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.


Introduction
The stability and attractivity of impulsive differential equations have been deeply investigated in the monographs of Baȋnov and Simeonov [1], Lakshmikantham et al. [6], Samoȋlenko and Perestyuk [16], Borisenko et al. [3].The recent work has provided a full discussion of this subject for impulsive delay differential equations (see, e.g., Yan and Shen [19], Liu and Ballinger [10], Liu et al. [12], Yu [20], Zhang and Sun [21], etc.).Most of these results on asymptotic behavior are valid locally in the neighborhood of the equilibrium state, but do not estimate the range of the stable region and domain of attraction (referring to the definition given by Lakshmikantham and Leela [7], Šiljak [17], Kolmanovskii and Nosov [5]).That is, we do not know how far initial conditions can be allowed to vary without disrupting the asymptotic properties of the equilibrium state.Furthermore, it may be difficult to know whether the equilibrium state exists in nonlinear impulsive delay systems.In this case, it should be important and interesting to estimate the region attracting solutions of the impulsive systems.Therefore, a general problem of the attractivity is to discuss the attracting set and attracting basin for the impulsive systems.Some significant progress has been made in the techniques and methods of determining the attracting set and attracting basin (domain of attraction) for the continuous systems described by ordinary differential equations [7,17] and functional differential equations [5,9,13,15,18].However, so far the corresponding problems for impulsive delay differential equations have not been considered.
In this paper, by developing an impulsive delay differential inequality and introducing the concept of nonlinear measure, we study the attractivity for a class of nonlinear impulsive delay differential equations.The criteria present a feasible and effective approach to estimate the attracting set, attracting basin, and asymptotically stable region of the impulsive systems by solving an algebraic equation.Examples and their simulations are given to demonstrate the effectiveness of our results.

Preliminaries
Let N be the set of all positive integers, let R n be the real n-dimensional vector space with a norm • , and let R m×n be the set of m × n real matrices.R + = [0,+∞) and I denotes the n × n unit matrix.
Let τ > 0 be the upper bound of time delays and let be the fixed points with lim k→∞ t k = ∞ (called impulsive moments).

C[X,Y
] denotes the space of continuous mappings from the topological space X to the topological space Y .Especially, let for all but at most a finite number of points t ∈ (−τ,0]}.PC is a space of piecewise right-hand continuous functions with the norm φ τ = sup −τ≤s≤0 { φ(s) }, for φ ∈ PC.
In this paper, we will consider the following nonlinear impulsive delay differential equations: where as t ≥ t 0 , and satisfies (2.2) with the initial condition Z. Yang and D. Xu 3 Throughout the paper, we always assume that for any φ ∈ PC, system (2.2) has at least one solution through (t 0 ,φ), denoted by x(t,t 0 ,φ) or x t (t 0 ,φ), where x t (t 0 ,φ)(s) = x(t + s,t 0 ,φ), s ∈ [−τ,0].Clearly, x t (t 0 ,φ) ∈ PC for t ≥ t 0 .For more details on the existence of solutions of impulsive delay differential equations, one refers to Liu and Ballinger [11], Baȋnov and Stamova [2].Definition 2.2.A set S ⊂ PC is called an attracting set of (2.2) and D ⊂ PC is called an attracting basin of S if for any initial value φ ∈ D, the solution x t (t 0 ,φ) converges to S as t → +∞.That is, dist x t t 0 ,φ ,S −→ 0, as t −→ +∞, (2.4) where dist(ϕ,S) Especially, the set S is called a global attracting set of (2.2) if D = PC.The set D is called a domain of attraction if x = 0 is a solution of (2.2) and the zero solution (i.e., S = {0}) attracts solutions x(t,t 0 ,φ) for all φ ∈ D.Moreover, if the zero solution is stable, we call D an asymptotically stable region of (2.2).
In order to introduce the concept of the nonlinear measure, we recall the matrix norms A and the matrix measure μ(A) introduced by the vector norm • as follows: Now, based on (2.5), we define the nonlinear measure as follows.
Definition 2.3.For a function F : R n → R n , call the nonlinear measure of F.
, where μ(A) is the matrix measure.Therefore, the concept of the nonlinear measure actually is an extension of the matrix measure (see also, Kolmanovskii and Myshkis [4], Qiao et al. [14]).According to the definition, we easily verify the following.
The following result on the impulsive delay differential inequality is an extension of the continuous case of Lakshmikantham and Leela [8, Theorem 6.9.1], and will play an important role in the qualitative analysis of impulsive delay differential equations in Section 3. (2.7) Proof.We will first prove that (2.8) In view of (2.9) and the monotonic character of F, we have (2.11) This contradicts the inequality (2.10), and so (2.8) holds.Suppose that for k = 1,2,...,m Employing the similar process of the proof of (2.8), we have u(t) ≤ v(t), for t ∈ [t m ,t m+1 ).By the induction, the conclusion holds and the proof is complete.
Z. Yang and D. Xu 5
Remark 3.2.The above conclusion remains valid even when the inequality (3.1) holds for z ∈ (z 1 ,z 2 ).In fact, for an enough small > 0, the inequality (3.1) holds when )} is an attracting basin of S 1 .Letting → 0 + , we can obtain the conclusion.
According to Theorem 3.1 and Remark 3.2, we have the following corollaries.
Proof.It is obvious that According to the above results, S 1 = {φ ∈ PC | φ τ ≤ z 1 } is a globally attracting set of (2.2).
Proof.According to Theorem 3.1 and Remark 3.2, the zero solution attracts solutions x(t,t 0 ,φ) for all φ ∈ D 1 and D 1 is a domain of attraction.Furthermore, for any given z ∈ (0,z 2 ] and φ ∈ D = {φ ∈ PC | φ τ < αz }, we can refer to the proof of (3.13) and obtain that x t,t 0 ,φ < z , t ≥ t 0 . (3.25) This implies that the zero solution is stable.Thus, the zero solution is asymptotically stable and D 1 = {φ ∈ PC | φ τ < αz 2 } is an asymptotically stable region of (2.2).The proof is complete.
Z. Yang and D. Xu 9 Similarly, we can obtain the following results for the case with α ≥ 1.
The rest of the proof is similar to one of the proof in Theorem 3.1 and we omit it here.
According to Theorem 3.5, we have the following.

Illustrative examples
Example 4.1.Consider a scalar nonlinear impulsive delay system: where We discuss the attractiveness of (4.1) for the following cases.

Figure 4 .
2 shows the attractivity of the impulsive delay system (4.4) under the different initial conditions.