THE $(S)_+$-CONDITION FOR VECTOR EQUILIBRIUM PROBLEMS

Abstract

In this paper, we generalize the $(S)_+$-condition to bifunctions with values in an oredered Hausdorff topological vector space $\mathcal{Z}$, and define a weak $(S)_+$-condition for the bifunctions. These conditions extend naturally to operators from nonempty subsets of a topological vector space $X$ into the set $\mathcal{L}(X,\mathcal{Z})$ of all continuous linear mappings from $X$ into $\mathcal{Z}$. Then we derive some existence results for vector equilibrium problems and vector variational inequalities associated with bifunctions or operators satisfying the weak $(S)_+$-condition.