Nonlinear Impulsive Delay Differential Equations Research Articles

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.

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.

Stability criteria of nonlinear impulsive differential equations with infinite delays

In this paper, we consider the uniform stability and uniformly asymptotical stability of nonlinear impulsive infinite delay differential equations. Instead of putting all components of the state variable x in one Lyapunov function, several Lyapunov-Razumikhin functions of partial components of the state variable x are used so that the conditions ensuring that stability are simpler and less restrictive; moreover, examples are discussed to illustrate the advantage of the results obtained.

Interval oscillation criteria for nonlinear impulsive delay differential equations with damping term

In this paper, a class of second order nonlinear impulsive delay differential equations with damping term is studied. Three factors-impulse, delay and damping that affect the interval qualitative properties of solutions of equations are taken into account together. Several interval oscillation criteria which generalize some known ones, such as, in Huang and Feng (2010), Xing and Zheng (2008) and Liu and Xu (2007), are established. Moreover, examples are also given to illustrate the effectiveness and non-emptiness of our results.

Asymptotic behaviors of nonlinear neutral impulsive delay differential equations with forced term

In this paper, we study the asymptotic behavior of solutions of a class of nonlinear neutral impulsive delay differential equations with forced term of the form $$\left\{\begin{array}{lll}[x(t)+c(t)x(t-\tau)]' +p(t)f(x(t-\delta))=q(t),& t\geq t_0, t\neq t_k,\\ x(t_k)=b_kx(t_k^-)+(1-b_k)\int_{t_k-\delta}^{t_k}p(s+\delta)f(x(s))ds\\ \qquad +(b_k-1)\int_{t_k}^\infty q(s)ds,& k\in{\mathbf{Z_+}}\end{array}\right.$$ Sufficient conditions are obtained for every solution of the equations that tends to a constant as t → ∞.

A survey on oscillation of impulsive delay differential equations

This paper presents a systematic development of the oscillatory results for the linear and nonlinear impulsive delay differential equations. We also discuss linearized oscillation theory, and give some applications to mathematical biology.

Positive solutions of periodic boundary value problems for nonlinear first-order impulsive differential equations

Motivated by Liu [Y. Liu, Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations, J. Math. Anal. Appl. 327 (2007) 435–452], Li et al. [X. Li, X. Lin, D. Jiang, X. Zhang, Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects, Nonlinear Anal. 62 (2005) 683–701] and Liu and Ge [Y. Liu, W. Ge, Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients, Nonlinear Anal. 57 (2004) 363–399], this paper is concerned with the periodic boundary value problem (PBVP for short) for the nonlinear impulsive differential equation { x ′ ( t ) + a ( t ) x ( t ) = f ( t , x ( t ) ) , a.e. t ∈ [ 0 , T ] ∖ { t 1 , … , t p } , Δ x ( t k ) = x ( t k + ) − x ( t k ) = I k ( x ( t k ) ) , k = 1 , … , p , x ( T ) = x ( 0 ) . We obtain new sufficient conditions for the existence of at least one, two or three positive solutions of the above PBVP, respectively. Examples are presented to illustrate the main results.

Existence of Periodic Solutions of a Type of Nonlinear Impulsive Delay Differential Equations with a Small Parameter1

The Banach fixed point theorem is used to prove the existence of a unique ω periodic solution of a new type of nonlinear impulsive delay differential equation with a small parameter.

Attractivity of nonlinear impulsive delay differential equations

The attractivity of nonlinear differential equations with time delays and impulsive effects is discussed. We obtain some criteria to determine the attracting set and attracting basin of the impulsive delay system by developing an impulsive delay differential inequality and introducing the concept of nonlinear measure. Examples and their simulations illustrate the effectiveness of the results and different asymptotical behaviors between the impulsive system and the corresponding continuous system.

Oscillatory and Asymptotic Behavior of Solutions for Nonlinear Impulsive Delay Differential Equations

The oscillatory and asymptotic behavior of the solutions for third order nonlinear impulsive delay differential equations are investigated. Some novel criteria for all solutions to be oscillatory or be asymptotic are established. Three illustrative examples are proposed to demonstrate the effectiveness of the conditions.

Asymptotic behavior of solutions of nonlinear impulsive delay differential equations with positive and negative coefficients

This paper is concerned with the nonlinear impulsive delay differential equations with positive and negative coefficients (*) { x ′ ( t ) + p ( t ) f ( x ( t − τ ) ) − q ( t ) f ( x ( t − σ ) ) = 0 , t ≥ t 0 , t ≠ t k , x ( t k ) = b k x ( t k − ) + ( 1 − b k ) ( ∫ t k − τ t k p ( s + τ ) f ( x ( s ) ) d s − ∫ t k − σ t k q ( s + σ ) f ( x ( s ) ) d s ) , k = 1 , 2 , 3 , … . Sufficient conditions are obtained for every solution of equation (*) tending to a constant as t → ∞ .

Oscillation properties of nonlinear impulsive delay differential equations and applications to population models

Comparison theorem and explicit sufficient conditions are obtained for oscillation and nonoscillation of solutions of nonlinear impulsive delay differential equations which can be utilized to population dynamic models. Our results in this paper generalize and improve several known results.

Existence and exponential stability of periodic solution for impulsive delay differential equations and applications

In this paper, we consider a class of nonlinear impulsive delay differential equations. By establishing an exponential estimate for delay differential inequality with impulsive initial condition and employing Banach fixed point theorem, we obtain several sufficient conditions ensuring the existence, uniqueness and global exponential stability of a periodic solution for nonlinear impulsive delay differential equations. Furthermore, the criteria are applied to analyze dynamical behavior of impulsive delay Hopfield neural networks and the results show different behavior of impulsive system originating from one continuous system.

Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients

We consider the following nonlinear impulsive delay differential equation with variable coefficients and forcing term: (∗) x′(t)+a(t)x(t)+F(t,x(·))=e(t), t⩾0, t≠t k, x(t k +)−x(t k)=I k(x(t k)), k∈N. Using new methods different from those in Tang et al. (Appl. Math. Comput. 131 (2002) 373), Nieto (J. Comput. Appl. Math. Lett. 15 (2002) 489), Franco and Nieto (J. Comput. Appl. Math. 88 (1998) 144), Nieto (J. Math. Anal. Appl. 205 (1997) 423), He and Yu (J. Math. Anal. Appl. 272 (2002) 67), we establish stability theorems for (∗) when e( t)≡0 under the assumption that there is { b k } such that b k x 2⩽ x( x+ I k ( x))⩽ x 2 for k∈ N and x∈ R (Theorems 2.1–2.4), the existence results for three positive periodic solutions and nonexistence results for periodic solution of (∗) when e( t)≡0 under the assumption exp ∫ 0 T a(u) du ≠∏ 0⩽t k<T (1+b k) and b k >−1 for all k (Theorems 3.1–3.2) and the existence results for periodic solution of (∗) at resonance, which is caused by impulses i.e. exp ( ∫ 0 T a(u) du)=∏ 0⩽t k<T (1+b k) , under the assumption b k >−1 for all k (Theorems 4.2–4.3). Some results obtained improve and generalize the known theorems and some other results are new. Examples are presented to illustrate the main results.

Oscillation and stability of nonlinear neutral impulsive delay differential equations

In this paper, oscillation and stability of nonlinear neutral impulsive delay differential equation are studied. The main result of this paper is that oscillation and stability of nonlinear impulsive neutral delay differential equations are equivalent to oscillation and stability of corresponding nonimpulsive neutral delay differential equations. At last, two examples are given to illustrate the importance of this study.

Linearized oscillation of nonlinear impulsive delay differential equations

The main result of this paper is to show oscillations of nonlinear impulsive delay differential equations are equivalent to those of corresponding linear impulsive delay differential equations. The results of this paper generalize some well-known results in the literature.

Monotone iterative technique for impulsive delay differential equations

In this paper, by proving a new comparison result, we present a result on the existence of extremal solutions for nonlinear impulsive delay differential equations.

Oscillation for Nonlinear Delay Differential Equations with Impulses

Conditions for the oscillation of the solutions of nonlinear impulsive delay differential equations are found. The results given by D. D. Bainov et al. (1998, Rocky Mountain J. Math.28, 25–40) are improved.