About one strengthening of the Massera’s existence theorem of periodic solutions of linear differential periodic systems
https://doi.org/10.29235/1561-8323-2024-68-3-188-195
Abstract
According to Massera’s theorem, an ordinary differential linear nonhomogeneous periodic system has a periodic solution with a period coinciding with that of the system if and only if this system has a bounded solution. We introduce the class L of vector functions called growing slower than a linear function. This class contains the class B of bounded vector functions in as its own subclass. It has been proved that Massera’s above-mentioned theorem will remain true if in its formulation a bounded solution is replaced by a slower growing solution than a linear function. It is shown that the set B in the metric space (L, distc ), where distc is the uniform convergence metric vector functions on intervals, has Baer’s first category, i. e. almost everything in the sense of the category of space vector functions (L, distc ) are not bounded. This fact shows the significance of the obtained strengthening of Massera’s theorem.
About the Authors
A. K. Demenchuk
Institute of Mathematics of the National Academy of Science of Belarus
Belarus
Belarus
Demenchuk Aleksandr K. – D. Sc. (Physics and Mathematics), Professor, Chief Researcher
11, Surganov Str., 220072, Minsk
A. V. Konuh
Belarusian State Economic University
Belarus
Belarus
Konuh Aleksandr V. – Ph. D. (Physics and Mathematics), Associate Professor
26, Partizanski Ave., 220070, Minsk
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