Through researching the operating condition of data transmission in a BitTorrent-like P2P file sharing system, we obtain two mathematical models for the BitTorrent-like P2P file sharing system, one (call it (a)) relates to the system's data transmission mechanism and the other (call it (b)) involves the system's optimal management. Then we study the solutions of (I), the general form of (a). Our study results show that the solutions of (a) and (b) depend on all minimal solutions of (I). Finally, by simplifying (I), we present an algorithm to search for minimal solutions of (I).