  The paper presents a fully explicit algorithm for monotone variational
inequalities. The method uses variable stepsizes that are computed using two
previous iterates as an approximation of the local Lipschitz constant without
running a linesearch. Thus, each iteration of the method requires only one
evaluation of a monotone operator $F$ and a proximal mapping $g$. The operator
$F$ need not be Lipschitz-continuous, which also makes the algorithm
interesting in the area of composite minimization where one cannot use the
descent lemma. The method exhibits an ergodic $O(1/k)$ convergence rate and
$R$-linear rate, if $F, g$ satisfy the error bound condition. We discuss
possible applications of the method to fixed point problems. We discuss
possible applications of the method to fixed point problems as well as its
different generalizations.
