Periodic solutions of the differential delay equation\dot x(t) = - f(x(t - 1))

For an odd function 1/2(x) defined only on a finite interval, this paper deals with the existence of periodic solutions and the number of simple periodic solutions of the differential delay equation (DDE) . By use of the method of qualitative analysis combined with the constructing of special solutions a series of interesting results are obtained on these problems.Supported by the National Natural Science Foundation of China.