Generalized Quasi-Variational Inequalities in Infinite-Dimensional Normed Spaces

In this paper, we deal with the following problem: given a real normed space E with topological dual E*, a closed convex set X⊑E, two multifunctions Γ:X→2X and $$\Phi :X \to 2^{E^* } $$ , find $$(\hat x,\hat \phi ) \in X \times E^* $$ such that $$\hat x \in \Gamma (\hat x),\hat \phi \in \Phi (\hat x),{\text{ and }}\mathop {{\text{sup}}}\limits_{y \in \Gamma (\hat x)} \left\langle {\hat \phi ,\hat x - y} \right\rangle \leqslant 0.$$ We extend to the above problem a result established by Ricceri for the case Γ(x)≡X, where in particular the multifunction Φ is required only to satisfy the following very general assumption: each set Φ(x) is nonempty, convex, and weakly-star compact, and for each y∈X−:X the set $$\{ x \in X:\inf _{\phi \in \Phi (x)} \left\langle {\phi ,y} \right\rangle \leqslant 0\} $$ is compactly closed. Our result also gives a partial affirmative answer to a conjecture raised by Ricceri himself.