Uniqueness and Stability of Slowly Oscillating Periodic Solutions of Delay Equations with Unbounded Nonlinearity

We study uniqueness and stability problems of slowly oscillating periodic solutions of delay equations with small parameters. If the nonlinearity decays to a negative number at − ∞ and blows up at + ∞ or vice versa, we show that, for sufficiently small parameters, the slowly oscillating periodic solutions are unique and asymptotically stable provided that the decay rate can dominate the growth rate in an appropriate sense. This result particularly implies that Wright′s equation has a unique and asymptotically stable slowly oscillating periodic solution for large parameter α.