Generalized vector equilibrium problems with pairs of bifunctions and some applications
Hung Bui The
TL;DR
The paper addresses the generalized vector equilibrium problem $GVEP(F, D, K)$ in real topological vector spaces and proves existence results under minimal structure on the constraint set $K$, including results holding on self-segment-dense subsets. It develops a two-bifunction framework with $F$ and $G$ satisfying KKM-type and coercivity conditions and leverages Ky Fan's lemma to obtain solutions, even when $K$ is noncompact or lacks algebraic structure. By connecting GVEP to variational relation problems, vector equilibrium problems, and common fixed point theorems, the work unifies and extends solvability results across these areas under weak assumptions. The findings broaden the applicability of vector equilibrium theory to nonlinear analysis and optimization, particularly in nondense or nonconvex settings.
Abstract
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 $F: D\times K\rightrightarrows Y$, can be formulated as the problem, find $\bar x\in D$ such that $$\mbox{GVEP}(F, D, K)\,\,\,\,\,\,θ\in F(\bar x, y)\ \mbox{for all}\ y\in K.$$ 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.