With the help of the NEYMAN-PEARSON lemma there are given general assertions about the form of the solutions of the minimax problem (3), (4), (5) with the restrictions (A), (B1), (B2) and the additional assumptions (C), (D). If the additional assumption (E) is fulfilled, too, the solutions can be calculated by means of the VOLTERA integral equations (34), (35). In part II of the paper a class of optimal control problems will be solved by this method.