Existence of solutions for lower semicontinuous quasi-equilibrium problems

In this paper, we deal with the following quasi-equilibrium problem: given a nonempty subset C of a topological vector space X, a nonempty set D, two functions T : C → D, f : X × D → R and a multifunction S : C → 2X, find such that , and for all . Such a problem, recently introduced in [1], extends and unifies several variational inequality problems, generalized complementarity problems and Nash equilibrium problems. We prove several existence results in which the upper semicontinuity of the multifunction S is not assumed.