Open Access

An Extragradient Method for Fixed Point Problems and Variational Inequality Problems

Journal of Inequalities and Applications20072007:038752

DOI: 10.1155/2007/38752

Received: 11 September 2006

Accepted: 10 December 2006

Published: 7 February 2007

Abstract

We present an extragradient method for fixed point problems and variational inequality problems. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for monotone mapping.

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Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University
(2)
Department of Information Management, Cheng Shiu University
(3)
Department of Applied Mathematics, National Sun Yat-sen University

References

  1. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967,20(2):197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar
  2. Liu F, Nashed MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Analysis 1998,6(4):313–344. 10.1023/A:1008643727926MathSciNetView ArticleMATHGoogle Scholar
  3. Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
  4. Yao J-C: Variational inequalities with generalized monotone operators. Mathematics of Operations Research 1994,19(3):691–705. 10.1287/moor.19.3.691MathSciNetView ArticleMATHGoogle Scholar
  5. Yao J-C, Chadli O: Pseudomonotone complementarity problems and variational inequalities. In Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optim. Appl.. Volume 76. Edited by: Crouzeix JP, Haddjissas N, Schaible S. Springer, New York, NY, USA; 2005:501–558. 10.1007/0-387-23393-8_12View ArticleGoogle Scholar
  6. Zeng LC, Schaible S, Yao J-C: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Optimization Theory and Applications 2005,124(3):725–738. 10.1007/s10957-004-1182-zMathSciNetView ArticleMATHGoogle Scholar
  7. Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003,118(2):417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar
  8. Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2006,128(1):191–201. 10.1007/s10957-005-7564-zMathSciNetView ArticleMATHGoogle Scholar
  9. Korpelevič GM: An extragradient method for finding saddle points and for other problems. Èkonomika i Matematicheskie Metody 1976,12(4):747–756.Google Scholar
  10. Zeng L-C, Yao J-C: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2006,10(5):1293–1303.MathSciNetMATHGoogle Scholar
  11. Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970,149(1):75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView ArticleMATHGoogle Scholar
  12. Osilike MO, Igbokwe DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Computers & Mathematics with Applications 2000,40(4–5):559–567. 10.1016/S0898-1221(00)00179-6MathSciNetView ArticleMATHGoogle Scholar
  13. Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar
  14. Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar
  15. Yao Y, Yao J-C: On modified iterative method for nonexpansive mappings and monotone mappings. Applied Mathematics and Computation 2007.Google Scholar

Copyright

© Yonghong Yao et al. 2007

This article is published under license to BioMed Central Ltd. 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.