11N05, 11N37.

In this paper, a class of generalized general mixed quasi variational inequalities is introduced and studied. We prove the existence of the solution of the auxiliary problem for the generalized general mixed quasi variational inequalities, suggest a predictor-corrector method for solving the generalized general mixed quasi variational inequalities by using the auxiliary principle technique. If the bi-function involving the mixed quasi variational inequalities is skew-symmetric, then it is shown that the convergence of the new method requires the partially relaxed strong monotonicity property of the operator, which is a weak condition than cocoercivity. Our results can be viewed as an important extension of the previously known results for variational inequalities. ;