In this paper, we introduce a new class of bifunction variationalinequalities, which is called the multivalued mixed quasi bifunction variationalinequalities. Using the auxiliary principle technique, we suggest and analyze aproximal point method for solving the multivalued mixed quasi bifunction variationalinequalities. It is shown that the convergence of the proposed methodrequires only pseudomonotonicity, which is a weaker condition than monotonicity.Our results represent an improvement and renement of previouslyknown results. Since the multivalued mixed quasi bifunction variational inequalitiesinclude bifunction variational inequalities and related optimizationproblems as special cases, results proved in this paper continue to hold forthese problem
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