Existence of positive periodic solutions for scalar delay differential equations with and without impulses (Q2318456)

The periodic impulsive system of the form \[ \begin{aligned} y^\prime(t)+a(t) y(t)&=g(t,y_t),\quad 0\le t\ne t_k,\\ y(t_k^+)-y(t_k) &= b_k y(t_k),\quad k\in\mathbb N, \end{aligned} \] is considered under the following assumptions: \begin{itemize} \item[\((A_1)\)] The functions \(a(t), g(t,\varphi)\) are continuous, non-negative, and \(\omega\)-periodic in \(t\), \(g\) is bounded on bounded sets of \(\mathbb R\times PC([-\tau,0], \mathbb R)\); \item[\((A_2)\)] For the impulsive sequence \(\{t_k\}\) there is a positive integer \(p\) such that \(0<t_1<t_2<\ldots< t_p\le\omega\) and \(t_{k+p}=t_k+\omega, b_{k+p}=b_k,\; k\in\mathbb N;\) \item[\((A_3)\)] The constants \(b_1,\ldots,b_p\) satisfy \(b_k>-1\); \item[\((A_4)\)] \(\prod_{k=1}^{p} (1+b_k)<e^{\int_0^{\omega}a(t)\,dt}.\) \end{itemize} A number of sufficient conditions for the existence of positive \(\omega\)-periodic solutions is established. The principal tools of the proof are the Schauder and Krasnoselskii fixed point theorems. The derived results generalize and improve some of the prior known results including those for systems without impulses (\(b_k=0\;\forall k\)). They are applied to several mathematical models describing biological processes, such as Mackey-Glass type equations, generalized Nicholson's blowflies equations, and others.