1. Introduction

In recent years, the delayed cellular neural networks (DCNNs) have been extensively studied because of their immense potentials of application perspective in different areas such as pattern recognition, optimization, and signal and image processing [1–3]. Hence, they have been the object of intensive analysis by numerous authors, and some interesting results on the existence and stability of periodic and almost periodic solutions have been obtained [4–12]. To our knowledge, few authors have considered global stability of periodic solutions for the neural networks with bounded and unbounded time-varying delays. In theory and application, global stability of periodic solutions of DCNNs is of great importance since the global stability of equilibrium points can be considered as a special case of periodic solution with zero period [8]. Hence, in this paper, we will study the existence and global exponential stability of periodic solutions of the following general neural networks with time-varying delays:where is the state of neuron, , and are connection matrices, is the input function, and is the activation function of the neurons.

DCNNs in [4–12] and the references cited therein are special cases of (1.1). In particular, when are constants, the authors of [9] considered the existence and global exponential stability for (1.1) with periodic impulses. The methods used in [9] are Mawhin's coincidence degree theorem [13] and Lyapunov functions. In [14], by using Mawhin's coincidence degree theorem [13], the authors investigated the global existence of positive periodic solutions of mutualism systems with bounded and unbounded time-varying delays, and some sufficient conditions are obtained. In [12], the authors considered (1.1) when are constants.

We assume what follows.

(H₁) are continuous -periodic functions, and are continuous -periodic in the first variable. .(H₂) There exist positive constants and such that , and , for all (H₃). There are positive constants , , such that , for all (H₄) The delay kernels are continuous, integrable, and satisfy(H₅) There exists a constant such that

The organization of this paper is 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 system (1.1) by using the continuation theorem of coincidence degree theorem proposed by Gains and Mawhin [13]. In Section 4, by constructing Lyapunov function we will derive sufficient conditions for the global exponential stability of the periodic solution of system (1.1). At last, an example is employed to illustrate the feasible results of this paper.

2. Preliminaries

For convenience, we use to denote the maximums of , respectively. We also use symbols to denote a column vector, in which the symbol denotes the transpose of a vector. denotes the identity matrix of size . A matrix or vector means that all entries of are greater than or equal to zero (resp., ). For matrices or vectors and , (resp., ) means that (resp., ).

The initial condition of (1.1) is of the formwhere , are continuous -periodic solutions.

Definition 2.1. Let be an -periodic solution of (1.1) with initial value . If there exist constants and such that for every solution of (1.1) with initial value ,where , then is said to be globally exponentially stable.

Definition 2.2. (See [15, 16]). A real matrix is said to be a nonsingular -matrix if , and , where denotes the inverse of .

Lemma 2.3 (See [15, 16]). Let with . Then the following statements are equivalent:
(1) is a nonsingular -matrix,(2) there exists a vector such that ,(3) there exists a vector such that .

Lemma 2.4 (See [16]). Let be an matrix and , then there exists a vector such that , where denotes the spectral radius of .

To end this section, we introduce Mawhin's continuation theorem [13, page 40] as follows.

Consider an abstract equation in a Banach space , where is a Fredholm operator with index-zero, and is a parameter. Let and denote two projectors, and such that .

Lemma 2.5. Let be a Banach space. Suppose that is a Fredholm operator with index-zero, let be an open bounded set, and let be a continuous operator which is -compact on . Moreover, assume that the following conditions are satisfied:
(a) for each ,(b) for each ,(c). Then, has at least one solution in .

3. Existence of Periodic Solutions

Theorem 3.1. Let hold. Assume that the following condition is satisfied:
there exists a vector such that where
Then, (1.1) has at least one -periodic solution.

Proof. For convenience, we introduce the following notations: In order to use Lemma 2.5, we take ; then is a Banach space with the norm Set Obviously, and So, is closed in . It is easy to show that and are continuous projectors satisfying Hence, is a Fredholm mapping of index-zero. Furthermore, through an easy computation, we find that the generalized inverse of has the formThus, Clearly, and are continuous. Using the Arzela-Ascoli theorem, it is not difficult to show that , are relatively compact for any open bounded set . Therefore, is -compact on for any open bounded set .
Now, we reach the position to search for an appropriate open bounded subset for the application of Lemma 2.5. Corresponding to the operator equation (2.3), we haveLet be a solution of system (3.7) for some . Then, for any are all continuously differentiable. Thus, there exist such that . Hence, From (3.7), we haveIn view of , we have Set . Clearly, (3.9) implies thatThus,together with , we haveTherefore,Again from , it follows from Lemma 2.3 that is a nonsingular -matrix, and there exists a vector such that , which implies that we can choose a constant such that and and We takewhich satisfies Condition (a) of Lemma 2.5. If , then is a constant vector in , and there exists some such that . We claim thatBy way of contradiction, suppose that then there exists some such thatwhich implies thathenceThis implies that , which contradicts (3.14). Therefore, (3.16) holds, and hence, Condition of Lemma 2.5 is satisfied.
Furthermore, we define a continuous function byfor all and . It follows that If , then is a constant vector in , and there exists some such that . We claim thatBy way of contradiction, suppose that then there exists some such thatthat is,Now, we will consider the following two cases.
Case 1. If , from , we have