*(English)*Zbl 0788.65074

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 $T:H\to H$ is a nonlinear strongly monotone operator, $K$ is a closed convex subset of a real Hilbert space $H$ and $f\in H$. The presented methods are based on the equivalence between (1), the fixed point problem $u={P}_{k}(u-\rho (T\left(u\right)-f))$, $\rho >0$ and the Wiener-Hopf equation $T\left({P}_{k}\left(v\right)\right)+{\rho}^{-1}{Q}_{k}\left(v\right)=f$, where ${P}_{k}$ is the projection of $H$ onto $K$, ${Q}_{k}=I-{P}_{k}$.

2) The sensitivity analysis of quasivariational inequalities

with a parameter $\lambda \in H$. The main result of this section establishes those conditions under which (2) has a locally unique solution $u\left(\lambda \right)$ and the function $\lambda \to u\left(\lambda \right)$ 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 $g:H\to H$ 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).

##### MSC:

65K10 | Optimization techniques (numerical methods) |

47J20 | Inequalities involving nonlinear operators |

49M05 | Numerical methods in calculus of variations based on necessary conditions |

49J40 | Variational methods including variational inequalities |

49J27 | Optimal control problems in abstract spaces (existence) |