Summary: In this paper, by using a fixed point theorem for expansive set-valued mappings with noncompact and nonconvex domains and ranges in topological spaces due to the author [

*X. P. Ding*, Nonlinear Anal., Theory Methods Appl. 70, No. 2, A, 881–889 (2009;

Zbl 1170.54015)], we first prove a collective fixed point theorem and an existence theorem of equilibrium points for a generalized game. As applications, some new existence theorems of solutions for systems of generalized quasi-variational inclusion problems are established in noncompact topological spaces. Our results are different from known results in the literature.