The generalized quasi-equilibrium problem of type II is formulated and some sufficient conditions on the existence of its solutions are shown. As special cases, we obtain several results on the existence of solutions of ideal quasivariational inclusion problems, quasivariational relation problems of type II, generalized quasi-KKM theorems etc. As corollaries, we show several results on the existence of solutions to other problems in the vector optimization theory concerning multivalued mappings.