一般多值混合隐拟变分不等式的解的存在性与算法

Existence and Algorithm of Solutions for General Multivalued Mixed Implicit Quasi-Variational Inequalities

引入了实Hilbert空间中一类新的一般多值混合隐拟变分不等式· 它概括了丁协平教授引入与研究过的熟知的广义混合隐拟变分不等式类成特例· 运用辅助变分原理技巧来解这类一般多值混合隐拟变分不等式· 首先,定义了具真凸下半连续的二元泛函的新的辅助变分不等式,并选取了一适当的泛函,使得其唯一的最小值点等价于此辅助变分不等式的解· 其次,利用此辅助变分不等式,构造了用于计算一般多值混合隐拟变分不等式逼近解的新的迭代算法· 在此,等价性保证了算法能够生成一列逼近解· 最后,证明了一般多值混合隐拟变分不等式解的存在性与逼近解的收敛性· 而且,给算法提供了新的收敛判据· 因此,结果对M.A.Noor提出的公开问题给出了一个肯定答案,并推广和改进了关于各种变分不等式与补问题的早期与最近的结果,包括最近文献中涉及单值与集值映象的有关混合变分不等式。

A new class of general multivalued mixed implicit quasi_variational inequalities in a real Hilbert space was introduced,which includes the known class of generalized mixed implicit quasi_variational inequalities as a special case,introduced and studied by Ding Xie_ping.The auxiliary variational principle technique was applied to solve this class of general multivalued mixed implicit quasi_variational inequalities.Firstly,a new auxiliary variational inequality with a proper convex,lower semicontinuous,binary functional was defined and a suitable functional was chosen so that its unique minimum point is equivalent to the solution of such an auxiliary variational inequality.Secondly,this auxiliary variational inequality was utilized to construct a new iterative algorithm for computing approximate solutions to general multivalued mixed implicit quasi_variational inequalities.Here,the equivalence guarantees that the algorithm can generate a sequence of approximate solutions.Finally,the existence of solutions and convergence of approximate solutions for general multivalued mixed implicit quasi_variational nequalities are proved.Moreover,the new convergerce criteria for the algorithm were provided.Therefore,the results give an affirmative answer to the open question raised by M.A.Noor,and extend and improve the earlier and recent results for various variational inequalities and complementarity problems including the corresponding results for mixed variational inequalities,mixed quasi_variatoinal inequalities and quasi_complementarity problems involving the single_valued and set_valued mappings in the recent literature.