Algorithms for common solutions of generalized mixed equilibrium problems and system of variational inclusion problems

Volume 9, Issue 5, pp 3397--3423 http://dx.doi.org/10.22436/jnsa.009.05.125

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Authors

Lu-Chuan Ceng - Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China. A. Latif - Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. A. E. Al-Mazrooei - Department of Mathematics, University of Jeddah, P. O. Box 80327, Jeddah 21589, Saudi Arabia.

Abstract

In this paper, we introduce a multi-step iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inclusions for maximal monotone and inverse strong monotone mappings, the set of solutions of general system of variational inequalities and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. This multi-step iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, projection method, and strongly positive bounded linear operator and W-mapping approaches. We establish the strong convergence of the sequences generated by the proposed algorithm to a common element of above mentioned problems under appropriate assumptions, which also solves some optimization problem. The result presented in this paper improves and extends some corresponding ones in the earlier and recent literature.

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Keywords

• Multi-step iterative algorithm
• mixed equilibrium problem
• Variational inclusion
• nonexpansive mapping
• maximal monotone mapping
• strong convergence.

MSC

•  49J30
•  47H09
•  47J20
•  49M05

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