This paper studies three pursuit differential game problems with many pursuers and a solitary evader in the Hilbert space ℓ₂ . In each problem, the dynamics of the pursuers obey certain second-order differential equations with pre-known initial positions and initial speeds. Whereas the lone evader changes position in conformity with a certain differential equation of first order with a pre-known initial position. In the first problem, integral constraints are considered on the control functions of both pursuers and the evader. For the second problem, the geometric constraint is applied to the control function of the evader and each of the pursuers. For the last problem, integral and geometric constraints are imposed on the control functions of the pursuers and the evader, respectively. For each game problem, we find conditions sufficient for the pursuers to finish the game in the ℓ -catch sense and introduce piece-wise pursuers’ strategies.