Nonlinear Impulsive Differential Equations Research Articles

Boundary value problems on half‐line for second‐order nonlinear impulsive differential equations

We obtain sufficient conditions for existence and uniqueness of solutions of boundary value problems on half‐line for a class of second‐order nonlinear impulsive differential equations. Our technique is different than the traditional ones, as it is based on asymptotic integration method involving principal and nonprincipal solutions. Examples are provided to illustrate the relevance of the results.

On the oscillation of impulsive vector partial differential equations with distributed deviating arguments

AbstractIn this paper, we consider a class of nonlinear impulsive neutral partial functional differential equations with continuous distributed deviating arguments. For this class, we establish sufficient conditions for theH-oscillation of the solutions, using impulsive differential inequalities and an averaging technique with two different boundary conditions. We provide an example to illustrate the main result.

Asymptotic representation of solutions for second-order impulsive differential equations

We obtain sufficient conditions which guarantee the existence of a solution of a class of second order nonlinear impulsive differential equations with fixed moments of impulses possessing a prescribed asymptotic behavior at infinity in terms of principal and nonprincipal solutions. An example is given to illustrate the relevance of the results.

Oscillation of a nonlinear impulsive differential equation system with piecewise constant argument

We deal with a nonlinear impulsive differential equation system with piecewise constant argument. We prove the existence and uniqueness of a solution. Moreover, we obtain sufficient conditions for the oscillation of the solution.

Existence and stability of periodic solutions of an impulsive differential equation and application to CD8 T-cell differentiation.

Unequal partitioning of the molecular content at cell division has been shown to be a source of heterogeneity in a cell population. We propose to model this phenomenon with the help of a scalar, nonlinear impulsive differential equation (IDE). To study the effect of molecular partitioning at cell division on the effector/memory cell-fate decision in a CD8 T-cell lineage, we study an IDE describing the concentration of the protein Tbet in a CD8 T-cell, where impulses are associated to cell division. We discuss how the degree of asymmetry of molecular partitioning can affect the process of cell differentiation and the phenotypical heterogeneity of a cell population. We show that a moderate degree of asymmetry is necessary and sufficient to observe irreversible differentiation. We consider, in a second part, a general autonomous IDE with fixed times of impulse and a specific form of impulse function. We establish properties of the solutions of that equation, most of them obtained under the hypothesis that impulses occur periodically. In particular, we show how to investigate the existence of periodic solutions and their stability by studying the flow of an autonomous differential equation. Then we apply those properties to prove the results presented in the first part.

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

Abstract This 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.

BOUNDEDNESS OF SOLUTIONS FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH INTEGRAL JUMP CONDITIONS

The boundedness of solutions for certain nonlinear impulsive differential equations are obtained, the jumping conditions at discontinuous points are related to the integral of the past states, rather than a left hand limit at the discontinuous points. These results are obtained by new built impulsive integral inequalities with integral jumping conditions using the method of successive iteration.

A class of nonlinear non-instantaneous impulsive differential equations involving parameters and fractional order

In this article, we study asymptotic and smooth properties of solutions to nonlinear non-instantaneous impulsive differential equations involving parameters of integer order and fractional order. We introduce the concept of continuous dependence and differentiability of solutions and establish sufficient conditions to guarantee the solution depends continuously and is differentiable on the initial condition, impulsive parameters and junction parameters. Finally, two models of non-instantaneous impulsive logistic equations are given to illustrate our results.

Application of Morse Theory for Some Damped Dirichlet Nonlinear Impulsive Differential Equations

In this paper, we consider the multiplicity of solutions for some damped nonlinear impulsive differential equations by using Morse theory in combination with the minimax arguments. Under some assumptions, we get some new results on the existence of multiple nontrivial solutions for the problems. Thus, we improve and extend recent results.

Sturm‐Picone comparison theorems for nonlinear impulsive differential equations with discontinuous solutions

The nonlinear versions of Sturm‐Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.

Asymptotic properties of the solutions of nonlinear non-instantaneous impulsive differential equations

In this article, we investigate asymptotic properties of solutions, continuous dependence and stability, of integer order and fractional order nonlinear non-instantaneous impulsive differential equations (IDEs). We introduce the concept of continuous dependence and stability of solutions to integer order and fractional order non-instantaneous impulsive Cauchy problems (ICPs) and establish sufficient conditions to guarantee that the solutions of both the original and the perturbed non-instantaneous ICPs are close to each other in a certain sense. Finally, examples are given to illustrate our results.

Asymptotic integration of second-order impulsive differential equations

We initiate a study of the asymptotic integration problem for second-order nonlinear impulsive differential equations. It is shown that there exist solutions asymptotic to solutions of an associated linear homogeneous impulsive differential equation as in the case for equations without impulse effects. We introduce a new constructive method that can easily be applied to similar problems. An illustrative example is also given.

Interval oscillation criteria for forced mixed nonlinear impulsive differential equations with variable delay

In this paper, the interval qualitative properties of a class of mixed nonlinear impulsive differential equations are studied. Under the hypothesis of delay σ(t) being variable, the ratio of functions x(t−σ(t)) and x(t) on each considered interval is estimated, then Riccati transformation and \omega functions are applied to obtain interval oscillation criteria. The results gained by Özbekler and Zafer (2011) for delay σ(t)=0 and by Guo et al. (2012) for delay σ(t) being nonnegative constant are developed.

Boundary value problems of the nonlinear multiple base points impulsive fractional differential equations with constant coefficients

In this paper, the nonlinear multiple base points boundary value problems of the impulsive fractional differential equations are studied. By using the fixed point theorem and the Mittag-Leffler functions, the sufficient conditions for the existence of the solutions to the given problems are formulated. An example is presented to illustrate the result.

Oscillation of Even Order Impulsive Neutral Partial Differential Equations with Distributed Deviating Arguments

In this paper, we will consider a class of boundary value problems associated with even order nonlinear impulsive neutral partial functional differential equations with continuous distributed deviating arguments. Sufficient conditions are obtained for the oscillation of solutions using impulsive differential inequalities and integral averaging scheme with pair of boundary conditions. Examples are specified to point up our important results.

High order Runge–Kutta methods for impulsive delay differential equations

The purpose of this paper is to obtain high order Runge–Kutta methods for nonlinear impulsive delay differential equations (IDDEs). In order to achieve the purpose, continuity and smoothness of the exact solutions of IDDEs are analyzed. And then the discontinuous points and non-smooth points are chosen as a part of the nodes of the Runge–Kutta methods. Furthermore, it is proved that the pth order Runge–Kutta method for IDDEs is also convergent of order p. And some simple numerical examples are given to demonstrate the theoretical results.

Impulsive stabilization of chaos in fractional-order systems

This paper considers a class of nonlinear impulsive Caputo differential equations of fractional order, which models chaotic systems. Computer-assisted proof of chaos suppression by stabilizing the unstable system equilibria is provided. A nonexistence result of periodic solutions is presented, and the commensurate fractional-order Lorenz system is simulated for illustration.

Existence of solutions for the nonlinear multiple base points impulsive fractional differential equations with the three-point boundary conditions

In this paper, we study a class of nonlinear multiple base points impulsive fractional differential equations involving the three-point boundary conditions. A new result on the existence of a solution is established by using fixed point theorems. An example is presented to illustrate the result.

Stability Analysis of Analytical and Numerical Solutions to Nonlinear Delay Differential Equations with Variable Impulses

A stability theory of nonlinear impulsive delay differential equations (IDDEs) is established. Existing algorithm may not converge when the impulses are variable. A convergent numerical scheme is established for nonlinear delay differential equations with variable impulses. Some stability conditions of analytical and numerical solutions to IDDEs are given by the properties of delay differential equations without impulsive perturbations.

EXTREMAL SOLUTIONS FOR A NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH MULTI-ORDERS FRACTIONAL DERIVATIVES

In this paper, by employing the lower and upper solutions method, we give an existence theorem for the extremal solutions for a nonlinear impulsive differential equations with multi-orders fractional derivatives and integral boundary conditions. A new comparison result is also established.