Existence Of Projected Solutions For Generalized Mixed Quasi-variational Inequalities

The existence of projected solutions for generalized mixed quasi-variational inequality is investigated in this dissertation.When the constraint map is non-self,the(classical)solution of the generalized mixed quasi-variational inequality may not exist.In particular,when the image of the constraint map does not intersect its domain,the(classical)solution of the generalized mixed quasi-variational inequality does not exist.In the present dissertation,the concept of the projected solution of the generalized mixed quasi-variational inequality is introduced.By using the concepts of φ-pseudomonotonicity and φ-lower sign-continuity,the existence theorems of projected solutions of generalized mixed quasi-variational inequalities are established.The main results obtained in this dissertation are specifically divided into the following two parts:1.The existence of projected solutions for a class of generalized mixed quasi-variational inequalities with non-self constrained map is considered in finite dimensional space.First,we prove some lemmas describing the relationship between solution sets of generalized mixed variational inequality.Then,we generalize the nonemptiness of solution set from generalized Minty variational inequality to generalized Minty mixed variational inequality,where the mixed term has general properties.Finally,under the conditions of φ-pseudomonotonicity and φ-lower sign-continuity,the existence of projected solutions for generalized mixed quasivariational inequality is obtained by using these lemmas.2.The existence of projected solutions for a class of generalized mixed quasi-variational inequalities with non-self constrained map is considered in Banach space.A key argument to prove our result is to consider a suitable parametric generalized mixed variational inequality problem and establish some regularity properties of the solutions map.Furthermore,a Kakutani factorizable function is constructed and the existence of projected solutions for generalized mixed quasi-variational inequality is proved by the fixed theorem.This work generalizes the existence results of projected solutions for generalized quasi-variational inequality and improves in the existence results for projected solutions in Banach space.