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

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.