Theorem 3.1.

Suppose holds and , , then system (1.5) has a periodic solution.

Proof.

Take and denote

Equipped with the norm , is a Banach space. For any , because of the periodicity, it is easy to check that

Let

where for any , we identify it as the constant function in *X* with the value vector . Define given by

Then, system (1.5) can be reduced to the operator equation . It is easy to see that

and *P*, *Q* are continuous projectors such that

It follows that *L* is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) is given by

Thus,

Clearly, and are continuous. For any bounded open subset , is obviously bounded. Moreover, applying the Arzela-Ascoli theorem, one can easily show that is compact. Note that is a compact operator and is bounded, therefore, is -compact on with any bounded open subset . Since , we take the isomorphism of onto to be the identity mapping. Corresponding to equation , we have

Now we reach the position to search for an appropriate open bounded subset for the application of the Lemma 2.2. Assume that is a solution of system (3.9) for some . Then, the components of are continuously differentiable. Thus, there exists such that . Hence, . This implies

Since

we get

Set , , we find that

Now, we choose a constant number and take

where , . We will show that satisfies all the requirements given in Lemma 2.2. In fact, we will prove that if then for . Therefore, it means that is uniformly bounded with respect to when the initial value function belongs to . It follows from (3.12) and (3.13) that

. Therefore,

Clearly, , are independent of . It is easy to see that there are no and such that . If , then is a constant vector in with for . Note that , we have

We claim that

Contrarily, suppose that there exists some such that , that is,

So, we have

this is a contradiction. Therefore, (3.17) holds, and hence,

Now, consider the homotopy , defined by

where and . When and , is a constant vector in with for . Thus

We claim that

Contrarily, suppose that , then,

Thus,

This is impossible. Thus, (3.22) holds. From the property of invariance under a homotopy, it follows that

We have shown that satisfies all the assumptions of Lemma 2.2. Hence, has at least one -periodic solution on . This completes the proof.

Corollary 3.2.

Suppose that there exist positive constants such that for . Then system (1.5) has at least an -periodic solution.

Proof.

As () implies that , hence, the conditions in Theorem 2.1 are all satisfied.

Remark 3.3.

From Corollary 3.2, we can find that the condition in [4, Theorem 2.1] can be dropped out. Therefore, Theorem 3.1 is greatly generalized results to [4, Theorem 2.1]. Furthermore, we should point out that the Theorem 3.1 is different from the the existing work in [5] when without assuming the boundedness, monotonicity, and differentiability of activation functions. In fact, the explicit presence of in Theorem 2.1 (see [5]) may impose a very strict constraint on the coefficients of (1.5) (e.g., when is very large or small). Since our results are presented independent of , it is more convenient to design a neural network with delays.

Theorem 3.4.

Suppose that satisfy the hypotheses . If , , then system (1.5) has exactly one periodic solution . Moreover, it is globally exponentially stable.

Proof.

From , we can conclude is true. It is obvious that all the hypotheses in Theorem 2.1 hold with Thus, system (1.5) has at least one periodic solution, say . Let be an arbitrary solution of system (1.5). For , a direct calculation of the upper left derivative of along the solutions of system (1.5) leads to

where denotes the upper left derivative, . Let . Then, (3.27) can be transformed into

From , and (3.13), we have

Then, there exists a constant such that

Thus, we can choose a constant such that

Now, we choose a constant such that

Set

for . From (3.32) and (3.34), we obtain

In view of (3.33) and (3.34), we have

We claim that

Contrarily, there must exist and such that

It follows that

From (3.29), (3.35), and (3.38), we obtain

which contradicts (3.39). Hence, (3.37) holds. Letting and , from (3.34) and (3.37), we have

This completes the proof.

Remark 3.5.

From Theorem 3.4, it is easy to see that our results independent of and , we have dropped out the condition (b) of Theorem 3.1 in [4] and the condition: of Theorem 3.1 in [5]. Therefore, it is generalized the corresponding results in [4, 5].