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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 \[ \langle T(u), v-u\rangle\geq \langle f, v-u\rangle\quad \forall v\in K\;(u\in K),\tag{1} \] 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 \(u= P_ k(u-\rho(T(u)-f))\), \(\rho>0\) and the Wiener-Hopf equation \(T(P_ k(v))+\rho^{-1} Q_ k(v)= f\), where \(P_ k\) is the projection of \(H\) onto \(K\), \(Q_ k= I-P_ k\).
2) The sensitivity analysis of quasivariational inequalities \[ \langle T(u;\lambda), v-u\rangle\geq 0\quad \forall v\in K_ \lambda(u)\quad (u\in K_ \lambda(u))\tag{2} \] with a parameter \(\lambda\in H\). The main result of this section establishes those conditions under which (2) has a locally unique solution \(u(\lambda)\) and the function \(\lambda\to u(\lambda)\) is continuous or Lipschitz continuous.
3) The constructing of iterative methods for solving generalized variational inequalities \[ \langle T(u),g(v)-g(u)\rangle\geq \langle f,g(v)- g(u)\rangle\quad\forall g(v)\in K\;(u\in H, g(u)\in K)\tag{3} \] 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 Numerical optimization and variational techniques
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49M05 Numerical methods based on necessary conditions
49J40 Variational inequalities
49J27 Existence theories for problems in abstract spaces
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