Export citation

References

  • [1.] L.-C. Ceng and J.-C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, Journal of Computational and Applied Mathematics, 214 (2008), 186-201.
  • [2.] S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Analysis, (2008), doi:10.10.1016/j.na.2008.02.042
  • [3.] S. D. Flam and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program, 78 (1997), 29-41.
  • [4.] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145.
  • [5.] K. Goebel and W. A. Kirk, Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990.
  • [6.] P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136.
  • [7.] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331 (2006), 506-515.
  • [8.] Y. Su, M. Shang and X. Qin, An iterative method of solutions for equilibrium and optimization problems, Nonlinear Analysis (2007), doi:10.1016/j.na.2007.08.045.
  • [9.] A. Tada and W. Takahashi, Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem, J. Optim. Theory Appl., 133 (2007), 359-370.
  • [12.] H. H. Bauschke and P. L. Combettes, Construction of best Bregman approximations in reflexive Banach spaces, Proc. Amer. Math. Soc., 131 (2003), 3757-3766.
  • [12.] R. S. Burachik, J. O. Lopes, and B. F. Svaiter, An outer approximation method for the variational inequality problem, SIAM J. Control Optim., 43 (2005), 2071-2088.
  • [13.] P. L. Combettes, Strong convergence of block-iterative outer approximation methods for convex optimization, SIAM J. Control Optim., 38 (2000), 538-565.
  • [14.] K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372-379.
  • [15.] M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program., 87 (2000), 189-202.
  • [16.] M. Kikkawa and W. Takahashi, Approximating Fixed Points of Infinite Nonexpansive Mappings by the Hybrid Method, J. Optim. Theory Appl., 117 (2003), 93-101.
  • [17.] H. Iiduka and W. Takahashi, Strong convergence theorem by a hybrid method for nonlinear mappings of nonexpansive and monotone type and applications, Adv. Nonlinear Var. Inequal., 9 (2006), 1-10.
  • [18.] G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), 747-756.
  • [19.] B.-S. He, Z.-H. Yang and X.-M. Yuan, An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl., 300 (2004), 362- 374.
  • [20.] R. G$\acute{\rm a}$rciga Otero and A. Iuzem, Proximal methods with penalization effects in Banach spaces, Numer. Funct. Anal. Optim., 25 (2004), 69-91.
  • [21.] M. V. Solodov and B. F. Svaiter, An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions, Math. Oper. Res.,25 (2000), 214-230.
  • [22.] M. V. Solodov, Convergence rate analysis of iteractive algorithms for solving variational inequality problem, Math. Program., 96 (2003), 513-528.
  • [23.] L. C. Zeng and J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwan. J. Math., 10 (2006), 1293-1303.
  • [24.] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.
  • [25.] Y. Yao and J.-C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput., 186(2) (2007), 1551-1558.
  • [26.] S. Plubtieng and R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Applied Mathematics and Computation, 197 (2008), 548-558.
  • [27.] N. Nadezhkina and W. Takahashi, Strong Convergence Theorem by a Hybrid Method for Nonexpansive Mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16(4) (2006), 1230-1241.
  • [28.] L. C. Ceng, N. Hadjisavvas and J. C. Yao, Strong Convergence Theorem by a Hybrid Extragradient-like Approximation Method for Variational Inequalities and Fixed point Problems, Submitted.
  • [29.] Z. Opial, Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 561-597.
  • [30.] R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), 75-88.
  • [31.] K. Fan, A generalization of Tychonoff's fixed-point theorem, Math. Ann., 142 (1961), 305-310.
  • [32.] K. Goebel and S. Reich, Uniform convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, Inc., 1984.
  • [33.] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.