Variational-like inequalities for multivalued maps (Q1283625)

The authors investigate the variational-like inequality Problem (P) To find \(x_0\in K\) such that for each \(y\in K\) there exists \(u_0\in T(x_0)\;\) fulfilling \[ \langle M(x_0,u_0),\eta(y,x_0)\rangle + b(x_0,y)-b(x_0,x_0)\geq 0, \] where \(K\) is a nonempty subset of a reflexive Banach space \(X,\) \(T:K\to 2^C\) a multivalued map, \(C\) a nonempty subset of a dual \(X^*,\;M:K\times C\to X^*,\;\eta:K\times K\to X,\;b:K\times K\to \mathbb R,\;\langle.,.\rangle\) is the duality pairing between \(X^*\) and \(X.\) They generalize the monotonicity and hemicontinuity properties for multivalued maps. The main result is the existence theorem for Problem (P). The special case \(h(x,y)=b(y)\) is also considered.