Open Access Research Article

An Extragradient Method for Fixed Point Problems and Variational Inequality Problems

Yonghong Yao1*, Yeong-Cheng Liou2 and Jen-Chih Yao3

Author Affiliations

1 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

2 Department of Information Management, Cheng Shiu University, Niaosong Township, Kaohsiung 833, Taiwan

3 Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan

For all author emails, please log on.

Journal of Inequalities and Applications 2007, 2007:038752  doi:10.1155/2007/38752


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2007/1/038752


Received:11 September 2006
Accepted:10 December 2006
Published:7 February 2007

© 2007 Yao et al.

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.

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.

References

  1. Browder, FE, Petryshyn, WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications. 20(2), 197–228 (1967). Publisher Full Text OpenURL

  2. Liu, F, Nashed, MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Analysis. 6(4), 313–344 (1998). Publisher Full Text OpenURL

  3. Takahashi, W: Nonlinear Functional Analysis,p. iv+276. Yokohama Publishers, Yokohama, Japan (2000)

  4. Yao, J-C: Variational inequalities with generalized monotone operators. Mathematics of Operations Research. 19(3), 691–705 (1994). Publisher Full Text OpenURL

  5. Yao, J-C, Chadli, O: Pseudomonotone complementarity problems and variational inequalities. In: Crouzeix JP, Haddjissas N, Schaible S (eds.) Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optim. Appl., vol. 76, pp. 501–558. Springer, New York, NY, USA (2005).

  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. 124(3), 725–738 (2005). Publisher Full Text OpenURL

  7. Takahashi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications. 118(2), 417–428 (2003). Publisher Full Text OpenURL

  8. Nadezhkina, N, Takahashi, W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications. 128(1), 191–201 (2006). Publisher Full Text OpenURL

  9. Korpelevič, GM: An extragradient method for finding saddle points and for other problems. Èkonomika i Matematicheskie Metody. 12(4), 747–756 (1976)

  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. 10(5), 1293–1303 (2006)

  11. Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society. 149(1), 75–88 (1970). Publisher Full Text OpenURL

  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. 40(4-5), 559–567 (2000). PubMed Abstract | Publisher Full Text OpenURL

  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. 305(1), 227–239 (2005). Publisher Full Text OpenURL

  14. Xu, H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications. 298(1), 279–291 (2004). Publisher Full Text OpenURL

  15. Yao, Y, Yao, J-C: On modified iterative method for nonexpansive mappings and monotone mappings. Applied Mathematics and Computation (2007)