As the previous Nash equilibrium for set-valued mappings is unconstrained, in order to solve this problem, a concept of Generalized Nash equilibrium with constraints for set-valued mappings is proposed, which includes usual Nash equilibrium and Loose Nash equilibrium as special cases. Firstly, by using the equivalent form of KKM Theorem, the existence theorem of generalized Nash equilibria for set-valued mappings is obtained. Secondly, in order to discuss the stability of generalized Nash equilibria for set-valued mappings, the sufficient and necessary conditions of Levitin-Polyak well-posedness are proved by defining the Levitin-Polyak approximating sequence. On this basis, the results of Levitin-Polyak well-posedness of generalized Nash equilibria for set-valued mappings are obtained. In addition, the results of existence and Levitin-polyak well-posedness of generalized Nash equilibria for set-valued mappings are verified by practical examples. It is shown that the most of generalized Nash equilibria for set-valued mappings are stable. Similarly, when the mappings of corresponding degenerates to a singleton valued function, the results of existence and Levitin-Polyak well-posedness still hold.