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A class of random variational inequalities and simple random unilateral boundary value problems: Existence, discretization, finite element approximation. (English) Zbl 0974.60048
At first the author proves the existence of a unique measurable solution for the following random variational inequality: Find for every elementary event $\omega$ an element $\stackrel{^}{y}\in 𝒦$ depending on $\omega$ such that $\beta \left(\omega ,\stackrel{^}{y},z-\stackrel{^}{y}\right)\ge \lambda \left(\omega ,z-\stackrel{^}{y}\right)$, $z\in 𝒦$, where $𝒦$ is a closed convex non-empty subset of a separable Hilbert space and for every $\omega$, $\lambda \left(\omega ,·\right)$ and $\beta \left(\omega ,·,·\right)$ being a linear form, respectively, bilinear form. It is assumed that $\beta$ is nonnegative. Then the author considers in detail a more specialized inequality, where the data decompose in deterministic data and given real-valued random variables. Existence and uniqueness results are given, various approximation procedures and its convergence are studied. The general theory is applied to a Helmholtz-like elliptic equation with Signorini boundary conditions.

##### MSC:
 60H25 Random operators and equations