Generalized vector equilibrium problems with pairs of bifunctions and some applications

In this paper, we deal with the following generalized vector equilibrium problem: Let $X, Y$ be topological vector spaces over reals, D be a nonempty subset of X, K be a nonempty set and θ be origin of Y. Given multi-valued mapping F: D× K Y, can be formulated as the problem, find x∈ D such that GVEP(F, D, K)\,\,\,\,\,\,θ∈ F( x, y)\ for all\ y∈ K. We prove several existence theorems for solutions to the generalized vector equilibrium problem when K is an arbitrary nonempty set without any algebraic or topological structure. Furthermore, we establish that some sufficient conditions ensuring the existence of a solution for the considered conditions are imposed not on the entire domain of the bifunctions but rather on a self-segment-dense subset. We apply the obtained results to variational relation problems, vector equilibrium problems, and common fixed point problems.