Uniqueness of the periodic solution for differential delay equations (Q1916005)

The author considers the differential delay equation (1) \(\dot x(t) = - \mu x(t) - f(x(t - \alpha))\) where \(\mu \geq 0\), \(\alpha> 0\) are constants, \(f:\mathbb{R}\to\mathbb{R}\) is a continuous function satisfying \(f(0) = 0\). A periodic solution \(x\) of (1) with period \(q\) is called a slowly oscillating periodic solution (SOP) if there exists \(\rho > \alpha\), such that \(q - p > \alpha\) and \(x(t) > 0\), \(t \in (0,P)\), \(x(t) < 0\), \(t \in (p,q)\). A result on the uniqueness of the slowly oscillating periodic solution of (1) (without assuming \(\mu = 0\) and \(f\) being odd) is presented.