Generalization of an Existence Theorem for Variational Inequalities |
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Authors: | Z.H. Huang |
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Affiliation: | (1) Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Institute of Applied Mathematics, Beijing, PRC |
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Abstract: | By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known Moré's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result. |
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Keywords: | Variational inequalities complementarity problems exceptional family of elements existence theorem |
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