EXISTENCE AND STABILITY OF PERIODIC SOLUTIONS FOR A CLASS OF GENERALIZED NONAUTONOMOUS NEURAL NETWORKS WITH DISTRIBUTED DELAYS

By using the continuation theorem of coincidence degree theory and Lyapunov functions, we study the existence and global stability of periodic solutions for a class of generalized nonautonomous neural networks with distributed delays.


Introduction
The study of the dynamics of neural networks has greatly attracted the attention of the scientific community because of their promising potential for the tasks of classification, associative memory, and parallel computations, and their ability to solve difficult optimization problems.Many papers [1,4,5,8,9,10,12,13,14,15] have been devoted to discussing the stability of neural networks with delays.Recently, the authors of [7] have studied the globally exponential stability of the trivial solution for the following generalized neural networks with distributed delays ẋi (t) = − d i x i (t) + n j=1 ω i j x 1 (t),...,x n (t) f j x j (t) + n j=1 ω τ i j x 1 (t),...,x n (t) where x i is the state of the i-neuron at time t, A = (ω i j ) and B = (ω τ i j ) are n × n interconnection matrices, respectively, f j is an activation function.However, under some practical circumstances, the connection weights, the activation functions, and the rate functions of most neural network models (i.e., ω i j , ω τ i j , f j , and d i in system (1.1)) depend not only on the state x i (t) but also on the time t, so the nonautonomous system can be applied in wider fields.In this paper, we are concerned with the following nonautonomous neural network system ẋi It is well known that studies on neural network dynamical systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior, almost-periodic oscillatory properties, chaos, and bifurcation [11], and to the best of our knowledge, few authors considered the existence of periodic solutions for the model (1.2).Our purpose of this paper is to prove the existence and stability of periodic solutions of (1.2).Throughout this paper, we assume that (H1) for each i = 1,2,...,n, ,R) are bounded and f i , ω τ i j , and ω i j are T-periodic with respect to their first arguments, respectively.The organization of this paper is as follows.In the second section, we prove the existence of periodic solutions of system (1.2) by applying the continuation theorem of coincidence degree theory.In the third section, some sufficient conditions are obtained to show the global asymptotic stability of periodic solutions of system (1.2).

Existence of positive periodic solutions
In this section, based on the Mawhin's continuation theorem, we will study the existence of at least one positive periodic solution of (1.2).First, we will make some preparations.
Let X,Y be normed vector spaces, L : DomL ⊂ X → Y a linear mapping, and N : X → Y a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dimKer L = codim Im L < +∞ and ImL is closed in Y .If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X and Q : Y → Y such that Im P = Ker L, KerQ = Im L = Im(I − Q), it follows that mapping L| DomL∩Ker P : (I − P)X → Im L is invertible.We denote the inverse of that mapping by K P .If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω if QN(Ω) is bounded and there exists an isomorphism J : ImQ → Ker L. Now, we introduce Mawhin's continuation theorem [2, page 40] as follows.
Lemma 2.1.Let Ω ⊂ X be an open bounded set and let N :

Assume that (H1)-(H3) hold. Then the system (1.2) has at least one Tperiodic solution.
Proof.In order to apply the continuation theorem of coincidence degree theory to establish the existence of a T-periodic solution of (1.2), we take Yongkun Li et al. 999 and denote then X is a Banach space.Set where (2.4) Define two projectors P and Q as ..,n} is closed in X and dim Ker L = codim Im L = n.Hence, L is a Fredholm mapping of index 0. Furthermore, similar to the proof of [6, Theorem 1], one can easily show that N is L-compact on Ω with any open bounded set Ω ⊂ X.
Proof.We consider the Lyapunov function where sign u i (t) ui (t) ) This completes the proof.