Taiwanese Journal of Mathematics

STRONG CONVERGENCE THEOREM BY AN EXTRAGRADIENT METHOD FOR FIXED POINT PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS

Lu-Chuan Zeng and Jen-Chih Yao

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Abstract

In this paper we introduce an iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. The iterative process is based on so-called extragradient method. We obtain a strong convergence theorem for two sequences generated by this process.

Article information

Source
Taiwanese J. Math., Volume 10, Number 5 (2006), 1293-1303.

Dates
First available in Project Euclid: 20 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500557303

Digital Object Identifier
doi:10.11650/twjm/1500557303

Mathematical Reviews number (MathSciNet)
MR2253379

Zentralblatt MATH identifier
1110.49013

Subjects
Primary: 49J30: Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47J20: Variational and other types of inequalities involving nonlinear operators (general) [See also 49J40]

Keywords
extragradient method fixed point monotone mapping nonexpansive mapping variational inequality

Citation

Zeng, Lu-Chuan; Yao, Jen-Chih. STRONG CONVERGENCE THEOREM BY AN EXTRAGRADIENT METHOD FOR FIXED POINT PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS. Taiwanese J. Math. 10 (2006), no. 5, 1293--1303. doi:10.11650/twjm/1500557303. https://projecteuclid.org/euclid.twjm/1500557303


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References

  • [1.] 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.
  • [2.] G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), 747-756.
  • [3.] F. Liu and M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6 (1998), 313-344.
  • [4.] K. Goebel and W. A. Kirk, Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990.
  • [5.] R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), 75-88.
  • [6.] H. K. Xu and T. H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theory Appl., 119 (2003), 185-201.
  • [7.] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
  • [8.] W. Takahashi and M. Toyoda, Weak convergence theorems for nonepxansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.
  • [9.] 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.), Kluwer Academic Publishers, Dordrecht, Holland, 2001.
  • [10.] 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.