Abstract. We present a modified extragradient method with dynamic stepsize adjustment to solve variational inequalities with monotone operators acting in a Hilbert space. In addition, we consider a variant of the method that finds a solution of a variational inequality that is also a fixed point of a given quasi-nonexpansive mapping. We establish the weak convergence of the methods without any Lipschitzian continuity assumption on operators.

Keywords: variational inequality, monotone operator, Hilbert space, extragradient method, weak convergence.