# The existence of Berge equilibrium in differential game with «separated» dynamics

The notion of «Berge equilibrium» (BE) appeared in Russia in 1994-1995 in the thesis written by Vaisman (post-graduate student of Zhukovskiy V.I. at that time; he died at the

age of 36 in 1998). Then the notion BE was exported by Moussa Larbani and Radjef Muchamed Said (Algerian post-graduate students of Zhukovskiy V.I. at that time) beyond the bounds of Russia and became widespread in the whole world. BE publications member over 100 titles. The survey showed that in mathematical theory of BE the period of accumulation of facts had been completed and the stage of evolutionary development is being arised.

The suggested article corresponds to the second stage and it is pioneer in construction of a new mathematical direction: the investigation of questions of Berge equilibrium existence in N-persons positional differential games. In will be established the existence of situation BE in one non-cooperative positional differential game of N-persons with «separated» dynamics. First the proof is based on the application of Germeier convolution of payoff functions, second, on the theory of antagonistic positional differential games, namely, the application of mathematical formalization of quasi-motions and finally, third, on the method of control with the help of a quide offered.

Key words: positional differential games, Germeier convolution, saddle point, strategies, quasi-motion, Nash and Berge equilibrium.