Abstract
In this paper, a new theorems of the derived numbers method to estimate the number of periodic solutions of first-order ordinary differential equations are formulated and proved. Approaches to estimate the number of periodic solutions of ordinary differential equations are considered. Conditions that allow us to determine both upper and lower bounds for these solutions are found. The existence and stability of periodic problems are considered.
Highlights
In this paper a method to study periodic solutions of first-order ordinary differential equations is developed based on the ideas of functional analysis
Theorems of the derived numbers method to estimate the number of periodic solutions of first-order ordinary differential equations are formulated and proved
The analysis problems for periodic solutions of differential equations arise in classical mechanics, celestial mechanics [4,5,6,7,8,9,10,11,12,13,14,15], space robotics [16,17,18,19,20,21,22,23,24], in modeling of economic processes [25,26,27,28,29]
Summary
In this paper a method to study periodic solutions of first-order ordinary differential equations is developed based on the ideas of functional analysis. The basic method to prove the existence of periodic solutions of differential equations is the Poincare-Andronov point mapping method, the method of directing functions, variational methods, the topological method, the Krylov-Bogolyubov averaging and so on. Note that these methods are difficult enough to be applied. The problem of estimating the number of periodic solutions of first-order differential equations is solved on the basis of the results of studies [30,31] and with the use of derived numbers theory [32,33]
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