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Zbl 0788.65074
Noor, Muhammad Aslam; Noor, Khalida Inayat; Rassias, Themistocles M.
Some aspects of variational inequalities. (English)
[J] J. Comput. Appl. Math. 47, No.3, 285-312 (1993). ISSN 0377-0427

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.\par 1) Iterative methods for solving variational inequalities of the form $$\langle T(u), v-u\rangle\ge \langle f, v-u\rangle\quad \forall v\in K\ (u\in K),\tag1$$ 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 $u= P\sb k(u-\rho(T(u)-f))$, $\rho>0$ and the Wiener-Hopf equation $T(P\sb k(v))+\rho\sp{-1} Q\sb k(v)= f$, where $P\sb k$ is the projection of $H$ onto $K$, $Q\sb k= I-P\sb k$.\par 2) The sensitivity analysis of quasivariational inequalities $$\langle T(u;\lambda), v-u\rangle\ge 0\quad \forall v\in K\sb \lambda(u)\quad (u\in K\sb \lambda(u))\tag2$$ 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.\par 3) The constructing of iterative methods for solving generalized variational inequalities $$\langle T(u),g(v)-g(u)\rangle\ge \langle f,g(v)- g(u)\rangle\quad\forall g(v)\in K\ (u\in H, g(u)\in K)\tag3$$ by transforming (3) to the fixed point problem or the general Wiener-Hopf equation. Here $g: H\to H$ is a continuous operator.\par 4) Variational inequalities for fuzzy mappings and iterative methods for solving such inequalities.\par 5) Finite element approximation and error estimation for (1).
[M.Yu.Kokurin (Yoshkar-Ola)]

MSC 2000:: *65K10 Optimization techniques (numerical methods)
47J20 Inequalities involving nonlinear operators
49M05 Methods of successive approximation based on necessary conditions
49J40 Variational methods including variational inequalities
49J27 Optimal control problems in abstract spaces (existence)

Keywords: iterative methods; finite element; variational inequality; nonlinear strongly monotone operator; Hilbert space; Wiener-Hopf equation; sensitivity analysis; quasivariational inequalities; fixed point problem; fuzzy mappings; error estimation

Cited in: Zbl 1140.49006 Zbl 1139.49012

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