The existence of periodic solutions of the equation \(x'(t)=-f(x(t),x(t- {\tau{}}))\) (Q1191851)

The existence of periodic solutions to the equation (1) \(x'(t)=-f(x(t),\;x(t-\tau))\) is considered, where \(f(x,y)\) is continuous on \(R^ 2\) and \(\tau>0\). Sufficient conditions under which the equation (1) has at least \(k\) non-constant oscillatory periodic solutions with period \(4/(1+4n)\), \(n=0,1,\dots,k-1\), are established.