Generalized quasi-variational inequalities in infinite-dimensional normed spaces

In this paper, we deal with the following problem: given a real normed space E with topological dual E*, a closed convex set X subset of or equal to E, two multifunctions Gamma:X --> 2(X) and Phi:X --> 2(E*), find ((x) over cap, <(phi)over cap>) is an element of X x E* such that (x) over cap is an element of Gamma((x) over cap), <(phi)over cap> is an element of Phi((x) over cap), and (y is an element of Gamma((x) over cap)sup [<(phi)over cap>, (x) over cap - y] less than or equal to 0. We extend to the above problem a result established by Ricceri for the case Gamma(x) = X, where in particular the multifunction Phi is required only to satisfy the following very general assumption: each set Phi(x) is nonempty, convex, and weakly-star compact, and for each y is an element of X-X the set {x is an element of X: inf(phi is an element of Phi(x)) [phi, y] less than or equal to 0} is compactly closed. Our result also gives a partial affirmative answer to a conjecture raised by Ricceri himself.