Using Double Inertial Steps Into the Single Projection Method with Non-monotonic Step Sizes for Solving Pseudomontone Variational Inequalities

In this paper, we propose a new modified algorithm for finding an element of the set of solutions of a pseudomonotone, Lipschitz continuous variational inequality problem in real Hilbert spaces. Using the technique of double inertial steps into a single projection method we give weak and strong convergence theorems of the proposed algorithm. The weak convergence does not require prior knowledge of the Lipschitz constant of the variational inequality mapping and only computes one projection onto a feasible set per iteration as well as without using the sequentially weak continuity of the associated mapping. Under additional strong pseudomonotonicity and Lipschitz continuity assumptions, the R-linear convergence rate of the proposed algorithm is presented. Finally, some numerical examples are given to illustrate the effectiveness of the algorithms.