Discontinuous generalized quasi-variational inequalities with application to fixed points

We consider the following generalized quasi-variational inequality problem introduced in [7]: given a real normed space X with topological dual X*, two sets C, D subset of X and two multifunctions S : C -> 2(D) and T :C -> 2(X)*, find ((x) over cap(phi) over cap) is an element of C x X* such that (x) over cap is an element of S ((x)over cap>), (phi) over cap is an element of T ((x) over cap) and <(phi) over cap, (x) over cap - y > <= 0 for all y is an element of S((x)over cap>). We prove an existence theorem where T is not assumed to have any continuity or monotonicity property, improving some aspects of the main result of [7]. In particular, the coercivity assumption is meaningfully weakened. As an application, we prove a theorem of the alternative for the fixed points of a Hausdorff lower semicontinuous multifunction. In particular, we obtain sufficient conditions for the existence of a fixed point which belongs to the relative boundary of the corresponding value.