scholarly journals Stability analysis for $ (\omega, c) $-periodic non-instantaneous impulsive differential equations

2022 ◽  
Vol 7 (2) ◽  
pp. 1758-1774
Author(s):  
Kui Liu ◽  
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Exponential Stability ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Cauchy Matrix ◽  
Impulsive Problems ◽  
The Stability ◽  
Homogeneous Linear

<abstract><p>In this paper, the stability of $ (\omega, c) $-periodic solutions of non-instantaneous impulses differential equations is studied. The exponential stability of homogeneous linear non-instantaneous impulsive problems is studied by using Cauchy matrix, and some sufficient conditions for exponential stability are obtained. Further, by using Gronwall inequality, sufficient conditions for exponential stability of $ (\omega, c) $-periodic solutions of nonlinear noninstantaneous impulsive problems are established. Finally, some examples are given to illustrate the correctness of the conclusion.</p></abstract>

scholarly journals ALMOST PERIODIC SOLUTIONS OF IMPULSIVE INTEGRO‐DIFFERENTIAL NEURAL NETWORKS

2010 ◽  
Vol 15 (4) ◽  
pp. 505-516 ◽  
Author(s):  
Gani Tr. Stamov ◽  
Jehad O. Alzabut
Keyword(s):  
Neural Networks ◽  
Periodic Solutions ◽  
Contraction Mapping ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Contraction Mapping Principle ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Cauchy Matrix ◽  
Bellman’S Inequality

In this paper, sufficient conditions are established for the existence of almost periodic solutions for system of impulsive integro‐differential neural networks. Our approach is based on the estimation of the Cauchy matrix of linear impulsive differential equations. We shall employ the contraction mapping principle as well as Gronwall‐Bellman's inequality to prove our main result. The research of Gani Tr. Stamov is partially supported by the Grand 100ni087–16 from Technical University–Sofia

scholarly journals Variational Approach to Impulsive Differential Equations with Singular Nonlinearities

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Naima Daoudi-Merzagui ◽  
Abdelkader Boucherif
Keyword(s):  
Differential Equations ◽  
Variational Method ◽  
Periodic Solutions ◽  
Variational Approach ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Singular Nonlinearities ◽  
Existence Of Periodic Solutions

We discuss the existence of periodic solutions for nonautonomous second order differential equations with singular nonlinearities. Simple sufficient conditions that enable us to obtain many distinct periodic solutions are provided. Our approach is based on a variational method.

scholarly journals Positivity of Fundamental Matrix and Exponential Stability of Delay Differential System

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Alexander Domoshnitsky ◽  
Roman Shklyar ◽  
Mikhail Gitman ◽  
Valery Stolbov
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Delay System ◽  
Necessary Condition ◽  
Cauchy Matrix ◽  
Linear Ordinary Differential Equations ◽  
Delay Differential System ◽  
Delay Differential ◽  
Necessary And Sufficient

The classical Wazewski theorem established that nonpositivity of all nondiagonal elementspij  (i≠j,  i,j=1,…,n)is necessary and sufficient for nonnegativity of the fundamental (Cauchy) matrix and consequently for applicability of the Chaplygin approach of approximate integration for system of linear ordinary differential equationsxi′t+∑j=1n‍pijtxjt=fit,   i=1,…,n.Results on nonnegativity of the Cauchy matrix for system of delay differential equationsxi′t+∑j=1n‍pijtxjhijt=fit,   i=1,…,n,which were based on nonpositivity of all diagonal elements, were presented in the previous works. Then examples, which demonstrated that nonpositivity of nondiagonal coefficientspijis not necessary for systems of delay equations, were found. In this paper first sufficient results about nonnegativity of the Cauchy matrix of the delay system without this assumption are proven. A necessary condition of nonnegativity of the Cauchy matrix is proposed. On the basis of these results on nonnegativity of the Cauchy matrix, necessary and sufficient conditions of the exponential stability of the delay system are obtained.

scholarly journals Variational Approach to Impulsive Problems: A Survey of Recent Results

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Fang-fang Liao ◽  
Juntao Sun
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Periodic Solutions ◽  
Variational Approach ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Homoclinic Solutions ◽  
Nontrivial Solutions ◽  
Impulsive Problems

We present a survey on the existence of nontrivial solutions to impulsive differential equations by using variational methods, including solutions to boundary value problems, periodic solutions, and homoclinic solutions.

scholarly journals Poincaré Map and Periodic Solutions of First-Order Impulsive Differential Equations on Moebius Stripe

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yefeng He ◽  
Yepeng Xing
Keyword(s):  
Periodic Orbit ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Poincare Map ◽  
First Order ◽  
Existence And Stability ◽  
Stability And Bifurcations

This paper is mainly concerned with the existence, stability, and bifurcations of periodic solutions of a certain scalar impulsive differential equations on Moebius stripe. Some sufficient conditions are obtained to ensure the existence and stability of one-side periodic orbit and two-side periodic orbit of impulsive differential equations on Moebius stripe by employing displacement functions. Furthermore, double-periodic bifurcation is also studied by using Poincaré map.

scholarly journals A Note on the Periodic Solutions for a Class of Third Order Differential Equations

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 31
Author(s):  
Zouhair Diab ◽  
Juan L. G. Guirao ◽  
Juan A. Vera
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Monodromy Matrix ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Averaging Theory ◽  
Third Order ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Necessary And Sufficient

The aim of the present work is to study the necessary and sufficient conditions for the existence of periodic solutions for a class of third order differential equations by using the averaging theory. Moreover, we use the symmetry of the Monodromy matrix to study the stability of these solutions.

scholarly journals Stability and Stabilization of Impulsive Stochastic Delay Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Kaining Wu ◽  
Xiaohua Ding
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Almost Sure Exponential Stability ◽  
Stability And Stabilization ◽  
The Stability ◽  
Delay Differential

We consider the stability and stabilization of impulsive stochastic delay differential equations (ISDDEs). Using the Lyapunov-Razumikhin method, we obtain the sufficient conditions to guarantee thepth moment exponential stability of ISDDEs. Then the almost sure exponential stability is considered and the sufficient conditions of the almost sure exponential stability are obtained. Moreover, the stabilization problem of ISDDEs is studied and the criterion on impulsive stabilization of ISDDEs is established. At last, examples are presented to illustrate the correctness of our results.

scholarly journals Practical Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
S. G. Hristova ◽  
A. Georgieva
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Practical Stability ◽  
Time Interval ◽  
Comparison Results ◽  
Two Measures ◽  
The Stability ◽  
The Given ◽  
Stability Concepts

The object of investigations is a system of impulsive differential equations with “supremum.” These equations are not widely studied yet, and at the same time they are adequate mathematical model of many real world processes in which the present state depends significantly on its maximal value on a past time interval. Practical stability for a nonlinear system of impulsive differential equations with “supremum” is defined and studied. It is applied Razumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied to both the given system and the comparison scalar equation. An example illustrates the usefulness of the obtained sufficient conditions.

On the stability of a system of two linear hybrid functional differential systems with aftereffect

2020 ◽  
pp. 299-306
Author(s):  
Pyotr M. Simonov
Keyword(s):  
Differential Equations ◽  
Matrix Function ◽  
Sufficient Conditions ◽  
Hybrid Functional ◽  
Cauchy Matrix ◽  
Linear Ordinary Differential Equations ◽  
Functional Differential Systems ◽  
Functional Differential ◽  
Linear Differential ◽  
The Stability

We consider a system of two hybrid vector equations containing linear difference (defined on a discrete set) and functional differential (defined on a half-axis) parts. To study it, a model system of two vector equations is chosen, one of which is linear difference with aftereffect (LDEA), and the other is a linear functional differential with aftereffect (LFDEA). Two equivalent representations of this system are shown: the first representation in the form of LFDEA, the second — in the form of LDEA. This allows us to study the stability issues of the system under consideration using the well-known results on the stability of LFDEA and LDEA. Using the results of the article [Gusarenko S. A. On the stability of a system of two linear differential equations with delayed argument // Boundary value problems. Interuniversity collection of scientific papers. Perm: PPI, 1989. P. 3–9], two examples are shown when a joint system of four equations will be stable with respect to the right side. In the first example, we use the LFDEA for which sufficient conditions for the sign-definiteness of the elements of the 2 Ч 2 Cauchy matrix function are known (in terms of the LFDEA coefficients). In the second example, LFDEA is given such that LFDEA is a system of linear ordinary differential equations (LODE) of the second order. In both cases, estimates of the components of the Cauchy matrix function are known. An exponential estimate with a negative exponent is given for the components of the Cauchy matrix function of LDEA.

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