Almost Periodic Solutions of First-Order Ordinary Differential Equations

Seifedine Kadry,Artem Sharlay,Gennady Alferov,Gennady Ivanov

doi:10.3390/math6090171

Seifedine Kadry, Artem Sharlay + Show 2 more

Open Access

https://doi.org/10.3390/math6090171

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Journal: Mathematics	Publication Date: Sep 17, 2018
Citations: 9	License type: CC BY 4.0

Affiliation: Beirut Arab University, St Petersburg University

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

Introduction

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

Upper Bound for Number of Almost Periodic Solutions

A Lower Bound for the Number of Almost Periodic Solutions

Stability of Almost Periodic Solutions

Conclusions