On positive periodic solutions for nonlinear delayed differential equations (Q305812)

Positive \(\omega-\)periodic solutions of the equation \[ \dot x(t) = a(t) g(x(t))x(t) - \lambda b(t) f[x(t-\tau(t))] \] are obtained under the assumption that \(a, b\) and \(\tau\) are nonnegative and \(\omega\)-periodic, and under less restrictive assumptions on the nonlinear functions \(f,g\) than in previous papers. Important methods are differential inequalities, integration over one period, and a fixed point theorem in cones. Writing `\(u\)' instead of `\(x\)' in the equation, where \(x\) appears nonlinearly, one obtains a linear equation in \(x\) which can be uniquely solved by an \(\omega\)-periodic function. This defines an operator \( u \mapsto x = T(u)\); fixed points of \(T\) are solutions of the original equation. The arguments are detailed and can be well followed.