^{1}

^{2}

^{1}

^{2}

This paper investigates a class of delay differential systems with feedback control. Sufficient conditions are obtained for the existence and uniqueness of the positive periodic solution by utilizing some results from the mixed monotone operator theory. Meanwhile, the dependence of the positive periodic solution on the parameter

As is known to all, the periodic environment changes and the unpredictable forces play an important role in many biological and ecological systems. Therefore, several different periodic models with feedback control have been studied by many authors (see [

However, as we know, there are few results on the uniqueness and parameter dependence of the positive periodic solution for delay differential systems with feedback control. Motivated by this fact, this paper is devoted to investigating the uniqueness and parameter dependence of the positive periodic solution for the following nonlinear nonautonomous delay differential system with feedback control:

The main features here are as follows. On one hand, by utilizing the mixed monotone operator theory, the existence and uniqueness of the positive periodic solution of the delay differential system (

The rest of this paper is organized as follows. Section

For convenience, let us first list some conditions.

There exists an

Denote

Consider

Now, we convert the system (

For the sake of using a fixed point theorem on mixed monotone operators, choose a fixed constant

For any

The proof of this lemma is similar to Theorem 2.1 in [

Next, we recall some results from the monotone operator theory. The following results are well known (see [

Assume that

Assume that

Suppose that (H1) and (H2) hold. Then,

For any

Therefore,

Assume that (H1) and (H2) hold. Then,

For any

In addition, for any

To sum up, the proof of this lemma is completed.

Finally, we present the main results of this paper.

Suppose that (H1) and (H2) hold. Then, for any

It is easy to see from Lemmas

Consequently, Lemmas

Assume that (H1) and (H2) hold. In addition, suppose that

Suppose that

Since

Define

Therefore,

From (

Thus, Conclusion (i) holds.

Next, let us prove Conclusion (ii).

In (

Similarly, let

Finally, we prove Conclusion (iii).

For any fixed

To sum up, the proof of this theorem is completed.

In this section, we give an illustrative example to show how to use our new results.

Consider the following nonlinear nonautonomous delay differential system with feedback control:

It is easy to see that

Since

Now, we check (H2). As a matter of fact,

Hence, Theorem

Let us set

Next, to illustrate Theorem

The graphs of

The paper is supported by NSF of Shandong (ZR2009AM006), the Key Project of Chinese Ministry of Education (no: 209072), the Science & Technology Development Funds of Shandong Education Committee (J08LI10), and Graduate Independent Innovation Foundation of Shandong University (yzc10064).