Let be Hausdorff topological spaces, be a topological vector space and be closed and such that . Given two multifunctions and , the “parametric vector quasiequilibrium problem” consists in finding, given and , some such that for every .
Assuming that the solution set is nonempty in a neighborhood of , the present paper gives necessary conditions for the multifunction to be lower semicontinuous, or upper semicontinuous. Also, a “strong” version of the quasiequilibrium problem is investigated, and sufficient conditions are given for its solution set to be equal to . These results generalize and sometimes improve previously known results on quasivariational inequalities.