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Abstract. In this paper, using a new nonlinear mapping called demimetric and the shrinking projection method, we prove a strong convergence theorem for finding a common element of the set of common fixed points for a finite family of these new demimetric mappings and the set of common solutions of variational inequality problems for a finite family of inverse strongly monotone mappings in a Hilbert space. Using the result, we obtain well-known and new strong convergence theorems in a Hilbert space.

We investigate the problem of finding a common solution of a general system of variational inequalities, a variational inclusion, and a fixed-point problem of a strictly pseudocontractive mapping in a real Hilbert space. Motivated by Nadezhkina and Takahashi's hybrid-extragradient method, we propose and analyze new hybrid-extragradient iterative algorithm for finding a common solution. It is proven that three sequences generated by this algorithm converge strongly to the same common solution under very mild conditions. Based on this result, we also construct an iterative algorithm for finding a common fixed point of three mappings, such that one of these mappings is nonexpansive, and the other two mappings are strictly pseudocontractive mappings. 0 ∈ Φ x * Mx * .

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