Sufficient conditions for the stability of linear periodic impulsive differential equations

2019 ◽  
Vol 210 (11) ◽  
pp. 1511-1530 ◽  
Author(s):  
V. O. Bivziuk ◽  
V. I. Slyn’ko
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
The Stability

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 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.

scholarly journals Solution of Some Impulsive Differential Equations via Coupled Fixed Point

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 501
Author(s):  
Ahmed Boudaoui ◽  
Khadidja Mebarki ◽  
Wasfi Shatanawi ◽  
Kamaleldin Abodayeh
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Coupled Fixed Point ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
System Of Differential Equations ◽  
Coupled Fixed Points

In this article, we employ the notion of coupled fixed points on a complete b-metric space endowed with a graph to give sufficient conditions to guarantee a solution of system of differential equations with impulse effects. We derive recisely some new coupled fixed point theorems under some conditions and then apply our results to achieve our goal.

scholarly journals Second-order impulsive differential systems with mixed and several delays

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shyam Sundar Santra ◽  
Apurba Ghosh ◽  
Omar Bazighifan ◽  
Khaled Mohamed Khedher ◽  
Taher A. Nofal
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillation Theory ◽  
Second Order ◽  
Neutral Delay ◽  
Differential Systems ◽  
Impulsive Differential Systems ◽  
Several Delays ◽  
Necessary And Sufficient

AbstractIn this work, we present new necessary and sufficient conditions for the oscillation of a class of second-order neutral delay impulsive differential equations. Our oscillation results complement, simplify and improve recent results on oscillation theory of this type of nonlinear neutral impulsive differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

scholarly journals New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 934
Author(s):  
Shyam Sundar Santra ◽  
Khaled Mohamed Khedher ◽  
Kamsing Nonlaopon ◽  
Hijaz Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Discrete Equation ◽  
Qualitative Behavior ◽  
Differential Systems ◽  
Impulsive Differential Systems

The oscillation of impulsive differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of impulsive differential equations. In this work, several sufficient conditions are established for oscillatory or asymptotic behavior of second-order neutral impulsive differential systems for various ranges of the bounded neutral coefficient under the canonical and non-canonical conditions. Here, one can see that if the differential equations is oscillatory (or converges to zero asymptotically), then the discrete equation of similar type do not disturb the oscillatory or asymptotic behavior of the impulsive system, when impulse satisfies the discrete equation. Further, some illustrative examples showing applicability of the new results are included.

scholarly journals Bounded Oscillation Theorem for Unstable-type Neutral Impulsive Differential Equations of the Second Order

2019 ◽  
pp. 1-9
Author(s):  
U. A. Abasiekwere ◽  
E. Eteng ◽  
I. O. Isaac ◽  
Z. Lipcsey
Keyword(s):  
Differential Equations ◽  
Transmission Lines ◽  
High Speed ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Central Place ◽  
Oscillatory Solution ◽  
Second Order ◽  
Oscillation Theorem ◽  
Neutral Differential Equations

The oscillations theory of neutral impulsive differential equations is gradually occupying a central place among the theories of oscillations of impulsive differential equations. This could be due to the fact that neutral impulsive differential equations plays fundamental and significant roles in the present drive to further develop information technology. Indeed, neutral differential equations appear in networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits).   In this paper, we study the behaviour of solutions of a certain class of second-order linear neutral differential equations with impulsive constant jumps.  This type of equation in practice is always known to have an unbounded non-oscillatory solution.  We, therefore, seek sufficient conditions for which all bounded solutions are oscillatory and provide an example to demonstrate the applicability of the abstract result.

Approximate controllability of non-instantaneous impulsive semilinear measure driven control system with infinite delay via fundamental solution

2020 ◽  
Author(s):  
Surendra Kumar ◽  
Syed Mohammad Abdal
Keyword(s):  
Fundamental Solution ◽  
Differential Equations ◽  
Control Systems ◽  
Approximate Controllability ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Existence Of A Solution ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
New Class

Abstract This article investigates a new class of non-instantaneous impulsive measure driven control systems with infinite delay. The considered system covers a large class of the hybrid system without any restriction on their Zeno behavior. The concept of measure differential equations is more general as compared to the ordinary impulsive differential equations; consequently, the discussed results are more general than the existing ones. In particular, using the fundamental solution, Krasnoselskii’s fixed-point theorem and the theory of Lebesgue–Stieltjes integral, a new set of sufficient conditions is constructed that ensures the existence of a solution and the approximate controllability of the considered system. Lastly, an example is constructed to demonstrate the effectiveness of obtained results.

scholarly journals Bifurcation of Bounded Solutions of Impulsive Differential Equations

2016 ◽  
Vol 26 (14) ◽  
pp. 1650242 ◽  
Author(s):  
Kevin E. M. Church ◽  
Xinzhi Liu
Keyword(s):  
Differential Equation ◽  
Integral Operator ◽  
Differential Equations ◽  
Impulsive Differential Equation ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Necessary Condition ◽  
Bounded Solutions ◽  
Pitchfork Bifurcations ◽  
Nonlinear Integral Operator

In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov–Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This leads to sufficient conditions for the presence of fold, transcritical and pitchfork bifurcations. Additionally, we provide a computable necessary condition for bifurcation in nonlinear scalar impulsive differential equations. Several examples are provided illustrating the results.

scholarly journals Interval Oscillation Criteria of Second Order Mixed Nonlinear Impulsive Differential Equations with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Zhonghai Guo ◽  
Xiaoliang Zhou ◽  
Wu-Sheng Wang
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Second Order ◽  
Oscillation Criteria ◽  
Nonnegative Constant ◽  
Interval Oscillation ◽  
Equations With Delay

We study the following second order mixed nonlinear impulsive differential equations with delay(r(t)Φα(x′(t)))′+p0(t)Φα(x(t))+∑i=1npi(t)Φβi(x(t-σ))=e(t),t≥t0,t≠τk,x(τk+)=akx(τk),x'(τk+)=bkx'(τk),k=1,2,…, whereΦ*(u)=|u|*-1u,σis a nonnegative constant,{τk}denotes the impulsive moments sequence, andτk+1-τk>σ. Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.

Sign in / Sign up

Export Citation Format

Share Document