On random variational inequalities (Q1353578)

Let \((H,(.,.))\) be a real separable Hilbert space with the Borel \(\sigma\)-algebra \({\mathcal B}(H).\) Let \((\Omega,{\mathcal X})\) be a measurable space. A mapping \(T:\;\Omega\times H\to H\) is called a random operator, if for any given \(x\in H,\;T(t,x)=y(t)\) is measurable. The author proves the existence of a unique solution \(u:\;\Omega \to H,\;\;u(t)\in K\), for the following random variational inequality problem \[ a(u(t),v-u(t))+b(u(t),v)-b(u(t),u(t))\leq A((t,u(t)),v-u(t)) \] for all \((t,v)\in \Omega\times K.\) Here \(A\) is a random operator and \(K\) is a closed convex set in \(H.\) The bilinear form \(a\) is coercive and bounded and the form \(b\) is linear in the first argument and bounded. This variational problem includes various known inequalities considered by M. A. Noor, G. Dvaut, J. Lions and G. Stampacchia.