Taiwanese Journal of Mathematics

WEAK AND STRONG CONVERGENCE THEOREMS FOR VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS WITH TSENG’S EXTRAGRADIENT METHOD

Fenghui Wang and Hong-Kun Xu

Full-text: Open access

Abstract

The paper is concerned with the problem of finding a common solution of a variational inequality problem governed by Lipschitz continuous monotone mappings and of a fixed point problem of nonexpansive mappings. To solve this problem, we introduce two new iterative algorithms which are based on Tseng's extragradient method. Moreover we prove the weak and strong convergence of these new algorithms to a solution of the above-stated problem.

Article information

Source
Taiwanese J. Math., Volume 16, Number 3 (2012), 1125-1136.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406682

Digital Object Identifier
doi:10.11650/twjm/1500406682

Mathematical Reviews number (MathSciNet)
MR2917259

Zentralblatt MATH identifier
06062768

Subjects
Primary: 47J20: Variational and other types of inequalities involving nonlinear operators (general) [See also 49J40] 49J40: Variational methods including variational inequalities [See also 47J20]
Secondary: 47H05: Monotone operators and generalizations 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.

Keywords
Lipschitz continuity nonexpansive mapping variational inequality problem iterative algorithms projection fixed point extragradient method

Citation

Wang, Fenghui; Xu, Hong-Kun. WEAK AND STRONG CONVERGENCE THEOREMS FOR VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS WITH TSENG’S EXTRAGRADIENT METHOD. Taiwanese J. Math. 16 (2012), no. 3, 1125--1136. doi:10.11650/twjm/1500406682. https://projecteuclid.org/euclid.twjm/1500406682


Export citation

References

  • J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, MA, 1990.
  • H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review, 38 (1996), 367-426.
  • K. Goebel and W. A. Kirk, Topics on Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.
  • H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly-monotone mappings, Nonlinear Anal., 61 (2005), 341-350.
  • 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.
  • G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonomika i Matematicheskie Metody, 12 (1976), 747-756.
  • 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.
  • N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16 (2006), 1230-1241.
  • K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372-379.
  • R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970) 75-88.
  • W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.
  • P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (1998), 431-446.
  • F. Wang and H. K. Xu, Strongly convergent iterative algorithms for solving a class of variational inequalities, J. Nonlinear Convex Anal., to appear.
  • L. C. Zeng and J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math., 10 (2006), 1293-1303.