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The purpose of this paper is to investigate the problem of finding a common element of the solution set of a general system of variational inequalities, the solution set of a variational inequality problem and the fixed point set of a strict pseudocontraction in a real Hilbert space. Based on the well-known viscosity approximation method, extragradient method and Mann's iteration method, we propose and analyze a relaxed projection-viscosity approximation method for computing a common element. Under very mild assumptions, we obtain a strong convergence theorem for three sequences generated by the proposed method. Our proposed method is quite general and flexible and develops some iterative methods considered in the earlier and recent literature.

In this paper, we introduce a modified extragradient method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone and Lipschitz continuous mapping. Our modified extragradient method is a variant of the so-called extragradient method. We obtain a weak convergent theorem for two sequences generated by this modified extragradient method.