Variational and generalized variational inequalities with discontinuous mappings (Q1323886)

The authors discuss the variational problem: given \(f: X\subset{\mathcal R}^ n\to {\mathcal R}^ n\), to find \(x\in X\) such that \[ \langle f(x), u- x\rangle\geq 0,\qquad\text{for all}\quad u\in X\tag{P}\(_ 1\) \] and its multivalued version: given a multivalued function \(F: X\to 2^{{\mathcal R}^ n}\), to find \(x\in X\), \(y\in F(x)\) such that \[ \langle y,u- x\rangle\geq 0,\qquad\text{for all}\quad u\in X.\tag{P}\(_ 2\) \] More precisely, they consider functions \(f\) and \(F\) which are not necessarily continuous. Several existence results are proved. Furthermore, they provide uniqueness results for problem \((\text{P}_ 2)\) and the complementary one: given a closed convex cone \(K\subset {\mathcal R}^ n\) to find \(x\in K\), \(y\in F(x)\) such that \(y\in K^*\), \(\langle y,x\rangle= 0\), where \(K^*\) denotes the polar cone. Applications to minimization problems for pseudo-convex functions are also investigated.