Abstract
Approaches to estimate the number of almost periodic solutions of ordinary differential equations are considered. Conditions that allow determination for both upper and lower bounds for these solutions are found. The existence and stability of almost periodic problems are studied. The novelty of this paper lies in the fact that the use of apparatus derivatives allows for the reduction of restrictions on the degree of smoothness of the right parts. In our work, regarding the number of periodic solutions of equations first order, we don’t require a high degree of smoothness and no restriction on the smoothness of the second derivative of the Schwartz equation. We have all of these restrictions lifted. Our new form presented also emphasizes this novelty.
Highlights
In the works of Lebedeva [1], regarding the number of periodic solutions of equations first order, they required a high degree of smoothness
Interest has arisen in the study of almost periodic solutions of differential equations and differential equations with almost periodic coefficients [6,7,8,9,10,11,12,13,14]
Equation (11) has a bounded solution, which implies that Equation (11) has at least one almost periodic solution taking into account Theorem 12
Summary
In the works of Lebedeva [1], regarding the number of periodic solutions of equations first order, they required a high degree of smoothness. The process, which consists of the sum of two periodic oscillations with incommensurate frequencies, is an almost periodic oscillation. The theory of almost periodic oscillations began to develop in the works of the Latvian mathematician P.G. Bol, the Danish mathematician H.A. Bohr, and others. Harald Bohr’s scientific papers relate mainly to functions theory. He made a great contribution to development of the almost periodic functions theory [4]. In this connection, interest has arisen in the study of almost periodic solutions of differential equations and differential equations with almost periodic coefficients [6,7,8,9,10,11,12,13,14]. The question of studying almost periodic functions in robotics [15,16,17,18,19,20,21], dynamic systems [22,23,24,25,26], stability theory [27,28,29,30,31], control systems for space objects [32,33,34], and economy problems [35,36,37,38] arose significantly
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