Quasi-variational inequalities with non-compact sets

In this paper, we first generalize a foundational quasi-variational inequality (Theorem 3) which plays a key role throughout this paper by relaxing the compactness condition. Then we set up general forms of (generalized) quasi-variational inequalities and obtain a series of existence theorems without the compactness assumption. Also, since many other quasi-variational inequalities in the literature are special cases of ours, they can be generalized by our results.