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 $\theta$ be origin of $Y$. Given multi-valued mapping $F: D\times K\rightrightarrows Y$, can be formulated as the problem, find $\bar x\in D$ such that $$\mbox{GVEP}(F, D, K)\,\,\,\,\,\,\theta\in F(\bar x, y)\ \mbox{for all}\ y\in K.$$ 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.