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The subject of investigation is game problems for the object control under the conditions of counteractions. The paper suggests that the object dynamics is described by the system of difference-differential equations. We consider the approach problem with fixed time. In the course of the game the information on the initial function and the prehistory of the evader's control is used. We suggest the way to solution of problems with fixed time. The game is completed when an integral of some numeral function describing the game course turns into a unit. An approach to solution of this game is based on the method of Resolving Functions. The research technique is based on using Minkovskii inverse functional of multi-valued maps, closely related to the given conflict-controlled process, and on constructing the resolving functions. At the heart of the method's scheme is the L.S. Pontryagin condition making it possible to choose a pursuers' control in the form of Borel measurable selections of special multi-valued map. We show that for objects with different inertial Pontryagin's Condition fails on some time interval. Moreover, the modification of the Pontryagin's Condition is proposed. Specifically, we solve the similar problem to the one in "The Boy and the Crocodile" with lag. [ABSTRACT FROM AUTHOR]