Abstract
As is proved earlier (the Massera theorem), the first-order scalar periodic ordinary differential equation does not have strongly irregular periodic solutions (solutions with a period incommensurable with the period of the equation). For difference equations with discrete time, strong irregularity means that the equation period and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the mentioned result has no complete analog.The purpose of this paper is to investigate the possibility of realizing an analog of the Massera theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a linear nonhomogeneous non-stationary periodic discrete equation of the first order does not have strongly irregular non-stationary periodic solutions.
Highlights
The purpose of this paper is to investigate the possibility of realizing an analog of the Massera theorem for certain classes of difference equations
We consider the class of linear difference equations
It is proved that a linear nonhomogeneous non-stationary periodic discrete equation of the first order does not have strongly irregular non-stationary periodic solutions
Summary
The purpose of this paper is to investigate the possibility of realizing an analog of the Massera theorem for certain classes of difference equations. Функция y ∈ S 1 называется периодической с периодом ω∈ (ω-периодической), если для любого n ∈ выполняется равенство yn+ω= yn. Проблеме существования и построения периодических решений дискретных уравнений посвящено достаточно большое число работ [1,2,3,4,5] и др., где в основном изучались решения, период которых совпадает с периодом самого уравнения. Как известно [6,7,8,9,10,11], системы обыкновенных периодических дифференциальных уравнений могут допускать периодические решения, период которых несоизмерим по отношению к периоду системы.
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