Dynamics of Cohen-Grossberg Neural Networks with Mixed Delays and Impulses

Impulsive Cohen-Grossberg neural networks with bounded and unbounded delays i.e., mixed delays are investigated. By using the Leray-Schauder fixed point theorem, differential inequality techniques, and constructing suitable Lyapunov functional, several new sufficient conditions on the existence and global exponential stability of periodic solution for the system are obtained, which improves some of the known results. An example and its numerical simulations are employed to illustrate our feasible results.


Introduction
In the recent years, dynamics of the Cohen-Grossberg neural networks CGNNs 1 has been extensively studied because of their immense potentials of application perspective in different areas such as pattern recognition, parallel computing, associative memory, combinational optimization, and signal and image processing 2-6 .The authors of 7-11 have studied the stability of equilibrium point or periodic solution of CGNNs with timevarying delays due to the transmission delays during the communication between neurons which will affect the dynamical behavior of neural networks.Considering the distant past also has influence on the recent behavior of the state, the authors of 12 investigated the stability of equilibrium point for CGNNs with continuously distributed delays.Impulsive effects are also likely to exist in the neural networks, that is, the state of the networks is subject to instantaneous perturbations and experiences abrupt change at certain moments.Authors of 13-16 have studied the stability of equilibrium point for impulsive delay CGNNs.However, the activation functions of CGNNs in 12-16 are bounded, and the restrictions on impulses in 13, 17 are very strong, which limit CGNNs' applications 7 .
In theory and applications, global stability of periodic solution of CGNNs is of great importance since the global stability of equilibrium points can be considered as a special case of periodic solution with random period.Moreover, CGNNs model is one of the most popular and typical neural network models.Some other models, such as Hopfield-type neural networks, cellular neural networks, and bidirectional associative memory neural networks, are special cases of the model 14 .To our best knowledge, few authors have considered the existence and global exponential stability of periodic solutions for CGNNs with mixed delays and impulses.Therefore, it is necessary to consider the existence and global exponential stability of periodic solution for impulsive CGNNs with mixed delays.
The main methods used in this paper are Leray-Schauder's fixed point theorem, differential inequality techniques, and Lyapunov functional.Several new sufficient conditions are obtained for the existence and global exponential stability of periodic solution for impulsive CGNNs with mixed delays.Moreover, we discharge some restrictive conditions on the activation functions of the neurons, such as boundedness, monotonicity, and differentiable, we offer the precise convergence index, and give the weak conditions on impulses.
The rest of this paper is organized as follows.In Section 2, we introduce some notations and definitions and state some preliminary results needed in later sections.We then study, in Section 3, the existence of periodic solutions of impulsive CGNNs with mixed delays by using Leray-Schauder's fixed point theorem.In Section 4, with the help of Lyapunov functional, we will derive sufficient conditions for the global exponential stability of the periodic solution.At last, an example and its numerical simulations are employed to illustrate the feasible results of this paper.

Preliminaries
Consider the following CGNNs with mixed delays and impulses: where Δu i t k i 1, 2, . . ., n are the impulses at moments t k , u i t is left continuous at time t k , and the right limit exists at t k , that is, where q i a i l i ; H 8 there exists a constant α 0 > 0 such that ∞ 0 k ij s e α 0 s ds < ∞, i,j 1, 2, . . ., n.

2.4
From H 2 , the antiderivative of 1/a i u i exists.We choose an antiderivative h i u i of 1/a i u i that satisfies h i 0 0. Obviously, d/du i h i u i 1/a i u i .By a i u i > 0, we obtain that h i u i is strictly monotone increasing about u i .In view of derivative theorem for inverse function, the inverse function Substituting these equalities into system 2.1 , we get which can be rewritten as where The existence and global exponential stability of periodic solution for system 2.1 are equivalent to the existence and global exponential stability of periodic solution for system 2.5 or 2.6 .So, we investigate the the existence and global exponential stability of periodic solution for system 2.6 .
From the definition of h −1 i u , using Lagrange mean value theorem, one gets where ξ is between x and y.Moreover, we have

Existence of periodic solutions
Lemma 3.1.Suppose that (H 1 -(H 5 hold and let x t be an ω-periodic solution of system 2.6 .Then,

3.1
where Proof.Let t q ≤ t < t q 1 , q ≤ p. From the first expression of 2.6 , we have x j s ds − I i t e t 0 d i x i s ds .

6
Abstract and Applied Analysis Integrating 3.3 on intervals 0, t − 1 , t 1 , t − 2 , . . ., t q , t and adding all of them, by the second formula of 2.6 , we have

3.4
Since x i ω x i 0 , from 3.4 , we obtain
In order to use Lemma 2.3, we take PC J, R n {x t | x : J 0, ω → R n as piecewise continuous periodic solution at t / t k , and x t k , x t − k x t k exist at t t k , k 1, 2, . . ., p}.Then, PC J, R n is a Banach space with the norm x max 1≤i≤n x i 0 , x i 0 sup 0≤t≤ω x i t , i 1, 2, . . ., n.

3.6
Set a mapping Φ : PC J, R n → PC J, R n by setting where

7
It is easy to know the fact that the existence of ω-periodic solution of 2.6 is equivalent to the existence of fixed point of the mapping Φ in PC J, R n .Lemma 3.2.Suppose that (H 1 -(H 7 hold.Then, Φ : PC J, R n → PC J, R n is completely continuous.
Proof.First, we show that Φ : PC J, R n → PC J, R n is continuous.For any ε > 0, we take 0 < δ < ε/max 1≤i≤n {θ p k 1 η ik / 1 − e −q i ω }, where η ik a i /a i |1 γ ik | 1.Then, for all x, y ∈ PC J, R n and x − y < δ, we have 3.9 Hence, Φ is continuous.Next, we show Φ maps bounded set into bounded set.For any x ∈ PC J, R n with x < D, where D is some positive constant, we have

3.10
where b ij max t∈ 0,ω |b ij t |, c ij max t∈ 0,ω |c ij t |, I i max t∈ 0,ω |I i t |.Equation 3.10 implies that Φx is uniformly bounded for any x < D. Hence, {Φx | x ∈ PC J, R n } is a family of uniformly bounded and equicontinuous functions on R. By using the Arzela-Ascoli theorem, Φ : PC J, R n → PC J, R n is compact.Therefore, Φ : PC J, R n → PC J, R n is completely continuous.This completes the proof.Theorem 3.3.Suppose that (H 1 -(H 7 hold.Then, system 2.6 has an ω-periodic solution.

3.11
If x is a solution of 3.11 , for t ∈ J, we obtain

3.12
where this and max 1≤i≤n {θ

3.13
This shows that x of 3.11 is bounded, which is independent of λ ∈ 0, 1 .In view of Lemma 2.3, we obtain that Φ has a fixed point.Hence, system 2.6 has one ω-periodic solution with x ≤ R.This completes the proof.

Global exponential stability of periodic solution
In this section, we will construct some suitable Lyapunov functionals to derive sufficient conditions which ensure that 2.6 has a unique ω-periodic solution, and all solutions of 2.6 exponentially converge to its unique ω-periodic solution.
Theorem 4.1.Assume that (H 1 -(H 8 hold and , η is a positive constant, α is given in the proof of this theorem, t 0 t p − ω.
Then, system 2.6 has exactly one ω-periodic solution, which is globally exponentially stable with the convergence index α − η.

4.3
Clearly, h i λ , i 1, 2, . . ., n are continuous functions on R. Since and h i ∞ ∞, hence h i λ , i 1, 2, . . ., n are strictly monotone increasing functions.Therefore, for any i ∈ {1, 2, . . ., n} and t ≥ 0, there is a unique λ t such that . ., n.Now, we will prove that λ * i > 0, i 1, 2, . . ., n. Suppose this is not true, from 4.2 , there exists a positive constant η such that inf t≥0, 1≤i≤n Pick small ε > 0, then there exists t i ≥ 0 such that Let us recall the inequality e x < 1 1.5x for sufficiently small x > 0, then, we obtain which is a contradiction, and hence, Obviously, for all t ≥ 0, we have k ij s e αs ds ≤ 0, i 1, 2, . . ., n. 4.9 It is obvious that k ij t − s e α t−s V j s ds .

4.17
Similar to the steps of 4.9 -4.16 , we can also prove that

4.18
When t t 2 , again, from the second expression of 4.1 , we have

4.19
By repeating the same procedure, we can deduce the following general result:

4.20
By H 9 , we have M k ≤ e η t k −t k−1 , which implies that

Application
In this section, we give an example to illustrate that our results are feasible.Consider the following Cohen-Grossberg neural networks with mixed delays and impulses: where  ln M k / t k − t k−1 ln 2.04 0.7129 < 1. Submitting λ 1 into 4.9 , we have h 1 λ < −3.9 < 0, h 2 λ < −4.9 < 0, see Figures 1 a and 1 b .This implies that ln 2.04 < 1 < α.Hence, all the conditions needed in Theorem 4.1 are satisfied.Therefore, system 5.1 has a unique 4periodic solution, which is globally exponentially stable see Figures 1 c -

Figure 1 :
Figure 1: a The trajectory of h 1 λ .b The trajectory of h 2 λ .c Numeric simulation of u 1 , u 2 .d Numeric simulation of u 1 , u 2 without impulses.e Numeric simulation of u 1 , u 2 with impulses.f Numeric simulation of u 1 , u 2 without impulses.g Numeric simulation of u 1 , u 2 with impulses.
1−t 0 e η t 2 −t 1 , ..., e η t k −t k−1 ≤ e η ω−t p e ηt , t ∈ t k , t k 1 .In view of Definition 2.2, the ω-periodic solution x * t of system 2.6 is globally exponentially stable with the convergence index α − η.This completes the proof.Remark 4.2.Note that we have dropped the restriction: the activation functions are bounded and |1 γ ik | ≤ 1, which is indispensable in 13, 17 .To the best of our knowledge, most of the existing research papers only give the existence of the convergence index, while we offer the precise convergence index in Theorem 4.1.