Uniqueness and stability of slowly oscillating periodic solutions of delay equations with unbounded nonlinearity (Q685819)

The paper deals with uniqueness and stability problems of slowly oscillating periodic solutions of the differential delay equation (*) \(- \varepsilon \dot x(t)=\sigma x(t)+f(x(t-1))\), where \(\varepsilon>0\) and \(\sigma\geq 0\) are parameters. The main condition on \(f\) is that it decays to a negative real number at \(-\infty\) and tends to \(+\infty\) at \(+\infty\). Under some additional (rather technical) assumptions on \(f\) it is shown that if the decay rate can dominate the growth rate in a certain sense, then a unique slowly oscillating periodic solution of (*) exists. Also, by estimating the Floquet multipliers it is proved that such a periodic solution is asymptotically stable for all small values of the parameters involved.