Theory and Modern Applications

# Existence and uniqueness of periodic solutions for a system of differential equations via operator methods

## Abstract

This article investigates the existence and uniqueness of periodic solutions for a new system of differential equations. By employing fixed point theorems for increasing φ-$$(h,\tau )$$-concave operators, we establish the existence of unique periodic solution for our differential system and then give a monotone iterative scheme to approximate the unique periodic solution. Some examples are presented in the end to illustrate the validity of our main results.

## Introduction

In this article, we investigate the existence and uniqueness of periodic solutions or positive periodic solutions for the following system of differential equations:

$$\textstyle\begin{cases} x'(t)=a(t)x(t)-f_{1}(t, x(t), y(t))+g_{1}(t), \\ y'(t)=-b(t)y(t)+f_{2}(t, x(t), y(t))-g_{2}(t),\end{cases}$$
(1.1)

where $$a, b\in C(\mathbf{R}, \mathbf{R}_{+})$$ are ω-periodic for some $$\omega >0$$, $$f_{1}(t,x,y), f_{2}(t,x,y) \in C(\mathbf{R}\times \mathbf{R}_{+}\times \mathbf{R}_{+}, \mathbf{R}_{+})$$ and $$g_{1}(t), g_{2}(t)\in C(\mathbf{R}, \mathbf{R}_{+})$$ are ω-periodic functions in t with $$g_{i}(t)\leq 1$$, $$i=1,2$$. Here we remark (1.1) is a new system in the context of one-order differential equations. By using recent fixed point theorems for increasing φ-$$(h,\tau )$$-concave operators, we not only get the existence and uniqueness of periodic solutions for (1.1), but also we can give convergent sequences which can approximate the unique solution. This is a significant improvement compared with some results in the literature. Different from other articles, we discuss a differential equation system by new operator methods.

During the past decades, many people have studied the theories of differential equations in economical, population dynamics, control, ecology and epidemiology; see the monographs  for example. Owing to its theoretical and practical significance, the study of periodic solutions for differential equations has been paid much attention to and it has a fast development in ordinary and partial differential equations; see the papers  and the references therein. In these papers, some good results have been established on the existence of periodic solutions. Very recently, there were some articles reported on the existence of periodic solutions for several systems of differential equations; see  for example. In , Radu Precup discussed the existence of multiple positive periodic solutions for the following differential system:

$$\textstyle\begin{cases} u'_{1}(t)=-a_{1}(t)u_{1}(t)+\epsilon _{1}f_{1}(t, u_{1}(t), u_{2}(t)), \\ u'_{2}(t)=-a_{2}(t)u_{2}(t)+\epsilon _{2}f_{2}(t, u_{1}(t), u_{2}(t)),\end{cases}$$
(1.2)

where for $$i \in \{1, 2\}: a_{i}\in C(\mathbf{R}, \mathbf{R})$$, $$\int ^{\omega }_{0} a_{i} \,dt \neq 0$$, $$\epsilon _{i}=\operatorname{sign} \int ^{\omega }_{0} a_{i}(t)\,dt$$, $$f_{i} \in C(\mathbf{R}\times \mathbf{R}^{2}_{+}, \mathbf{R}_{+})$$, and $$a_{i}$$, $$f_{i}(\cdot , u_{1}, u_{2})$$ are ω-periodic functions for some $$\omega > 0$$. The method used to resolve (1.2) is a different version of Krasnosel’skii’s fixed point theorem in cones.

In , the authors studied the following system of differential equations:

$$\textstyle\begin{cases} u'_{i}(t)=u_{i}(t)[a_{i}(t)-f_{i}(t, u(t), v(t))],\quad i=1,2,\ldots, n, \\ v'_{j}(t)=v_{j}(t)[b_{j}(t)+g_{j}(t, u(t), v(t))], \quad j=1,2,\ldots, m,\end{cases}$$
(1.3)

where $$u(t)=(u_{1}(t),u_{2}(t),\ldots,u_{n}(t))^{T}$$, $$v(t)=(v_{1}(t),v_{2}(t),\ldots,v_{m}(t))^{T}$$, and $$f_{i}$$, $$g_{j}$$ for $$i=1,2, \ldots, n$$, $$j=1,2,\ldots, m$$, are ω-periodic functions in t. By applying a fixed point theorem, they gave the existence of positive periodic solutions for system (1.3).

However, there are still few papers that studied periodic solutions for systems of differential equations and the uniqueness of solutions is seldom obtained in literature. Motivated by some recently published articles , we will study the uniqueness of periodic solutions for system (1.1). We will give the existence and uniqueness of periodic solutions or positive periodic solutions for system (1.1) and construct an iterative to approximate the unique solution. Also, we can get the existence and uniqueness of periodic solutions or positive periodic solutions for the following system:

$$\textstyle\begin{cases} x'(t)=-a(t)x(t)+f_{1}(t, x(t), y(t))-g_{1}(t), \\ y'(t)=b(t)y(t)-f_{2}(t, x(t), y(t))+g_{2}(t),\end{cases}$$
(1.4)

where a, b, $$f_{1}(t,x,y)$$, $$f_{2}(t,x,y)$$ and $$g_{1}(t)$$, $$g_{2}(t)$$ are the same as in (1.1).

From the articles mentioned above, we know that $$f_{i}-g_{i}$$ ($$i=1,2$$) in (1.1) and (1.4) may be nonnegative or negative, while the $$f_{i}$$ ($$i=1,2,\ldots,n$$), $$g_{j}$$ ($$j=1,2,\ldots,m$$) in (1.2) and (1.3) are always nonnegative. So systems (1.1), (1.4) are different from (1.2), (1.3) and other ones in the literature. Moreover, our results indicate that the unique periodic solution exists in a product set, and can be approximated by making an iterative sequence for any initial point in the product set.

## Preliminaries

We shall find a unique periodic solution for system (1.1) and for this purpose we use operator methods as in . For ω-periodic functions $$a, b\in C(\mathbf{R}, \mathbf{R}_{+})$$ and $$f_{1}, f_{2}\in C(\mathbf{R}, \mathbf{R})$$, from , the unique ω-periodic solution $$(x, y)$$ of the system

$$\textstyle\begin{cases} x'(t)=a(t)x(t)- f_{1}(t), \\ y'(t)=-b(t)y(t)+f_{2}(t), \end{cases}$$
(2.1)

can be written as

$$\textstyle\begin{cases} x(t)=\int _{t}^{t+\omega } H_{1}(t, s)f_{1}(s)\,ds, \\ y(t)=\int _{t}^{t+\omega } H_{2}(t, s)f_{2}(s)\,ds , \end{cases}$$
(2.2)

where

$$H_{1}(t,s)= \frac{e^{-\int ^{s}_{t} a(\xi )\,d\xi }}{1-e^{-\int ^{\omega }_{0} a(\xi )\,d\xi }}, \quad\quad H_{2}(t,s)= \frac{e^{\int ^{s}_{t} b(\xi )\,d\xi }}{e^{\int ^{\omega }_{0} b(\xi )\,d\xi }-1}, \quad (t,s)\in (\mathbf{R}, \mathbf{R}).$$
(2.3)

Set

\begin{aligned}& m_{1}= \min_{t \in [0, \omega ]} \int ^{t+\omega }_{t}H_{1}(t,s)\,ds, \quad\quad m_{2}= \min_{t \in [0, \omega ]} \int ^{t+\omega }_{t}H_{2}(t,s)\,ds, \\& M_{1}= \max_{t \in [0, \omega ]} \int ^{t+\omega }_{t}H_{1}(t,s)\,ds, \quad\quad M_{2}= \max_{t \in [0, \omega ]} \int ^{t+\omega }_{t}H_{2}(t,s)\,ds. \end{aligned}

Clearly, $$(x, y)$$ is a periodic solution of system (1.1) if and only if $$(x, y)$$ is a solution of the following integral equation system:

$$\textstyle\begin{cases} x(t)=\int _{t}^{t+\omega } H_{1}(t, s)f_{1}(s,x(s), y(s))\,ds-\int _{t}^{t+ \omega } H_{1}(t, s)g_{1}(s)\,ds, \\ y(t)=\int _{t}^{t+\omega } H_{2}(t, s)f_{2}(s, x(s), y(s))\,ds-\int _{t}^{t+ \omega } H_{2}(t, s)g_{2}(s)\,ds, \end{cases}$$
(2.4)

which can be regarded as an operator equation.

Now we present some notations, concepts and lemmas which have already become known in previous work; see  and the references therein. Let $$(E, \Vert \cdot \Vert )$$ be a real Banach space which is partially ordered by a cone $$P\subset E$$. For any $$x, y\in E$$, $$x\sim y$$ means that there are $$\alpha > 0$$ and $$\beta >0$$ such that $$\alpha x \leq y \leq \beta x$$. Take $$h> \theta$$ (i.e., $$h\geq \theta$$ and $$h\neq \theta$$), we consider a set $$P_{h}=\{x\in E\mid x\sim h\}$$. Clearly, $$P_{h}\subset P$$. Take another element $$\tau \in P$$ with $$\theta \leq \tau \leq h$$, we define $$P_{h,\tau }=\{x\in E\mid x+\tau \in P_{h}\}$$.

Next we list the definition of φ-$$(h,\tau )$$-concave operators and fixed point theorems for such operators, which are fundamental to our proofs of our results.

### Definition 2.1

(See )

Suppose that N: $$P_{h,\tau }\rightarrow E$$ is an operator which satisfies: for any $$x \in P_{h,\tau }$$ and $$\lambda \in (0, 1)$$, there exists $$\varphi (\lambda ) > \lambda$$ such that $$N(\lambda x +(\lambda - 1)\tau )\geq \varphi (\lambda )Nx+(\varphi ( \lambda )-1)\tau$$. Then we call N a φ-$$(h,\tau )$$-concave operator.

### Lemma 2.1

(See )

Assume thatPis a normal cone andNis an increasingφ-$$(h,\tau )$$-concave operator satisfying$$Nh\in P_{h,\tau }$$. ThenNhas a unique fixed point$$x^{*}$$in$$P_{h,\tau }$$. In addition, for any$$w_{0} \in P_{h,\tau }$$, constructing the sequence$$w_{n}=Nw_{n-1}$$, $$n=1,2,\ldots$$ , then$$\Vert w_{n}-x^{*} \Vert \rightarrow 0$$as$$n\rightarrow \infty$$.

### Lemma 2.2

(See )

Assume thatPis normal andNis an increasingφ-$$(h,\theta )$$-concave operator satisfying$$Nh\in P_{h}$$. ThenNhas a unique fixed point$$x^{*}$$in$$P_{h}$$. In addition, for any$$w_{0} \in P_{h}$$, constructing the sequence$$w_{n}=Nw_{n-1}$$, $$n=1,2,\ldots$$ , then$$\Vert w_{n}-x^{*} \Vert \rightarrow 0$$as$$n\rightarrow \infty$$.

For $$h_{1},h_{2}\in P$$ with $$h_{1},h_{2} \neq \theta$$. Let $$h=(h_{1},h_{2})$$, then $$h \in \overline{P}:=P\times P$$. Take $$\theta \leq \tau _{1} \leq h_{1}$$, $$\theta \leq \tau _{2} \leq h_{2}$$, and denote $$\overline{\theta }=(\theta , \theta )$$, $$\tau =(\tau _{1},\tau _{2})$$. Then $$\overline{\theta }=(\theta , \theta ) \leq (\tau _{1},\tau _{2}) \leq (h_{1},h_{2})=h$$. That is, $$\overline{\theta } \leq \tau \leq h$$. If P is normal, then $$\overline{P}=P\times P$$ is normal (see ).

### Lemma 2.3

()

$$\overline{P}_{h}=P_{h_{1}}\times P_{h_{2}}$$.

### Lemma 2.4

()

$$\overline{P}_{h,\tau }=P_{h_{1},\tau _{1}}\times P_{h_{2},\tau _{2}}$$.

## Existence and uniqueness of periodic solutions

In this section, we will prove the existence and uniqueness of periodic solutions for system (1.1). Let $$E=\{x\in C(\mathbf{R}, \mathbf{R}): x(t)=(t+\omega )\text{ for every }t \in \mathbf{R} \}$$, then E is a Banach space under the norm

$$\Vert x \Vert _{\infty }= \max_{t \in [0, \omega ]} \bigl\vert x(t) \bigr\vert .$$

We will discuss (1.1) in $$E\times E$$. For $$(x,y)\in E\times E$$, let $$\Vert (x, y) \Vert = \Vert x \Vert _{\infty } + \Vert y \Vert _{\infty }$$. Then $$(E\times E, \Vert (x, y) \Vert )$$ is also a Banach space. Moreover, let

$$\overline{P}=\bigl\{ (x, y) \in E\times E: x(t)\geq 0, y(t)\geq 0, t \in \mathbf{R} \bigr\} , \quad\quad P= \bigl\{ x \in E: x(t)\geq 0, t \in \mathbf{R} \bigr\} ,$$

then $$\overline{P}\subset E\times E$$ and $$\overline{P}=P\times P$$ is normal and $$E\times E$$ has a partial order: $$(x_{1},y_{1})\leq (x_{2},y_{2}) \Leftrightarrow x_{1}(t)\leq x_{2}(t)$$, $$y_{1}(t)\leq y_{2}(t)$$, $$t\in \mathbf{R}$$.

For $$(x,y)\in E\times E$$, we define an operator $$N=(N_{1} ,N_{2})$$ with

\begin{aligned}& N_{1}(x, y) (t) = \int _{t}^{t+\omega } H_{1}(t, s)f_{1} \bigl(s, x(s), y(s)\bigr)\,ds- \int _{t}^{t+\omega } H_{1}(t, s)g_{1}(s)\,ds, \\& N_{2}(x, y) (t) = \int _{t}^{t+\omega } H_{2}(t, s)f_{2} \bigl(s, x(s), y(s)\bigr)\,ds- \int _{t}^{t+\omega } H_{2}(t, s)g_{2}(s)\,ds. \end{aligned}

Then $$N_{1}, N_{2}: E\times E\rightarrow E$$ and $$N: E\times E\rightarrow E\times E$$. From (2.4), $$(x,y)$$ is an ω-periodic solution of system (1.1) if and only if $$(x,y)$$ is a fixed point of operator N.

To obtain our results, we first define several functions:

\begin{aligned}& \tau _{1}(t)= \int _{t}^{t+\omega } H_{1}(t, s)g_{1}(s)\,ds, \quad\quad \tau _{2}(t)= \int _{t}^{t+\omega } H_{2}(t, s)g_{2}(s)\,ds, \end{aligned}
(3.1)
\begin{aligned}& h_{1}(t)= \int _{t}^{t+\omega } H_{1}(t, s)\,ds, \quad\quad h_{2}(t)= \int _{t}^{t+ \omega } H_{2}(t, s)\,ds. \end{aligned}
(3.2)

### Remark 3.1

From (2.3), we can prove that $$\tau _{1}(t)$$, $$\tau _{2}(t)$$, $$h_{1}(t)$$ and $$h_{2}(t)$$ are ω-periodic functions. Moreover, it is easy to show $$\tau _{1}, \tau _{2}, h_{1}, h_{2} \in P$$.

### Theorem 3.1

Let$$\tau _{1}$$, $$\tau _{2}$$, $$h_{1}$$, $$h_{2}$$be given as in (3.1) and (3.2). Moreover, for$$i =1,2$$,

$$(H_{1})$$:

$$f_{i}(t,x,y)$$: $$\mathbf{R}\times [-\tau ^{\ast }_{1}, +\infty ) \times [-\tau ^{\ast }_{2}, +\infty )\rightarrow \mathbf{R}$$isω-periodic with respect to first variable, and increasing with respect to the second, third variables, where$$\tau ^{\ast }_{i}=\max \{\tau _{i}(t): t\in [0, \omega ] \}$$;

$$(H_{2})$$:

for$$\lambda \in (0, 1)$$, there exists$$\varphi (\lambda ) >\lambda$$such that

\begin{aligned}& f_{i}\bigl(t, \lambda x_{1}+(\lambda -1)x_{2}, \lambda y_{1}+(\lambda -1)y_{2}\bigr) \geq \varphi (\lambda )f_{i}(t,x_{1},y_{1}), \\& \quad t, x_{1}, y_{1}\in \mathbf{R}, x_{2}\in \bigl[0,\tau _{1}^{\ast }\bigr], y_{2}\in \bigl[0,\tau _{2}^{ \ast }\bigr]; \end{aligned}
$$(H_{3})$$:

$$f_{i}(t,0,0)\geq 0$$with$$f_{i}(t,0,0) \not \equiv 0$$for$$t\in [0, \omega ]$$.

Then:

1. (1)

system (1.1) has a unique periodic solution$$(x^{\ast }, y^{\ast })$$in$$\overline{P}_{h,\tau }$$, where

$$\tau (t)=\bigl(\tau _{1}(t), \tau _{2}(t)\bigr), \quad\quad h(t)=\bigl(h_{1}(t),h_{2}(t)\bigr), \quad t \in [0, \omega ];$$
2. (2)

for any point$$(x_{0}, y_{0})\in \overline{P}_{h,\tau }$$, we construct the following sequences:

\begin{aligned}& x_{n+1}(t) = \int _{t}^{t+\omega } H_{1}(t, s)f_{1} \bigl(s, x_{n}(s), y_{n}(s)\bigr)\,ds- \int _{t}^{t+\omega } H_{1}(t, s)g_{1}(s)\,ds, \\& y_{n+1}(t) = \int _{t}^{t+\omega } H_{2}(t, s)f_{2} \bigl(s, x_{n}(s), y_{n}(s)\bigr)\,ds- \int _{t}^{t+\omega } H_{2}(t, s)g_{2}(s)\,ds, \end{aligned}

$$n=0,1,2,\ldots$$ , and then we obtain$$x_{n+1}(t)\rightarrow x^{\ast }(t)$$, $$y_{n+1}(t)\rightarrow y^{ \ast }(t)$$as$$n\rightarrow \infty$$.

### Proof

From Remark 3.1, $$\tau =(\tau _{1}, \tau _{2})\in \overline{P}$$, $$h=(h_{1},h_{2})\in \overline{P}$$. Due to $$g_{i}(t)\leq 1$$, $$i=1, 2$$. For $$t\in \mathbf{R}$$,

\begin{aligned}& \tau _{1}(t) = \int _{t}^{t+\omega } g_{1}(s)H_{1}(t,s) \,ds \leq \int _{t}^{t+ \omega } H_{1}(t,s)\,ds = h_{1}(t), \\& \tau _{2}(t) = \int _{t}^{t+\omega } g_{2}(s)H_{2}(t,s) \,ds \leq \int _{t}^{t+ \omega } H_{2}(t,s)\,ds = h_{2}(t). \end{aligned}

So we get $$\tau _{1}\leq h_{1}$$, $$\tau _{2}\leq h_{2}$$ and thus $$\tau =(\tau _{1}, \tau _{2})\leq (h_{1},h_{2})=h$$.

Now we show that operator $$N:\overline{P}_{h,\tau }\rightarrow E\times E$$ is a φ-$$(h, \tau )$$-concave operator. For $$(x,y)\in \overline{P}_{h,\tau }$$ and $$\lambda \in (0,1)$$, we get

\begin{aligned} N\bigl(\lambda (x,y)+(\lambda -1)\tau \bigr) (t) =&N\bigl(\lambda (x,y)+(\lambda -1) \tau \bigr) (t) \\ =& \bigl(N_{1}\bigl(\lambda (x,y)+(\lambda -1)\tau \bigr), N_{2}\bigl(\lambda (x,y)+( \lambda -1)\tau \bigr) \bigr) (t). \end{aligned}

Hence we need to discuss $$N_{1}(\lambda (x,y)+(\lambda -1)\tau )(t)$$ and $$N_{2}(\lambda (x,y)+(\lambda -1)\tau )(t)$$, respectively. By considering $$(H_{2})$$,

\begin{aligned}& N_{1}\bigl(\lambda (x,y)+(\lambda -1)\tau \bigr) (t) \\& \quad = N_{1}\bigl(\lambda x+( \lambda -1)\tau _{1}, \lambda y+( \lambda -1)\tau _{2}\bigr) (t) \\& \quad = \int _{t}^{t+\omega } H_{1}(t,s)f_{1} \bigl(s, \lambda x(s)+(\lambda -1) \tau _{1}(s), \lambda y(s)+(\lambda -1)\tau _{2}(s)\bigr)\,ds-\tau _{1}(t) \\& \quad \geq \varphi (\lambda ) \int _{t}^{t+\omega } H_{1}(t,s)f_{1} \bigl(s,x(s),y(s)\bigr)\,ds- \tau _{1}(t) \\& \quad = \varphi (\lambda ) \biggl[ \int _{t}^{t+\omega } H_{1}(t,s)f_{1} \bigl(s,x(s),y(s)\bigr)\,ds- \tau _{1}(t) \biggr]+\varphi (\lambda ) \tau _{1}(t)-\tau _{1}(t) \\& \quad = \varphi (\lambda )N_{1}(x,y) (t)+\bigl[\varphi (\lambda )-1\bigr] \tau _{1}(t). \end{aligned}

Similarly,

\begin{aligned}& N_{2}\bigl(\lambda (x,y)+(\lambda -1)\tau \bigr) (t) \\& \quad = N_{2}\bigl(\lambda x+( \lambda -1)\tau _{1}, \lambda y+( \lambda -1)\tau _{2}\bigr) (t) \\& \quad = \int _{t}^{t+\omega } H_{2}(t,s)f_{2} \bigl(s, \lambda x(s)+(\lambda -1) \tau _{1}(s), \lambda y(s)+(\lambda -1)\tau _{2}(s)\bigr)\,ds-\tau _{2}(t) \\& \quad \geq \varphi (\lambda ) \int _{t}^{t+\omega } H_{2}(t,s)f_{2} \bigl(s,x(s),y(s)\bigr)\,ds- \tau _{2}(t) \\& \quad = \varphi (\lambda ) \biggl[ \int _{t}^{t+\omega } H_{2}(t,s)f_{2} \bigl(s,x(s),y(s)\bigr)\,ds- \tau _{2}(t) \biggr]+\varphi (\lambda )\tau _{2}(t)-\tau _{2}(t) \\& \quad = \varphi (\lambda )N_{2}(x,y) (t)+\bigl[\varphi (\lambda )-1\bigr] \tau _{2}(t). \end{aligned}

Hence,

\begin{aligned}& N\bigl(\lambda (x,y)+(\lambda -1)\tau \bigr) (t) \\& \quad \geq \bigl(\varphi ( \lambda )N_{1}(x,y) (t)+\bigl[\varphi (\lambda )-1\bigr]\tau _{1}(t), \varphi ( \lambda )N_{2}(x,y) (t)+\bigl[\varphi ( \lambda )-1\bigr]\tau _{2}(t) \bigr) \\& \quad = \bigl(\varphi (\lambda )N_{1}(x,y) (t), \varphi (\lambda )N_{2}(x,y) (t) \bigr)+ \bigl(\bigl(\varphi (\lambda )-1\bigr)\tau _{1}(t), \bigl(\varphi ( \lambda )-1\bigr)\tau _{2}(t) \bigr) \\& \quad = \varphi (\lambda ) \bigl(N_{1}(x,y) (t),N_{2}(x,y) (t) \bigr)+\bigl( \varphi (\lambda )-1\bigr) \bigl(\tau _{1}(t), \tau _{2}(t) \bigr) \\& \quad = \varphi (\lambda )N(x,y) (t)+\bigl(\varphi (\lambda )-1\bigr)\tau (t). \end{aligned}

That is,

$$N\bigl(\lambda (x,y)+(\lambda -1)\tau \bigr) \geq \varphi (\lambda )N(x,y)+ \bigl[ \varphi (\lambda )-1\bigr]\tau , \quad (x,y)\in \overline{P}_{h,\tau }, \lambda \in (0,1).$$

Therefore, we find that N is a φ-$$(h,\tau )$$-concave operator.

In the following, we prove that $$N: \overline{P}_{h,\tau }\rightarrow E\times E$$ is increasing. For $$(x,y)\in \overline{P}_{h,\tau }$$, one has $$(x,y)+\tau \in \overline{P}_{h}$$. From Lemma 2.3, $$(x+\tau _{1}, y+\tau _{2})\in P_{h_{1}}\times P_{h_{2}}$$, which means that there exist $$\lambda _{1}, \lambda _{2} >0$$ such that

$$x(t)+\tau _{1}(t)\geq \lambda _{1}h_{1}(t),\quad\quad y(t)+\tau _{2}(t)\geq \lambda _{2}h_{2}(t),\quad t \in\mathbf{R}.$$

Consequently, $$x(t)\geq \lambda _{1}h_{1}(t)-\tau _{1}(t) \geq -\tau _{1}(t) \geq - \tau ^{\ast }_{1}$$, $$y(t)\geq \lambda _{2}h_{2}(t)-\tau _{2}(t) \geq - \tau _{2}(t) \geq -\tau ^{\ast }_{2}$$. By considering $$(H_{1})$$ and the definitions of $$N_{1}$$, $$N_{2}$$, we know that $$N:\overline{P}_{h,\tau } \rightarrow E\times E$$ is increasing.

Now we show that the important condition $$Nh \in \overline{P}_{h,\tau }$$ is also satisfied. That is, we need to prove $$Nh+\tau \in \overline{P}_{h}$$. For any $$t \in \mathbf{R}$$,

\begin{aligned} Nh(t)+\tau (t) =& N(h_{1}, h_{2}) (t)+\tau (t) \\ =& \bigl(N_{1}(h_{1},h_{2}) (t),N_{2}(h_{1},h_{2}) (t)\bigr)+\bigl(\tau _{1}(t), \tau _{2}(t)\bigr) \\ =& \bigl(N_{1}(h_{1},h_{2}) (t)+\tau _{1}(t), N_{2}(h_{1},h_{2}) (t)+\tau _{2}(t)\bigr). \end{aligned}

Clearly, we need to discuss $$N_{1}(h_{1},h_{2})(t)+\tau _{1}(t)$$, $$N_{2}(h_{1},h_{2})(t)+\tau _{2}(t)$$, respectively. For convenience, we set

$$r_{i}=\min_{t \in [0,\omega ]}\bigl\{ f_{i}(t, m_{1}, m_{2})\bigr\} , \quad\quad R_{i}= \min _{t \in [0,\omega ]}\bigl\{ f_{i}(t, M_{1}, M_{2})\bigr\} , \quad i=1,2.$$

By $$(H_{1})$$ and $$(H_{3})$$, $$R_{1}\geq r_{1} >0$$, $$R_{2}\geq r_{2} > 0$$. Note that $$m_{i} \leq h_{i}(t) \leq M_{i}$$, $$i={1,2}$$, and from $$(H_{1})$$,

\begin{aligned} N_{1}(h_{1},h_{2}) (t)+\tau _{1}(t) =& \int _{t}^{t+\omega } H_{1}(t, s)f_{1} \bigl(s, h_{1}(s), h_{2}(s)\bigr)\,ds \\ \geq & \int _{t}^{t+\omega } H_{1}(t, s)f_{1}(s, m_{1}, m_{2})\,ds \\ =& r_{1} \int _{t}^{t+\omega } H_{1}(t, s)\,ds= r_{1}h_{1}(t) \end{aligned}

and

\begin{aligned} N_{1}(h_{1},h_{2}) (t)+\tau _{1}(t) =& \int _{t}^{t+\omega } H_{1}(t, s)f_{1} \bigl(s, h_{1}(s), h_{2}(s)\bigr)\,ds \\ \leq & \int _{t}^{t+\omega } H_{1}(t, s)f_{1}(s, M_{1}, M_{2})\,ds \\ =& R_{1} \int _{t}^{t+\omega } H_{1}(t, s)\,ds= R_{1}h_{1}(t). \end{aligned}

So we get $$r_{1}h_{1} \leq N_{1}(h_{1},h_{2})+\tau _{1} \leq R_{1}h_{1}$$, i.e., $$N_{1}(h_{1},h_{2})+\tau _{1} \in P_{h_{1}}$$. Similarly, we can obtain $$N_{2}(h_{1},h_{2})+\tau _{2} \in P_{h_{2}}$$. Hence, by Lemma 2.4,

$$N(h_{1},h_{2})+\tau =\bigl(N_{1}(h_{1},h_{2})+ \tau _{1}, N_{2}(h_{1},h_{2})+ \tau _{2}\bigr) \in P_{h_{1}}\times P_{h_{2}}= \overline{P_{h}}.$$

Finally, an application of Lemma 2.1 implies that N has a unique fixed point $$(x^{\ast }, y^{\ast }) \in \overline{P}_{h,\tau }$$. And, for any given $$(x_{0},y_{0})\in \overline{P}_{h,\tau }$$, the sequence

$$(x_{n},y_{n})=N(x_{n-1},y_{n-1})=\bigl(N_{1}(x_{n-1},y_{n-1}), N_{2}(x_{n-1},y_{n-1})\bigr), \quad n=1, 2,\ldots,$$

converges to $$(x^{\ast }, y^{\ast })$$ as $$n\rightarrow \infty$$. Therefore, system (1.1) has a unique periodic solution $$(x^{\ast }, y^{\ast })$$ in $$\overline{P}_{h,\tau }$$; and choosing any initial point $$(x_{0},y_{0})\in \overline{P}_{h, \tau }$$, we have the following sequences:

\begin{aligned}& x_{n}(t) = \int _{t}^{t+\omega } H_{1}(t, s)f_{1} \bigl(s, x_{n-1}(s), y_{n-1}(s)\bigr)\,ds- \int _{t}^{t+\omega } H_{1}(t, s)g_{1}(s)\,ds, \\& y_{n}(t) = \int _{t}^{t+\omega } H_{2}(t, s)f_{2} \bigl(s, x_{n-1}(s), y_{n-1}(s)\bigr)\,ds- \int _{t}^{t+\omega } H_{2}(t, s)g_{2}(s)\,ds, \end{aligned}

$$n=1,2,\ldots$$ , satisfying $$x_{n+1}\rightarrow x^{\ast }$$, $$y_{n+1}\rightarrow y^{\ast }$$ as $$n \rightarrow \infty$$. □

Let

$$H_{1}(t,s)= \frac{e^{\int ^{s}_{t} a(\xi )\,d\xi }}{e^{\int ^{\omega }_{0} a(\xi )\,d\xi }-1}, \quad\quad H_{2}(t,s)= \frac{e^{-\int ^{s}_{t} b(\xi )\,d\xi }}{1-e^{-\int ^{\omega }_{0} b(\xi )\,d\xi }}, \quad (t,s)\in (\mathbf{R}, \mathbf{R}).$$
(3.3)

Similar to the proof of Theorem 3.1, we can obtain the following conclusion.

### Theorem 3.2

Let$$\tau _{1}$$, $$\tau _{2}$$, $$h_{1}$$, $$h_{2}$$be given as in (3.1) and (3.2) with$$H_{1}$$, $$H_{2}$$are replaced by (3.3). Assume that the conditions$$(H_{1})$$$$(H_{3})$$hold. Then:

1. (1)

system (1.4) has a unique periodic solution$$(x^{\ast }, y^{\ast })$$in$$\overline{P}_{h,\tau }$$, where

$$\tau (t)=\bigl(\tau _{1}(t), \tau _{2}(t)\bigr), \quad\quad h(t)=\bigl(h_{1}(t),h_{2}(t)\bigr),\quad t \in [0, \omega ];$$
2. (2)

for any point$$(x_{0}, y_{0})\in \overline{P}_{h,\tau }$$, construct the following sequences:

\begin{aligned}& x_{n+1}(t) = \int _{t}^{t+\omega } H_{1}(t, s)f_{1} \bigl(s, x_{n}(s), y_{n}(s)\bigr)\,ds- \int _{t}^{t+\omega } H_{1}(t, s)g_{1}(s)\,ds, \\& y_{n+1}(t) = \int _{t}^{t+\omega } H_{2}(t, s)f_{2} \bigl(s, x_{n}(s), y_{n}(s)\bigr)\,ds- \int _{t}^{t+\omega } H_{2}(t, s)g_{2}(s)\,ds, \end{aligned}

$$n=0,1,2,\ldots$$ , and then one has$$x_{n+1}(t)\rightarrow x^{\ast }(t)$$, $$y_{n+1}(t)\rightarrow y^{ \ast }(t)$$as$$n\rightarrow \infty$$.

When $$g_{1}(t)=g_{2}(t)\equiv 0$$, we can get the uniqueness of positive periodic solutions for systems (1.1) and (1.4) by using Lemma 2.2. The proofs are similar to Theorem 3.1.

### Corollary 3.1

Let$$h_{1}$$, $$h_{2}$$be given as in (3.2). Moreover, for$$i =1,2$$,

$$(H_{4})$$:

$$f_{i}(t,x,y)$$: $$\mathbf{R}\times \mathbf{R}_{+} \times \mathbf{R}_{+}\rightarrow \mathbf{R}_{+}$$isω-periodic with respect to first variable, and increasing with respect to the second, third variables;

$$(H_{5})$$:

for$$\lambda \in (0, 1)$$, there exists$$\varphi (\lambda ) >\lambda$$such that

$$f_{i}(t, \lambda x, \lambda y) \geq \varphi (\lambda )f_{i}(t,x,y), \quad t, x, y\in \mathbf{R};$$
$$(H_{6})$$:

$$f_{i}(t,0,0) \not \equiv 0$$for$$t\in [0, \omega ]$$.

Then system (1.1) has a unique positive periodic solution$$(x^{\ast }, y^{\ast })$$in$$\overline{P}_{h}$$, where$$h(t)=(h_{1}(t),h_{2}(t))$$, $$t\in \mathbf{R}$$. Further, for any point$$(x_{0}, y_{0})\in \overline{P}_{h}$$, make the following sequences:

\begin{aligned}& x_{n+1}(t) = \int _{t}^{t+\omega } H_{1}(t, s)f_{1} \bigl(s, x_{n}(s), y_{n}(s)\bigr)\,ds, \\& y_{n+1}(t) = \int _{t}^{t+\omega } H_{2}(t, s)f_{2} \bigl(s, x_{n}(s), y_{n}(s)\bigr)\,ds, \end{aligned}

$$n=0,1,2,\ldots$$ , and then we get$$x_{n+1}(t)\rightarrow x^{\ast }(t)$$, $$y_{n+1}(t)\rightarrow y^{ \ast }(t)$$as$$n\rightarrow \infty$$.

### Corollary 3.2

Let$$h_{1}$$, $$h_{2}$$be given as in (3.2) with$$H_{1}$$, $$H_{2}$$are replaced by (3.3). Assume that the conditions$$(H_{4})$$$$(H_{6})$$hold. Then system (1.4) has a unique positive periodic solution$$(x^{\ast }, y^{\ast })$$in$$\overline{P}_{h}$$, where$$h(t)=(h_{1}(t),h_{2}(t))$$, $$t\in \mathbf{R}$$. Further, for any point$$(x_{0}, y_{0})\in \overline{P}_{h}$$, put the following sequences:

\begin{aligned}& x_{n+1}(t) = \int _{t}^{t+\omega } H_{1}(t, s)f_{1} \bigl(s, x_{n}(s), y_{n}(s)\bigr)\,ds, \\& y_{n+1}(t) = \int _{t}^{t+\omega } H_{2}(t, s)f_{2} \bigl(s, x_{n}(s), y_{n}(s)\bigr)\,ds, \end{aligned}

$$n=0,1,2,\ldots$$ , and then we obtain$$x_{n+1}(t)\rightarrow x^{\ast }(t)$$, $$y_{n+1}(t)\rightarrow y^{ \ast }(t)$$as$$n\rightarrow \infty$$.

### Remark 3.2

The form of differential system (1.1) is more general. Our method is new to the study of nonlinear systems of differential equations, which gives the existence and uniqueness of periodic solutions. Moreover, the unique periodic solution can be approximated by an iteration.

### Remark 3.3

By using the same discussion as with Theorem 3.1 and Corollary 3.1, we can consider the following differential equation:

$$x'(t)=a(t)x(t)-f\bigl(t, x(t)\bigr)+g(t),$$

where $$a\in C(\mathbf{R}, \mathbf{R}_{+})$$ is ω-periodic for some $$\omega >0$$, $$f(t,x) \in C(\mathbf{R}\times \mathbf{R}_{+}, \mathbf{R}_{+})$$ and $$g(t)\in C(\mathbf{R}, \mathbf{R}_{+})$$ are ω-periodic functions in t with $$g(t)\leq 1$$. We can also give the existence and uniqueness of periodic solutions or positive periodic solutions. Moreover, the unique periodic solution can be also approximated by making an iterative sequence.

## Examples

In this section, we present two simple examples to illustrate the main results.

### Example 4.1

Consider the simple system of differential equations:

$$\textstyle\begin{cases} x'(t)=ax(t)-[y(t)+\frac{1}{4b}]^{\frac{1}{3}}\sin ^{2} t+\frac{1}{2}, \\ y'(t)=-by(t)+[x(t)+\frac{1}{2a}]^{\frac{1}{3}}\cos ^{2} t-\frac{1}{4},\end{cases}$$
(4.1)

where $$a,b>0$$. In this example, we let

$$f_{1}(t,y)=\biggl(y+\frac{1}{4b}\biggr)^{\frac{1}{3}}\sin ^{2}t, \quad\quad f_{2}(t,x)=\biggl(x+ \frac{1}{2a} \biggr)^{\frac{1}{3}}\cos ^{2}t, \quad\quad g_{1}(t)= \frac{1}{2}, \quad\quad g_{2}(t)=\frac{1}{4}$$

and they are π-periodic functions in t. By direct calculation,

\begin{aligned}& H_{1}(t,s)=\frac{e^{-a(s-t)}}{1-e^{-a\omega }}, \quad\quad H_{2}(t,s)= \frac{e^{b(s-t)}}{e^{b\omega } -1}, \\& \tau _{1}(t)= \int _{t}^{t+\pi } H_{1}(t, s)g_{1}(s)\,ds=\frac{1}{2a}, \quad\quad \tau _{2}(t)= \int _{t}^{t+\pi } H_{2}(t, s)g_{2}(s)\,ds=\frac{1}{4b}, \\& h_{1}(t)= \int _{t}^{t+\pi } H_{1}(t, s)\,ds= \frac{1}{a}, \quad\quad h_{2}(t)= \int _{t}^{t+\pi } H_{2}(t, s)\,ds= \frac{1}{b}, \end{aligned}

and thus $$\tau _{1}\leq h_{1}$$, $$\tau _{2}\leq h_{2}$$, $$\tau _{1}^{*}= \frac{1}{2a}$$, $$\tau _{2}^{*}=\frac{1}{4b}$$. It is easy to see that $$f_{1}(t,y):\mathbf{R}\times [-\frac{1}{4b}, +\infty )\rightarrow \mathbf{R}$$ and $$f_{2}(t,x):\mathbf{R}\times [-\frac{1}{2a}, +\infty )\rightarrow \mathbf{R}$$ are π-periodic with respect to first variable and increasing with respect to the second variable. In addition,

$$f_{1}(t,0)=\biggl(\frac{1}{4b}\biggr)^{\frac{1}{3}} \sin ^{2} t \not \equiv 0, \quad\quad f_{2}(t,0)=\biggl( \frac{1}{2a}\biggr)^{\frac{1}{3}}\cos ^{2} t\not \equiv 0, \quad t \in \mathbf{R}.$$

Hence, the conditions $$(H_{1})$$, $$(H_{3})$$ are satisfied.

In the following, we show that $$(H_{2})$$ holds. Let $$\varphi (\lambda )=\lambda ^{\frac{1}{3}}$$, then $$\varphi (\lambda )>\lambda$$ for $$\lambda \in (0,1)$$, and for $$x_{1},y_{1} \in \mathbf{R}$$, $$x_{2} \in [0, \frac{1}{2a}]$$, $$y_{2} \in [0, \frac{1}{4b}]$$,

\begin{aligned}& \begin{aligned} f_{1}\bigl(t,\lambda y_{1}+(\lambda -1)y_{2} \bigr) &= \biggl[\lambda y_{1}+(\lambda -1)y_{2}+ \frac{1}{4b}\biggr]^{\frac{1}{3}}\sin ^{2}t \\ &= \lambda ^{\frac{1}{3}}\biggl[y_{1}+\biggl(1-\frac{1}{\lambda } \biggr)y_{2}+ \frac{1}{\lambda }\frac{1}{4b}\biggr]^{\frac{1}{3}} \sin ^{2}t \\ &\geq \lambda ^{\frac{1}{3}}\biggl[y_{1}+\biggl(1- \frac{1}{\lambda }\biggr)\frac{1}{4b}+ \frac{1}{\lambda }\frac{1}{4b} \biggr]^{\frac{1}{3}}\sin ^{2}t \\ &= \lambda ^{\frac{1}{3}}\biggl[y_{1}+\frac{1}{4b} \biggr]^{\frac{1}{3}}\sin ^{2}t \\ &= \varphi (\lambda )f_{1}(t,y_{1}), \end{aligned} \\& \begin{aligned} f_{2}\bigl(t,\lambda x_{1}+(\lambda -1)x_{2} \bigr) &= \biggl[\lambda x_{1}+(\lambda -1)x_{2}+ \frac{1}{2a}\biggr]^{\frac{1}{3}}\cos ^{2}t \\ &= \lambda ^{\frac{1}{3}}\biggl[x_{1}+\biggl(1-\frac{1}{\lambda } \biggr)x_{2}+ \frac{1}{\lambda }\frac{1}{2a}\biggr]^{\frac{1}{3}} \cos ^{2}t \\ &\geq \lambda ^{\frac{1}{3}}\biggl[x_{1}+\biggl(1- \frac{1}{\lambda }\biggr)\frac{1}{2a}+ \frac{1}{\lambda }\frac{1}{2a} \biggr]^{\frac{1}{3}}\cos ^{2}t \\ &= \lambda ^{\frac{1}{3}}\biggl[x_{1}+\frac{1}{2a} \biggr]^{\frac{1}{3}}\cos ^{2}t \\ &= \varphi (\lambda )f_{2}(t,x_{1}). \end{aligned} \end{aligned}

So, the condition $$(H_{2})$$ is satisfied. By Theorem 3.1, system (4.1) has a unique periodic solution $$(x^{*},y^{*})$$ in $$\overline{P}_{h,\tau }$$, where

$$\tau (t)=\bigl(\tau _{1}(t), \tau _{2}(t)\bigr)=\biggl( \frac{1}{2a}, \frac{1}{4b}\biggr), \quad\quad h(t)= \bigl(h_{1}(t),h_{2}(t)\bigr)=\biggl( \frac{1}{a}, \frac{1}{b}\biggr).$$

Take any initial point $$(x_{0},y_{0})\in \overline{P}_{h,\tau }$$, making the sequences:

\begin{aligned} x_{n+1}(t) =& \int _{t}^{t+\pi }\frac{e^{-a(s-t)}}{1-e^{-a\omega }} \biggl\{ \biggl[y_{n}(s)+\frac{1}{4b}\biggr]^{\frac{1}{3}}\sin ^{2}t-\frac{1}{2} \biggr\} \,ds, \\ y_{n+1}(t) =& \int _{t}^{t+\pi }\frac{e^{b(s-t)}}{e^{b\omega } -1} \biggl\{ \biggl[x_{n}(s)+\frac{1}{2a}\biggr]^{\frac{1}{3}}\cos ^{2}t-\frac{1}{4} \biggr\} \,ds, \end{aligned}

$$n=0,1,2,\ldots$$ , one has $$x_{n+1}\rightarrow x^{*}(t)$$, $$y_{n+1}\rightarrow y^{*}(t)$$ as $$n \rightarrow \infty$$.

### Example 4.2

Consider the following system of differential equations:

$$\textstyle\begin{cases} x'(t)=a(t)x(t)-[x(t)+y^{2}(t)+1]^{\frac{1}{4}}\sin ^{2} t, \\ y'(t)=-b(t)y(t)-[x^{2}(t)+y^{3}(t)+2]^{\frac{1}{6}}\cos ^{2} t,\end{cases}$$
(4.2)

where $$a(t),b(t)\geq 0$$ and $$a(t)$$, $$b(t)$$ are π-periodic in t. In this example, we let

$$f_{1}(t,x,y)=\bigl[x(t)+y^{2}(t)+1\bigr]^{\frac{1}{4}}\sin ^{2} t, \quad\quad f_{2}(t,x,y)=\bigl[x^{2}(t)+y^{3}(t)+2 \bigr]^{\frac{1}{6}}\cos ^{2} t,$$

and they are π-periodic functions in t. Moreover,

$$f_{1}(t,0,0)=\sin ^{2} t \not \equiv 0, \quad\quad f_{2}(t,0,0)=2\cos ^{2} t \not \equiv 0, \quad t \in \mathbf{R}.$$

So the $$f_{i}(t,x,y)$$: $$\mathbf{R}\times \mathbf{R}_{+} \times \mathbf{R}_{+}\rightarrow \mathbf{R}_{+}$$ satisfy $$(H_{4})$$ and $$(H_{6})$$ in Corollary 3.1. Next, we prove that the condition $$(H_{5})$$ also holds. Let $$\varphi (\lambda )=\lambda ^{\frac{1}{2}}$$, then $$\varphi (\lambda )>\lambda$$, $$\lambda \in (0,1)$$. And for $$\lambda \in (0,1)$$ and $$x,y \geq 0$$,

\begin{aligned}& f_{1}(t,\lambda x, \lambda y)= \bigl(\lambda x+ \lambda ^{2} y^{2}+1\bigr)^{\frac{1}{4}}\sin ^{2}t\geq \lambda ^{\frac{1}{2}}\bigl(x+y^{2}+1\bigr)\sin ^{2}t= \varphi (\lambda )f_{1}(t,x,y), \\& f_{2}(t,\lambda x, \lambda y)= \bigl(\lambda ^{2} x^{2}+ \lambda ^{3} y^{3}+2\bigr)^{\frac{1}{6}} \cos ^{2}t\geq \lambda ^{\frac{1}{2}}\bigl(x^{2}+y^{3}+2 \bigr)\cos ^{2}t= \varphi (\lambda )f_{2}(t,x,y). \end{aligned}

Hence the condition $$(H_{5})$$ is satisfied. By Corollary 3.1, system (4.2) has a unique positive periodic solution $$(x^{\ast }, y^{\ast })$$ in $$\overline{P}_{h}$$, where $$h(t)=(h_{1}(t), h_{2}(t))$$, $$h_{1}(t) = \int _{t}^{t+\pi } H_{1}(t,s)\,ds$$, $$h_{2}(t) = \int _{t}^{t+\pi } H_{2}(t,s)\,ds$$ with

$$H_{1}(t,s)= \frac{e^{-\int ^{t}_{s}a(s)\,ds}}{1-e^{-\int ^{\pi }_{0}a(s)\,ds}}, \quad\quad H_{2}(t,s)= \frac{e^{\int ^{t}_{s}a(s)\,ds}}{e^{-\int ^{\pi }_{0}b(s)\,ds}-1}.$$

Further, for any point $$(x_{0},y_{0}) \in \overline{P}_{h}$$, put the following sequences:

\begin{aligned}& x_{n+1}(t) = \int _{t}^{t+\pi } H_{1}(t, s) \bigl[x_{n}(s)+y^{2}_{n}(s)+1\bigr]^{\frac{1}{4}} \sin ^{2} s\,ds, \\& y_{n+1}(t) = \int _{t}^{t+\pi } H_{2}(t, s) \bigl[x^{2}_{n}(s)+y_{n}^{3}(s)+2 \bigr]^{\frac{1}{6}}\cos ^{2} s\,ds, \end{aligned}

$$n=0,1,2,\ldots$$ , and then we get $$x_{n+1}(t)\rightarrow x^{*}(t)$$, $$y_{n+1}(t)\rightarrow y^{*}(t)$$ as $$n\rightarrow \infty$$.

## References

1. Gopalsamy, K.: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic, Boston (1992)

2. Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001)

3. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)

4. Murray, J.D.: Mathematical Biology I: An Introduction. Springer, New York (2002)

5. Padhi, S., Graef, J.R., Srinivasu, P.D.N.: Periodic Solutions of First-Order Functional Differential Equations in Population Dynamics. Springer, Berlin (2014)

6. Jiang, D., Wei, J.: Existence of positive periodic solutions of nonautonomous functional differential equations. Chin. Ann. Math., Ser. A 20(6), 715–720 (1999)

7. Cheng, S., Zhang, G.: Existence of positive periodic solutions for nonautonomous functional differential equations. Electron. J. Differ. Equ. 2001, 59 (2001)

8. Liu, X., Li, W.: Existence and uniqueness of positive periodic solutions of functional differential equations. J. Math. Anal. Appl. 293, 28–39 (2004)

9. Wang, H.: Positive periodic solutions of functional differential equations. J. Differ. Equ. 202, 354–366 (2004)

10. Bai, D., Xu, Y.: Periodic solutions of first order functional differential equations with periodic deviations. Comput. Math. Appl. 53, 1361–1366 (2007)

11. Li, Y., Fan, X., Zhao, L.: Positive periodic solutions of functions differential equations with impulses and a parameter. Comput. Math. Appl. 56, 2556–2560 (2008)

12. Wang, J., Xiang, X., Wei, W., Chen, Q.: Existence and global asymptotical stability of periodic solution for the T-periodic logistic system with time-varying generating operators and $$T_{0}$$-periodic impulsive perturbations on Banach spaces. Discrete Dyn. Nat. Soc. 2008, Article ID 524945 (2008)

13. Weng, A., Sun, J.: Positive periodic solutions of first-order functions differential equations with parameter. J. Comput. Appl. Math. 229, 327–332 (2009)

14. Kang, S., Shi, B., Wang, G.: Existence of maximal and minimal periodic solutions for first-order functions differential equations. Appl. Math. Lett. 23, 22–25 (2010)

15. Liu, B., Liu, L., Wu, Y.: Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem. Nonlinear Anal. 72, 3337–3345 (2010)

16. Cao, F., Han, Z., Sun, S.: Existence of periodic solutions for p-Laplacian equations on time scales. Adv. Differ. Equ. 2010, Article ID 584375 (2010)

17. Ma, R., Chen, R., He, Z.: Positive periodic solutions of second-order differential equations with weak singularities. Appl. Math. Comput. 232, 97–103 (2014)

18. Ma, R., Lu, Y.: Existence of positive periodic solutions for second-order functional differential equations. Monatshefte Math. 173, 67–81 (2014)

19. Yang, C., Zhai, C., Hao, M.: Existence and uniqueness of positive periodic solutions for a first-order functional differential equation. Adv. Differ. Equ. 2015, 5 (2015)

20. Hai, D.D., Qian, C.: On positive periodic solutions for nonlinear delayed differential equations. Mediterr. J. Math. 13, 1641–1651 (2016)

21. Kang, S.: Existence and uniqueness of positive periodic solutions for a class of integral equations with mixed monotone nonlinear terms. Appl. Math. Lett. 71, 24–29 (2017)

22. Lv, L., Cheng, Z.: Positive periodic solution to superlinear neutral differential equation with time-dependent parameter. Appl. Math. Lett. 98, 271–277 (2019)

23. Precup, R.: A vector version of Krasnosel’skii’s fixed point theorem in cones and positive periodic solutions of nonlinear systems. J. Fixed Point Theory Appl. 2, 141–151 (2007)

24. Lv, X., Lu, S., Yan, P.: Existence and global attractivity of positive periodic solutions of Lotka–Volterra predator–prey systems with deviating arguments. Nonlinear Anal., Real World Appl. 11, 574–583 (2010)

25. Pati, S., Graef, J.R., Padhi, S.: Positive periodic solutions to a system of nonlinear differential equations with applications to Lotka–Volterra-type ecological models with discrete and distributed delays. J. Fixed Point Theory Appl. 21, 80 (2019)

26. Prados, C.L., Precup, R.: Positive periodic solutions for Lotka–Volterra systems with a general attack rate. Nonlinear Anal. 52, 103024 (2020)

27. Yang, C., Zhai, C., Zhang, L.: Local uniqueness of positive solutions for a coupled system of fractional differential equations with integral boundary conditions. Adv. Differ. Equ. 2017, 282 (2017)

28. Zhai, C., Jiang, R.: Unique solutions for a new coupled system of fractional equations. Adv. Differ. Equ. 2018, 1 (2018)

29. Zhai, C., Wang, W., Li, H.: A uniqueness method to a new Hadamard fractional differential system with four-point boundary conditions. J. Inequal. Appl. 2018, 207 (2018)

30. Zhai, C., Ren, J.: The unique solution for a fractional q-difference equation with three-point boundary conditions. Indag. Math. New Ser. 29, 948–961 (2018)

31. Zhai, C., Zhu, X.: Unique solution for a new system of fractional differential equations. Adv. Differ. Equ. 2019, 394 (2019)

32. Ren, J., Zhai, C.: Unique solutions for fractional q-difference boundary value problems via a fixed point method. Bull. Malays. Math. Sci. Soc. 42, 1507–1521 (2019)

33. Wang, L., Zhai, C.: Unique solutions for new fractional differential equations with p-Laplacian and infinite-point boundary conditions. Int. J. Dyn. Syst. Differ. Equ. 9(1), 1–13 (2019)

34. Zhai, C., Wang, W.: Solutions for a system of Hadamard fractional differential equations with integral conditions. Numer. Funct. Anal. Optim. 41(2), 209–229 (2020)

35. Zhai, C., Wang, L.: φ-$$(h, e)$$-concave operators and applications. J. Math. Anal. Appl. 454, 571–584 (2017)

36. Zhai, C., Wang, F.: Properties of positive solutions for the operator equation $$Ax = \lambda x$$ and applications to fractional differential equations with integral boundary conditions. Adv. Differ. Equ. 2015, 366 (2015)

37. Zhai, C., Ren, J.: Some properties of sets, fixed point theorems in ordered product spaces and applications to a nonlinear system of fractional differential equations. Topol. Methods Nonlinear Anal. 49(2), 625–645 (2017)

### Acknowledgements

The authors wish to thank the anonymous referees for their valuable suggestions.

Not applicable.

## Funding

This paper was supported financially by Shanxi Province Science Foundation (201901D111020) and Graduate Science and Technology Innovation Project of Shanxi (2019BY014).

## Author information

Authors

### Contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Chengbo Zhai.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

## Rights and permissions 