In this paper, we introduced a new double inertial subgradient extragradient method for solving a variational inequality problem in Hilbert space. In our method, the mapping needed not to satisfy any assumption of monotonicity and two different self-adaptive step sizes were used for avoiding the need of Lipschitz constant of the mapping. The strong convergence of the proposed method was proved under some new conditions. Finally, some numerical examples were presented to illustrate the convergence of our method and compare with some related methods in the literature.

In this paper, we proposed a modified projection and contraction method for solving a variational inequality problem in Hilbert spaces. In the existing projection and contraction methods for solving the variational inequality problem, the sequence $\{{\beta }_n\}$ has a similar computation manner, which is computed in a self-adaptive manner, but the sequence $\{{\beta }_n\}$ in our method is a sequence of numbers in (0, 1) given in advance, which is the main difference of our method with the existing projection and contraction methods. A line search is used to deal with the unknown Lipschitz constant of the mapping. The strong convergence of the proposed method is proved under certain conditions. Finally, some numerical examples are presented to illustrate the effectiveness of our method and compare the computation results with some related methods in the literature. The numerical results show that our method has an obvious competitive advantage compared with the related methods.