非强制混合向量变分不等式解的存在性研究

Existence Results of Solutions for Noncoercive Mixed Vector Variational Inequalities

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结果;我们研究了当算子为余正仿射算子时,给出混合仿射向量变分不等式不存在例外簇的充分条件,得到混合仿射向量变分不等式弱有效解的存在性,给出了混合仿射向量变分不等式的弱有效解集为非空紧致集的充分条件.将Iusem等人(2019)在有限维空间中标量混合变分不等式解的存在性结果推广到自反Banach空间中混合向量变分不等式.

By employing the notion of exceptional family of elements,we establish some existence results for weakly efficient solutions for mixed vector variational inequality problems.We show that the nonexistence of an exceptional family of elements is a necessary condition for the solvability of mixed vector variational inequality problems.By using the asymptotic mappings of vector-valued mappings,we present a sufficient condition for the nonexistence of an exceptional family of elements for mixed vector variational inequality problems in reflexive Banach spaces,and obtain some existence results for weakly efficient solutions for mixed vector variational inequality problems.When the operator is affine and copositive,we present a sufficient condition for the nonexistence of an exceptional family of elements for mixed affine vector variational inequality problems,establish some existence results for weakly efficient solutions of mixed affine vector variational inequality problems,and provide some sufficient conditions for solution sets of mixed affine vector variational inequality problems to be nonempty and compact.The existence results of solutions for scalar mixed variational inequality problems in finite dimensional spaces established by Iusem et al.(2019),are generalized to mixed vector variational inequality problems in reflexive Banach spaces.