Suppose holds and , , then system (1.5) has a periodic solution.
Take and denote
Equipped with the norm , is a Banach space. For any , because of the periodicity, it is easy to check that
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
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
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
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,
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.
Suppose that there exist positive constants such that for . Then system (1.5) has at least an -periodic solution.
As () implies that , hence, the conditions in Theorem 2.1 are all satisfied.
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  when without assuming the boundedness, monotonicity, and differentiability of activation functions. In fact, the explicit presence of in Theorem 2.1 (see ) 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.
Suppose that satisfy the hypotheses . If , , then system (1.5) has exactly one periodic solution . Moreover, it is globally exponentially stable.
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
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.
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  and the condition: of Theorem 3.1 in . Therefore, it is generalized the corresponding results in [4, 5].