Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems

The purpose of this paper is to study and analyze two different kinds of extragradient-viscosity-type iterative methods for finding a common element of the set of solutions of the variational inequality problem for a monotone and Lipschitz continuous operator and the set of fixed points of a demicontractive mapping in real Hilbert spaces. Although the problem can be translated to a common fixed point problem, the algorithm’s structure is not derived from algorithms in this field but from the field of variational inequalities and hence can be computed quite easily. We extend several results in the literature from weak to strong convergence in real Hilbert spaces and moreover, the prior knowledge of the Lipschitz constant of cost operator is not needed. We prove two strong convergence theorems for the sequences generated by these new methods. Primary numerical examples illustrate the validity and potential applicability of the proposed schemes.