Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces

Abstract

This paper deals with the existence of $\omega$-periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E. $L_{n}u\left(t\right)=f(t,u\left(t\right),u(t-\tau )),t\in \mathbb{R},$ where $L_{n}u\left(t\right):=u^{\left(n\right)}\left(t\right)+{\displaystyle \sum _{i=0}^{n-1}}{a}_i{u}^{\left(i\right)}\left(t\right)$, $a_i\in \mathbb{R}$, $i=0,1,\cdots ,n-1$, are constants, $f(t,x,y):\mathbb{R}\times E\times E⟶E$ is continuous and $\omega$-periodic with respect to t, $\tau >0$. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions.

1. Introduction

The properties of periodic solutions of differential equations are significant problems in application science. A great number of works have focused on the existence of periodic solutions of differential equations, but they mainly studied the self-adjoint equations. For the case of non-self-adjoint differential equations, the researches are seldom because of their complex spectral structure. Since the nth-order differential equations are typical non-self-adjoint differential equations, it is very important both in theory and practice to prove the existence theorems of periodic solutions for nth-order ordinary differential equations. Recently, there are many beautiful results are obtained, for instance, see Cabada [1,2,3], Li [4,5,6], Liu [7] and V. Seda [8] and the references therein. The higher-order differential equation and its application in optimization and control theory were also studied, see [9,10,11] and the references therein. In some publications, the maximum principle is essential in the proof of main results. In [4], by using the obtained maximum principle, Li extended the results of Cabada in [1,2,3] and proved some existence results for the nth-order periodic boundary value problem of ordinary differential equations. Later, Li in [5] discussed the existence as well as the uniqueness of solutions for the nth-order periodic boundary value problem under spectral conditions. The maximum principle was also used in [6] to deal with the periodic boundary value problem of nth-order ordinary differential equation

$$\left\{\begin{array}{c}{L}_{n}u\left(t\right)=f(t,u\left(t\right)),0\le t\le \omega ,\hfill \\ u^{\left(i\right)}\left(0\right)=u^{\left(i\right)}\left(\omega \right),i=0,1,2,\cdots ,n-1,\hfill \end{array}\right.$$

where $L_{n}u\left(t\right):=u^{\left(n\right)}\left(t\right)+{\displaystyle \sum _{i=0}^{n-1}}{a}_i{u}^{\left(i\right)}\left(t\right)$, $a_i\in \mathbb{R}$, $i=0,1,\cdots ,n-1$, are constants, $f:[0,\omega ]\times \mathbb{R}⟶\mathbb{R}$ is a continuous mapping. By using the obtained maximum principle, the author proved some existence and uniqueness theorems. In [7], Liu investigated the existence results of periodic solutions for the two special cases of nth-order delay differential equation by applying the coincidence degree theory, but the above mentioned literatures did not consider the periodic solutions for the general delayed differential equations in abstract spaces.

In the present work, we consider the existence as well as the uniqueness of $\omega$-periodic solutions for nth-order ordinary differential equation involving delay in Banach space E

$$L_{n}u\left(t\right)=f(t,u\left(t\right),u(t-\tau )),t\in \mathbb{R},$$

(1)

where $f(t,x,y):\mathbb{R}\times E\times E⟶E$ is a continuous mapping and it is $\omega$-periodic with respect to t and $\tau >0$. Firstly, we establish the maximum principle to the corresponding linear delayed equation

$$L_{n}u\left(t\right)+bu(t-\tau )=h\left(t\right),\forall t\in \mathbb{R},$$

where $h:\mathbb{R}\to E$ is an $\omega$-periodic continuous function and $b\ge 0$ is a constant. Then, by applying the obtained maximum principle, some existence and uniqueness theorems are proved by applying the fixed point approach and monotone iterative technique.

The next

Table 1. Symbols used in this paper.

E	the ordered and separable Banach space
$\mathbb{R}$	$(-\infty ,+\infty )$
${\mathbb{R}}^{+}$	$[0,+\infty )$
$\mathbb{N}$	the set of natural number
$\mathbb{C}$	the complex plane

describes several symbols which will be later used within the body of the manuscript.

2. Preliminaries

Let $J:=[0,\omega ]$ and $C_{\omega }(\mathbb{R},\mathbb{R})$ be the set of all continuous and $\omega$-periodic functions. Then $C_{\omega }(\mathbb{R},\mathbb{R})$ is a Banach space equipped with norm ${\parallel u\parallel }_C:=\underset{t\in [0,\omega ]}{max}\left|u\left(t\right)\right|$ and $C(J,\mathbb{R})$ is also the Banach space. In general, $C^n(J,\mathbb{R})$ is the Banach space of nth-order continuous and differentiable functions.

For all $h\in C(J,\mathbb{R})$, we know that the linear periodic boundary value problem(LPBVP)

$$\left\{\begin{array}{c}{L}_{n}u\left(t\right)=h\left(t\right),t\in J,\hfill \\ u^{\left(i\right)}\left(0\right)=u^{\left(i\right)}\left(\omega \right)\hfill \end{array}\right.$$

(2)

possesses a unique solution $u\in C^n(J,\mathbb{R})$:

$$u\left(t\right)={\int }_0^{\omega }{G}_n(t,s)h\left(s\right)ds,$$

where

$$G_n(t,s)=\left\{\begin{array}{c}{r}_n(t-s),0\le s\le t\le \omega ,\hfill \\ r_n(\omega +t-s),0\le t\le s\le \omega ,\hfill \end{array}\right.$$

where $r_n\left(t\right)\in C^{\infty }(I,\mathbb{R})$ is the unique solution of the LBVP

$$\left\{\begin{array}{c}{L}_{n}v\left(t\right)=0,t\in J,\hfill \\ v^{\left(i\right)}\left(0\right)=v^{\left(i\right)}\left(\omega \right),i=0,1,\dots \dots ,n-2,\hfill \\ v^{(n-1)}\left(0\right)-v^{(n-1)}\left(\omega \right)=-1.\hfill \end{array}\right.$$

(3)

Let $P_n\left(\lambda \right)$ be the characteristic polynomial of $L_n$ defined by

$$P_n\left(\lambda \right)={\lambda }^n+a_{n-1}{\lambda }^{n-1}+\dots +a_0.$$

(4)

And let $\mathcal{N}\left(P_n\left(\lambda \right)\right)$ be the set of null points of $P_n\left(\lambda \right)$ in $\mathbb{C}$. For the LBVP (3), we assume the following hypothesis.

Hypothesis 1 (H1).

$\mathcal{N}\left(P_n\left(\lambda \right)\right)\cap \{\frac{2k\pi }{\omega }i:k=0,\pm 1,\pm 2,\dots \}=\varnothing$.

Lemma 1.

If the Hypothesis 1 (H1) holds, then the LBVP (3) possesses a unique solution $r_n\left(t\right)\in C^{\infty }(I,\mathbb{R})$.

Proof of Lemma 1. 

Denote by $U\left(t\right):={(u\left(t\right),u^{{}^{\prime }}\left(t\right),\dots u^{(n-1)}\left(t\right))}^T$ and $B:={(0,0,\dots 0,1)}^T$. Then the LBVP (3) equivalents to the linear system

$$\left\{\begin{array}{c}{U}^{{}^{\prime }}\left(t\right)=AU\left(t\right),t\in I,\hfill \\ U\left(0\right)-U\left(\omega \right)=B,\hfill \end{array}\right.$$

(5)

where A is defined by

$$A=\left(\begin{array}{ccccccc}0& 1& 0& .& .& .& 0\\ 0& 0& 1& .& .& .& 0\\ .& .& .& .& .& .& .\\ .& .& .& .& .& .& .\\ .& .& .& .& .& .& .\\ 0& 0& 0& .& .& .& 1\\ -a_0& -a_1& -a_2& .& .& .& -a_{n-1}\end{array}\right).$$

If we take $U\left(0\right)$ as the initial value, the first equation of (5) has a unique solution expressed by

$$U\left(t\right)=e^{tA}U\left(0\right).$$

Then $U\left(t\right)$ satisfies $U\left(0\right)-U\left(\omega \right)=B$ if and only if

$$(I-e^{\omega A})U\left(0\right)=B.$$

By the Hypothesis 1 (H1), we know that ${(I-e^{\omega A})}^{-1}$ exists and

$$U\left(0\right)={(I-e^{\omega A})}^{-1}B.$$

This implies that the linear system (5) has a unique solution

$$V_0\left(t\right)=e^{tA}\left({(I-e^{\omega A})}^{-1}B\right).$$

(6)

The first component of $V_0\left(t\right)$ is denoted by $r_n\left(t\right),$ then it follows from (6) that $r_n\left(t\right)\in C^{\infty }(I,\mathbb{R})$ and it is a unique solution of the LBVP (3). □

Remark 1.

Clearly, for each $h\in C(J,\mathbb{R})$, the LPBVP (2) possesses a unique solution if and only if the LBVP(3) has a unique solution.

Lemma 2.

If the Hypothesis 1 (H1) holds, then for each $h\in C_{\omega }(\mathbb{R},\mathbb{R})$ and $t\in \mathbb{R}$, the linear equation

$$L_{n}u\left(t\right)=h\left(t\right)$$

(7)

possesses a unique solution $u:=T_{n}h\in C_{\omega }^n(\mathbb{R},\mathbb{R})$, and $T_n:C_{\omega }(\mathbb{R},\mathbb{R})⟶C_{\omega }(\mathbb{R},\mathbb{R})$ is a bounded linear operator satisfying $\parallel T_n\parallel =\frac{1}{|a_0|}$ when $a_0\ne 0$.

Proof of Lemma 2. 

If $h\in C_{\omega }(\mathbb{R},\mathbb{R})$, since the $\omega$-periodic solution of (7) is equivalent to the solution of the LPBVP (2), by Lemma 1, the linear Equation (7) possesses a unique $\omega$-periodic solution

$$u\left(t\right)={\int }_{t-\omega }^t{G}_n(t,s)h\left(s\right)ds:=\left(T_{n}h\right)\left(t\right),t\in \mathbb{R}.$$

Clearly, $T_n:C_{\omega }(\mathbb{R},\mathbb{R})⟶C_{\omega }^n(\mathbb{R},\mathbb{R})\hookrightarrow C_{\omega }(\mathbb{R},\mathbb{R})$, and

$$\begin{array}{ccc}\hfill \parallel T_{n}h\left(t\right)\parallel & \le & \parallel {\int }_0^{\omega }{G}_n{(t,s)ds\parallel \parallel h\parallel }_C\hfill \\ & =& \parallel {\int }_0^{\omega }{r}_n{\left(s\right)ds\parallel \parallel h\parallel }_C\hfill \\ & =& \frac{1}{|a_0|}{\parallel h\parallel }_C,\hfill \end{array}$$

that is, $\parallel T_n\parallel \le \frac{1}{|a_0|}$.

On the other hand, let $h_0\left(t\right)\equiv 1$. Then $h_0\in C_{\omega }(\mathbb{R},\mathbb{R})$ and $\parallel h_0{\parallel }_C=1$. Thus,

$$\begin{array}{ccc}\hfill \parallel T_n{h}_0\left(t\right)\parallel & =& \parallel {\int }_{t-\omega }^t{G}_n(t,s)h_0\left(s\right)ds\parallel \hfill \\ & =& \parallel {\int }_0^{\omega }{G}_n(t,s)ds\parallel \hfill \\ & =& \frac{1}{|a_0|},\hfill \end{array}$$

that is, $\parallel T_n\parallel \ge \frac{1}{|a_0|}$. Consequently, we obtain $\parallel T_n\parallel =\frac{1}{|a_0|}$. □

Let $(E,\parallel ·\parallel ,\le )$ be an ordered and separable Banach space, $K:=\{x\in E:x\ge \theta \}$ be a positive cone of E, where $\theta$ denotes the zero element of E. Then K is a normal cone with the constant N. Denote by $C_{\omega }(\mathbb{R},E)$ the set of E-valued continuous and $\omega$-periodic functions. Then $C_{\omega }(\mathbb{R},E)$ is a Banach space whose norm is defined by ${\parallel u\parallel }_C:=\underset{t\in [0,\omega ]}{max}\parallel u\left(t\right)\parallel$ for every $u\in C_{\omega }(\mathbb{R},E)$. Let $K_C:=C_{\omega }(\mathbb{R},K)=\{u\in C_{\omega }(\mathbb{R},E):u\left(t\right)\in K,\forall t\in \mathbb{R}\}$. Then $K_C$ is also a normal cone with the same constant of cone K, and $C_{\omega }(\mathbb{R},E)$ is an ordered Banach space. Generally, $C_{\omega }^n(\mathbb{R},E)$ is the Banach space of all $\omega$-periodic and nth-order continuous differentiable functions for $n\in \mathbb{N}$.

Now, for any $h\in C_{\omega }(\mathbb{R},E)$, we consider the linear delayed differential equation(LDDE)

$$L_{n}u\left(t\right)+bu(t-\tau )=h\left(t\right),t\in \mathbb{R},$$

(8)

where $b\ge 0$ and $\tau >0$ are constants.

For $G_n(t,s)$, if $r_n\left(t\right)>0$, $G_n(t,s)>0$. Hence, by Lemma 2, when $r_n\left(t\right)>0$ and the Hypothesis 1 (H1) holds, the operator $T_n:C_{\omega }(\mathbb{R},E)\to C_{\omega }(\mathbb{R},E)$ is a positive operator. Let $m_n:=\underset{t\in I}{min}r{}_n\left(t\right)$ and $M_n:=\underset{t\in I}{max}{r}_n\left(t\right)$. It is clear that $0<m_n\le r_n\left(t\right)\le M_n$. By Lemma 2, we obtain the following lemma.

Lemma 3.

Let the Hypothesis 1 (H1) hold, $a_0>0,0\le b<\frac{m_n{a}_0}{M_n}$. Then for any $h\in C_{\omega }(\mathbb{R},E)$, the LDDE (8) possesses a unique ω-periodic solution $u:=Th\in C_{\omega }(\mathbb{R},E)$ satisfying $\parallel T\parallel \le \frac{1}{a_0-b}$. Furthermore, if $r_n\left(t\right)>0$, $T:C_{\omega }(\mathbb{R},E)\to C_{\omega }(\mathbb{R},E)$ is a linear bounded and positive operator.

Proof of Lemma 3. 

By Lemma 2, it is easy to see that the LDDE (8) possesses a solution

$$u\left(t\right)={\int }_{t-\omega }^t{G}_n(t,s)[h\left(s\right)-bu(s-\tau )]ds.$$

(9)

Define $B_1:C_{\omega }(\mathbb{R},E)⟶C_{\omega }(\mathbb{R},E)$ by

$$B_{1}u\left(t\right)=bu(t-\tau ).$$

(10)

Obviously, $B_1:C_{\omega }(\mathbb{R},E)⟶C_{\omega }(\mathbb{R},E)$ is a linear operator and $\parallel B_1\parallel \le b$. Then (9) and (10) yield

$$\begin{array}{ccc}\hfill u\left(t\right)& =& {\int }_{t-\omega }^t{G}_n(t,s)h\left(s\right)ds-{\int }_{t-\omega }^t{G}_n(t,s)B_{1}u\left(s\right)ds\hfill \\ & =& \left(T_{n}h\right)\left(t\right)-\left(T_n{B}_{1}u\right)\left(t\right).\hfill \end{array}$$

This implies that

$$(I+T_n{B}_1)u\left(t\right)=\left(T_{n}h\right)\left(t\right).$$

(11)

Since $\parallel T_n{B}_1\parallel \le \parallel T_n\parallel ·\parallel B_1\parallel \le \frac{b}{a_0}<1$, the perturbation theorem yields that ${(I+T_n{B}_1)}^{-1}$ exists and

$${(I+T_n{B}_1)}^{-1}=\sum _{i=0}^{\infty }{(-1)}^i{\left(T_n{B}_1\right)}^i=\sum _{i=0}^{\infty }{\left(T_n{B}_1\right)}^{2i}(I-T_n{B}_1),$$

(12)

which implies

$$\parallel {(I+T_n{B}_1)}^{-1}\parallel \le \frac{1}{1-\parallel T_n{B}_1\parallel }\le \frac{a_0}{a_0-b}.$$

Hence, by (11), we conclude that

$$u\left(t\right)={(I+T_n{B}_1)}^{-1}\left(T_{n}h\right)\left(t\right):=\left(Th\right)\left(t\right),t\in \mathbb{R}.$$

(13)

Consequently, $u\left(t\right)$ is an $\omega$-periodic solution of the LDDE (8). It follows from (13) that

$$\begin{array}{ccc}\hfill \parallel Th\left(t\right)\parallel & =& \parallel {(I+T_n{B}_1)}^{-1}{T}_{n}h\left(t\right)\parallel \hfill \\ & \le & \parallel {(I+T_n{B}_1)}^{-1}\parallel \parallel T_n{\parallel \parallel h\parallel }_C\hfill \\ & \le & \frac{a_0}{a_0-b}·\frac{1}{a_0}·{\parallel h\parallel }_C\hfill \\ & =& \frac{1}{a_0-b}{\parallel h\parallel }_C.\hfill \end{array}$$

Hence

$$\parallel T\parallel \le \frac{1}{a_0-b}.$$

Next, we prove that $T:C_{\omega }(\mathbb{R},E)\to C_{\omega }(\mathbb{R},E)$ is a positive operator when $r_n\left(t\right)>0$. By (12), for any $h\in C_{\omega }(\mathbb{R},K)$, we have

$$Th\left(t\right)=\sum _{i=0}^{\infty }{\left(T_n{B}_1\right)}^{2i}(I-T_n{B}_1)\left(T_{n}h\right)\left(t\right),t\in \mathbb{R}.$$

Form the above equality, it remains to prove the positivity of $(I-T_n{B}_1)T_n$. Since

$$\left(T_{n}h\right)\left(t\right)={\int }_{t-\omega }^t{G}_n(t,s)h\left(s\right)ds\ge m_n{\int }_0^{\omega }h\left(s\right)ds$$

and

$$\left(T_{n}h\right)\left(t\right)\le M_n{\int }_0^{\omega }h\left(s\right)ds,$$

it follows that

$$\begin{array}{ccc}\hfill (I-T_n{B}_1)\left(T_{n}h\right)\left(t\right)& =& T_{n}h\left(t\right)-\left(T_n{B}_1\right)\left(T_{n}h\right)\left(t\right)\hfill \\ & \ge & (m_n-\frac{b{M}_n}{a_0}){\int }_0^{\omega }h\left(s\right)ds.\hfill \end{array}$$

The condition $h\in C_{\omega }(\mathbb{R},K)$ implies $h\left(t\right)\not\equiv 0$ for $t\in \mathbb{R}$. Then there exist a small interval $[c,d]\subset [0,\omega ]$ and a constant $\epsilon >0$ such that

$$h\left(t\right)>\epsilon ,t\in [c,d].$$

Hence, ${\int }_0^{\omega }h\left(s\right)ds\ge {\int }_c^{d}h\left(s\right)ds>\epsilon (d-c)>0$ and

$$(I-T_n{B}_1)\left(T_{n}h\right)\left(t\right)\ge (m_n-\frac{b{M}_n}{a_0})\epsilon (d-c)>0.$$

Consequently, the operator $T:C_{\omega }(\mathbb{R},E)⟶C_{\omega }(\mathbb{R},E)$ is positive. □

In Lemma 3, the condition $r_n\left(t\right)>0$ is essential. We now introduce a condition to guarantee $r_n\left(t\right)>0$ for all $t\in \mathbb{R}$:

Hypothesis 2 (H2).

$\mathcal{N}\left(P_n\left(\lambda \right)\right)\subset \{\xi \in \mathbb{C}:|Im\xi |<\frac{\pi }{\omega }\}$. See Theorem 1.5 of [6] for more detail.

Lemma 4.

Assume that $a_0>0$ and $P_n\left(\lambda \right)$, defined by (4), satisfies the Hypothesis 2 (H2). Then the Hypothesis 1 (H1) holds and the LBVP (3) possesses a unique solution $r_n\left(t\right)>0$ for all $t\in \mathbb{R}$.

Hence, from Lemmas 3 and 4, the following lemma is easy to obtain.

Lemma 5.

Let the Hypothesis 2 (H2) hold and $a_0>0,0\le b<\frac{m_n{a}_0}{M_n}$. If $u\in C_{\omega }(\mathbb{R},E)$ satisfies

$$L_{n}u\left(t\right)+bu(t-\tau )\ge \theta ,t\in \mathbb{R},$$

then, for any $t\in \mathbb{R}$, $u\left(t\right)\ge \theta$.

Proof of Lemma 5. 

Let $h\left(t\right):=L_{n}u\left(t\right)+bu(t-\tau )\ge \theta$. Then $h\in C_{\omega }(\mathbb{R},K)$. So, for any $t\in \mathbb{R}$, Lemma 3 yields $u\left(t\right)\ge \theta$. □

Let ${\beta }_E(·)$ and ${\beta }_C(·)$ denote the Kuratowski’s measure of non-compactness(MNC) of bounded subsets in E and $C_{\omega }(\mathbb{R},E)$, respectively. For every bounded subset $D\subset C_{\omega }(\mathbb{R},E)$, ${\beta }_E\left(D\left(t\right)\right)\le {\beta }_C\left(D\right)$ for all $t\in \mathbb{R}$, where $D\left(t\right):=\left\{u\right(t):u\in D\}$. For more detail of the MNC, we refer to [12,13] and the references therein. The following lemmas can be found in [12,14], which are more useful in our arguments.

Lemma 6.

Let D be a equicontinuous and bounded subset of $C(J,E)$. Then ${\beta }_E\left(D\left(t\right)\right)$ is continuous and

$${\beta }_C\left(D\right)=\underset{t\in J}{max}{\beta }_E\left(D\left(t\right)\right).$$

Lemma 7.

Let D be bounded in E. Then there is a countable subset $D_0$ in D such that

$${\beta }_E\left(D\right)\le 2{\beta }_E\left(D_0\right).$$

Lemma 8.

Let E be a separable Banach space and $D=\left\{u_n\right\}$ be a countable and bounded subset of $C(J,E)$. Then ${\beta }_E\left(D\left(t\right)\right)$ is Lebesgue integrable on J and

$${\beta }_E\left(\left\{{\int }_J{u}_n\left(t\right)dt\right\}\right)\le 2{\int }_J{\beta }_E\left(D\left(t\right)\right)dt.$$

By Lemma 3, we present the definition of $\omega$-periodic solution of Equation (1) as follows.

Definition 1.

A function $u\in C_{\omega }(\mathbb{R},E)$ is called an ω-periodic solution of Equation (1) if it satisfies the integral equation

$$u\left(t\right)={(I+T_n{B}_1)}^{-1}{\int }_{t-\omega }^t{G}_n(t,s)[f(s,u\left(s\right),u(s-\tau ))+bu(s-\tau )]ds,t\in \mathbb{R},$$

where $b\ge 0$ is a constant and $B_1:C_{\omega }(\mathbb{R},E)⟶C_{\omega }(\mathbb{R},E)$ is defined as in (10).

To end this section, we introduce the definitions of lower and upper $\omega$-periodic solutions of Equation (1).

Definition 2.

If $v_0\in C_{\omega }^n(\mathbb{R},E)$ satisfies

$$L_n{v}_0\left(t\right)\le f(t,v_0\left(t\right),v_0(t-\tau )),\forall t\in \mathbb{R},$$

(14)

then it is called the lower ω-periodic solution of Equation (1). If we inverse the inequality in (14), then it is called the upper ω-periodic solution of Equation (1).

3. The Method of Upper and Lower Solutions and the Monotone Iterative Technique

In this section, by utilizing the Sadovskii’s fixed point theorem, we first consider the existence of $\omega$-periodic solutions of Equation (1) between the lower and upper $\omega$-periodic solutions. Then the monotone iterative technique is applied to study the existence as well as the uniqueness of $\omega$-periodic solutions of Equation (1). At last, A sufficient condition is established for the existence of lower and upper $\omega$-periodic solutions of the Equation (1).

At first, we make the following assumptions:

Hypothesis 3 (H3).

There is $b\in [0,\frac{m_n{a}_0}{M_n})$ such that

$$f(t,x,y)-f(t,v_0\left(t\right),v_0(t-\tau ))\ge b(v_0(t-\tau )-y)$$

and

$$f(t,x,y)-f(t,w_0\left(t\right),w_0(t-\tau ))\le b(w_0(t-\tau )-y)$$

for all $t\in \mathbb{R}$, $v_0\left(t\right)\le x\le w_0\left(t\right)$ and $v_0(t-\tau )\le y\le w_0(t-\tau )$.

Hypothesis 4 (H4).

There is $L_1\in (0,\frac{a_0-b}{8})$ such that

$${\beta }_E(f(t,D_1,D_2)+b{D}_2)\le L_1({\beta }_E\left(D_1\right)+{\beta }_E\left(D_2\right)),t\in \mathbb{R},$$

for any countable subsets $D_i\subset E,i=1,2$.

Theorem 1.

Let the Hypothesis 2 (H2) hold and $a_0>0$. If Equation (1) possesses lower and upper ω-periodic solutions $v_0$ and $w_0$ satisfying $v_0\le w_0$, and the Hypothesis 3 (H3) and Hypothesis 4 (H4) are satisfied, then Equation (1) possesses at least one ω-periodic solution on $\mathbb{R}$.

Proof of Theorem 1. 

Since Equation (1) can be rewritten as

$$L_{n}u\left(t\right)+bu(t-\tau )=f(t,u\left(t\right),u(t-\tau ))+bu(t-\tau ),t\in \mathbb{R},$$

by Lemma 3 and Definition 1, we define $Q:C_{\omega }(\mathbb{R},E)⟶C_{\omega }(\mathbb{R},E)$ by

$$\left(Qu\right)\left(t\right)={(I+T_n{B}_1)}^{-1}{\int }_{t-\omega }^t{G}_n(t,s)\left[f(s,u(s),u(s-\tau \left)\right)+bu(s-\tau )\right]ds,t\in \mathbb{R}.$$

(15)

Let $D=[v_0,w_0]$. It is obvious that $D\subset C_{\omega }(\mathbb{R},E)$ is nonempty bounded, convex and closed. We will apply the approach of fixed point to discuss the existence of fixed points of Q in D. These fixed points are the $\omega$-periodic solutions of Equation (1) between $v_0$ and $w_0$ due to Lemma 3 and Definition 1.

First of all, we prove $Q\left(D\right)\subset D$. Let $u\in D$. Then

$$v_0\left(t\right)\le u\left(t\right)\le w_0\left(t\right)$$

and

$$v_0(t-\tau )\le u(t-\tau )\le w_0(t-\tau )$$

for all $t\in \mathbb{R}$. By the Hypothesis 3 (H3), we have

$$f(t,u\left(t\right),u(t-\tau ))+bu(t-\tau )\ge f(t,v_0\left(t\right),v_0(t-\tau ))+b{v}_0(t-\tau ),t\in \mathbb{R}$$

and

$$f(t,u\left(t\right),u(t-\tau ))+bu(t-\tau )\le f(t,w_0\left(t\right),w_0(t-\tau ))+b{w}_0(t-\tau ),t\in \mathbb{R}.$$

For any $u\in D$, let $v=Qu$, then

$$\begin{array}{ccc}& & L_n(v-v_0)\left(t\right)+b(v-v_0)(t-\tau )\hfill \\ & =& L_{n}v\left(t\right)+bv(t-\tau )-(L_n{v}_0\left(t\right)+b{v}_0(t-\tau ))\hfill \\ & \ge & f(t,u\left(t\right),u(t-\tau ))+bu(t-\tau )-f(t,v_0\left(t\right),v_0(t-\tau ))-b{v}_0(t-\tau )\hfill \\ & \ge & 0\hfill \end{array}$$

and

$$\begin{array}{ccc}& & L_n(w_0-v)\left(t\right)+b(w_0-v)(t-\tau )\hfill \\ & =& L_n{w}_0\left(t\right)+b{w}_n(t-\tau )-(L_{n}v\left(t\right)+bv(t-\tau ))\hfill \\ & \ge & f(t,w_0\left(t\right),w_0(t-\tau ))+b{w}_0(t-\tau )-f(t,u\left(t\right),u(t-\tau ))-bu(t-\tau )\hfill \\ & \ge & 0.\hfill \end{array}$$

By Lemma 5, it follows that $v-v_0\ge \theta$ and $w_0-v\ge \theta$. Therefore, $v=Qu\in [v_0,w_0]$, that is, $Q\left(D\right)\subset D$ is continuous.

Secondly, we prove the equi-continuity of $Q\left(D\right)$. For any $u\in D$ and $0\le t_1\le t_2\le \omega$, since $r_n\in C^{\infty }(I,\mathbb{R})$, by the definition of $G_n(t,s)$, we have

$$|G_n(t_2,s)-G_n(t_1,s)|\to 0,(t_2-t_1\to 0).$$

Together this fact with the definition of Q, we obtain that

$$\parallel \left(Qu\right)\left(t_2\right)-\left(Qu\right)\left(t_1\right)\parallel \to 0$$

as $t_2-t_1\to 0$ independent of $u\in D$. Therefore, the set $Q\left(D\right)$ is equi-continuous.

It remains to prove that $Q:D⟶D$ is a condensing mapping. By Lemma 7, since $Q\left(D\right)$ is bounded, there is a countable subset $D_0\subset D$ such that

$${\beta }_E\left(Q\left(D\right)\left(t\right)\right)\le 2{\beta }_E\left(Q\left(D_0\right)\left(t\right)\right).$$

Hence, Lemma 8 and the Hypothesis 4 (H4) yield

$$\begin{array}{ccc}\hfill {\beta }_E\left(Q\left(D_0\right)\left(t\right)\right)& =& {\beta }_E\left({(I+T_n{B}_1)}^{-1}{\int }_{t-\omega }^t{G}_n(t,s)[f(s,D_0\left(s\right),D_0(s-\tau ))+b{D}_0(s-\tau )]ds\right)\hfill \\ & \le & \parallel {(I+T_n{B}_1)}^{-1}\parallel {\beta }_E\left({\int }_{t-\omega }^t{G}_n(t,s)[f(s,D_0\left(s\right),D_0(s-\tau ))+b{D}_0(s-\tau )]ds\right)\hfill \\ & \le & \frac{2a_0}{a_0-b}{\int }_{t-\omega }^t{G}_n(t,s){\beta }_E\left(f(s,D_0\left(s\right),D_0(s-\tau ))+b{D}_0(s-\tau )\right)ds\hfill \\ & \le & \frac{2a_0}{a_0-b}{\int }_{t-\omega }^t{G}_n(t,s)\left[L_1({\beta }_E\left(D_0\left(s\right)\right)+{\beta }_E\left(D_0(s-\tau )\right))\right]ds\hfill \\ & \le & \frac{4a_0{L}_1}{a_0-b}{\int }_0^{\omega }{G}_n(t,s){\beta }_C\left(D_0\right)ds\hfill \\ & \le & \frac{4L_1}{a_0-b}{\beta }_C\left(D\right).\hfill \end{array}$$

By the equi-continuity and boundedness of $Q\left(D\right)$, we have

$${\beta }_C\left(Q\left(D\right)\right)=\underset{t\in I}{max}{\beta }_E\left(Q\left(D\right)\left(t\right)\right)\le 2\underset{t\in I}{max}{\beta }_E\left(Q\left(D_0\right)\left(t\right)\right)\le \frac{8L_1}{a_0-b}{\beta }_C\left(D\right).$$

Since $\frac{8L_1}{a_0-b}<1$, it follows that $Q:D⟶D$ is a condensing operator. Therefore, the Sadovskii’s fixed point theorem guarantees that there is at least one fixed point of Q in D. So, the Equation (1) possesses at least one $\omega$-periodic solution in D. □

If we replace the conditions Hypothesis 3 (H3) and Hypothesis 4 (H4) in Theorem 1 by

Hypothesis 5 (H5).

There is a constant $b\in [0,\frac{m_n{a}_0}{M_n})$ such that

$$f(t,x_2,y_2)-f(t,x_1,y_1)\ge b(y_1-y_2),\forall t\in \mathbb{R}$$

for any $v_0\left(t\right)\le x_1\le x_2\le w_0\left(t\right)$ and $v_0(t-\tau )\le y_1\le y_2\le w_0(t-\tau )$.

Hypothesis 6 (H6).

There is a constant $L_2\in (0,\frac{a_0-b}{4})$ such that

$${\beta }_E\left(\{f(t,u_n\left(t\right),u_n(t-\tau ))+b{u}_n(t-\tau )\}\right)\le L_2({\beta }_E\left(\left\{u_n\left(t\right)\right\}\right)+{\beta }_E\left(\left\{u_n(t-\tau )\right\}\right),\forall t\in \mathbb{R}$$

for every monotonous sequence $\left\{u_n\right\}\subset [v_0,w_0]$. Then we can obtain the following theorem by utilizing the monotone iterative technique.

Theorem 2.

Let the Hypothesis 2 (H2) hold and $a_0>0$. If Equation (1) possesses lower and upper ω-periodic solutions $v_0$ and $w_0$ satisfying $v_0\le w_0$, and the conditions Hypothesis 5 (H5) and Hypothesis 6 (H6) are satisfied, then there exist minimal and maximal ω-periodic solutions $\underline{u},\overline{u}$ of Equation (1) between $v_0$ and $w_0$. Moreover, $\underline{u}$ and $\overline{u}$ can be derived by iterative sequences starting from $v_0$ and $w_0$, respectively.

Proof of Theorem 2. 

We first prove that Q has properties:

(i)

$v_0\le Q{v}_0,Q{w}_0\le w_0$,

(ii)

$Q{u}_1\le Q{u}_2$ for all $u_1,u_2\in [v_0,w_0]$ satisfying $u_1\le u_2$,

where the operator Q is defined as in (15).

Let $v_1:=Q{v}_0$. Then

$$L_n{v}_1\left(t\right)+b{v}_1(t-\tau )=f(t,v_0\left(t\right),v_0(t-\tau ))+b{v}_0(t-\tau ),t\in \mathbb{R}.$$

Hence Definition 2 yields

$$\begin{array}{ccc}& & L_n(v_1-v_0)\left(t\right)+b(v_1-v_0)(t-\tau )\hfill \\ & =& L_n{v}_1\left(t\right)+b{v}_1(t-\tau )-(L_n{v}_0\left(t\right)+b{v}_0(t-\tau ))\hfill \\ & \ge & f(t,v_0\left(t\right),v_0(t-\tau ))+b{v}_0(t-\tau )-f(t,v_0\left(t\right),v_0(t-\tau ))+b{v}_0(t-\tau )\hfill \\ & =& 0.\hfill \end{array}$$

It follows from Lemma 5 that $v_0\left(t\right)\le v_1\left(t\right)=\left(Q{v}_0\right)\left(t\right)$ for each $t\in \mathbb{R}$. On the other hand, let $w_1:=Q{w}_0$. Then

$$L_n{w}_1\left(t\right)+b{w}_1(t-\tau )=f(t,w_0\left(t\right),w_0(t-\tau ))+b{w}_0(t-\tau ),t\in \mathbb{R}.$$

Hence, we have

$$\begin{array}{ccc}& & L_n(w_0-w_1)\left(t\right)+b(w_0-w_1)(t-\tau )\hfill \\ & \ge & f(t,w_0\left(t\right),w_0(t-\tau ))+b{w}_0(t-\tau )-f(t,w_0\left(t\right),w_0(t-\tau ))-b{w}_0(t-\tau )\hfill \\ & =& 0.\hfill \end{array}$$

Lemma 5 implies $\left(Q{w}_0\right)\left(t\right)=w_1\left(t\right)\le w_0\left(t\right)$ for all $t\in \mathbb{R}$. Hence, $\left(i\right)$ is satisfied.

For any $u_1,u_2\in [v_0,w_0]$ with $u_1\le u_2$, owing to Hypothesis 5 (H5), we have

$$f(t,u_1\left(t\right),u_1(t-\tau ))+b{u}_1(t-\tau )\le f(t,u_2\left(t\right),u_2(t-\tau ))+b{u}_2(t-\tau ),t\in \mathbb{R}.$$

By (15), $Q{u}_1\le Q{u}_2$. Hence, $\left(ii\right)$ holds.

Secondly, let

$$v_n=Q{v}_{n-1},w_n=Q{w}_{n-1},n=1,2,\cdots .$$

(16)

Then, we deduce from $\left(i\right)$ and $\left(ii\right)$ that

$$v_0\le v_1\le \cdots \le v_n\le \cdots \le w_n\le \cdots \le w_1\le w_0.$$

(17)

By the countability and boundedness of $\left\{v_n\right\}$, we conclude from Lemma 8 and Hypothesis 6 (H6) that

$$\begin{array}{ccc}& & {\beta }_E\left(\left\{v_n\left(t\right)\right\}\right)={\beta }_E\left(\left\{\left(Q{v}_{n-1}\right)\left(t\right)\right\}\right)\hfill \\ & =& {\beta }_E\left(\left\{{(I+T_n{B}_1)}^{-1}{\int }_{t-\omega }^t{G}_n(t,s)[f(s,v_{n-1}\left(s\right),v_{n-1}(s-\tau ))+b{v}_{n-1}(s-\tau )]ds\right\}\right)\hfill \\ & \le & \frac{2a_0}{a_0-b}{\int }_{t-\omega }^t{G}_n(t,s){\beta }_E\left(\{f(s,v_{n-1}\left(s\right),v_{n-1}(t-\tau )+b{v}_{n-1}(t-\tau )\}\right)ds\hfill \\ & \le & \frac{2a_0{L}_2}{a_0-b}{\int }_{t-\omega }^t{G}_n(t,s)\left[{\beta }_E\left(\left\{v_{n-1}\left(s\right)\right\}\right)+{\beta }_E\left(\left\{v_{n-1}(t-\tau )\right\}\right)\right]ds\hfill \\ & \le & \frac{4L_2}{a_0-b}{\beta }_C\left(\left\{v_n\right\}\right).\hfill \end{array}$$

Furthermore, $\left\{v_n\right\}$ is equi-continuous, by Lemma 6, we get

$$0\le {\beta }_C\left(\left\{v_n\right\}\right)=\underset{t\in I}{max}{\beta }_E\left(\left\{v_n\left(t\right)\right\}\right)\le \frac{4L_2}{a_0-b}{\beta }_C\left(\left\{v_n\right\}\right).$$

Hence $\beta \left(\left\{v_n\right\}\right)=0$ due to $\frac{4L_2}{a_0-b}<1$. Similarly, we obtain ${\beta }_C\left(\left\{w_n\right\}\right)=0$. Hence, the sets $\left\{v_n\right\}$ and $\left\{w_n\right\}$ have convergent subsequences due to their relative compactness in $C_{\omega }(\mathbb{R},E)$. Since the cone $K_C$ is normal and $\left\{v_n\right\}$, $\left\{w_n\right\}$ are monotone, we assume that $\left\{v_n\right\}$ and $\left\{w_n\right\}$ are convergent. That is, there exist $\underline{u}$ and $\overline{u}$ belong to $C_{\omega }(\mathbb{R},E)$ such that

$$\underline{u}=\underset{n\to \infty }{lim}{v}_n,\overline{u}=\underset{n\to \infty }{lim}{w}_n.$$

Putting $n\to \infty$ in (16), we get

$$\underline{u}=Q\underline{u},\overline{u}=Q\overline{u}.$$

This means that $\underline{u}$ and $\overline{u}$ are all the fixed points of Q. Consequently, $\underline{u}$ and $\overline{u}$ are $\omega$-periodic solutions of Equation (1).

Let $\tilde{u}$ be any fixed point of Q between $v_0$ and $w_0$. Then $v_0\left(t\right)\le \tilde{u}\left(t\right)\le w_0\left(t\right)$ for each $t\in \mathbb{R}$. By (17), we deduce that

$$v_1\left(t\right)=\left(Q{v}_0\right)\left(t\right)\le \left(Q\tilde{u}\right)\left(t\right)\le \left(Q{w}_0\right)\left(t\right)=w_1\left(t\right),t\in \mathbb{R}.$$

So, $v_1\left(t\right)\le \tilde{u}\left(t\right)\le w_1\left(t\right)$ for each $t\in \mathbb{R}$. Generally, for every $t\in \mathbb{R}$, we conclude that

$$v_n\left(t\right)\le \tilde{u}\left(t\right)\le w_n\left(t\right).$$

(18)

Taking $n\to \infty$ in (18), we get

$$\underline{u}\le \tilde{u}\le \overline{u}.$$

Therefore, $\underline{u}$ and $\overline{u}$ are minimal and maximal $\omega$-periodic solutions of Equation (1). □

The MNC conditions are necessary in Theorems 1 and 2, but they are not easy to verify in application. The next theorem establishes sufficient conditions to guarantee the existence as well as the uniqueness of $\omega$-periodic solution of Equation (1), where the nonlinearity f is not asked to satisfy the MNC condition.

Theorem 3.

Let the Hypothesis 2 (H2) hold and $a_0>0$. If Equation (1) possesses lower and upper ω-periodic solutions $v_0$ and $w_0$ sarisfying $v_0\le w_0$, and the nonlinearity f satisfies the Hypothesis 5 (H5) and Hypothesis 7 (H7).

Hypothesis 7 (H7).

there is a constant $L_3$ satisfying $max\{2b-a_0,0\}<L_3<b$ such that

$$f(t,x_2,y_2)-f(t,x_1,y_1)\le -L_3(y_2-y_1),\forall t\in \mathbb{R},$$

where $v_0\left(t\right)\le x_1\le x_2\le w_0\left(t\right)$ and $v_0(t-\tau )\le y_1\le y_2\le w_0(t-\tau )$, then there is a unique ω-periodic solution of Equation (1) between $v_0$ and $w_0$.

Proof of Theorem 3. 

Define a mapping $\Phi$ by

$$\Phi \left(u\right)(·)=f(·,u(·),u(·-\tau \left)\right)+bu(·-\tau ),$$

then $\Phi :C_{\omega }(\mathbb{R},E)\to C_{\omega }(\mathbb{R},E)$ is a continuous mapping. By Lemma 3, for any $h\in C_{\omega }(\mathbb{R},E)$, the linear equation

$$L_{n}u\left(t\right)+bu(t-\tau )=\Phi \left(h\right)\left(t\right),t\in \mathbb{R}$$

(19)

has a unique $\omega$-periodic solution, which is given by

$$u\left(t\right)=T(\Phi (h\left)\right(t\left)\right):=\left(Qh\right)\left(t\right),t\in \mathbb{R}.$$

Then $Q:C_{\omega }(\mathbb{R},E)\to C_{\omega }(\mathbb{R},E)$ is a continuous operator. It follows from (19) that the fixed point of operator Q is the $\omega$-periodic solution of Equation (1).

From the proof of Theorem 2, the operator Q satisfies the properties:

(i)

$v_0\le Q{v}_0,Q{w}_0\le w_0$;

(ii)

$Q{u}_1\le Q{u}_2$ for every $u_1,u_2\in [v_0,w_0]$ satisfying $u_1\le u_2$.

Let $\left\{v_n\right\}$ and $\left\{w_n\right\}$ be two sequences defined by (16). The properties $\left(i\right)$ and $\left(ii\right)$ yield that (17) holds. Then for $t\in \mathbb{R}$, we have

$$\begin{array}{ccc}\hfill \theta & \le & w_n\left(t\right)-v_n\left(t\right)=\left(Q{w}_{n-1}\right)\left(t\right)-\left(Q{v}_{n-1}\right)\left(t\right)\hfill \\ & =& T\left(\mathcal{F}\left(w_{n-1}\right)\left(t\right)-\mathcal{F}\left(v_{n-1}\right)\left(t\right)\right)\hfill \\ & =& T\left(f(t,w_{n-1}\left(t\right),w_{n-1}(t-\tau ))-f(t,v_{n-1}\left(t\right),v_{n-1}(t-\tau ))+b{w}_{n-1}\left(t\right)-b{v}_{n-1}\left(t\right)\right)\hfill \\ & \le & (b-L_3)T\left(w_{n-1}\left(t\right)-v_{n-1}\left(t\right)\right)\hfill \\ & \le & \cdots \hfill \\ & \le & {(b-L_3)}^n{T}^n\left(w_0\left(t\right)-v_0\left(t\right)\right).\hfill \end{array}$$

The normality of cone K yields

$$\parallel w_n\left(t\right)-v_n\left(t\right)\parallel \le {(b-L_3)}^n{\parallel T\parallel }^{n}N\parallel w_0\left(t\right)-v_0\left(t\right)\parallel ,t\in \mathbb{R},$$

which implies that

$$\parallel w_n-v_n{\parallel }_C\le {(b-L_3)}^n{\parallel T\parallel }^{n}N{\parallel w_0-v_0\parallel }_C.$$

Since $max\{2b-a_0,0\}<L_3<b$, it follows that $0<\frac{b-L_3}{a_0-b}<1$. Thus,

$${(b-L_3)}^n{\parallel T\parallel }^n\le {\left(\frac{b-L_3}{a_0-b}\right)}^n\to 0(n\to \infty ).$$

Therefore, there exists a unique function $\tilde{u}$ belongs to ${\cap }_{n=1}^{\infty }[v_n,w_n]$ such that

$$v_n\to \tilde{u},w_n\to \tilde{u}$$

as $n\to \infty$. Since $Q{v}_{n-1}=v_n\le \tilde{u}\le w_n\le Q{w}_n$, taking $n\to \infty$ we get $\tilde{u}=Q\tilde{u}$. This implies that Equation (1) possesses unique $\omega$-periodic solution. □

In Theorems 1–3, we always suppose that Equation (1) possesses lower and upper $\omega$-periodic solutions $v_0$ and $w_0$ satisfying $v_0\le w_0$, but it is still a problem whether Equation (1) possesses lower and upper $\omega$-periodic solutions. Next, we will prove that Equation (1) possesses a pair of lower and upper $\omega$-periodic solutions.

Theorem 4.

Let the Hypothesis 2 (H2) hold and $a_0>0$. If f satisfies the condition Hypothesis 8 (H8) then Equation (1) possesses lower and upper ω-periodic solutions $v_0$ and $w_0$ satisfying $v_0\le w_0$.

Hypothesis 8 (H8).

There exist $L^{*}\in [0,\frac{m_n{a}_0}{M_n})$ and $h\in C_{\omega }(\mathbb{R},K)$ such that

$$f(t,u,v)\le L^{*}v+h\left(t\right),u\ge \theta$$

and

$$f(t,u,v)\ge L^{*}v-h\left(t\right),u\le \theta$$

for any $u,v\in E$ and $t\in \mathbb{R}$.

Proof of Theorem 4. 

By Lemma 4, if the condition Hypothesis 2 (H2) holds and $a_0>0$, the LBVP (3) possesses a unique solution $r_n\left(t\right)>0$ for $t\in \mathbb{R}$. By the definition of $G_n(t,s)$ and Lemma 2, we know that $T_n:C_{\omega }(\mathbb{R},E)\to C_{\omega }(\mathbb{R},E)$ is a positive linear bounded operator with $\parallel T_n\parallel =\frac{1}{a_0}$. If the condition Hypothesis 8 (H8) holds, we consider the linear differential equation

$$L_{n}u\left(t\right)=L^{*}u(t-\tau )+h\left(t\right),t\in \mathbb{R}.$$

(20)

Let

$$B_{2}u\left(t\right)=L^{*}u(t-\tau ),t\in \mathbb{R}.$$

Then $B_2:C_{\omega }(\mathbb{R},E)\to C_{\omega }(\mathbb{R},E)$ is positive and linear bounded, and $\parallel B_2\parallel \le L^{*}$. Lemma 2 yields that Equation (20) possesses a unique $\omega$-periodic solution

$$\widehat{u}\left(t\right)=\left(T_n{B}_2\right)\widehat{u}\left(t\right)+T_{n}h\left(t\right),t\in \mathbb{R}.$$

(21)

Since $\parallel T_n{B}_2\parallel \le \frac{L^{*}}{a_0}<\frac{m_n}{M_n}\le 1$, we get that ${(I-T_n{B}_2)}^{-1}$ exists and ${(I-T_n{B}_2)}^{-1}={\displaystyle \sum _{i=0}^{\infty }}{\left(T_n{B}_2\right)}^i$ is a positive linear operator. Hence, from (21), $\widehat{u}\left(t\right)$ is given by

$$\widehat{u}\left(t\right)={(I-T_n{B}_2)}^{-1}{T}_{n}h\left(t\right)t\in \mathbb{R}$$

and $\widehat{u}\left(t\right)\ge 0$ for any $t\in \mathbb{R}$ owing to $h\left(t\right)\in K$. Let $v_0=-\widehat{u}$ and $w_0=\widehat{u}$, by the Hypothesis 8 (H8), we have

$$\begin{array}{ccc}\hfill L_n{v}_0\left(t\right)& =& L_n(-\widehat{u}\left(t\right))=-L_n\widehat{u}\left(t\right)=L^{*}(-\widehat{u}(t-\tau ))-h\left(t\right)\hfill \\ & \le & f(t,-\widehat{u}\left(t\right),-\widehat{u}(t-\tau ))\hfill \\ & =& f(t,v_0\left(t\right),v_0(t-\tau ))\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill L_n{w}_0\left(t\right)& =& L_n\left(\widehat{u}\left(t\right)\right)=L^{*}\left(\widehat{u}(t-\tau )\right)+h\left(t\right)\hfill \\ & \ge & f(t,\widehat{u}\left(t\right),\widehat{u}(t-\tau ))\hfill \\ & =& f(t,w_0\left(t\right),w_0(t-\tau )).\hfill \end{array}$$

Hence, the Equation (1) possesses lower and upper $\omega$-periodic solutions $v_0$ and $w_0$ satisfying $v_0\le w_0$. □

Example 1.

Consider the following fourth-order ordinary differential equation in Banach space E

$$u^{\left(4\right)}\left(t\right)+u^{\prime \prime \prime }\left(t\right)+u^{\prime \prime }\left(t\right)+u^{\prime }\left(t\right)+u\left(t\right)=F(t,u\left(t\right),u(t-\tau )),t\in \mathbb{R},$$

(22)

where $u\in C_{\omega }(\mathbb{R},E)$ and $F(t,u,v):\mathbb{R}\times E\times E\to E$ is continuous and ω-periodic with respect to t. We suppose that the following conditions hold.

Hypothesis 9 (P1).

$\{\lambda \in \mathbb{C}:{\displaystyle \sum _{i=0}^4}{\lambda }^i=0\}\subset \{\xi \in \mathbb{C}:|Im\xi |<\frac{\pi }{\omega }\}$.

Hypothesis 10 (P2).

There exist $L^{*}\in [0,\frac{m_n}{M_n})$ and $h\in C_{\omega }(\mathbb{R},K)$ such that

$$F(t,u,v)\le L^{*}v+h\left(t\right),u\ge \theta$$

and

$$F(t,u,v)\ge L^{*}v-h\left(t\right),u\le \theta$$

for any $u,v\in E$ and $t\in \mathbb{R}$.

Then the condition Hypothesis 9 ( P1) implies Hypothesis 2 (H2). if we choose $f(t,u(t),u(t-\tau \left)\right)=F(t,u(t),u(t-\tau \left)\right)$, the Hypothesis 10 ( P2) yields Hypothesis 8 (H8). Thus, by Theorem 4, the fourth-order ordinary differential Equation (22) possesses lower and upper $\omega$-periodic solutions $v_0$ and $w_0$ satisfying $v_0\le w_0$.

4. Conclusions

In this work, the maximum principle of the linear problem involving delay term is first established. Then the approach of upper and lower solutions and the monotone iterative technique are applied to consider the existence as well as the uniqueness of $\omega$-periodic solutions for the nth-order ordinary differential Equation (1) by using the obtained maximum principle. The existence of lower and upper $\omega$-periodic solutions of Equation (1) is also discussed in this paper. The results extend and improve some existing works.