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 Lp-equivalence of a linear and a nonlinear impulsive differential equation in a Banach space

1993 ◽  
Vol 36 (1) ◽  
pp. 17-33 ◽  
Author(s):  
D. D. Bainov ◽  
S. I. Kostadinov ◽  
P. P. Zabreiko
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Impulsive Differential Equation ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations

In the present paper by means of the Schauder-Tychonoff principle sufficient conditions are obtained for Lp-equivalence of a linear and a nonlinear impulsive differential equations.

scholarly journals Sufficient conditions for oscillations of all solutions of a class of impulsive differential equations with deviating argument

1996 ◽  
Vol 9 (1) ◽  
pp. 33-42 ◽  
Author(s):  
D. D. Bainov ◽  
M. B. Dimitrova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Impulsive Differential Equation ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Deviating Argument ◽  
All Solutions

Sufficient conditions are found for oscillation of all solutions of impulsive differential equation with deviating argument.

scholarly journals Existence and Uniqueness of Mild Solutions for Nonlinear Stochastic Impulsive Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
L. J. Shen ◽  
J. T. Sun
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Stochastic Analysis ◽  
Impulsive Differential Equation ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Analysis Technique

This paper investigates the existence and uniqueness of mild solutions to the general nonlinear stochastic impulsive differential equations. By using Schaefer's fixed theorem and stochastic analysis technique, we propose sufficient conditions on existence and uniqueness of solution for stochastic differential equations with impulses. An example is also discussed to illustrate the effectiveness of the obtained results.

Oscillatory Properties of Solutions of Impulsive Differential Equations with Several Retarded Arguments

1998 ◽  
Vol 5 (3) ◽  
pp. 201-212
Author(s):  
D. D. Bainov ◽  
M. B. Dimitrova ◽  
V. A. Petrov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Impulsive Differential Equation ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
All Solutions ◽  
Oscillatory Properties

Abstract The impulsive differential equation with several retarded arguments is considered, where pi (t) ≥ 0, 1 + bk > 0 for i = 1, . . . , m, t ≥ 0, k ∈ ℕ. Sufficient conditions for the oscillation of all solutions of this equation are found.

scholarly journals Nonlinear Impulsive Differential Equations with Weighted Exponential or Ordinary Dichotomous Linear Part in a Banach Space

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Hristo Kiskinov ◽  
Andrey Zahariev
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Differential Equations ◽  
Linear Part ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Bounded Solutions ◽  
Fixed Point Principle

We consider nonlinear impulsive differential equations withψ-exponential andψ-ordinary dichotomous linear part in a Banach space. By the help of Banach’s fixed-point principle sufficient conditions are found for the existence ofψ-bounded solutions of these equations onRandR+.

CONTINUOUS DEPENDENCE ON A PARAMETER OF THE SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE

1993 ◽  
Vol 03 (04) ◽  
pp. 477-483
Author(s):  
D.D. BAINOV ◽  
S.I. KOSTADINOV ◽  
NGUYEN VAN MINH ◽  
P.P. ZABREIKO
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Small Parameter ◽  
Continuous Dependence ◽  
Impulsive Differential Equation ◽  
Impulsive Differential Equations ◽  
Right Hand ◽  
Dependence On A Parameter ◽  
The Right

Continuous dependence of the solutions of an impulsive differential equation on a small parameter is proved under the assumption that the right-hand side of the equation and the impulse operators satisfy conditions of Lipschitz type.

scholarly journals On Nonlinear Forced Impulsive Differential Equations under Canonical and Non-Canonical Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2066
Author(s):  
Shyam Sundar Santra ◽  
Hammad Alotaibi ◽  
Samad Noeiaghdam ◽  
Denis Sidorov
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Impulsive System ◽  
Bounded Solutions ◽  
Oscillatory Behaviour ◽  
Impulsive Systems ◽  
Forcing Term

This study is connected with the nonoscillatory and oscillatory behaviour to the solutions of nonlinear neutral impulsive systems with forcing term which is studied for various ranges of of the neutral coefficient. Furthermore, sufficient conditions are obtained for the existence of positive bounded solutions of the impulsive system. The mentioned example shows the feasibility and efficiency of the main results.

scholarly journals Positive Solutions for Impulsive Differential Equations with Mixed Monotonicity and Optimal Control

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Lingling Zhang ◽  
Noriaki Yamazaki ◽  
Rui Guo
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Control Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Impulsive Differential Equation ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Initial Value ◽  
Mixed Monotone Operator

We consider positive solutions and optimal control problem for a second order impulsive differential equation with mixed monotone terms. Firstly, by using a fixed point theorem of mixed monotone operator, we study positive solutions of the boundary value problem for impulsive differential equations with mixed monotone terms, and sufficient conditions for existence and uniqueness of positive solutions will be established. Also, we study positive solutions of the initial value problem for our system. Moreover, we investigate the control problem of positive solutions to our equations, and then, we prove the existence of an optimal control and its stability. In addition, related examples will be given for illustrations.

scholarly journals Variational analysis to fourth-order impulsive differential equations

2018 ◽  
Vol 38 (1) ◽  
pp. 151-163
Author(s):  
Saeid Shokooh
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Critical Point ◽  
Fourth Order ◽  
Variational Analysis ◽  
Impulsive Differential Equation ◽  
Impulsive Differential Equations ◽  
Three Solutions ◽  
One Dimensional ◽  
Critical Point Theorems

Applying two critical point theorems, we prove the existence of atleast three solutions for a one-dimensional fourth-order impulsive differential equation with two real parameters.

Bounded oscillations under the effect of retardations for differential equations of arbitrary order

1977 ◽  
Vol 77 (1-2) ◽  
pp. 129-136 ◽  
Author(s):  
V. A. Staikos ◽  
I. P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Arbitrary Order ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Bounded Solutions ◽  
Image Position

SynopsisThis paper is concerned with the effect of the delays on the bounded solutions of the nth order n ≧ 1) differential equationSufficient conditions involving the retardations τJ(j = 1,2, …, m) which insure that every bounded solution of the considered equation is oscillatory are given. The results obtained generalise recent ones in [3 and 4].

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