On approximating solutions to quasimonotone variational inequality problems: An approach through a new modified extragradient method

In this paper, we investigate quasimonotone and Lipschitz continuous variational inequalities in real Hilbert spaces. To solve this problem, we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method. Some very recent papers have considered different inertial algorithms which allow the inertial factor to be chosen in $ [0, 1] $. The purpose of this work is to continue working in this direction, so we propose another inertial extragradient method in that the inertial factor can be chosen in a special case to be $ 1 $. Under suitable mild conditions, we establish the weak convergence of the proposed algorithm. Moreover, linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.