Summary: In this paper, we use the power, upward power and the downward power preorders on the power sets of topological vector spaces to define the split generalized vector variational inequality problems for set-valued mappings on topological vector spaces. By using the Fan-KKM theorem, we prove an existence theorem for solutions to some split generalized vector variational inequality problems for set-valued mappings on topological vector spaces. Consequently, we prove the solvability of some generalized vector variational inequality problems for set-valued mappings. As applications, we study the existence of efficient pair of initial social activity profiles for some split generalized social utility optimization problems.