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 x D --> R and a multifunction S : C --> 2(X), find (x) over cap is an element of X such that (x) over cap is an element of C, (x) over cap is an element of S((x) over cap) and f((x) over cap, T (x) over cap) less than or equal to f(x, T (x) over cap) for all x is an element of S((x) over cap). 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.