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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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