The paper written in an expository style provides a brief review of some modern trends and achievements in the variational inequality theory. In particular the following aspects of variational inequalities are considered.
1) Iterative methods for solving variational inequalities of the form
where is a nonlinear strongly monotone operator, is a closed convex subset of a real Hilbert space and . The presented methods are based on the equivalence between (1), the fixed point problem , and the Wiener-Hopf equation , where is the projection of onto , .
2) The sensitivity analysis of quasivariational inequalities
with a parameter . The main result of this section establishes those conditions under which (2) has a locally unique solution and the function is continuous or Lipschitz continuous.
3) The constructing of iterative methods for solving generalized variational inequalities
by transforming (3) to the fixed point problem or the general Wiener-Hopf equation. Here is a continuous operator.
4) Variational inequalities for fuzzy mappings and iterative methods for solving such inequalities.
5) Finite element approximation and error estimation for (1).