In this paper, we apply new results on variational relation problems obtained by D.T. Luc to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of its solutions of generalized quasi-equilibrium problems are shown. As special cases, we obtain several results on the existence of solutions of  generalized Pareto  and weak quasi-equilibrium problems concerning $C$-pseudomonotne multivalued mappings.    As applications, we shall show some results on the existence of solutions to generalized vector quasivariational inequalities with  multivalued mappings.