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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.

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