令 T 是一個從拓樸向量空間 X 的子集映至 X* 的多值函數。此文章主要是討論 T 滿足 (S)_+ 條件和 T 滿足 (S)_+^1 條件之間的關係。為了整合單值與多值函數之 (S)_+ 條件的定義，我們將 [9] 中對於單值函數定義的 (S)_+^w 條件延拓至多值函數的情形。上列這些條件對於映至 L(X,Z) 之函數亦可定義，其中 Z 是賦序拓樸向量空間。利用這些條件，我們推導了一些廣義向量變分不等式與廣義變分不等式的存在性結果。

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.

1. Introduction 2
2. Preliminary 4
3. The (S)_+ and (S)_+^1 Conditions 7
4. Generalized Vector Variational Inequalities 19
5. Appendix 27

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