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Positive periodic solutions of functional discrete systems and population models. (English) Zbl 1111.39008
The authors study the \(\omega\)-periodic functional difference equation \[ x(n+1)=A(n)x(n)+f(n,x_n).\tag{\({*}\)} \] Here, \(A(n)\) is a real invertible \(k\times k\)-matrix and \(f\colon{\mathbb Z}\times X\to{\mathbb R}^k\), where \(X\) denotes the space of \(\omega\)-periodic sequences in \({\mathbb R}^k\) and \(x_n(\theta)\colon x(n+\theta)\) for \(\theta \in \mathbb Z\). The authors considers the existence of positive periodic solutions to \((*)\) under stability assumptions on the linear part and continuity of the function \(f\). The main methods are a reformulation of \((*)\) via the variation-of-constants formula and a cone-theoretical fixed point theorem due to Krasnosel’skii. Finally, the authors apply their results to two dynamic population models described by Volterra difference equations.

39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
92D25 Population dynamics (general)
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