AMO - Advanced Modeling and Optimization, Volume 14, Number 3, 2012 Some developments in general mixed quasi variational inequalities Abdellah Bnouhachem a,b 1 ,Mohamed Khalfaoui c and Hafida Benazza c a School of Management Science and Engineering, Nanjing University, Nanjing, 210093, P.R. China. b Ibn Zohr University, ENSA, BP 32/S, Agadir, Morocco. c Ecole Sup´ erieure de Technologie de Sal´ e, Mohamed 5 University, Agdal-Rabat, Morocco. Abstract. In this paper, we use the resolvent operator to suggest and analyze two new numerical methods for solving general mixed quasi variational inequalities coupled with new directions and new step sizes. Under certain conditions, the global convergence of the both methods is proved. Our results can be viewed as significant extensions of the previously known results for general mixed quasi variational inequalities. Key word. General mixed quasi variational inequalities, self-adaptive rules, pseu- domonotone operators, resolvent operator. 1 Introduction Variational inequality has become a rich of inspiration in pure and applied mathematics. In recent years, classical variational inequality problems have been extended and generalized to study a large variety of problems arising in structural analysis, economics, optimization, operations research and engineering sciences, see [1-39] and the references therein. The pro- jection and contraction method and its invariant forms represent an important tool for finding the approximation solution of various types of variational inequalities and complementarity problems. In recent years variational inequalities have been extended in various directions 1 Corresponding author. E-mail: babedallah@yahoo.com. *AMO - Advanced Modeling and Optimization. ISSN: 1841-4311 451