In this paper, we discuss several extragradient-like algorithms for solving variational inequalities over the fixed point set of a nonexpansive mapping. Under the assumptions that the involving mapping is pseudomonotone and Lipschitz continuous, the sequence generated by our algorithms converges to a desired solution. Compared with the original extragradient algorithm, the new ones have an advantage: they do not require to compute any projection onto the feasible set. This feature helps to reduce the computational cost of our methods. Moreover, to implement the new algorithms, we do not need to know the Lipschitz constant of the involving mapping. Also, we extend these results to the equilibrium problem and present some numerical experiments to verify the efficiency of the new algorithms.
