On discontinuous quasi-variational inequalities

In this paper, we derive a general theorem concerning the quasi-variational inequality problem: find x̅ ∈ C and y̅ ∈ T(x̅) such that x̅ ∈ S(x̅) and ⟨y̅,z-x̅⟩ ≥ 0, ∀ z ∈ S(x̅), where C,D are two closed convex subsets of a normed linear space X with dual X*, and ${\displaystyle T:X\to 2^{X\cdot }}$ and ${\displaystyle S:C\to 2^D}$ are multifunctions. In fact, we extend the above to an existence result proposed by Ricceri [12] for the case where the multifunction T is required only to satisfy some general assumption without any continuity. Under a kind of Karmardian's condition, we give a partial affirmative answer to an unbounded quasi-variational inequality problem.