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.

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.

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 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 Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1613
Author(s):  
Mun-Jin Bae ◽  
Chan-Ho Park ◽  
Young-Ho Kim
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Exponential Estimates ◽  
Analysis Theory ◽  
Linear Growth Condition ◽  
Uniqueness Of The Solution ◽  
The Uniqueness Theorem

The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufficient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose a weakened Ho¨lder condition and a weakened linear growth condition. Stochastic results are obtained for the theory of the existence and uniqueness of the solution. We first show that the conditions guarantee the existence and uniqueness; then, we show some exponential estimates for the solutions.

scholarly journals Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
T. E. Govindan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Local Lipschitz Condition ◽  
Functional Differential ◽  
Special Case ◽  
Partial Functional Differential Equation

This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.

scholarly journals Some Existence Results for Systems of Impulsive Stochastic Differential Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sliman Mekki ◽  
Tayeb Blouhi ◽  
Juan J. Nieto ◽  
Abdelghani Ouahab
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Impulsive Systems ◽  
Uniqueness Of Solutions

Abstract In this paper we study a class of impulsive systems of stochastic differential equations with infinite Brownian motions. Sufficient conditions for the existence and uniqueness of solutions are established by mean of some fixed point theorems in vector Banach spaces. An example is provided to illustrate the theory.

scholarly journals Existence and uniqueness of solutions for nonlinear impulsive differential equations with three-point boundary conditions

2018 ◽  
Vol 1 (1) ◽  
pp. 21-36 ◽  
Author(s):  
Mısır J. Mardanov ◽  
Yagub A. Sharifov ◽  
Kamala E. Ismayilova
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Point Boundary ◽  
Uniqueness Of Solutions

AbstractThis paper is devoted to a system of nonlinear impulsive differential equations with three-point boundary conditions. The Green function is constructed and considered original problem is reduced to the equivalent impulsive integral equations. Sufficient conditions are found for the existence and uniqueness of solutions for the boundary value problems for the first order nonlinear system of the impulsive ordinary differential equations with three-point boundary conditions. The Banach fixed point theorem is used to prove the existence and uniqueness of a solution of the problem and Schaefer’s fixed point theorem is used to prove the existence of a solution of the problem under consideration. We illustrate the application of the main results by two examples.

scholarly journals Periodic Solutions for Stochastic Differential Equations Driven by General Counting Processes: Application to Malaria

2021 ◽  
Vol 13 (4) ◽  
pp. 1
Author(s):  
KOUAME Yao Simplice ◽  
NZI Modeste
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Markov Processes ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Uniqueness Result ◽  
Poisson Processes ◽  
Counting Processes ◽  
Periodic Environment

In this paper, a class of periodic stochastic differential equations driven by general counting processes (SDEsGp) is studied. First, an existence-uniqueness result for the solution of general SDEsGp based on Poisson processes with т-periodic stochastic intensity of time t has been given, for some  т> 0. Then, using the properties of periodic Markov processes, sufficient conditions for the existence and uniqueness of a periodic solution of the considered equations are obtained. We will then apply the obtained results to the propagation of malaria in a periodic environment.

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