Discontinuous implicit generalized quasi-variational inequalities in Banach spaces

We consider the following implicit quasi-variational inequality problem: given two topological vector spaces E and F, two nonempty sets X subset of E and C F, two multifunctions Gamma : X -> 2(X) and Phi: X -> 2(C), and a single-valued map psi : X x C x X -> IR, find a pair ((x) over cap, (z) over cap) is an element of X x C such that (x) over cap is an element of Gamma((x) over cap), (z) over cap is an element of Gamma((x) over cap) Phi((x) over cap), and psi((x) over cap, (z) over cap, y) <= 0 for all y is an element of Gamma((x) over cap). We prove an existence theorem in the setting of Banach spaces where no continuity or monotonicity assumption is required on the multifunction Phi. Our result extends to non-compact and infinite-dimensional setting a previous results of the authors (Theorem 3.2 of Cubbiotti and Yao [15] Math. Methods Oper. Res. 46, 213-228 (1997)). It also extends to the above problem a recent existence result established for the explicit case (C = E* and psi(x, z, y) = [z, x - y]).