Almost Periodic Solutions of First-Order Ordinary Differential Equations

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 the works of Lebedeva [1], regarding the number of periodic solutions of equations first order, they required a high degree of smoothness. Franco et al. required the smoothness of the second derivative of the Schwartz equation [2]. We have all of these restrictions lifted. Our new form presented also emphasizes this novelty.

Design of PID Controller for Magnetic Levitation System using Harris Hawks Optimization

“…Most of the real systems will exhibit the non-linear and unstable behavior and based on the operating region, it can work as complete or semi-stable system. When the operating region is varied, its stability also will be affected and hence, it is necessary to implement an optimally designed PID to provide essential stability to the system even though the conditions of the system are changing due to uncertainties [16][17][18].…”

Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

“…Neto in [2] states that for equation (1), we are unable to have upper bound for number of periodic solutions until some coefficients are restricted. Kadry in [3] gives conditions that allow determination for both upper and lower bounds. Our basic focus is to acquire highest periodic solutions of any class of the type (1); our main concern in this paper is this nearby question of bifurcation.…”

A New Method to Study the Periodic Solutions of the Ordinary Differential Equations Using Functional Analysis

“…Periodic solutions take an important part in the qualitative theory of differential equations and in applied problems [3]. 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]. However, there is no general approach to study periodic solutions of differential equations.…”

“…A more general result in this direction was obtained by V.M. Lebedeva, having shown that an equation in which the right-hand side is a 4th order polynomial, can have any preassigned number of periodic solutions [5,33]. Our idea in this paper is to reject the requirement of polynomiality of the right-hand side of the equation under study and proposed a different approach for obtaining the upper and lower estimates for the number of periodic solutions.…”