^{1, 2}

^{2}

^{2}

^{1}

^{2}

By using a fixed point theorem of strict-set-contraction, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for

Many systems in physics, chemistry, biology, and information science have impulsive dynamical behavior due to abrupt jumps at certain instants during the evolving processes. This complex dynamical behavior can be modeled by impulsive differential equations. Impulsive differential equations have become more important in recent years in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics; see [

In this paper, we consider more general neutral delay functional differential equation with impulse:

In 1993, Kuang in [

The main purpose of this paper is to establish new criteria to guarantee the existence of positive periodic solutions of the system (

For convenience, we introduce the notation

Throughout this paper, we assume the following.

There exist

We assume that

We assume that

We assume that

The paper is organized as follows. In the next section, we give some definitions and lemmas to prove the main results of this paper. In Section

In order to obtain the existence of a periodic solution of system (

A function

for each

Let

Let

Let

The set

The set

Assume that

Thus,

If

Let

In order to apply Lemma

Assume that

If

If

For any

Therefore, for

Furthermore, for

Now, we show that

On the other hand, from (

It follows from (

(ii) In view of the proof of (i), we only need to prove that

Assume that

If

If

where

We only need to prove (i), since the proof of (ii) is similar. It is easy to see that

Hence,

Assume that

If

If

In this section, we will study the existence of positive

Assume that

If

If

We only need to prove (i), since the proof of (ii) is similar. Let

First, we prove that

Finally, we prove that

In addition, for any

In this section, we apply the result obtained in the previous section to some periodic population models with impulses which are mentioned in the first section.

First, we consider a general neutral delay model of single-species population growth with impulse:

Assume

If

If

The proof is similar to that of Theorem

Second, we consider a general neutral delay model of single-species population growth with impulse:

Assume

If

If

The proof is similar to that of Theorem

We apply the main result obtained in the previous section to study some examples which have some biological implications. A very basic and important ecological problem associated with the study of population is that of the existence of a positive periodic solution which plays the role played by the equilibrium of the autonomous models and means that the species is in an equilibrium state. From Theorems

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by NSF of China (nos. 10971229, 11161015, and 11371367), PSF of China (nos. 2012M512162 and 2013T60934), NSF of Hunan province (nos. 11JJ900, 12JJ9001, and 13JJ4098), the Education Foundation of Hunan province (nos. 12C0541, 12C0361, and 13C084), and the construct program of the key discipline in Hunan province.