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We apply the Krasnoselskii’s fixed point theorem to study the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with infinite delay and two parameters. In particular, the presented criteria improve and generalize some related results in the literature. As an application, we study some special cases of systems, which have been studied extensively in the literature.

First, we give the following definitions. Let

Impulsive differential equations are suitable for the mathematical simulation of evolutionary process whose states are subject to sudden changes at certain moments. Equations of this kind are found in almost every domain of applied sciences; numerous examples are given in [

Models of forms (

Throughout the paper, we make the following assumptions.

The delay kernel

In addition, the parameters in this paper are assumed to be not identically equal to zero.

To conclude this section, we summarize in the following a few concepts and results that will be needed in our arguments.

Let

Let

or

Then,

For convenience in the following discussion, we introduce the following notations:

The paper is organized as follows. In Section

We establish the existence of positive periodic solutions of (

A function

for each

Let

Assume that

Assume that

Assume that

Therefore,

If

Assume that

From (

Assume that

We first show that

Next we show that

Our main results of this paper are as follows.

Assume that

there exists a

First, we define

Likewise, in view of

Assume that

There exists a

Assume that

there exists a

We define

Assume that

there exists a

Assume that

Without loss of generality, we may assume that

In addition to

In view of

Assume that

In view of

Assume that

From

On the other hand, from

In addition to

Assume that

From

On the other hand, from

In addition to

Assume that

From

Assume that

Assume that

From

On the other hand, if

Assume that

Similarly, one can prove the following theorems and corollaries.

Assume that

Assume that

Assume that

Assume that

Now, we are in a position to attack the existence of positive periodic solutions of (

For

Assume that

Assume that

Assume that

Assume

Assume

Assume

Assume

Assume

Assume

In this section, as some applications of our main results, we will consider some special cases of systems (

there exists a

there exists a

By applying theorems in Sections

Assume

Assume

Assume

Assume

Assume

Assume

there exists a

there exists a

By applying theorems in Sections

Assume

Assume

Assume

Assume

Assume

Assume

there exists a

there exists a

We obtain the following theorems.

Assume

Assume

Assume

Assume

Assume

Assume

Hence, our results generalize and improve the corresponding results of [

Assume that

We let

Assume that

then, (

We let

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, 11371367, and 11361012), PSF of China (nos. 2012M512162 and 2013T60934), NSF of Hunan Province (nos. 11JJ900, 12JJ9001, and 13JJ4098), the Education Foundation of Hunan Province (nos. 12C0541 and 13C084), the Science Foundation of Hengyang Normal University (no. 11B36), and the construct program of the key discipline in Hunan Province.