Existence and uniqueness of positive periodic solutions for a class of differential delay equations (Q928483)

Consider the delay differential equation \[ x'(t)=-a(t)x(t)+b(t)f(x(t-c(t))), \tag{\(*\)} \] where \(a,b\) and \(c\) are continuous \(\omega\)-periodic real-valued functions with \[ \int^\omega_0a(t)dt>0, \int^\omega_0b(t)dt>0\text{ and }b(t)\geq 0\text{ for }t\in \mathbb{R}. \] The authors observe that solving \((*)\) is equivalent to finding a continuous \(\omega\)-periodic function \(x(t)\) such that \[ x(t)=\int^{t+\omega}_tG(t,s)b(s)f(x(s-c(s)))ds, \tag{\(**\)} \] where \[ G(t,s)=\frac{e^{\int^s_ta(u)du}}{e^{\int^\omega_0a(u)du}-1}, \quad 0\leq t\leq \omega,\quad t\leq s\leq t+\omega. \] Equation \((**)\) is a fixed-point equation of the map \[ Ax(t)=\int^{t+\omega}_tG(t,s)b(s)f(x(s-c(s)))ds, \] The idea is to introduce Hilbert's projective metric on an appropriate cone in which the contraction mapping principle can be applied.