It is proved that if a linear inhomogeneous periodic system $ x(n+1)=P(n)x(n) +f(n),$ in which the matrices $P(n)$ and $f(n)$ are $\omega$-periodic $(\omega\neq 1)$ and $\det P(n)\neq 0$, has an $\Omega$-periodic solution $(\Omega\neq1)$ whose period is coprime with the period of the system, then there is $n_{0}\in \mathbb Z_{+}$ such that $ \det(P(n)-P(n_{0}))=0 $ for all $n\in\mathbb Z_{+}$. An example of a linear inhomogeneous second-order system is given in which the matrices $P(n)$ and $f(n)$ have a common period 3, and the system has a 2-periodic solution. In the two-dimensional and three-dimensional cases, we construct nonlinear $\omega$-periodic ($\omega\neq 1$) systems of difference equations that have a solution whose period $\Omega\neq 1$ is coprime with $\omega$, but the system does not have $\omega$-periodic solutions. It is proved that the discrete nonautonomous logistic equation $$ x_{n+1}=x_{n}\exp\Big(r_{n}\Big(1-\frac{x_{n}}{K_{n}}\Big)\Big), $$ where $\{r_{n}\}$ and $\{K_{n}\}$ are positive periodic sequences with a common period $\omega$ ($\omega\neq 1$), cannot have an $\Omega$-periodic solution ($\Omega\neq 1$) whose period is coprime with $\omega$.
Keywords: periodic solutions of a periodic system of difference equations, coprimality of periods
Received October 8, 2024
Revised November 16, 2024
Accepted November 25, 2024
Alexander Vasilievich Lasunsky, Dr. Phys.-Math. Sci., Yaroslav the Wise Novgorod State University, Veliky Novgorod, 173003 Russia, e-mail: Alexandr.Lasunsky@novsu.ru
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Cite this article as: A.V. Lasunsky. On periodic solutions of a system of difference equations whose period is coprime with the period of the system. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 110–118.
