Theory and Algorithms of Variational Inequality and Equilibrium Problems, and Their Applications (original) (raw)
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In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of a general system of variational inequalities, the set of solutions of a mixed equilibrium problem and the set of common fixed points of a finite family of nonexpansive mappings in a real Hilbert space. Using the demi-closedness principle for nonexpansive mapping, we prove that the iterative sequence converges strongly to a common element of the above three sets under some control conditions. Our main result extends a recent result of Ceng, Wang and Yao [L.C. Ceng, C.Y. Wang and 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].
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In this paper, we introduce and study a new hybrid iterative method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of variational inequalities for a ξξ-Lipschitz continuous and relaxed (m,v)(m,v)-cocoercive mappings in Hilbert spaces. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm which solves some optimization problems under some suitable conditions. Our results extend and improve the recent results of Yao et al. [Y. Yao, M.A. Noor, S. Zainab and Y.C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl (2009). doi:10.1016/j.jmaa.2008.12.005] and Gao and Guo [X. Gao and Y. Guo, Strong convergence theorem of a modified iterative algorithm for Mixed equilibrium problems in Hilbert spaces, J. Inequal. Appl. (2008). doi:10.1155/2008/454181] and many others.
2012
The purpose of this work, we present a new iterative algorithm for finding a common of the set of solutions of a mixed equilibrium problem, the set of a variational inclusion and the set of fixed point of nonexpansive mapping in a real Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. The results presented in the paper improve some recent results of Y. C. Liou, [An Iterative Algorithm for Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities, Fixed Point Theory and Applications, Volume 2010, Article ID 564361, 15 pages. doi:10.1155/2010/564361].
Fixed-Point Theory, Variational Inequalities, and Its Approximation Algorithms
International Journal of Mathematics and Mathematical Sciences, 2011
The study of variational inequalities, fixed points and approximation algorithms constituted a topic of intensive research efforts, especially within the past 30 years. As of today, this remains one of the most active fields in mathematics, and its ground of application varies from game theory, economics, engineering, and natural sciences, among others. On the other hand, the nature of many practical problems suggests an iterative approach to the solution.