scholarly journals EVENTUAL STABILITY AND EVENTUAL BOUNDEDNESS FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH “SUPREMUM”

2011 ◽  
Vol 16 (1) ◽  
pp. 304-314 ◽  
Author(s):  
Ivanka Stamova
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lyapunov Functions ◽  
Impulsive Differential Equations ◽  
Comparison Method ◽  
Applied Method ◽  
Piecewise Continuous ◽  
Eventual Stability

Eventual stability and eventual boundedness for nonlinear impulsive differential equations with supremums are studied. The impulses take place at fixed moments of time. Piecewise continuous Lyapunov functions have been applied. Method of Razumikhin as well as comparison method for scalar impulsive ordinary differential equations have been employed.

scholarly journals Integralφ0-Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Peiguang Wang ◽  
Xiaojing Liu ◽  
Qing Xu
Keyword(s):  
Differential Equations ◽  
Lyapunov Functions ◽  
Comparative Method ◽  
Impulsive Differential Equations ◽  
Two Measures ◽  
Piecewise Continuous

This paper establishes a criterion on integralφ0-stability in terms of two measures for impulsive differential equations with “supremum” by using the cone-valued piecewise continuous Lyapunov functions, Razumikhin method, and comparative method. Meantime, an example is given to illustrate our result.

Nonuniqueness Theorem for a Singular Cauchy Problem

2000 ◽  
Vol 7 (2) ◽  
pp. 317-327 ◽  
Author(s):  
Josef Kalas
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lyapunov Functions ◽  
Singular Cauchy Problem ◽  
Vector Lyapunov Functions

Abstract A general nonuniqueness theorem is given for ordinary differential equations with singularities. The criterion uses vector Lyapunov functions and extends the previously known criteria ones. The applicability is illustrated by several examples.

scholarly journals "On Lyapunov- Schmidt Reduction to Reduce Differential Algebraic Equation "

2017 ◽  
Vol 4 (2) ◽  
pp. 20-22
Author(s):  
Eman A. Mansour
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Algebraic Equation ◽  
Lyapunov Functions ◽  
Differential Algebraic Equation

"In this paper , we transformation singularity differential – Algebraic equation to an ordinary differential equations by use Lyapunov– Schmidt reduction and constructing Lyapunov functions depending on Reiss and Geiss method."

De la Vallée Poussin inequality for impulsive differential equations

2021 ◽  
Vol 71 (4) ◽  
pp. 881-888
Author(s):  
Sibel Doğru Akgöl ◽  
Abdullah Özbekler
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Impulsive Differential Equations ◽  
Open Problems ◽  
Early Date ◽  
Impulse Effect ◽  
Appl Math ◽  
De La Vallée Poussin

Abstract The de la Vallée Poussin inequality is a handy tool for the investigation of disconjugacy, and hence, for the oscillation/nonoscillation of differential equations. The results in this paper are extensions of former those of Hartman and Wintner [Quart. Appl. Math. 13 (1955), 330–332] to the impulsive differential equations. Although the inequality first appeared in such an early date for ordinary differential equations, its improved version for differential equations under impulse effect never has been occurred in the literature. In the present study, first, we state and prove a de la Vallée Poussin inequality for impulsive differential equations, then we give some corollaries on disconjugacy. We also mention some open problems and finally, present some examples that support our findings.

scholarly journals On the Practical Stability of Impulsive Differential Equations with Infinite Delay in Terms of Two Measures

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Bo Wu ◽  
Jing Han ◽  
Xiushan Cai
Keyword(s):  
Differential Equations ◽  
Lyapunov Functions ◽  
Infinite Delay ◽  
Impulsive Differential Equations ◽  
Practical Stability ◽  
Stability Criteria ◽  
Razumikhin Technique ◽  
Two Measures

We consider the practical stability of impulsive differential equations with infinite delay in terms of two measures. New stability criteria are established by employing Lyapunov functions and Razumikhin technique. Moreover, an example is given to illustrate the advantage of the obtained result.

scholarly journals Vector Lyapunov functions and conditional stability for systems of impulsive differential-difference equations

2001 ◽  
Vol 42 (3) ◽  
pp. 341-353 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Difference Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Comparison Method ◽  
Conditional Stability ◽  
Continuous Vector ◽  
Vector Lyapunov Functions ◽  
Piecewise Continuous

AbstractBy means of piecewise continuous vector functions, which are analogues of the classical Lyapunov functions and via the comparison method, sufficient conditions are found for conditional, stability of the zero solution of a system of impulsive differential-difference equations.

scholarly journals Stability of the solutions of impulsive functional-differential equations by Lyapunov's direct method

2001 ◽  
Vol 43 (2) ◽  
pp. 269-278 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Differential Equations ◽  
Lyapunov Functions ◽  
Direct Method ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Continuous Functions ◽  
Piecewise Continuous ◽  
Functional Differential ◽  
The Stability

AbstractWe consider the stability of the zero solution of a system of impulsive functional-differential equations. By means of piecewise continuous functions, which are generalizations of classical Lyapunov functions, and using a technique due to Razumikhin, sufficient conditions are found for stability, uniform stability and asymptotical stability of the zero solution of these equations. Applications to impulsive population dynamics are also discussed.

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