Discontinuous quasivariational-like inequalities

In this note, we deal with the following problem: given X Rn, a multification gG : X → 2X, two (single-valued) maps f : X → Rn, η : X × X → Rn, find a point x* X such that x* Γ (x*) and f(x*), η(x,x*) ≥ 0 for all x Γ(x*). We prove an existence theorem in which, in particular, the function f is not supposed to be continuous.