1 Introduction

Difference equations usually appear in the investigation of systems with discrete time or in the numerical solution of systems with continuous time [1]. In recent years, the stability investigation of difference equations has been interesting to many investigators, and various advanced results on this problem have been reported [2, 3]. However, almost all available results have been focused on systems with discrete delays. In reality, difference systems with distributed delays become important because it is essential to formulate the discrete-time analogue of the continuous-time system with distributed delays when one wants to simulate or compute the continuous-time one after obtaining its dynamical characteristics. Fortunately, such an issue has been addressed in [4–7].

However, besides the delay effect, an impulsive effect likewise exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time, involving such fields as medicine, biology, economics, mechanics, electronics, and telecommunications. Recently, the asymptotic behaviors of impulsive difference equations have attracted considerable attention. Many interesting results on impulsive effect have been obtained [8–11].

It is well known that distributed delay differential equations with impulses or without impulses have been considered by many authors (see, for instance [12–14]). But, to the best of our knowledge, there is no concerning on the stability of impulsive difference equations with distributed delays in literature. Motivated by the above discussion, we here make a first attempt to arrive at results on the global exponential stability of impulsive difference equations with distributed delays.

2 Model description and preliminaries

Let be the space of n-dimensional (non-negative) real column vectors and denotes the set of m × n (non-negative) real matrices. Usually, E denotes an n × n unit matrix. For A, B ∈ Rm × nor A, B ∈ Rn, the notation A ≥ B (A > B) means that each pair of corresponding elements of A and B satisfies the inequality " ≥ (>)". Especially, A ∈ Rm × nis called a nonnegative matrix if A ≥ 0, and z ∈ Rn is called a positive vector if z > 0. Z denotes the integer set, Z∞ = {j ∈ Z |-∞ < j ≤ 0} and . C denotes the set of all bounded functions φ(j) ∈ Rn, j ∈ Z∞.

For x ∈ Rn, A ∈ Rn×n, φ ∈ C, we define

where , and introduce the corresponding norm for them as follows:

In this paper, we mainly consider the following impulsive difference equations with distributed delays

(1)

where 0 < i ≤ n and a i , b ij , c ij are constants. The fixed moments of time m k ∈ Z, and satisfy . The constants μ ij (k) satisfy the following convergence conditions:

where λ0 is a positive constant.

For convenience, we shall rewrite (1) in the vector form:

(2)

where x (m) = (x1 (m), ..., x n (m))T, A = diag{a1, ..., a n }, B = {b ij }n × n, C = {c ij }n × n, f (x) = (f1 (x1), ..., f n (x n ))T, g(x) = (g1(x1), ..., g n (x n ))T, μ(k) = (μ ij (k)) n × n , H m (x(m)) = (H1m(x(m)), ..., H nm (x(m)))T, φ ∈ C, and f(x), g(x), H m (x) ∈ C[Rn, Rn].

We will assume that there exists one solution of system (2) which is denoted by x(m, 0, φ), or, x(m), if no confusion occurs. We will also assume that g(0) = 0, f(0) = 0 and H m (0) = 0, m = m k , for the stability purpose of this paper. Then system (2) admits an equilibrium solution x(m) ≡ 0.

Definition 2.1. The zero solution of Equation 2 is called globally exponentially stable if there are positive constants λ and M ≥ 1 such that for any initial condition φ ∈ C,

Here λ is called the exponential convergence rate.

For , the spectral radius ρ(A) is an eigenvalue of A and its eigenspace is denoted by

which includes all positive eigenvectors of A provided that the non-negative matrix A has at least one positive eigenvector(see [15]).

Lemma 2.1. [16] Suppose that and ρ(M) < 1, then there exists a positive vector z such that

For and ρ(M) < 1, we denote

which is a nonempty set by Lemma 2.1, and satisfying that k1z1+k2z2 ∈ Ω ρ (M) for any scalars k1 > 0, k2 > 0 and vectors z1, z2 ∈ Ω ρ (M). So Ω ρ (M) is a cone without vertex in Rn, we call it a "ρ-cone."

Lemma 2.2. Suppose and Q(k) = (q ij (k)) n × n , where q ij (k) ≥ 0 and satisfy

where λ1 is a positive constant. Denote and let ρ(P + Q) < 1 and be a solution of the following inequality with the initial condition u(m0 + m) ∈ C, m ∈ Z∞,

(3)

Then

(4)

provided that the initial conditions satisfy

(5)

where z = (z1, z2, ..., z n )T ∈ Ω ρ (P + Q), m0 ∈ Z and the positive number λ ≤ λ1 is determined by the following inequality

(6)

Proof. Since ρ(P + Q) < 1 and , then, by Lemma 2.1, there exists a positive vector z ∈ Ω ρ (P + Q) such that (E - (P + Q))z > 0. Using continuity, there must be a sufficiently small constant λ > 0 such that , i.e., inequality (6) has at least one positive solution λ ≤ λ1.

Let

Then, from (5), we have

(7)

By (3), we have

(8)

Since , we derive that

(9)

We next show for any m ≥ m0

(10)

If this is not true, then there must be a positive constant m* ≥ m0 and some integer i such that

(11)

By (6), (9), and the second inequality of (11), we obtain that

which contradicts the first inequality of (11). Thus (10) holds for all m ≥ m0. Therefore, we have

and the proof is completed.

3 Main results

To obtain the global exponential stability of the zero solution of system (2), we introduce the following assumptions.

(A1) For any x ∈ Rn, there exist non-negative diagonal matrices U and V such that

(A2) For any x ∈ Rn, there exist non-negative matrices R k such that

(A3) Let , and ρ(P + Q) < 1.

(A4) The set is nonempty.

(A5) Let

(12)

and there exists a constant γ such that

(13)

where the positive number λ ≤ λ0 is determined by the following inequality

(14)

Theorem 3.1. Assume that the hypothesis (H) and Conditions (A1)-(A5) hold. Then the zero solution of (2) is globally exponentially stable and the exponential convergent rate equals λ - γ.

Proof. Since ρ(P + Q) < 1 and , then, by Lemma 2.1, there exists a positive vector z ∈ Ω ρ (P + Q) such that (E - (P + Q))z > 0. Using continuity and hypothesis (H), there must be a sufficiently small constant λ > 0 such that , i.e., inequality (14) has at least one positive solution λ ≤ λ0.

From (2), Conditions (A1) and (A3), we have

(15)

where m0 = 0.

For the initial conditions: x(s) = φ(s), -∞ < s ≤ 0, where φ ∈ C, we can get

(16)

where

By the property of "ρ-cone" and z ∈ Ω ⊆ Ω ρ (P + Q), we have d ||φ||∈ Ω ρ (P + Q). Then, all the conditions of Lemma 2.2 are satisfied by (15), (16), and Condition (A3), we derive that

(17)

Suppose for all q = 1, ..., k, the inequalities

(18)

hold, where γ0 = 1. Then, from Condition (A2) and (18), we have

(19)

Since d ∈ Ω ⊆ W ρ (R q ), we have R q d = ρ(R q )d. Therefore, from (12) and (19), we obtain

(20)

This, together with (18), leads to

(21)

By the property of "ρ-cone" again, the vector γ0 ⋯ γk-1γ k d ∈ Ω ρ (P + Q). It follows from (21) and Lemma 2.2 that

yielding, together with (18), that

By mathematical induction, we can conclude that

(22)

Noticing that by (13), we can use (22) to conclude that

which implies that the conclusions of the theorem hold.

Remark 3.1. In Theorem 3.1, we may properly choose the matrix R k in the condition (A2) such that Ω ≡ ∅ Especially, when R k = α k E (α k are non-negative constants), Ω is certainly nonempty. So, by using Theorem 3.1, we can easily obtain the following corollary.

Remark 3.2. The conditions (A1)-(A5) is conservative. For example, we get the absolute value of all coefficients of (2). Recently, the delay-fractioning or delay-partitioning approach [17, 18] is widely used that has shown the potential of reducing conservatism. We will combine delay-partitioning approach with difference inequality approach in our future work to reduce the conservatism.

Corollary 3.1. Assume that (H), (A1), (A3), and (A5) hold. For any x ∈ Rn, there exist non-negative constants α k such that

(23)

And let γ k ≥ {1, α k }, where the scalar 0 < λ < λ0 is determined by (14). Then the zero solution of (2) is globally exponentially stable and the exponential convergent rate equals λ - γ.

Proof. Noticing that (23) is a special case of Condition (A2). Since ρ(R k ) = α k , then W ρ (R k ) = Rn. So, we have . Since the "ρ-cone" Ω ρ (P + Q) is nonempty by Lemma 2.1, (A4) obviously holds. Thus we can deduce the conclusion in terms of Theorem 3.1.

Remark 3.3. If H k (x) = x, then Equation 2 becomes difference equations with distributed delays without impulses in vector form

(24)

which contains many popular models such as discrete-time Hopfield neural networks, discrete-time cellular neural networks, and discrete-time recurrent neural networks, and so on.

Corollary 3.2. Assume that (H), (A1), and (A3) hold. Then Equation 24 has exactly one equilibrium point, which is globally exponentially stable.

4 An illustrate example

In this section, we will give an example to illustrate the global exponential stability of Equation 1 further.

Example. Consider the following difference equation with distributed delays:

(25)

with

(26)

and m1 = 4, m k = mk-1+ k for k = 2, 3,.... One can check that all the properties given in (H) are satisfied provided that 0 < λ0 < 1.

Case 1. If for i = 1, 2 and k = 1, 2,..., then Equation 25 becomes difference equation with distributed delays without impulses. The parameters of Conditions (A1) and (A3) are as follows:

where . We can easily observe that ρ(P + Q) = 0.8345 < 1. By Corollary 3.2, Equation 25 has exactly one globally exponentially stable equilibrium (0, 0)T.

Case 2. Next we consider the case where

We can verify that point (0, 0)T is also an equilibrium point of the impulsive difference equation with distributed delays (25)-(26) and the parameters of Conditions (A2) and (A4) as follows:

So Ω = {(z1, z2)T > 0 |z2 = z1} is not empty. Let z = (1, 1)T ∈ Ω and λ = 0.05 which satisfies the inequality . We can obtain that for k = 1, 2,...

Clearly, all conditions of Theorem 3.1 are satisfied, so the equilibrium (0, 0)T is globally exponentially stable and the exponential convergent rate is equal to 0.01.

5 Conclusion

In this paper, we consider a class of impulsive difference equations with distributed delays. By establishing an impulsive delay difference inequality and using the properties of "ρ-cone" and eigenspace of the spectral radius of non-negative matrices, some new sufficient conditions for global exponential stability of the impulsive difference equations with distributed delays are obtained. The conditions (A1)-(A5) are conservative. For example, we get the absolute value of all coefficients of (2). We will combine delay-partitioning approach with difference inequality approach in our future work to reduce the conservatism.