Abstract: In this paper, we consider the nonlinear delay dynamic system
xΔ(t) = p(t)f1(y(t)), yΔ(t) = −q(t)f2(x(t − τ )).
We obtain some necessary and sufficient conditions for the existence of nonoscillatory solutions with special asymptotic properties of the system. We generalize the known results in the literature. One example is given to illustrate the results.