Existence theorem for the discontinuous generalized quasivariational inequality problem

We consider the following generalized quasivariational inequality problem: given a real Banach space E with topological dual E* and given two multifunctions G: X --> 2(X) and F: X --> 2(E*), find ((x) over cap, (φ) over cap) is an element of X x E* such that (x) over cap is an element of G((x) over cap), (φ) over cap is an element of F((x) over cap), [(φ) over cap, (x) over cap - y] less than or equal to 0, for all y is an element of G((x) over cap). We prove an existence theorem where F is not assumed to have any continuity or monotonicity property. Making use of a different technical construction, our result improves some aspects of a recent existence result ( Theorem 3.1 of Ref. 1). In particular, the coercivity assumption of this latter result is weakened meaningfully.