The objective of this paper is to study a pursuit differential game with finite or countably\nnumber of pursuers and one evader. The game is described by differential equations in l2-space,\nand integral constraints are imposed on the control function of the players. The duration of the game\nis fixed and the payoff functional is the greatest lower bound of distances between the pursuers and\nevader when the game is terminated. However, we discuss the condition for finding the value of\nthe game and construct the optimal strategies of the players which ensure the completion of the\ngame. An important fact to note is that we relaxed the usual conditions on the energy resources of\nthe players. Finally, some examples are provided to illustrate our result.
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