On Approximating Solutions to Non-monotone Variational Inequality Problems: An Approach Through the Modified Projection and Contraction Method

In this paper, we introduce a novel approach to approximate the solution of variational inequality problems without relying on the monotonicity assumption. We propose a two-step inertial modified projection and contraction method for solving quasi-monotone and without-monotone variational inequalities in real Hilbert spaces. We establish a weak convergence result for the proposed method under suitable conditions. Additionally, numerical examples and a network equilibrium flow problem are provided to illustrate the effectiveness of our method and compare it with recent related methods in the literature.