Periodic Solutions and Exponential Stability of a Class of Neural Networks with Time-Varying Delays

Employing fixed point theorem, we make a further investigation of a class of
neural networks with delays in this paper. A family of sufficient conditions is given for
checking global exponential stability. These results have important leading significance in
the design and applications of globally stable neural networks with delays. Our results
extend and improve some earlier publications.


Introduction
The stability of dynamical neural networks with time delay which have been used in many applications such as optimization, control, and image processing has received much attention recently see, e.g., 1-15 .Particularly, the authors 3, 8, 9, 14, 16 have studied the stability of neural networks with time-varying delays.
As pointed out in 8 , Global dissipativity is also an important concept in dynamical neural networks.The concept of global dissipativity in dynamical systems is a more general concept, and it has found applications in areas such as stability theory, chaos and synchronization theory, system norm estimation, and robust control 8 .Global dissipativity of several classes of neural networks was discussed, and some sufficient conditions for the global dissipativity of neural networks with constant delays are derived in 8 .
In this paper, without assuming the boundedness, monotonicity, and differentiability of activation functions, we consider the following delay differential equations: H ij t − s f j x j s ds J i t , i 1, 2, . . ., n, 1.1 where n denotes the number of the neurons in the network, x i t is the state of the ith neuron at time t, x t x 1 t , x 2 t , . . ., x n t T ∈ R n , f x t f 1 x 1 t , f 2 x 2 t , . . ., f n x n t T ∈ R n denote the activation functions of the jth neuron at time t, and the kernels H ij : 0, ∞ → 0, ∞ are piece continuous functions with A1 the time delays τ ij t ∈ C R, 0, ∞ are periodic functions with a common period ω > 0 for i, j 1, 2, . . ., n; The organization of this paper is as follows.In Section 2, problem formulation and preliminaries are given.In Section 3, some new results are given to ascertain the global robust dissipativity of the neural networks with time-varying delays.Section 4 gives an example to illustrate the effectiveness of our results.

Preliminaries and Lemmas
For the sake of convenience, two of the standing assumptions are formulated below as follows.
A3 |f j u | ≤ p j |u| q j for all u ∈ R, j 1, 2, . . ., n, where p j , q j are nonnegative constants.A4 There exist nonnegative constants p j , j 1, 2, . . ., n, such that The initial conditions associated with system 1.1 are of the form T is an equilibrium of system 1.1 , then we denote

2.3
Definition 2.1.The equilibrium x 0 x 0 1 , x 0 1 , . . ., x 0 n T is said to be globally exponentially stable, if there exist constants λ > 0 and m ≥ 1 such that for any solution x t x 1 t , x 2 t , . . ., x n t T of 1.1 , we have for t ≥ 0, where λ is called to be globally exponentially convergent rate.
, where E denotes the identity matrix of size n.

Periodic Solutions and Exponential Stability
We will use the coincidence degree theory to obtain the existence of a ω-periodic solution to systems 1.1 .For the sake of convenience, we briefly summarize the theory as follows.
Let X and Z be normed spaces, and let L : Dom L ⊂ X → Z be a linear mapping and be a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dimKer L codimIm L < ∞ and Im L is closed in Z.If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X → X and Q : Z → Z such that Im P Ker L and Im L Ker ImQ is isomorphic to Ker L, there exists an isomorphism J : ImQ → Ker L.
Let Ω ⊂ R n be open and bounded, f ∈ C 1 Ω, R n ∩ C Ω, R n and y ∈ R n \ f ∂Ω ∪ S f , that is, y is a regular value of f.Here, S f {x ∈ Ω : J f x 0}, the critical set of f, and J f is the Jacobian of f at x. Then the degree deg{f, Ω, y} is defined by with the agreement that the above sum is zero if f −1 y ∅.For more details about the degree theory, we refer to the book of Deimling 18 .Lemma 3.1 continuation theorem 19, page 40 .Let L be a Fredholm mapping of index zero, and let N be L-compact on Ω. Suppose that Then the equation Lx Nx has at least one solution lying in Dom L ∩ Ω.
For the simplicity of presentation, in the remaining part of this paper, for a continuous function g : 0, ω → R, we denote Proof.Take X Z {x t x 1 t , x 2 t , . . ., x n t T ∈ C R, R n : x t x t ω , for all t ∈ R}, and denote Equipped with the norms • , both X and Z are Banach spaces.Denote H ij t − s f j x j s ds J i t .

3.5
Since then, for any x t ∈ X, because of the periodicity, it is easy to check that

3.8
Here, for any W w 1 , w 2 , . . ., w n T ∈ R n , we identify it as the constant function in X or Z with the value vector W w 1 , w 2 , . . ., w n T .Then system 1.1 can be reduced to the operator equation Lx Nx.It is easy to see that 3.9 and P , Q are continuous projectors such that ImP ker L, Ker Q ImL Im I − Q .

3.10
It follows that L is a Fredholm mapping of index zero.Furthermore, the generalized inverse to L K p : ImL → Ker P ∩ Dom L is given by Then,

3.13
Assume that x x t ∈ X is a solution of system 1.1 for some λ ∈ 0, 1 .Integrating both sides of 3.13 over the interval 0, ω , we obtain Then

3.15
Noting that f j u ≤ p j |u| q j ∀u ∈ R, j 1, 2, . . ., n, 3.16 we get 3.17 It follows that Note that each x i t is continuously differentiable for i 1, 2, . . ., n, and it is certain that there exists t i ∈ 0, ω such that In view of ρ K < 1 and Lemma 2.2, we have E − K −1 D l l 1 , l 2 , . . ., l n T ≥ 0, where l i is given by Then, for t ∈ t i , t i ω , we have

3.24
On the contrary, suppose that there exists some i such that | QNu i | 0, that is,

3.25
Then, we have

3.27
Consider the homotopy 0, then, as before, we have

3.30
It follows from the property of invariance under a homotopy that

3.31
Thus, we have shown that Ω satisfies all the assumptions of Lemma 3.1.Hence, Lu Nu has at least one ω-periodic solution on Dom L ∩ Ω.This completes the proof.
When c ij 0, 1.1 turns into the following system: then system 1.1 has exactly one ω-periodic solution.Moreover, it is globally exponentially stable. Proof.
Hence, all the hypotheses in Theorem 3.2 hold with q j |f j 0 |, j 1, 2, . . ., n.Thus, system 1.1 has at least one ω-periodic solution, say x t x 1 t , x 2 t , . . ., x n t T .Let x t x 1 t , x 2 t , . . ., x n t T be an arbitrary solution of system 1.1 .For t ≥ 0, a direct calculation of the right derivative x j s − x j t .

3.33
Let z i t |x i t − x i t |.Then 3.33 can be transformed into

An Example
In this section, an example is used to demonstrate that the method presented in this paper is effective.
Example 4.1.Consider the following two state neural networks:

4.2
Therefore, by Theorem 3.4, the system 1.1 has an exponentially stable 2π-periodic solution.
1, 2, . . ., n.Moreover, we consider model 1.1 with τ ij t , d i t , a ij t , b ij t , c ij t , and J i t satisfying the following assumptions: s ds.Since ImQ Ker L, we take the isomorphism J of ImQ onto Ker L to be the identity mapping.Now, we reach the point to search for an appropriate open bounded set Ω for the application of the continuation theorem corresponding to the operator equation Lx λNx, λ ∈ 0, 1 , and we have p I − Q N are continuous.For any bounded open subset Ω ⊂ X, QN Ω is obviously bounded.Moreover, applying the ArzelaCAscoli theorem, one can easily show that K p I − Q N Ω is compact.Therefore, N is L-compact on with any bounded open subset Ω ∈ X.
h ij p j sup |a ij u | |b ij u | |c ij u h ij | p j du eWithout loss of generality, we let t 0 0. For t ≥ 0, t/ω denotes the largest integer less than or equal to t/ω.Noting t/ω ≥ t/ω − 1, andd i > n j 1 |a ij | |b ij | |c ij h ij | p j e d * i τ , we get t ≥ 0,3.41where m max 1≤i≤n {e d * i τ } and λ min 1≤i≤n{d i − n j 1 |a ij | |b ij | |c ij h ij | p j e d * i τ} are positive constants.From 3.41 , it is obvious that the periodic solution is global exponentially stable, and this completes the proof of Theorem 3.4.To the best of our knowledge, few authors have considered the existence of periodic solution and global exponential stability for model 1.1 with coefficients and delays all periodically varying in time.We only find model P in 20, 21 ; however, it is assumed in 20 that τ ij t ≥ 0 are constants and in 21 that a ij t , b ij t , J i t are continuous ωperiodic functions, and d i are positive constants.Especially, the authors of 21 suppose that τ ij t ≥ 0 are continuously differentiable ω-periodic functions and 0 ≤ τ ij t < 1, clearly, which implies that τ ij t are also constants.Obviously, our model is more general.Furthermore, in 20, 21 f i , i 1, 2, . . ., n, are assumed to be strictly monotone, and the explicit presence of the maximum value of the coefficients functions in Theorems 3.2 and 3.4 see 20, 21 may impose a very strict constraint on the model e.g., when some of the maximum value of the coefficients functions are very large .Therefore, our results are more convenient when designing a cellular neural network.