A special variational inequality and the implicit complementarity problem. (English) Zbl 0702.49008

We study the implicit complementarity problem (denoted by ICP) in infinite dimensional vector spaces, that is, the following problem: Let \(<E,E^*>\) be a dual system of locally convex spaces, \(K\subset E\) a closed convex cone and \(K^*\) the dual of K. Given two mappings T: \(K\to E^*\) and S: \(K\to E\), the problem studied in this paper is:
(I.C.P.(T,S,K) Find \(x_ 0\in K\) such that \(S(x_ 0)\in K\), \(T(x_ 0)\in K^*\) and \(<S(x_ 0),T(x_ 0)>=0.\)
This problem is important in stochastic optimal control. We define a special variational inequality which is used to obtain several existence theorem for the problem I.C.P.(T,S,K).
Reviewer: G.Isac


49J40 Variational inequalities
93E20 Optimal stochastic control
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)