We consider a pursuit-evasion differential game problem with countable number pursuers and one evader in the Hilbert space l 2 . Players' dynamic equations described by certain n th order ordinary differential equations. Control functions of the players subject to integral constraints. The goal of the pursuers is to minimize the distance to the evader and that of the evader is the opposite. The stoppage time of the game is fixed and the game payoff is the distance between evader and closest pursuer when the game is stopped. We study this game problem and find the value of the game. In addition to this, we construct players' optimal strategies.

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We study a simple motion pursuit differential game of many pursuers and one evader in a Hilbert space $l_{2}$. The control functions of the pursuers and evader are subject to integral and geometric constraints respectively. Duration of the game is denoted by positive number $\theta $. Pursuit is said to be completed if there exist strategies $u_{j}$ of the pursuers $P_{j}$ such that for any admissible control $v(\cdot)$ of the evader $E$ the inequality $\|y(\tau)-x_{j}(\tau)\|\leq l_{j}$ is satisfied for some $ j\in \{1,2, \dots\}$ and some time $\tau$. In this paper, sufficient conditions for completion of pursuit were obtained. Consequently strategies of the pursuers that ensure completion of pursuit are constructed.

We study pursuit differential game problem in which a countable number of pursuers chase one evader. The problem is formulated in a Hilbert space l2 with pursuers’ motions described by nth order differential equations and that of the evader by mth order differential equation. The control functions of the pursuers and evader are subject to integral and geometric constraints respectively.Duration of the game is denoted by the positive number?. Pursuit is said to be completed if there exist strategies uj of the pursuers Pj such that for any admissible control v(·) of the evader E the inequality ky(?) ? xj (?)k ? rj is satisfied for some j ? {1, 2, . . .}. In this paper, sufficient condition for completion of pursuit were obtained and also strategies of the pursuers that ensure completion of pursuit are constructed.

In the Hilbert space l2, we investigate a pursuit-evasion differential game involving countable number of pursuers and one evader. Players move in agreement with certain nth order ordinary differential equations with control functions of players satisfying integral constraints. The period of the game, which is denoted as θ, is fixed. During the game, pursuers want to minimize the distance to the evader and the evader want to maximize it. The game's payoff is the distance between evader and closest pursuer at time θ. Independent of relationship between energy resources of the players, we provide formula that defines value of the game and constructed players' optimal strategies.