In this paper, we obtain existence results for the Minty variational inequality associated to a monotone operator with noncompact domain. As a consequence, we derive the surjectivity of some classes of monotone set-valued operators in Hausdorff topological vector spaces. Using Ky Fan minimax inequality, sufficient optimally conditions for solvability of Minty variational inequality and consequently, monotonicity results of the involved operator are established. Finally, we provide a characterization of relaxed monotonicity (global hypomonotonicity) by means of the existence of solutions to relaxed Minty variational inequalities.
