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Positive periodic solutions of infinite delay functional differential equations. (English) Zbl 1080.34558

The paper deals with the functional-differential equation \[ \dot x(t)+A(t) x(t)=f(t,x_t),\quad t\in \mathbb{R}, \] where \(A(t)=\text{ diag\,}[a_1(t),a_2(t),\dots,a_n(t)]\), \(a_j\in C(\mathbb{R}, \mathbb{R})\) is \(\omega\)-periodic, \(j=1,2,\dots,n\), and \(\omega>0\) is a constant, \(f(t,x_t)\) is a function defined on \(\mathbb{R}\times BC\) (\(BC\) denotes the Banach space of bounded continuous functions \(\phi=(\phi_1,\dots,\phi^n)^T: \mathbb{R}\to \mathbb{R}^n\) being \(\omega\)-periodic whenever \(x\) is \(\omega\)-periodic, and \(x_t(\theta)=x(t+\theta)\), \(\theta\in R\). The main objective of the paper is to establish the existence of positive periodic solutions of the considered equation.

MSC:

34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
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