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WEAK CONVERGENCE OF AN ITERATIVE SCHEME FOR GENERALIZED EQUILIBRIUM PROBLEMS

Part of: Nonlinear operators and their properties Connections with other structures, applications

Published online by Cambridge University Press:  05 May 2009

JIAN-WEN PENG  and
JEN-CHIH YAO
JIAN-WEN PENG*
Affiliation:
College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, People’s Republic of China (email: [email protected])
JEN-CHIH YAO
Affiliation:
Department of Applied Mathematics, National Sun Yat-sen University Kaohsiung, Taiwan 804, R.O.C.
*
✉For correspondence; e-mail: [email protected]
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Abstract

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In this paper, we introduce an iterative scheme using an extragradient method for finding a common element of the set of solutions of a generalized equilibrium problem, 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 weak convergence theorem for three sequences generated by this process. Based on this result, we also obtain several interesting results. The results in this paper generalize and extend some well-known weak convergence theorems in the literature.

Keywords

generalized equilibrium problemextragradient methodnonexpansive mappingmonotone mappingvariational inequalityweak convergencefixed point

MSC classification

Secondary: 47H10: Fixed-point theorems 54H25: Fixed-point and coincidence theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 3 , June 2009 , pp. 437 - 453
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The first author was supported by the National Natural Science Foundation of China (grant number 10771228), the Science and Technology Research Project of Chinese Ministry of Education (grant number 206123), the Education Committee project Research Foundation of Chongqing (grant number KJ070816). The second author was partially supported by the grant NSC96-2628-E-110-014-MY3.

References

[1] Blum, E. and Oettli, W., ‘From optimization and variational inequalities to equilibrium problems’, Math. Stud. 63 (1994), 123–145.Google Scholar
[2] Browder, F. E. and Petryshyn, W. V., ‘Construction of fixed points of nonlinear mappings in Hilbert space’, J. Math. Anal. Appl. 20 (1967), 197–228.CrossRefGoogle Scholar
[3] Ceng, L. C. and Yao, J. C., ‘Viscosity relaxed-extragradient method for monotone variational inequalities and fixed point problems’, J. Math. Inequalities 1 (2007), 225–241.CrossRefGoogle Scholar
[4] Combettes, P. L. and Hirstoaga, S. A., ‘Equilibrium programming in Hilbert spaces’, J. Nonlinear Convex Anal. 6 (2005), 117–136.Google Scholar
[5] Flam, S. D. and Antipin, A. S., ‘Equilibrium programming using proximal-like algorithms’, Math. Program. 78 (1997), 29–41.CrossRefGoogle Scholar
[6] Gárciga Otero, R. and Iuzem, A., ‘Proximal methods with penalization effects in Banach spaces’, Numer. Funct. Anal. Optim. 25 (2004), 69–91.CrossRefGoogle Scholar
[7] Goebel, K. and Kirk, W. A., Topics on Metric Fixed-point Theory (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[8] He, B.-S., Yang, Z.-H. and Yuan, X.-M., ‘An approximate proximal-extragradient type method for monotone variational inequalities’, J. Math. Anal. Appl. 300 (2004), 362–374.CrossRefGoogle Scholar
[9] Korpelevich, G. M., ‘The extragradient method for finding saddle points and other problems’, Matecon 12 (1976), 747–756.Google Scholar
[10] Nadezhkina, N. and Takahashi, W., ‘Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings’, J. Optim. Theory Appl. 128 (2006), 191–201.CrossRefGoogle Scholar
[11] Opial, Z., ‘Weak convergence of the sequence of successive approximation for nonexpansive mappings’, Bull. Amer. Math. Soc. 73 (1967), 561–597.CrossRefGoogle Scholar
[12] Plubtieng, S. and Punpaeng, R., ‘A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings’, Appl. Math. Comput. 197 (2008), 548–558.Google Scholar
[13] Rockafellar, R. T., ‘On the maximality of sums of nonlinear monotone operators’, Trans. Amer. Math. Soc. 149 (1970), 75–88.CrossRefGoogle Scholar
[14] Schu, J., ‘Weak and strong convergence to fixed points of asymptotically nonexpansive mappings’, Bull. Austral. Math. Soc. 43 (1991), 153–159.CrossRefGoogle Scholar
[15] Solodov, M. V., ‘Convergence rate analysis of iteractive algorithms for solving variational inequality problem’, Math. Program. 96 (2003), 513–528.CrossRefGoogle Scholar
[16] Solodov, M. V. and Svaiter, B. F., ‘An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions’, Math. Oper. Res. 25 (2000), 214–230.CrossRefGoogle Scholar
[17] Su, Y., Shang, M. and Qin, X., ‘An iterative method of solutions for equilibrium and optimization problems’, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.08.045Google Scholar
[18] Tada, A. and Takahashi, W., ‘Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem’, J. Optim. Theory Appl. 133 (2007), 359–370.CrossRefGoogle Scholar
[19] Takahashi, W., Nonlinear Functional Analysis (Yokohama Publishers, Yokohama, Japan, 2000).Google Scholar
[20] Takahashi, S. and Takahashi, W., ‘Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces’, J. Math. Anal. Appl. 331 (2006), 506–515.CrossRefGoogle Scholar
[21] Takahashi, S. and Takahashi, W., ‘Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space’, Nonlinear Anal. (2008), doi:10.10.1016/j.na.2008.02.042CrossRefGoogle Scholar
[22] Takahashi, W. and Toyoda, M., ‘Weak convergence theorems for nonexpansive mappings and monotone mappings’, J. Optim. Theory Appl. 118 (2003), 417–428.CrossRefGoogle Scholar
[23] Yao, Y. and Yao, J.-C., ‘On modified iterative method for nonexpansive mappings and monotone mappings’, Appl. Math. Comput. 186(2) (2007), 1551–1558.Google Scholar
[24] Zeng, L. C. and Yao, J. C., ‘Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems’, Taiwan. J. Math. 10 (2006), 1293–1303.Google Scholar