Finite-dimensional quasi-variational inequalities associated with discontinuous functions

In this paper, given a nonempty closed convex setX ? ⁿ , a functionf: X→? ⁿ , and a multifunction Γ:X→2^X, we deal with the problem of finding a point (hat x) ∈X such that $$hat x in Gamma (hat x) and langle f(hat x), hat x - yrangle leqslant 0, for all y in Gamma (hat x).$$ For such problem, we establish a result where, in particular, the functionf is not assumed to be continuous. More precisely, we extend to the present setting a finite-dimensional version of a result by Ricceri on variational inequalities (Ref. 1).