The System Of Generalized Nonlinear Mixed Quasi-variational Inclusions

Much attention has been paid on the research of the variational inequality theory in the last decades. While the researche on the system of variational inequalities and the system of variational inclusions are quite indequate.In this paper, we introduce a new system of generalized nonlinear mixed quasi-variational inclusions and study considerably systematically the existence of the solutions, the sensitivity of the parametric solutions, and the iterative algorithms for solving the system of generalized nonlinear mixed quasi-variational inclusions, the convergence of the iterative sequence generated by the algorithms proposed are also given. The results generalize, unify and extend a large number of known results related to variational inequalities, variatioal inclusions , and the system of variatioal inequalities. Details are as follows.● The background, present research situation and mathematical models of the system of generalized nonlinear mixed quasi-variational inclusions are introduced briefly. Some basic definition and lemmas to be used in this paper are also presented.● Both the existence of the generalized nonlinear mixed quasi-variational inclusions and that of parametric generalized nonlinear mixed quasi-variational inclusions are discussed.● By using the implicit resolvent operator technique, we establish the equivalence between the system of generalized nonlinear mixed quasi-variational inclusions , a class of fixed point problems and the system of resolvent equations. Then we prove the existence of the solutions for the system of generalized nonlinear mixed quasi-variational inclusions. Finally, a number of iterative algorithms are given for solving the system of generalized nonlinear mixed quasi-variational inclusions . The convergence of the iterative sequence generated by the proposed iterative algorithms are also given.● The sensitivity analysis for the system of parametric generalized nonlinear mixed quasi-variational inclusions is discussed particularly.