Generalized Solutions of Multi-Valued Monotone Quasi-Variational Inequalities

An ill-posed quasi-variational inequality with multi-valued maps can be conveniently formulated as a parameter identification problem on the graph of a variational selection. Using elliptic regularization for parametric variational inequalities, it is possible to pose another parameter identification problem that gives a stable approximation procedure for the ill-posed problem. The results are quite general and are applicable to ill-posed variational inequalities, inverse problems, split-feasibility problem, among others.