Abstract
In 1950 J. Massera proved that a fi rst-order scalar periodic ordinary differential equation has no strongly ira proved that a first-order scalar periodic ordinary differential equation has no strongly irregular periodic solutions, that is, such solutions whose period of solution is incommensurable with the period of equation. For difference equations with discrete time, strong irregularity means that the period of the equation and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the above result of J. Massera has no complete analog.The purpose of this article is to investigate the possibility to realize Massera’s theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a first-order linear homogeneous non-stationary periodic discrete equation has no strongly irregular non-stationary periodic solutions.
Highlights
Massera p r o v e d t h a t a fi r s t - o r d e r s c a l a r p e r i o d i c o r d i n a ry d i ff e r e n ti a l e q u a ti o n h a s n o s t r o n g l y i r regular periodic solutions, that is, such solutions whose period of solution is incommensurable with the period of equation
For difference equations with discrete time, strong irregularity means that the period of the equation and the period of its solution are relatively prime numbers
We consider the class of linear difference equations
Summary
We consider the class of linear difference equations. Последовательность y ∈ S 1 называется периодической с периодом ω∈ ( ω -периодической), если для любого n ∈ выполняется равенство yn+ω= yn. Периодические последовательности при определенных условиях могут быть решениями дискретных (разностных) уравнений. Как известно [5; 6], нелинейное скалярное периодическое обыкновенное дифференциальное уравнение не имеет отличных от постоянных периодических решений таких, что периоды решения и уравнения несоизмеримы.
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