Relaxed inertial method for solving a class of bilevel variational inequalities

In this paper, we investigate a problem of monotone variational inequalities over the solution set of a split variational inequality problem in real Hilbert spaces.We propose a new two-step inertial iterative method with self-adaptive step sizes for approximating the solution of the problem.Our proposed algorithm does not require the co-coerciveness of the associated single-valued operators.Moreover, some parameters are relaxed to accommodate a larger range of values for the step sizes.We obtain strong convergence results under some mild conditions on the control parameters and without prior knowledge of the transformation operators and the monotone and Lipschitz continuous constants of the involved operators.Finally, we apply our result to the traffic network models and we present several numerical experiments to demonstrate the implementability of the proposed method.