In this paper we study ``Differential Variational Inequalities'' defined in 
$\BR^N$. First we establish the existence of extremal trajectories, then 
we show that those extremal trajectories are dense in the solution set of the 
original system. Afterwards we prove that the solution set is path connected. 
Also we view the solution set as multifunction of the initial datum and 
for this multifunction we show that we can construct a continuous selector. 
Using this selector we prove the existence of periodic trajectories. 
Finally we show that the solution set depends continuously (in both the 
Vietoris and Hausdorff hyperspace topologies) on the data of the problem.