The generalized vector quasi-variational-like inequalities
The following class of generalized vector quasi-variational-like inequalities in real Hausdorff topological spaces \(E,\;F\) is studied: \(\widetilde x \in G(\widetilde x)\), for any \(y \in G(\widetilde x)\) there is \(\widetilde u \in T(\widetilde x)\) such that \((\widetilde u,\eta(y,\widetilde x)) \notin -\operatorname {int} C(\widetilde x).\) Here \(T:X \to 2^{L(E,F)}\), \(G:X \to 2^X\), \(C:X \to 2^F\) are set-valued mappings (\(X\) is a nonempty convex subset of \(E\)), \(\eta:X\times X \to E\) is a single-valued mapping. It is supposed that \(C(x)\) is a closed convex cone, \(C(x) \cap -C(x) = \{\emptyset \}\)), \(\operatorname {int} C(x)\neq \emptyset\) for any \(x \in X\), \(G\) is hemicontinuous with compact values, \(T\) satisfies a certain monotonicity assumption. Some particular cases are mentioned. General existence results are established by using a scalarization technique.
generalized vector quasi-variational-like inequality
scientific article; zbMATH DE number 1391536
scientific article