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Some aspects of variational inequalities. (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

$〈T\left(u\right),v-u〉\ge 〈f,v-u〉\phantom{\rule{1.em}{0ex}}\forall v\in K\phantom{\rule{4pt}{0ex}}\left(u\in K\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

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 $u={P}_{k}\left(u-\rho \left(T\left(u\right)-f\right)\right)$, $\rho >0$ 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

$〈T\left(u;\lambda \right),v-u〉\ge 0\phantom{\rule{1.em}{0ex}}\forall v\in {K}_{\lambda }\left(u\right)\phantom{\rule{1.em}{0ex}}\left(u\in {K}_{\lambda }\left(u\right)\right)\phantom{\rule{2.em}{0ex}}\left(2\right)$

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

$〈T\left(u\right),g\left(v\right)-g\left(u\right)〉\ge 〈f,g\left(v\right)-g\left(u\right)〉\phantom{\rule{1.em}{0ex}}\forall g\left(v\right)\in K\phantom{\rule{4pt}{0ex}}\left(u\in H,g\left(u\right)\in K\right)\phantom{\rule{2.em}{0ex}}\left(3\right)$

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)