Stability of a periodic solution to a nonlinear delay differential equation (Q5951321)

The scalar differential equation with constant delay \[ \frac{dx(t)}{dt}=-\alpha f(x(t-1))\tag{1} \] is considered, where \(\alpha\) is a positive parameter and \(f\) is a continuously differentiable function on \((-\infty,+\infty)\) such that \(f\) is odd, i.e. \(f(-x)=-f(x)\), and satisfies the condition \(df/dx>0, ~ x\in(-\infty,+\infty)\). An odd periodic solution to (1) satisfying the condition \(x(t+2)=-x(t)\), \(t\in (-\infty,+\infty)\), is given. The stability of this solution is investigated.