This paper gives existence theorems for solutions of the problem of finding a point $(z_0, x_0, y_0)\in B(z_0, x_0)\times A(x_0)\times F(z_0, x_0, x_0)$ such that, for all $x \in A(x_0), F(z_0, x_0, x)- y_0 \not\subset C(z_0, x_0, x_0)$, where $A, B,C$ and $F$ are set-valued maps between topological vector spaces. Our results generalize some known existence theorems for quasivariational inequalities.