Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization
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Author list: Kumam W., Kumam P.
Publisher: Elsevier
Publication year: 2009
Journal: Nonlinear Analysis: Hybrid Systems (1751-570X)
Volume number: 3
Issue number: 4
Start page: 640
End page: 656
Number of pages: 17
ISSN: 1751-570X
eISSN: 1878-7460
Languages: English-Great Britain (EN-GB)
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Abstract
In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the solutions of the variational inequality problem for two inverse-strongly monotone mappings. We introduce a new viscosity relaxed extragradient approximation method which is based on the so-called relaxed extragradient method and the viscosity approximation method. We show that the sequence converges strongly to a common element of the above three sets under some parametric controlling conditions. Moreover, using the above theorem, we can apply to finding solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. The results of this paper extended, improved and connected with the results of Ceng et al., [L.-C. Ceng, C.-Y. Wang, J.-C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. 67 (2008), 375-390], Plubtieng and Punpaeng, [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput. 197 (2) (2008) 548-558] Su et al., [Y. Su, et al., An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. 69 (8) (2008) 2709-2719], Li and Song [Liwei Li, W. Song, A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces, Nonlinear Anal.: Hybrid Syst. 1 (3) (2007), 398-413] and many others. ฉ 2009 Elsevier Ltd. All rights reserved.
Keywords
Fixed point problems and Variational inequality
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