Let T be a multivalued mapping from a nonempty subset of a topological vector space into its topological dual. In this paper, we discuss the relationship between the multivalued mapping T satisfying the (S)_+ condition and T satisfying the (S)_+^1 condition. To unify the (S)_+ condition for single-valued and multivalued mappings, we introduce the weak (S)_+ condition for single-valued mappings defined in [9] to multivalued mappings. The above conditions extend naturally to mappings into L(X,Z), where Z is an ordered Hausdorff topological vector space. We also derive some existence results for generalized vector variational inequalities and generalized variational inequalities associated with mappings which satisfy the (S)_+, (S)_+^1 or weak (S)_+ condition.