Electronic Journal of Qualitative Theory of Differential Equations

This paper is concerned with a non-autonomous impulsive neutral integro-differential equation with time-varying delays. We establish a novel singular delay integro-differential inequality, which enables us to derive several sufficient criteria on the positive invariant set, global attracting set and stability. An example is given to demonstrate the efficiency of proposed r


Introduction
Due to the plentiful dynamical behaviors, integro-differential equations with delays have many applications in a variety of fields such as control theory, biology, ecology, medicine, etc [1,2].Especially, the effects of delays on the stability of integro-differential equations have been extensively studied in the previous literature (see [3]- [9] and references cited therein).
Besides delays, impulsive effect usually exist in many evolution processes in which the states exhibit abrupt changes at certain moments, such as threshold phenomena in biology, bursting rhythm models in medicine and frequency modulated systems, etc.In recent years, the theory of impulsive integro-differential equations with delays has attracted wide attention and lots of significant results on existence, initial (boundary) value problems and stability have been reported [10]- [20].Some results for impulsive neutral differential equations with delays have been published.For instance, in [21], the exponential stability for impulsive neutral differential equations with finite delays has been studied by using differential inequality technique.In [22,23], some stability conditions based on Lyapunov-Krasovkii functional method have been established for impulsive neutral differential equations with finite delays.In [24], authors studied the exponential stability for impulsive neutral integro-differential equations with delays by developing a singular integro-differential inequality.However, in general, the results about impulsive neutral differential equations with delays are still scarce due to some theoretical and technical difficulties.
Additionally, it worth noting that those results in previous literature [21]- [24] have only focused on the stability of the equilibrium point for autonomous impulsive neutral differential equations with delays.However, under impulsive perturbation, the equilibrium point sometimes does not exist in many real physical systems, especially in nonlinear and non-autonomous dynamical systems.Therefore, an interesting and more general issue is to discuss the invariant set and attracting set of non-autonomous impulsive systems.Some important progress has been made in the techniques and methods for determining the invariant and attracting sets of delay differential equations [25,26], impulsive differential equations with delays [27] and neutral differential equations [28].Until now the corresponding problems for impulsive neutral differential (or integro-differential) equations with delays have not been considered.
Motivated by the above discussion, we will investigate the asymptotic behaviors of solutions for a non-autonomous impulsive neutral integro-differential equation with time-varying delays in this paper.As shown in [20,21,24], differential inequalities are very important tools to investigate dynamical behaviors of differential equations.We shall develop a novel singular delay integrodifferential inequality in Section 3. Compared with those existing results such as (7) in [20], (8) in [21] and (16) in [24], the presented inequality (formulated by the later inequality ( 6)) has the following improvements.(a) All of those key inequalities established in [20,21,24] are autonomous.That is to say the involved coefficients are constants.However, in this paper, the presented singular integro-differential inequality is non-autonomous, which means the coefficients are time varying.(b) In the proposed inequality (6), the additional input term J is very novel and crucial for our studying.If J = 0, we can use the inequality to estimate the positive invariant set and global attracting set explicitly.If J = 0, inequality (6) can cover those inequalities in [20,21,24] and enable us to investigate the stability of the equilibrium point.
In Section 4, by using the transform technique similar to [21,24], we derive some sufficient criteria on the global attracting set, positive invariant set and stability.In Section 5, an example and its simulations are given.Finally, we make some conclusions.

Notations and Model Description
Let R n be the space of n-dimensional real column vectors and R m×n be the class of m × n matrices with real components.The inequality " ≤ " (" > ") between matrices or vectors such as A ≤ B (A > B) means that each pair of corresponding elements of A and B satisfies the inequality " ≤ " (" > ").A ∈ R m×n is called a nonnegative matrix if A ≥ 0 and x ∈ R n is called a positive vector if x > 0. x T and A −1 denote the transpose of a vector x and the inverse of a square matrix A, respectively.I denotes the identity matrix with appropriate dimensions.N = {1, 2, . . ., n}, For A ∈ R m×n and function C[X, Y ] denotes the space of continuous mappings from the topological space X to the topological space Y .
For any initial condition φ ∈ P C 1 , we always assume that (1) has a solution denoted by x(t, t 0 , φ) or x t (t 0 , φ) (simply x(t) or x t if no confusion occurs), where x t (t 0 , φ) = x(t + s, t 0 , φ), −∞ < s ≤ 0. We know x(t) is continuously differentiable for t ≥ t 0 and t = t k .Moreover x(t) has discontinuities of the first type at the fixed impulsive moments t k .Namely, x t ∈ P C 1 .For convenience, we denote 1) be transformed to an 2n-dimensional non-autonomous singular impulsive integro-differential equation as follows with the initial condition A • B is called the Hadamard product or Schur product of A and B. Definition 2.6.[29] A matrix A = (a ij ) n×n ∈ R n×n is called an M-matrix if A has non-positive off-diagonal elements (i.e., a ij ≤ 0 for i = j), and one of the following conditions holds: (i) there exists a positive vector z such that Az > 0; (ii) A −1 exists and A −1 ≥ 0.
For an M-matrix A, we define Obviously, Definition 2.6 leads to the following lemma.Lemma 2.1.[21] If A is an M-matrix, then Ω M (A) ⊂ R n is a nonempty cone without conical surface.

Singular Integro-differential Inequality
In what follows, we shall develop a novel non-autonomous singular delay integro-differential inequality, which is a useful tool to study impulsive delay differential equations.Theorem 3.1.Assume u ∈ C[[t 0 , b), R r ] satisfies the non-autonomous singular delay integrodifferential inequality with initial condition for i, j ∈ N * and J = (J 1 , . . ., J r ) T ≥ 0; (C 3 ) there exist a positive vector z ∈ R r and a positive constant σ such that If the initial condition then Denote z := (z 1 , • • • , z r ) T .We rewrite (7) as which implies for any i That is Consequently, by Definition 2.6, it is easy to deduce D is an M-matrix, and D −1 exists with D −1 ≥ 0.

Attracting Set and Invariant Set
In this section, we will present the main results for the global attracting set, positive invariant set and stability of (3) by using the improved non-autonomous singular delay integro-differential inequality in Section 3.
From the initial condition (4), we see that x t0 = φ ∈ P C 1 ⊂ P C, y t0 = φ ′ ∈ P C. That is Noting z > 0, T ≥ 0, it is easy to deduce Suppose that for any m = 1, 2, . . ., k, we have with We note that conditions (23) and (25) indicate R k zx ≤ ζ k zx and R k Tx ≤ ν k Tx , respectively.Then from assumption (33) it suffices to obtain Meanwhile, the fourth equation in (3) together with (A 1 ) implies that for all i ∈ N

Figure 2 : 6 Conclusions
Figure 2: Simulation for Case 2 (3)ark 2.2.Recalling the definition of P C 1 and the properties of derivative function, x t ∈ P C 1 implies that y(t) has discontinuities of the first type at the fixed impulsive moments t k and y(t) is continuous on [t k−1 , t k ) for k ∈ Z + .Therefore, studying the asymptotic behaviors of (1) in P C 1 is equivalent to those for(3)in P C[(−∞, 0], R 2n ].Some definitions and lemma will be employed in this paper.Definition 2.1.A set A ⊂ P C 1 is called a positive invariant set of (1), if for any initial condition φ ∈ A, the solution x t (t 0 , φ) ∈ A for t ≥ t 0 .Definition 2.2.A set B ⊂ P C 1 is called an attracting set of (1), if B possesses an open neighborhood U, such that for any initial condition φ ∈ U, the solution x(t, t 0 , φ) satisfies lim Definition 2.3.The zero solution of (1) is called to be globally asymptotically stable in P C 1 , if for any initial condition φ ∈ P C 1 , the solution x(t, t 0 , φ) satisfies lim Definition 2.4.The zero solution of (1) is called to be globally exponentially stable in P C 1 , if there exist positive constants α and λ, such that for any initial condition φ ∈ P C 1 , the solution x(t, t 0 , φ) satisfies ||x(t, t 0 , φ)|| ≤ α||φ|| 1∞ e −λ(t−t0) , t ≥ t 0 .