Abstract

In this paper, we combine the subgradient extragradient method and the hybrid method with inertial extrapolation step to introduce two strongly convergent methods for solving variational inequalities for which the underlying cost function is a pseudo-monotone operator in real Hilbert spaces. The first proposed method involves step-sizes bounded by the inverse of the Lipschitz constant of the cost function and the second proposed method involves line search approach when the cost function is uniformly continuous. Our strong convergence results are obtained under some relaxed conditions on the inertial factor and the iterative parameters. Our numerical implementations show that our proposed methods are competitive and efficient.