The periodic solutions of nonlinear differential difference equations (Q1332217)

The existence of slowly oscillating periodic solutions and rapidly oscillating solutions to a class of differential difference equations is considered. Sufficient conditions under which the equations have at least \(N\) or an infinite number of nonconstant periodic solutions are established. For example, it is shown that if \(ah > {\pi \over 2}\), then the equation \(x'(t) = - x(t - h) (a - bx^ 2(t) - cx^ 2 (t - h))\), \(h>0\), \(a>0\), \(b \geq 0\), \(c \geq 0\), \(b^ 2 + c^ 2 \neq 0\) has a nonconstant periodic solution with period of \(4h/1 + 4n\), \(n = 0,1, \dots, N - 1\), where \(N\) is the greatest integer which is less than \(ah - {3 \over 2} \pi/2 \pi\).