Positive Periodic Solutions for a First Order Singular Ordinary Differential Equation Generated by Impulses

2017 ◽  
Vol 17 (3) ◽  
pp. 637-650 ◽  
Author(s):  
Juan J. Nieto ◽  
José M. Uzal
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Positive Periodic Solutions ◽  
First Order ◽  
Singular Ordinary Differential Equation

scholarly journals Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Saima Akram ◽  
Allah Nawaz ◽  
Humaira Kalsoom ◽  
Muhammad Idrees ◽  
Yu-Ming Chu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
First Order ◽  
Multiple Periodic Solutions ◽  
Fine Focus ◽  
Nonautonomous Equation ◽  
Maximum Multiplicity

In this article, approaches to estimate the number of periodic solutions of ordinary differential equation are considered. Conditions that allow determination of periodic solutions are discussed. We investigated focal values for first-order differential nonautonomous equation by using the method of bifurcation analysis of periodic solutions from a fine focus Z=0. Keeping in focus the second part of Hilbert’s sixteenth problem particularly, we are interested in detecting the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For some classes like C7,7,C8,5,C8,6,C8,7, eight periodic multiplicities have been observed. The new formulas ξ10 and ϰ10 are constructed. We used our new formulas to find the maximum multiplicity for class C9,2. We have succeeded to determine the maximum multiplicity ten for class C9,2 which is the highest known multiplicity among the available literature to date. Another challenge is to check the applicability of the methods discussed which is achieved by presenting some examples. Overall, the results discussed are new, authentic, and novel in its domain of research.

scholarly journals STRONGLY IRREGULAR PERIODIC SOLUTIONS OF THE FIRST-ORDER LINEAR HOMOGENEOUS DISCRETE EQUATION

2018 ◽  
Vol 62 (3) ◽  
pp. 263-267 ◽  
Author(s):  
A. K. Demenchuk
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Prime Numbers ◽  
Discrete Equation ◽  
Linear Difference Equations ◽  
Order Linear ◽  
First Order ◽  
Discrete Equations

 In 1950 J. Massera proved that a fi rst-order scalar periodic ordinary differential equation has no strongly ira proved that a first-order scalar periodic ordinary differential equation has no strongly irregular periodic solutions, that is, such solutions whose period of solution is incommensurable with the period of equation. For difference equations with discrete time, strong irregularity means that the period of the equation and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the above result of J. Massera has no complete analog.The purpose of this article is to investigate the possibility to realize Massera’s theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a first-order linear homogeneous non-stationary periodic discrete equation has no strongly irregular non-stationary periodic solutions.

scholarly journals On strongly irregular periodic solutions of the linear nonhomogeneous discrete equation of the first order

2020 ◽  
Vol 56 (1) ◽  
pp. 30-35
Author(s):  
A. K. Demenchuk
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Discrete Time ◽  
Difference Equations ◽  
Prime Numbers ◽  
Discrete Equation ◽  
Linear Difference Equations ◽  
First Order ◽  
Discrete Equations

As is proved earlier (the Massera theorem), the first-order scalar periodic ordinary differential equation does not have strongly irregular periodic solutions (solutions with a period incommensurable with the period of the equation). For difference equations with discrete time, strong irregularity means that the equation period and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the mentioned result has no complete analog.The purpose of this paper is to investigate the possibility of realizing an analog of the Massera theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a linear nonhomogeneous non-stationary periodic discrete equation of the first order does not have strongly irregular non-stationary periodic solutions.

scholarly journals Existence and Lyapunov Stability of Positive Periodic Solutions for a Third-Order Neutral Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Jingli Ren ◽  
Zhibo Cheng ◽  
Yueli Chen
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Lyapunov Stability ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Neutral Differential Equation ◽  
Positive Periodic Solutions ◽  
Third Order

By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.

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