This paper is strictly related to Ref. 1. A pursuit-evasion game described in part by the system [(x)\dot] = f₁ (t,x,u)\dot x = f_1 (t,x,u) and [(y)\dot] = f₂ (t,y,v)\dot y = f_2 (t,y,v) is considered. The state variablesx andy are restricted, in the sense that (x(t),t) N ₁ and (y(t),t) N ₂. The existence of a value in the sense of Varaiya and Lin is proved under the assumption that the sets of all admissible trajectories for the two players are compact and the lower value is not greater than the upper value.