Pseudo almost periodic solutions for a class of nonlinear Duffing equations on time scales

Abstract

In this paper, we study the pseudo almost periodic solutions for a class of nonlinear Duffing equations with $S^p$-pseudo almost periodic coefficients and delays on time scales. For this purpose, we establish a result of the existence and uniqueness of pseudo almost periodic solution for an abstract linear equation with $S^p$-almost periodic coefficients and $S^p$-pseudo almost periodic forcing term. Meanwhile, to deal with the delay, we extend some concepts of functions from $\mathbb {T}\rightarrow \mathbb {R}$ to $\mathbb {T}\rightarrow \Pi$, where $\mathbb {T}$ is a time scale with translation set $\Pi$, and give some basic properties for these concepts. Then, applying these results, we obtain some results on the existence and uniqueness of pseudo almost periodic solutions for the Duffing equation. Moreover, some examples are given to illustrate our main results.

1 Introduction

In recent years, the dynamic behaviors of nonlinear Duffing equations have been widely investigated in Burton (1986); Hale (1977); Kuang (2012); Yoshizawa (1975) due to the potential use in the areas of physics, mechanics and other engineering technique fields. Among them, the existence of almost periodic solutions and pseudo almost periodic solutions have attracted many authors. Some results on the existence of almost periodic solutions were obtained in the literature (see e.g., Zhou and Liu (2009); Peng and Wang (2010); Xu (2012); Liu and Tunç (2015)).

Recently, Zhou and Liu (2009) considered the following model for a nonlinear Duffing equation with a deviating argument:

$$\begin{aligned} x''(t)-ax(t)+bx^m(t-\tau (t))=p(t), \end{aligned}$$

(1)

where $\tau (t)$ and p(t) are almost periodic functions on $\mathbb {R}$, $m>1$ is an integer, $a>0$ and $b\ne 0$ are constants. By setting

$$\begin{aligned} y=x'(t)+\delta x(t), \end{aligned}$$

where $\delta >1$ is a constant, (1) transforms into the following system:

$$\begin{aligned} \left\{ \begin{aligned}&x'(t)=-\delta x(t)+y(t),\\&y'(t)=\delta y(t)+(a-\delta ^2)x(t)-bx^m(t-\tau (t))+p(t). \end{aligned} \right. \end{aligned}$$

The authors gave some criteria for the existence of almost periodic solutions for (1).

Then, Peng and Wang (2010) considered the following model for a nonlinear Duffing equation with a deviating argument:

$$\begin{aligned} x''(t)+cx'(t)-ax(t)+bx^m(t-\tau (t))=p(t), \end{aligned}$$

(2)

where $\tau (t)$ and p(t) are almost periodic functions on $\mathbb {R}$, $m>1$ is an integer and a, b, c are constants. By the transformation

$$\begin{aligned} y(t)=x'(t)+\xi x(t)-Q_1(t),\quad Q_2(t)=p(t)+(\xi -c)Q_1(t)-Q_1'(t), \end{aligned}$$

where $\xi >1$ is a constant and $Q_1(t)$ is continuous and differentiable, (2) transforms into the following system:

$$\begin{aligned} \left\{ \begin{aligned}&x'(t)=-\xi x(t)+y(t)+Q_1(t),\\&y'(t)=-(c-\xi )y(t)+(a-\xi (\xi -c))x(t)-bx^m(t-\tau (t))+Q_2(t), \end{aligned} \right. \end{aligned}$$

(3)

and then proved the existence of positive almost periodic solutions of (2) and (3).

After that, system (3) has been naturally extended by Xu (2012) to the following system with time-varying coefficients and delays:

$$\begin{aligned} \left\{ \begin{aligned}&\frac{dx(t)}{dt}=-\delta _1(t) x(t)+y(t)+Q_1(t),\\&\frac{dy(t)}{dt}=\delta _2(t)y(t)+(a(t)-\delta _2^2(t))x(t)-b(t)x^m(t-\tau (t))+Q_2(t), \end{aligned} \right. \end{aligned}$$

(4)

where $a(t),b(t),\tau (t),\delta _1(t),\delta _2(t),Q_1(t),Q_2(t)$ are almost periodic functions on $\mathbb {R}$, $m>1$ is an integer and $a(t)>0,b(t)\ne 0$, and gave some sufficient conditions for the existence of almost periodic solutions of (4).

Based on the work of Xu (2012), Liu and Tunç (2015) considered the system (4) with $\delta _1,\delta _2\in AP(\mathbb {R};\mathbb {R})$, $a,b,\tau , Q_1,Q_2\in PAP(\mathbb {R};\mathbb {R})$, and $a>0,b\ne 0$ for $t\in \mathbb {R}$. They gave some sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions of (4). Their results improved the results in the literature (Peng and Wang 2010; Xu 2012).

Moreover, Yang and Li (2014) considered the Duffing equation on time scales:

$$\begin{aligned} (x^{\Delta })^{\Delta }(t)+c(t)x^{\Delta }(t)-a(t)x(t)+b(t)x^m(t-\tau (t))=p(t), \end{aligned}$$

(5)

where $\mathbb {T}$ is an invariant time scale, $t\in \mathbb {T},t-\tau (t)\in \mathbb {T}$ and $m>1$ is a constant, and presented the existence and global exponential stability of almost periodic solutions for (5).

To combine continuous and discrete issues, Hilger proposed the idea of time scales in his Ph.D. thesis (Hilger 1988) in 1988. Several mathematicians have been interested in this theory since it provides an efficient mathematical technique for studying economics, biomathematics, and quantum physics, among other subjects.

Motivated by the above works, in this paper, we study the pseudo almost periodic solutions for the nonlinear Duffing Eq. (5) with $S^p$-pseudo almost periodic coefficients and delays on time scales. For this purpose, we establish a result of the existence and uniqueness of pseudo almost periodic solution for an abstract linear equation with $S^p$-almost periodic coefficients and $S^p$-pseudo almost periodic forcing term (see Theorem 3.1). Meanwhile, to deal with the delay $\tau (t)$, we extend some concepts of functions from $\mathbb {T}\rightarrow \mathbb {R}$ to $\mathbb {T}\rightarrow \Pi$, where $\mathbb {T}$ is a time scale with translation set $\Pi$ (see Definition 2.9, 2.10, 2.13), and give some basic properties for these concepts including the composition result (see Lemma 2.4, 2.6, 2.7). Then applying these results and Banach fixed point theorem, we get the existence and uniqueness of the pseudo almost periodic solution for the Duffing Eq. (5) (see Theorem 3.2, Theorem 3.3). Moreover, some examples are given to illustrate our main results at the end of this work.

2 Preliminaries

We refer to the sets of positive integers, integers, real numbers and non-negative real numbers, respectively, as $\mathbb {N},\ \mathbb {Z},\ \mathbb {R}$ and $\mathbb {R}^+$ throughout this work. The space of all $n\times n$ real-valued matrices with matrix norm $\Vert \cdot \Vert$ is denoted by $\mathbb {R}^{n\times n}$, while the Euclidian space $\mathbb {R}^n$ or $\mathbb {C}^n$ with Euclidian norm $\vert \cdot \vert$ is denoted by $\mathbb {E}^n$.

2.1 Time scale

Let $\mathbb {T}\subset \mathbb {R}$ be a time scale, that is, $\mathbb {T}\ne \emptyset$ is closed. The forward and backward jump operators $\sigma ,\ \rho :\mathbb {T}\rightarrow \mathbb {T}$ and the graininess $\mu :\mathbb {T}\rightarrow \mathbb {R^+}$ are defined, respectively, by

$$\begin{aligned} \sigma (t)=\inf \{s\in \mathbb {T}:s>t\},\quad \rho (t)= \sup \{s\in \mathbb {T}:s<t\},\quad \mu (t)=\sigma (t)-t. \end{aligned}$$

If $\sigma (t)>t,$ we say t is right-scattered; otherwise, t is right-dense. Similarly, if $\rho (t)<t$, we say t is left-scattered; otherwise t is left-dense.

If $\mathbb {T}$ has a left-scattered maximum m, then $\mathbb {T}^{\kappa }=\mathbb {T}\setminus m$; otherwise $\mathbb {T}^{\kappa }=\mathbb {T}$.

Definition 2.1

A time scale $\mathbb {T}$ is called invariant under translations if

$$\begin{aligned} \Pi :=\{\tau \in \mathbb {R}:t\pm \tau \in \mathbb {T},\ t\in \mathbb {T}\}\ne \{0\}, \end{aligned}$$

and define

$$\begin{aligned} \mathcal {K}=\left\{ \begin{array}{ll} \inf \{\vert \tau \vert :\tau \in \Pi ,\tau \ne 0\}&{},\text { if } \mathbb {T}\ne \mathbb {R};\\ 1&{},\text { if } \mathbb {T}=\mathbb {R}.\\ \end{array} \right. \end{aligned}$$

In fact, if $\mathbb {T}\ne \mathbb {R}$, we have $\mathcal {K}>0$ and one can show that $\Pi =\mathcal {K}\mathbb {Z}$. We say $\Pi$ the translation set of $\mathbb {T}$ (see e.g, Tang and Li (2017)).

In this paper, we always assume that $\mathbb {T}$ is invariant under translations.

Definition 2.2

(Bohner and Peterson (2001))

• (i) A function $f:\mathbb {T}\rightarrow \mathbb {E}^n$ is continuous on $\mathbb {T}$ if f is continuous at every right-dense point and at every left-dense point.

• (ii) A function $f:\mathbb {T}\rightarrow \mathbb {E}^n$ is rd-continuous on $\mathbb {T}$ if it is continuous at all right-dense points in $\mathbb {T}$ and its left-sided limit exists at all left-dense points in $\mathbb {T}$.

For $t,s\in \mathbb {T},\ t<s,$ denote $(t,s),\ [t,s],\ (t,s],\ [t,s)$ the standard intervals in $\mathbb {R},$ and use the following symbols:

$$\begin{aligned} (t,s)_{\mathbb {T}}=(t,s)\cap \mathbb {T},\ [t,s]_{\mathbb {T}}=[t,s]\cap \mathbb {T},\ (t,s]_{\mathbb {T}}=(t,s]\cap \mathbb {T},\ [t,s)_{\mathbb {T}}=[t,s)\cap \mathbb {T}. \end{aligned}$$

Denote

$$\begin{aligned}&C(\mathbb {T};\mathbb {E}^n)=\left\{ f:\mathbb {T}\rightarrow \mathbb {E}^n:f \text{ is } \text{ continuous }\right\} ,\\&C(\mathbb {T}\times D;\mathbb {E}^n)=\left\{ f:\mathbb {T}\times D\rightarrow \mathbb {E}^n:f \text{ is } \text{ continuous }\right\} ,\\&BC(\mathbb {T};\mathbb {E}^n)=\left\{ f:\mathbb {T}\rightarrow \mathbb {E}^n:f \text{ is } \text{ bounded } \text{ and } \text{ continuous }\right\} ,\\&BUC(\mathbb {T};\mathbb {E}^n)=\left\{ f:\mathbb {T}\rightarrow \mathbb {E}^n:f \text{ is } \text{ uniformly } \text{ continuous } \text{ and } \text{ bounded }\right\} ,\\&BC(\mathbb {T}\times D;\mathbb {E}^n)=\left\{ f:\mathbb {T}\times D\rightarrow \mathbb {E}^n:f \text{ is } \text{ bounded } \text{ and } \text{ continuous }\right\} ,\\ \end{aligned}$$

where $D\subset \mathbb {E}^n$ is an open set.

Definition 2.3

(Bohner and Peterson (2001)) For $f:\mathbb {T}\rightarrow \mathbb {E}^n$ and $t\in \mathbb {T}^{\kappa }$, $f^{\Delta }(t)\in \mathbb {E}^n$ is called the delta derivative of f(t) if for a given $\varepsilon >0$, there exists a neighborhood U of t such that

$$\begin{aligned} \vert f(\sigma (t))-f(s)-f^{\Delta }(t)(\sigma (t)-s)\vert < \varepsilon \vert \sigma (t)-s\vert \end{aligned}$$

for all $s\in U$.

Lemma 2.1

(Cabada and Vivero (2006)) Fix a point $\omega \in \mathbb {T}$ and an interval $[\omega ,\omega +\mathcal {K})_{\mathbb {T}}$, there are at most countably many right-scattered points $\{t_i\}_{i\in I},\ I\subseteq \mathbb {N}$ in this interval. If we denote $t_{ij}=t_i+j\mathcal {K},i\in I,j\in \mathbb {Z},$ we get all the right-scattered points, and we have $\mu (t_{ij})=\mu (t_i).$

Let $\mathcal {F}_1=\{[t,s)_{\mathbb {T}}:t,s\in \mathbb {T}\ with\ t\leqslant s\}.$ Define a countably additive measure $m_1$ on $\mathcal {F}_1$ by assigning to every $[t,s)_{\mathbb {T}}\in \mathcal {F}_1$ its lengths, i.e.

$$\begin{aligned} m_1([t,s)_{\mathbb {T}})=s-t. \end{aligned}$$

Using $m_1,$ we can generate the outer measure $m_1^{*}$ on the power set $\mathcal {P}(\mathbb {T})$ of $\mathbb {T}$: for $E\in \mathcal {P}(\mathbb {T})$

$$\begin{aligned} m_1^*(E)=\left\{ \begin{array}{ll} \inf _{\mathcal {B}}\Big \{\sum \limits _{i\in I_{\mathcal {B}}} (s_i-t_i)\Big \}\in \mathbb {R}^+,&{}\ \beta \notin E;\\ +\infty ,&{}\ \beta \in E,\\ \end{array} \right. \end{aligned}$$

where $\beta =\sup \mathbb {T}$ and

$$\begin{aligned} \mathcal {B}=\Big \{\{[t_i,s_i)\in \mathcal {F}_1\}_{i\in I_{\mathcal {B}}}:I_{\mathcal {B}}\subset \mathbb {N},E\subset \bigcup \limits _{i\in I_{\mathcal {B}}}[t_i,s_i)_{\mathbb {T}}\Big \}. \end{aligned}$$

A set $A\subset \mathbb {T}$ is called $\Delta -$measurable if for $E\subset \mathbb {T},$ we have

$$\begin{aligned} m_1^*(E)=m_1^*(E\cap A)+m_1^*(E\cap (\mathbb {T}\setminus A)). \end{aligned}$$

Let $\mathcal {M}(m_1^*)=\{A:A \text{ is } \text{ a } \Delta -\text{ measurable } \text{ subset } \text{ in } \mathbb {T}\}.$ Restricting $m_1^*$ to $\mathcal {M}(m_1^*)$, we get the Lebesgue $\Delta -$measure, which is denoted by $\mu _{\Delta }.$

Definition 2.4

(Cabada and Vivero (2006))

• (i) A function $\mathcal {S}:\mathbb {T}\rightarrow \mathbb {E}^n$ is said to be simple if $\mathcal {S}$ takes a finite number of values $c_1,c_2,\cdots ,c_N.$ Let $E_j=\{s\in \mathbb {T}:\mathcal {S}(s)=c_j\},$ then

  $$\begin{aligned} \mathcal {S}=\sum \limits _{j=1}^{N}c_j\chi _{E_j}, \end{aligned}$$

  where $\chi _{E_j}$ is the characteristic function of $E_j,$ that is

  $$\begin{aligned} \chi _{E_j}(s)=\left\{ \begin{array}{ll} 1,&{} \text { if } s\in E_j;\\ 0,&{} \text { if } s\in \mathbb {T}\setminus E_j.\\ \end{array} \right. \end{aligned}$$

• (ii) Assume that E is a $\Delta -$measurable subset of $\mathbb {T}$ and $\mathcal {S}:\mathbb {T}\rightarrow \mathbb {E}^n$ is a $\Delta -$measurable simple function, then the Lebesgue $\Delta -$integral of $\mathcal {S}$ on E is defined as

  $$\begin{aligned} \int _{E}\mathcal {S}(s)\Delta s=\sum \limits _{j=1}^Nc_j\mu _{\Delta }(E_j\cap E). \end{aligned}$$

• (iii) A function $g:\mathbb {T}\rightarrow \mathbb {E}^n$ is a $\Delta -$integrable function if there exists a simple function sequence $\{g_k:k\in \mathbb {N}\}$ such that $g_k(s)\rightarrow g(s)\ a.e.\ in\ \mathbb {T}$, then the integral of g is defined as

  $$\begin{aligned} \int _{\mathbb {T}}g(s)=\lim \limits _{k\rightarrow \infty }\int _{\mathbb {T}}g_k(s)\Delta s. \end{aligned}$$

• (iv) For $p\geqslant 1,g:\mathbb {T}\rightarrow \mathbb {E}^n$ is called locally $L^p\ \Delta -$integrable if g is $\Delta -$measurable and for any compact $\Delta -$measurable set $E\subset \mathbb {T},$ the $\Delta -$integral

  $$\begin{aligned} \int _{E}\vert g(s)\vert ^p\Delta s<\infty . \end{aligned}$$

  The set of all $L^p\ \Delta -$integrable functions is denoted by $L_{loc}^p(\mathbb {T};\mathbb {X}).$

Definition 2.5

(Tang and Li (2018)) Define $\Vert \cdot \Vert _{S^p}:L_{loc}^p(\mathbb {T};\mathbb {E}^n)\rightarrow \mathbb {R}^+\cup \{{+\infty }\}$ as

$$\begin{aligned} \Vert g\Vert _{S^p}:=\sup \limits _{s\in \mathbb {T}}\left( \frac{1}{\mathcal {K}}\int _{s}^{s+\mathcal {K}}\vert g(r)\vert ^p\Delta r\right) ^{\frac{1}{p}}. \end{aligned}$$

where $\mathcal {K}$ is defined in Definition 2.1. A function $g\in L_{loc}^p(\mathbb {T};\mathbb {E}^n)$ is called $S^p$-bounded if $\Vert g\Vert _{S^p}<\infty .$ The space of all $S^p$-bounded functions is denoted by $BS^p(\mathbb {T};\mathbb {E}^n);$ if $\mathbb {T}=\mathbb {R},$ denote it by $BS^p(\mathbb {E}^n).$

2.2 Almost periodicity and pseudo almost periodicity on $\mathbb {T}$

Definition 2.6

(Wang and Agarwal (2015)) A set $A\subset \mathbb {T}$ is called relatively dense in $\mathbb {T}$ if there exists $l>0$ such that $[s,s+l]_{\mathbb {T}}\cap A\ne \emptyset ,s\in \mathbb {T},$ we call l the inclusion length.

Definition 2.7

(Li and Wang (2011))

• (i) A function $g\in C(\mathbb {T};\mathbb {X})$ is almost periodic on $\mathbb {T}$ if for $\varepsilon >0,$

  $$\begin{aligned} T(g,\varepsilon )=\{\tau \in \Pi :\Vert g(s+\tau )-g(s)\Vert <\varepsilon \text{ for } s\in \mathbb {T}\} \end{aligned}$$

  is a relatively dense set in $\Pi .$ We call $T(g,\varepsilon )$ the $\varepsilon$-translation set of g and $\tau$ the $\varepsilon -$translation period of g, and the set of all almost periodic functions on $\mathbb {T}$ is denoted by $AP(\mathbb {T};\mathbb {X}).$

• (ii) Let $D\subset \mathbb {E}^n$ be open. The set $AP(\mathbb {T}\times D;\mathbb {E}^n)$ consists of all functions $f:\mathbb {T}\times D\rightarrow \mathbb {E}^n$ such that $f(\cdot ,x)\in AP(\mathbb {T};\mathbb {E}^n)$ uniformly for each $x\in K$ where K is any compact subset of D.

Definition 2.8

(Li and Wang (2011)) A continuous function $g:\mathbb {T}\rightarrow \mathbb {\mathbb {E}}^n$ is said to be normal on $\Pi$ if for any sequence $\{\alpha _n'\}\subset \Pi$, there is a subsequence $\{\alpha _n\}\subset \{\alpha _n'\}$ such that $\{g(t+\alpha _n)\}$ converges uniformly for $t\in \mathbb {T}$.

Lemma 2.2

(Li and Wang (2011)) A continuous function $g:\mathbb {T}\rightarrow \mathbb {\mathbb {E}}^n$ is almost periodic on $\mathbb {T}$ if and only if it is normal on $\Pi$.

To ensure $t-\tau (t)\in \mathbb {T}$, we have to give a restriction: $\tau (t)\in \Pi$. So we extend some concepts of functions from $\mathbb {T}\rightarrow \mathbb {R}$ to $\mathbb {T}\rightarrow \Pi$ below.

Definition 2.9

A function $f:\mathbb {T}\rightarrow \Pi$ is continuous if f is continuous at every right-dense point and at every left-dense point.

Denote

$$\begin{aligned} C(\mathbb {T};\Pi )= & {} \left\{ f:\mathbb {T}\rightarrow \Pi :f \text{ is } \text{ continuous }\right\} ,\\ BC(\mathbb {T};\Pi )= & {} \left\{ f:\mathbb {T}\rightarrow \Pi :f \text{ is } \text{ bounded } \text{ and } \text{ continuous }\right\} . \end{aligned}$$

Definition 2.10

A function $g\in C(\mathbb {T};\Pi )$ is almost periodic on $\mathbb {T}$ if for $\varepsilon >0,$

$$\begin{aligned} T(g,\varepsilon )=\{\tau \in \Pi :\Vert g(s+\tau )-g(s)\Vert <\varepsilon \text{ for } s\in \mathbb {T}\} \end{aligned}$$

is a relatively dense set in $\Pi .$ We call $T(g,\varepsilon )$ the $\varepsilon$-translation set of g and $\tau$ the $\varepsilon -$translation period of g, and the set of all almost periodic functions on $\mathbb {T}$ is denoted by $AP_m(\mathbb {T};\Pi ).$

Remark 2.1

For $\mathbb {T}=\mathbb {R}$, we have $\Pi =\mathbb {R}$, $AP_m(\mathbb {T};\Pi )=AP(\mathbb {R};\mathbb {R})$.

Denote the set

$$\begin{aligned}{} & {} \begin{aligned} PAP_0(\mathbb {T};\mathbb {E}^n)&=\Big \{f\in BC(\mathbb {T};\mathbb {E}^n):\lim \limits _{r\rightarrow \infty }\frac{1}{2r}\int _{t_0-r}^{t_0+r}\vert f(s)\vert \Delta s=0,\\&\qquad \text{ where } t_0\in \mathbb {T},r\in \Pi \Big \},\\ \end{aligned} \\{} & {} \begin{aligned} PAP_0(\mathbb {T}\times D;\mathbb {E}^n)&=\Big \{f\in BC(\mathbb {T}\times D;\mathbb {E}^n):f(\cdot ,x)\in PAP_0(\mathbb {T};\mathbb {E}^n)\\&\qquad \text{ uniformly } \text{ in } x\in D\Big \},\\ PAP_0(\mathbb {T};\Pi )&=\Big \{f\in BC(\mathbb {T};\Pi ):\lim \limits _{r\rightarrow \infty }\frac{1}{2r}\int _{t_0-r}^{t_0+r}\vert f(s)\vert \Delta s=0,\\&\qquad \text{ where } t_0\in \mathbb {T},r\in \Pi \Big \}. \end{aligned} \end{aligned}$$

Definition 2.11

(Li and Wang (2012)) A closed subset C of $\mathbb {T}$ is said to be an ergodic zero set in $\mathbb {T}$ if

$$\begin{aligned} \frac{\mu _{\Delta }(C\cap ([t_0-r,t_0+r])\cap \mathbb {T})}{2r}\rightarrow 0 \text{ as } r\rightarrow \infty , \text{ for } t_0\in \mathbb {T}. \end{aligned}$$

Definition 2.12

(Li and Wang (2012))

• (i) A function $f\in BC(\mathbb {T};\mathbb {E}^n)$ is called pseudo almost periodic if $f=g+\phi$, where $g\in AP(\mathbb {T};\mathbb {E}^n)$ and $\phi \in PAP_0(\mathbb {T};\mathbb {E}^n)$. We denote by $PAP(\mathbb {T};\mathbb {E}^n)$ the set of all pseudo almost periodic functions.

• (ii) A function $f\in BC(\mathbb {T}\times D;\mathbb {E}^n)$ is called pseudo almost periodic if $f=g+\phi$, where $g\in AP(\mathbb {T}\times D;\mathbb {E}^n)$ and $\phi \in PAP_0(\mathbb {T}\times D;\mathbb {E}^n)$. We denote by $PAP(\mathbb {T}\times D;\mathbb {E}^n)$ the set of all pseudo almost periodic functions.

Definition 2.13

A function $f\in BC(\mathbb {T};\Pi )$ is called pseudo almost periodic if $f=g+\phi$, where $g\in AP_m(\mathbb {T};\Pi )$ and $\phi \in PAP_0(\mathbb {T};\Pi )$. We denote by $PAP(\mathbb {T};\Pi )$ the set of all pseudo almost periodic functions.

Lemma 2.3

(Li and Wang (2012))

• (i) If $f\in PAP(\mathbb {T};\mathbb {E}^n)$ and $\phi \in PAP_0(\mathbb {T};\mathbb {E}^n)$, then for any $\tau \in \Pi ,\ f(\cdot +\tau )\in PAP(\mathbb {T};\mathbb {E}^n)$ and $\phi (\cdot +\tau )\in PAP_0(\mathbb {T};\mathbb {E}^n)$.

• (ii) $PAP(\mathbb {T};\mathbb {E}^n)$ and $PAP_0(\mathbb {T};\mathbb {E}^n)$ are Banach spaces under the sup norm.

Lemma 2.4

Assume that $\mathbb {T}\ne \mathbb {R}$.

• (i) Let $f\in AP_m(\mathbb {T};\Pi )$, then f is periodic.

• (ii) $AP_m(\mathbb {T};\Pi )$ is a $\mathbb {Z}$-module.

Proof

(i) For $\varepsilon >0$, $T(f,\varepsilon )=\{\tau \in \Pi :\Vert f(\cdot +\tau )-f(\cdot ) \Vert <\varepsilon \text{ for } s\in \mathbb {T}\}$ is relatively dense in $\Pi$. Let $\tau \in \Pi$, $f(t+\tau )-f(t)\in \Pi =\mathcal {K}\mathbb {Z}$ for $t\in \mathbb {T}$. Let $\varepsilon <\mathcal {K}$, we can get that $\Vert f(\cdot +\tau )-f(\cdot )\Vert <\varepsilon$ if and only if $f(t+\tau )-f(t)=0$ for $t\in \mathbb {T}$. Thus, f is periodic.

(ii) Let $f_1,f_2\in AP_m(\mathbb {T};\Pi )$ with period $T_1=n_1\mathcal {K},\ T_2=n_2\mathcal {K}$, respectively. Then we have $f_1+f_2$ is of period $T=[n_1,n_2]\mathcal {K}$, where $[n_1,n_2]$ denotes the least common multiple of $n_1$ and $n_2$, we get that $f_1+f_2\in AP_m(\mathbb {T};\Pi )$ and thus $AP_m(\mathbb {T};\Pi )$ is an additive group. Then it is easy to check that $AP_m(\mathbb {T};\Pi )$ is a $\mathbb {Z}$-module.

$\square$

Remark 2.2

For $\mathbb {T}\ne \mathbb {R}$, obviously, $AP_{m}(\mathbb {T};\Pi )$ is not a vector space on $\mathbb {R}$.

Lemma 2.5

(Zhang (1995)) A function $\phi _0\in BC(\mathbb {R};\mathbb {R})$ is in $PAP_0(\mathbb {R};\mathbb {R})$ if and only if, for $\varepsilon >0$, the set $C_{\varepsilon }=\{t\in \mathbb {R}:\vert \phi _0(t)\vert \geqslant \varepsilon \}$ is an ergodic zero subset of $\mathbb {R}$.

Lemma 2.6

A bounded continuous function $\phi _0\in PAP_0(\mathbb {T};\Pi )$ if and only if for $\varepsilon >0$, the set $C_{\varepsilon }=\{t\in \mathbb {T}:\vert \phi _0(t)\vert \geqslant \varepsilon \}$ is an ergodic zero subset of $\mathbb {T}$.

Proof

If $\mathbb {T}=\mathbb {R}$, the conclusion follows from Lemma 2.5. Assume that $\mathbb {T}\ne \mathbb {R}$. Let $\phi _0\in PAP_0(\mathbb {T};\Pi )$, by contradiction, suppose that $C_{\varepsilon }$ is not an ergodic zero subset of $\mathbb {T}$. Then there exists a constant $\varepsilon _0>0$ such that

$$\begin{aligned} \limsup \limits _{r\rightarrow \infty }\frac{\mu _{\Delta }(C_{\varepsilon }\cap ([t_0-r,t_0+r])\cap \mathbb {T})}{2r}\geqslant \varepsilon _0, \text{ for } \text{ some } t_0\in \mathbb {T}. \end{aligned}$$

We can derive that

$$\begin{aligned} \lim \limits _{r\rightarrow \infty }\frac{1}{2r}\int _{t_0-r}^{t_0+r}\vert \phi _0(s)\vert \Delta s\geqslant \limsup \limits _{r\rightarrow \infty }\frac{1}{2r}\int _{C_{\varepsilon }\cap [t_0-r,t_0+r]}\vert \phi _0(s)\vert \Delta s\geqslant \varepsilon _0\varepsilon >0, \end{aligned}$$

which contradicts that $\phi _0\in PAP_0(\mathbb {T};\Pi )$ and then $C_{\varepsilon }$ is an ergodic zero subset of $\mathbb {T}$.

On the other hand, for $\varepsilon >0$ and $C_{\varepsilon }$ is an ergodic zero set. Without loss of generality, we can choose $\varepsilon <\mathcal {K}$, then we have $\phi _0(t)=0$ for $t\in \mathbb {T}\setminus C_{\varepsilon }$. Let $M=\sup \limits _{t\in \mathbb {T}}\vert \phi _0(t)\vert$, for $t_0\in \mathbb {T}$, we obtain that

$$\begin{aligned} \begin{aligned} \frac{1}{2r}\int _{t_0-r}^{t_0+r}\vert \phi _0(s)\vert \Delta s&=\frac{1}{2r}\int _{C_{\varepsilon }\cap [t_0-r,t_0+r]}\vert \phi _0(s)\vert \Delta s+\int _{([t_0-r,t_0+r]\cap \mathbb {T})\setminus C_{\varepsilon }}\vert \phi _0(s)\vert \Delta s\\&=\frac{1}{2r}\int _{C_{\varepsilon }\cap [t_0-r,t_0+r]}\vert \phi _0(s)\vert \Delta s\\&\leqslant M\cdot \frac{\mu _{\Delta }(C_{\varepsilon }\cap [t_0-r,t_0+r]\cap \mathbb {T})}{2r}\rightarrow 0,\ \text{ as }\ r\rightarrow \infty . \end{aligned} \end{aligned}$$

Thus, we have $\phi _0\in PAP_0(\mathbb {T};\Pi )$. $\square$

Lemma 2.7

(Liu and Tunç (2015)) Suppose that $F\in PAP(\mathbb {R};\mathbb {R})\cap BUC(\mathbb {R};\mathbb {R})$ and $\phi \in PAP(\mathbb {R};\mathbb {R})$. Then $F(\cdot -\phi (\cdot ))\in PAP(\mathbb {R};\mathbb {R})$.

Lemma 2.8

For $\mathbb {T}\ne \mathbb {R}$, suppose that $F\in PAP(\mathbb {T};\mathbb {E}^n)$ and $\phi \in PAP(\mathbb {T};\Pi )$. Then $F(\cdot -\phi (\cdot ))\in PAP(\mathbb {T};\mathbb {E}^n)$.

Proof

Let $F=F_1+F_0,\ \phi =\phi _1+\phi _0$ with $F_1\in AP(\mathbb {T};\mathbb {E}^n),\ F_0\in PAP_0(\mathbb {T};\mathbb {E}^n)$ and $\phi _1\in AP_m(\mathbb {T};\Pi ),\ \phi _0\in PAP_0(\mathbb {T};\Pi )$. Note that, for $t\in \mathbb {T}$,

$$\begin{aligned} \begin{aligned} F(t-\phi (t))&=F_1(t-\phi (t))+F_0(t-\phi (t))\\&=F_1(t-\phi _1(t))+(F_1(t-\phi (t))-F_1(t-\phi _1(t)))+F_0(t-\phi (t)). \end{aligned} \end{aligned}$$

We first prove the almost periodicity of $F_1(t-\phi _1(t))$. From Lemma 2.4, we know that $\phi _1(t)$ is periodic on $\mathbb {T}$, then for $\{\alpha _n''\}\subset \Pi$, there exists a subsequence $\{\alpha _n'\}\subset \{\alpha _n''\}$ such that $\phi _1(t+\alpha _n')=\phi _1(t+\alpha _m')=\tau _0$ for $n,m\in \mathbb {N}$. Since $F_1\in AP(\mathbb {T};\mathbb {E}^n)$, by Lemma 2.2, for $\{\alpha _n'\}$, we can extract a subsequence $\{\alpha _n\}$ such that $\{F_1(t+\alpha _n)\}$ converges uniformly for $t\in \mathbb {T}$. Thus, $F_1(t+\alpha _n-\phi _1(t+\alpha _n))=F_1(t+\alpha _n-\tau _0)$ converges uniformly for $t\in \mathbb {T}$, and $F_1(\cdot -\phi _1(\cdot ))$ is normal on $\Pi$. Hence, $F_1(\cdot -\phi _1(\cdot ))\in AP(\mathbb {T};\mathbb {E}^n)$ by Lemma 2.2 again.

Then we only need to show that $h=(F_1(\cdot -\phi (\cdot ))-F_1(\cdot -\phi _1(\cdot )))+F_0(\cdot -\phi (\cdot ))\in PAP_0(\mathbb {T};\mathbb {E}^n)$. First we show that $F_1(\cdot -\phi (\cdot ))-F_1(\cdot -\phi _1(\cdot ))\in PAP_0(\mathbb {T};\mathbb {E}^n)$. For $0<\delta <\mathcal {K}$, let $C_{\delta }=\{\vert \phi _0(\cdot )\vert \geqslant \delta \}$. By Lemma 2.6, we can get that $C_{\delta }$ is an ergodic zero set in $\mathbb {T}$. This means that for $\varepsilon >0$, there exists $T>0$ such that when $r>T$, $t_0\in \mathbb {T}$,

$$\begin{aligned} \frac{\mu _{\Delta }([t_0-r,t_0+r]\cap \mathbb {T}\cap C_{\delta })}{2r}<\frac{\varepsilon }{2\Vert F_1\Vert }. \end{aligned}$$

It is obvious that if $t\in [t_0-r,t_0+r]\cap \mathbb {T}{\setminus } C_{\delta }$, $\phi _0(t)=\phi (t)-\phi _1(t)=0$. So we have

$$\begin{aligned} \quad \frac{1}{2r}&\int _{t_0-r}^{t_0+r}\vert F_1(s-\phi (s))-F_1(s-\phi _1(s))\vert \Delta s\\&=\frac{1}{2r}\Bigg (\int \limits _{t\in [t_0-r,t_0+r]\cap \mathbb {T}\setminus C_{\delta }}\vert F_1(s-\phi (s))-F_1(s-\phi _1(s))\vert \Delta s\\&\quad \qquad +\int \limits _{t\in [t_0-r,t_0+r]\cap \mathbb {T}\cap C_{\delta }}\vert F_1(s-\phi (s))-F_1(s-\phi _1(s))\vert \Delta s\Bigg )\\&\leqslant 0+2\Vert F_1\Vert \cdot \frac{\mu _{\Delta }([t_0-r,t_0+r]\cap \mathbb {T}\cap C_{\delta })}{2r}<\varepsilon . \end{aligned}$$

Therefore, $F_1(\cdot -\phi (\cdot ))-F_1(\cdot -\phi _1(\cdot ))\in PAP_0(\mathbb {T};\mathbb {E}^n)$.

Next we show that $F_0(\cdot -\phi (\cdot ))\in PAP_0(\mathbb {T};\mathbb {E}^n)$. Since $\phi$ is bounded, $\phi (\mathbb {T})\subset \Pi =\mathcal {K}\mathbb {Z}$ is of finite number of values, denote them by $\{k_1,k_2,\ldots ,k_n\}$, where $k_i\in \Pi ,\ i=1,2,\ldots ,n$. By Lemma 2.3 (i), we have $F_0(\cdot -k_i)\in PAP_0(\mathbb {T};\mathbb {E}^n),\ i=1,2,...,n$. So for $\varepsilon >0$, there exists $T_1>0$ such that for $r>T_1$,

$$\begin{aligned} \frac{1}{2r}\int _{t_0-r}^{t_0+r}\vert F_0(s-k_i)\vert \Delta s<\frac{\varepsilon }{n},\ i=1,2,...,n. \end{aligned}$$

Then we can get

$$\begin{aligned} \begin{aligned} \frac{1}{2r}\int _{t_0-r}^{t_0+r}\vert F_0(s-\phi (s))\vert \Delta s&\leqslant \frac{1}{2r}\sum _{i=1}^{n}\int _{t_0-r}^{t_0+r}\vert F_0(s-k_i)\vert \Delta s\\&<n\cdot \frac{\varepsilon }{n}=\varepsilon . \end{aligned} \end{aligned}$$

This implies that $F_0(\cdot -\phi (\cdot ))\in PAP_0(\mathbb {T};\mathbb {E}^n)$. $\square$

2.3 $S^p$-almost periodic functions and $S^p$-pseudo almost periodic functions

Definition 2.14

(Tang and Li (2018)) A function $g\in L_{loc}^p(\mathbb {T};\mathbb {E}^n)$ is $S^p$-almost periodic on $\mathbb {T}$ if given $\varepsilon >0,$ the $\varepsilon$-translation set of g

$$\begin{aligned} T(g,\varepsilon )=\{\tau \in \Pi :\Vert g(\cdot +\tau )-g(\cdot )\Vert _{S^p}<\varepsilon \} \end{aligned}$$

is a relatively dense set in $\Pi .$ The space of all these functions is denoted by $S^pAP(\mathbb {T};\mathbb {E}^n)$ with norm $\Vert \cdot \Vert _{S^p}.$

Define the norm operator $\mathcal {N}_p$ on $BS^p(\mathbb {T};\mathbb {E}^n)$ as follows:

$$\begin{aligned} \mathcal {N}_p(f)(t):=\left( \frac{1}{\mathcal {K}}\int _{t}^{t+\mathcal {K}}\vert f(s)\vert ^p\Delta s\right) ^{\frac{1}{p}} \text{ for } f\in BS^p(\mathbb {T};\mathbb {E}^n),t\in \mathbb {T}. \end{aligned}$$

Lemma 2.9

(Tang and Li (2018)) The norm operator $\mathcal {N}_p$ maps $BS^p(\mathbb {T};\mathbb {E}^n)$ in to $BC(\mathbb {T};\mathbb {R})$ and maps $S^pAP(\mathbb {T};\mathbb {E}^n)$ into $AP(\mathbb {T};\mathbb {R})$. Moreover, for $f,g\in BS^p(\mathbb {T};\mathbb {E}^n),\ t\in \mathbb {T}$,

$$\begin{aligned} \Vert \mathcal {N}_p(f)\Vert _{\infty }=\Vert f\Vert _{S^p},\ \vert \mathcal {N}_p(f)(t)-\mathcal {N}_p(g)(t)\vert \leqslant \mathcal {N}_p(f\pm g)(t)\leqslant \mathcal {N}_p(f)(t)+\mathcal {N}_p(g)(t). \end{aligned}$$

Lemma 2.10

Let $f\in BS^p(\mathbb {T};\mathbb {E}^n),\ g\in BS^q(\mathbb {T};\mathbb {E}^n)$ with $\displaystyle p,q\geqslant 1,\frac{1}{p}+\frac{1}{q}=1$. Then we have, for $t\in \mathbb {T}$,

$$\begin{aligned} \mathcal {N}_1(f\cdot g)(t)\leqslant \mathcal {N}_p(f)(t)\cdot \mathcal {N}_q(g)(t)\leqslant \Vert f\Vert _{S^p}\cdot \mathcal {N}_q(g)(t). \end{aligned}$$

(6)

In addition, if f is bounded and continuous, we have

$$\begin{aligned} \mathcal {N}_1(f\cdot g)(t)\leqslant \mathcal {N}_q(f\cdot g)(t)\leqslant \Vert f\Vert \cdot \mathcal {N}_q(g)(t). \end{aligned}$$

(7)

Proof

If $p=1,q=+\infty$, it is obvious. Now suppose that $p,q>1$. By H${\ddot{\textrm{o}}}$lder inequality, for $t\in \mathbb {T}$, we have

$$\begin{aligned} \begin{aligned} \mathcal {N}_1(f\cdot g)(t)&=\frac{1}{\mathcal {K}}\int _{t}^{t+\mathcal {K}}\vert f(s)g(s)\vert \Delta s\\&\leqslant \frac{1}{\mathcal {K}}\left( \int _{t}^{t+\mathcal {K}}\vert f(s)\vert ^p\Delta s\right) ^{\frac{1}{p}}\cdot \left( \int _{t}^{t+\mathcal {K}}\vert g(s)\vert ^q\Delta s\right) ^{\frac{1}{q}}\\&=\left( \frac{1}{\mathcal {K}}\int _{t}^{t+\mathcal {K}}\vert f(s)\vert ^p\Delta s\right) ^{\frac{1}{p}}\cdot \left( \frac{1}{\mathcal {K}}\int _{t}^{t+\mathcal {K}}\vert g(s)\vert ^q\Delta s\right) ^{\frac{1}{q}}\\&=\mathcal {N}_p(f)(t)\cdot \mathcal {N}_q(g)(t)\leqslant \Vert f\Vert _{S^p}\cdot \mathcal {N}_q(g)(t). \end{aligned} \end{aligned}$$

If f is bounded and continuous, then we have

$$\begin{aligned} \begin{aligned} \mathcal {N}_1(f\cdot g)(t)&=\frac{1}{\mathcal {K}}\int _{t}^{t+\mathcal {K}}\vert f(s)g(s)\vert \Delta s\leqslant \frac{1}{\mathcal {K}}\left( \int _{t}^{t+\mathcal {K}}\vert f(s)g(s)\vert ^q\Delta s\right) ^{\frac{1}{q}}\cdot \left( \int _t^{t+\mathcal {K}}1^p\Delta s\right) ^{\frac{1}{p}}\\&=\left( \frac{1}{\mathcal {K}}\int _t^{t+\mathcal {K}}\vert f(s)g(s)\vert ^q\Delta s\right) ^{\frac{1}{q}}\leqslant \Vert f\Vert \cdot \left( \frac{1}{\mathcal {K}}\int _t^{t+\mathcal {K}}\vert g(s)\vert ^q\Delta s\right) ^{\frac{1}{q}}\\&=\Vert f\Vert \cdot \mathcal {N}_q(g)(t) \end{aligned} \end{aligned}$$

$\square$

Definition 2.15

(Tang and Li (2018)) A function $f\in BS^p(\mathbb {T};\mathbb {E}^n)$ is said to be ergodic if $\mathcal {N}_p(f)\in PAP_0(\mathbb {T};\mathbb {R})$. We denote by $S^pPAP_0(\mathbb {T};\mathbb {E}^n)$ the set of all ergodic functions from $\mathbb {T}$ to $\mathbb {E}^n$.

Definition 2.16

(Tang and Li (2018)) A function $f\in BS^p(\mathbb {T};\mathbb {E}^n)$ is called $S^p$-pseudo almost periodic if $f=g+\phi$, where $g\in S^pAP(\mathbb {T};\mathbb {E}^n)$ and $\phi \in S^pPAP_0(\mathbb {T};\mathbb {E}^n)$. We denote by $S^pPAP(\mathbb {T};\mathbb {E}^n)$ the set of all such functions f.

Lemma 2.11

(Tang and Li (2018))

• (i) $PAP(\mathbb {T};\mathbb {E}^n)\subset S^pPAP(\mathbb {T};\mathbb {E}^n)$.

• (ii) $S^qPAP(\mathbb {T};\mathbb {E}^n)\subset S^p(\mathbb {T};\mathbb {E}^n)$ for $1\leqslant p\leqslant q$.

• (iii) Assume that $f\in BS^p(\mathbb {T};\mathbb {E}^n)$. For $t_0\in \mathbb {T}$, we have $\displaystyle \int _{t_0}^{t_0+\mathcal {K}}\vert f(s)\vert \Delta s\leqslant \mathcal {K}\Vert f\Vert _{S^p}$.

Lemma 2.12

For $f=f_1+f_2\in PAP(\mathbb {T};\mathbb {E}^n)$ and $g=g_1+g_2\in S^pPAP(\mathbb {T};\mathbb {E}^n)$ with $f_1\in AP(\mathbb {T};\mathbb {E}^n),\ f_2\in PAP_0(\mathbb {T};\mathbb {E}^n),\ g_1\in S^pAP(\mathbb {T};\mathbb {E}^n)$ and $g_2\in S^pPAP_0(\mathbb {T};\mathbb {E}^n)$. Then $f\cdot g\in S^1PAP(\mathbb {T};\mathbb {E}^n)$.

Proof

For convenience, we denote $f^{\tau }(\cdot )=f(\cdot +\tau )$ in the proof. In fact, we have $f\cdot g=f_1\cdot g_1+f_2\cdot g_1+f\cdot g_2$. Now we prove that $f\cdot g\in S^1PAP(\mathbb {T};\mathbb {E}^n)$ by the following 3 steps.

Step 1: We prove that $f_1\cdot g_1\in S^1AP(\mathbb {T};\mathbb {E}^n)$. For $\varepsilon >0$, choose $\tau \in T(f_1,\varepsilon )\cap T(g_1,\varepsilon )$, by Lemma 2.9 and (7), we can get that

$$\begin{aligned} \begin{aligned} \Vert f_1(\cdot +\tau )g_1(\cdot +\tau )-f_1(\cdot )g_1(\cdot )\Vert _{S^p}&=\sup \limits _{t\in \mathbb {T}}\mathcal {N}_p((f_1\cdot g_1)^{\tau }-f_1\cdot g_1)(t)\\&=\sup \limits _{t\in \mathbb {T}}\mathcal {N}_p(f_1^{\tau }\cdot (g_1^{\tau }-g_1)+(f_1^{\tau }-f_1)\cdot g_1)(t)\\&\leqslant \sup \limits _{t\in \mathbb {T}}\mathcal {N}_p(f_1^{\tau }\cdot (g_1^{\tau }-g_1))(t)+\sup \limits _{t\in \mathbb {T}}\mathcal {N}_p((f_1^{\tau }-f_1)\cdot g_1)(t)\\&\leqslant \Vert f_1\Vert \cdot \sup \limits _{t\in \mathbb {T}}\mathcal {N}_p(g_1^{\tau }-g_1)(t)+\Vert f_1^{\tau }-f_1\Vert \cdot \sup \limits _{t\in \mathbb {T}}\mathcal {N}_p(g_1)(t)\\&\leqslant (\Vert f_1\Vert +\Vert g_1\Vert _{S^p})\varepsilon , \end{aligned} \end{aligned}$$

which means that $f_1\cdot g_1\in S^pAP(\mathbb {T};\mathbb {E}^n)$ and by Lemma 2.11 (ii), we have $f_1\cdot g_1\in S^1AP(\mathbb {T};\mathbb {E}^n)$.

Step 2: We prove that $f\cdot g_2\in S^1PAP_0(\mathbb {T};\mathbb {E}^n)$. By Lemma 2.11 (iii) and (6), we have

$$\begin{aligned} \frac{1}{2r}\int _{t_0-r}^{t_0+r}\mathcal {N}_1(f\cdot g_2)(t)\Delta t\leqslant \Vert f\Vert \cdot \frac{1}{2r}\int _{t_0-r}^{t_0+r}\mathcal {N}_p(g_2)(t)\Delta t, \end{aligned}$$

(8)

for a fixed $t_0\in \mathbb {T}$ and $r\in \Pi$. Let $r\rightarrow \infty$ in (8) we derive that

$$\begin{aligned} \lim \limits _{r\rightarrow \infty }\frac{1}{2r}\int _{t_0-r}^{t_0+r}\mathcal {N}_p(f\cdot g_2)\Delta t=0, \end{aligned}$$

since $g_2\in S^pPAP_0(\mathbb {T};\mathbb {E}^n)$. Thus, $f\cdot g_2\in S^1PAP_0(\mathbb {T};\mathbb {E}^n)$.

Step 3: We prove that $f_2\cdot g_1\in S^1PAP_0(\mathbb {T};\mathbb {E}^n)$. By Lemma 2.11 (i) we can get that $f_2\in S^qPAP_0(\mathbb {T};\mathbb {E}^n)$ where $\displaystyle \frac{1}{q}+\frac{1}{p}=1$. For a fixed $t_0\in \mathbb {T},\ r\in \Pi$, by (6), we have

$$\begin{aligned} \begin{aligned} \lim \limits _{r\rightarrow \infty }\frac{1}{2r}\int _{t_0-r}^{t_0+r}\mathcal {N}_1(f_2\cdot g_1)(t)\Delta t&\leqslant \lim \limits _{r\rightarrow \infty }\frac{1}{2r}\int _{t_0-r}^{t_0+r}\mathcal {N}_q(f_2)(t)\cdot \mathcal {N}_p(g_1)(t)\Delta t\\&\leqslant \Vert g_1\Vert _{S^p}\cdot \lim \limits _{r\rightarrow \infty }\frac{1}{2r}\int _{t_0-r}^{t_0+r}\mathcal {N}_q(f_2)(t)\Delta t=0. \end{aligned} \end{aligned}$$

Thus, we get $f_2\cdot g_1\in S^1PAP_0(\mathbb {T};\mathbb {E}^n)$. $\square$

2.4 Exponential functions

For a function $p:\mathbb {T}\rightarrow \mathbb {R}$, if we have $1+\mu (t)p(t)\ne 0,\ t\in \mathbb {T}^{\kappa }$, we say that p is regressive. Denote the set of all regressive and rd-continuous function $p:\mathbb {T}\rightarrow \mathbb {R}$ by $\mathcal {R}=\mathcal {R}(\mathbb {T})=\mathcal {R}(\mathbb {T};\mathbb {R})$ and define the set $\mathcal {R}^+=\mathcal {R}^+(\mathbb {T};\mathbb {R})=\{p\in \mathcal {R}: 1+\mu (t)p(t)>0 \text{ for } t\in \mathbb {T}\}$. We can see that the set $\mathcal {R}(\mathbb {T};\mathbb {R})$ is an Abelian group with addition $\oplus$ defined by $p\oplus q=p+q+\mu (t)pq$, and the additive inverse in this Abelian group is defined by $\displaystyle \ominus p=-\frac{p}{1+\mu (t)p}$.

Definition 2.17

(Bohner and Peterson (2001)) For $p\in \mathcal {R}$, the exponential function is defined by

$$\begin{aligned} e_{p}(t,s)=\text{ exp }\left( \int _{s}^t \xi _{\mu (\tau )}(p(\tau ))\Delta \tau \right) , \end{aligned}$$

for $t,s\in \mathbb {T}$ with the cylinder transformation

$$\begin{aligned} \xi _{h}(z)=\left\{ \begin{array}{ll} \frac{1}{h}\mathrm{{Log}}(1+hz)&{}, \text{ if } h\ne 0;\\ z &{}, \text{ if } h=0, \end{array} \right. \end{aligned}$$

where Log is the principal logarithm.

Definition 2.18

(Bohner and Peterson (2001)) For a matrix-valued function $A:\mathbb {T}\rightarrow \mathbb {R}^{n\times n}$, we say that $A(\cdot )$ is regressive if $I+\mu (t)A(t)$ is invertible for $t\in \mathbb {T}^{\kappa }$, and denote the set of all such regressive and rd-continuous functions by $\mathcal {R}(\mathbb {T};\mathbb {R}^{n\times n})$.

Definition 2.19

(Bohner and Peterson (2001)) Let $A\in \mathcal {R}(\mathbb {T};\mathbb {R}^{n\times n})$. The initial value problem

$$\begin{aligned} X(t)^{\Delta }=A(t)X(t), X(t_0)=I,\ t,t_0\in \mathbb {T} \end{aligned}$$

has a unique solution which is denoted by $e_A(\cdot ,t_0)$. We say that $e_A(\cdot ,t_0)$ is the matrix exponential function at $t_0$.

Lemma 2.13

(Bohner and Peterson (2001)) Let $t,s\in \mathbb {T}$.

• (i) $e_p(t,t)=1,\ e_{A}(t,t)=I.$

• (ii) $e_p(\sigma (t),s)=(1+\mu (t)p(t))e_p(t,s).$

• (iii) $e_p(t,s)e_p(s,r)=e_p(t,r),e_A(t,s)e_A(s,r)=e_A(t,r).$

Lemma 2.14

(Tang and Li (2018)) Let $\alpha >0$ be a constant and $t,s\in \mathbb {T}.$

• (i) $e_{\ominus \alpha }(t,s)\leqslant 1$ if $t>s$.

• (ii) $e_{\ominus \alpha }(t+\tau ,s+\tau )=e_{\ominus \alpha }(t,s)$ for $\tau \in \Pi$.

• (iii) There exists $N_{\alpha }>0$ depending on $\alpha$ such that $n_{ts}\mathcal {K}e_{\ominus \alpha }(t,s)\leqslant N_{\alpha }$ for $t\geqslant s$, where $(n_{ts}-1)\mathcal {K}\leqslant t-s<n_{ts}\mathcal {K}$.

• (iv) The series $\sum \limits _{j=1}^\infty e_{\ominus \alpha }(t,\sigma (t)-(j-1)\mathcal {K})$ converges uniformly for $t\in \mathbb {T}$. Moreover, for all $t\in \mathbb {}T$,

  $$\begin{aligned} \sum \limits _{j=1}^\infty e_{\ominus \alpha }(t,\sigma (t)-(j-1)\mathcal {K})\leqslant \lambda _{\alpha }= \left\{ \begin{array}{ll} \frac{1}{1-e^{-\alpha }}&{}, \text{ if } \mathbb {T}=\mathbb {R};\\ 2+\alpha \bar{\mu }+\frac{1}{\alpha \bar{\mu }}&{}, \text{ if } \mathbb {T}\ne \mathbb {R}, \end{array} \right. \end{aligned}$$

  where $\bar{\mu }=\sup \limits _{t\in \mathbb {T}}\mu (t)$.

Lemma 2.15

Assume that $A\in \mathcal {R}(\mathbb {T};\mathbb {R}^{n\times n})$ is $S^p$-almost periodic and

$$\begin{aligned} \Vert e_{A}(t,s)\Vert \leqslant Ce_{\ominus \alpha }(t,s),t\geqslant s, \end{aligned}$$

(9)

where C and $\alpha$ are positive real numbers. Let $M=\left\{ \begin{aligned}&C^2(1+\alpha \mathcal {K})N_{\alpha }, \text{ if } \mathbb {T}\ne \mathbb {R},\\&C^2N_{\alpha }, \text{ if } \mathbb {T}=\mathbb {R} \end{aligned} \right.$ with $N_{\alpha }$ the constant in Lemma 2.14 (iii), and for $\varepsilon >0$,

$$\begin{aligned} \varUpsilon (\varepsilon )=\{r\in \Pi : \Vert e_A(t+r,\sigma (s)+r)-e_A(t,\sigma (s))\Vert <\varepsilon , t,s\in \mathbb {T},t\geqslant \sigma (s)\}. \end{aligned}$$

Then $T(A,\varepsilon /M)\subset \varUpsilon (\varepsilon ).$

Proof

For $\varepsilon >0$, let $r\in T(A,\varepsilon /M)$ and $U(t,\sigma (s))= e_A(t+r,\sigma (s)+r)-e_A(t,\sigma (s))$. Differentiate U with respect to t and denote by $\displaystyle \frac{\partial _{\Delta }U}{\partial _{\Delta }t}$ the partial derivative, then

$$\begin{aligned} \begin{aligned} \frac{\partial _{\Delta }U}{\partial _{\Delta }t}&=A(t+r)e_A(t+r,\sigma (s)+r)-A(t)e_A(t,\sigma (s))\\&=A(t)U(t,\sigma (s))+(A(t+r)-A(t))e_A(t+r,\sigma (s)+r). \end{aligned} \end{aligned}$$

Note that $U(\sigma (s),\sigma (s))=0$, then by the variation of constants formula,

$$\begin{aligned} U(t,\sigma (s))=\int _{\sigma (s)}^{t}e_A(t,\sigma (\tau ))(A(\tau +r)-A(\tau ))e_A(\tau +r,\sigma (s)+r)\Delta \tau . \end{aligned}$$

For $\mathbb {T}=\mathbb {R}$,

$$\begin{aligned} \begin{aligned} \Vert U(t,s)\Vert&\leqslant \int _s^t \Vert e_A(t,\tau )\Vert \cdot \Vert A(\tau +r)-A(\tau )\Vert \cdot \Vert e_A(\tau +r,s+r)\Vert d\tau \\&\leqslant C^2\int _s^t e^{-\alpha (t-\tau )}e^{-\alpha (\tau -s)}\Vert A(\tau +r)-A(\tau )\Vert d\tau \\&=C^2e^{-\alpha (t-s)}\int _s^t\Vert A(\tau +r)-A(\tau )\Vert d\tau \\&\leqslant C^2e^{-\alpha (t-s)}\int _{t-n_{ts}}^t\Vert A(\tau +r)-A(\tau )\Vert d\tau \\&=C^2e^{-\alpha (t-s)}\sum \limits _{j=1}^{n_{ts}}\int _{t-j}^{t-(j-1)}\Vert A(\tau +r)-A(\tau )\Vert d\tau \\&\leqslant C^2n_{ts}e^{-\alpha (t-s)}\Vert A(\cdot +r)-A(\cdot )\Vert _{S^p}\\&\leqslant C^2N_{\alpha }\varepsilon /M=\varepsilon . \end{aligned} \end{aligned}$$

For $\mathbb {T}\ne \mathbb {R}$, by Lemma 2.11, 2.13, 2.14 and the fact that $\mu (\tau )\leqslant \mathcal {K},\tau \in \mathbb {T}$, for $t,s\in \mathbb {T},t\geqslant \sigma (s)$,

$$\begin{aligned} \Vert U(t,\sigma (s))\Vert&\leqslant \int _{\sigma (s)}^t \Vert e_A(t,\sigma (\tau ))\Vert \cdot \Vert A(\tau +r)-A(\tau )\Vert \cdot \Vert e_A(\tau +r,\sigma (s)+r)\Vert \Delta \tau \\&\leqslant C^2\int _{\sigma (s)}^t e_{\ominus \alpha }(t,\sigma (\tau ))e_{\ominus \alpha }(\tau +r,\sigma (s)+r)\Vert A(\tau +r)-A(\tau )\Vert \Delta \tau \\&=C^2e_{\ominus \alpha }(t,\sigma (s))\int _{\sigma (s)}^t e_{\ominus \alpha }(\tau ,\sigma (\tau ))\Vert A(\tau +r)-A(\tau )\Vert \Delta \tau \\&\leqslant C^2(1+\alpha \bar{\mu })e_{\ominus \alpha }(t,\sigma (s))\int _{\sigma (s)}^t\Vert A(\tau +r)-A(\tau )\Vert \Delta \tau \\&\leqslant C^2(1+\alpha \mathcal {K})e_{\ominus \alpha }(t,\sigma (s))\int _{t-n_{ts}\mathcal {K}}^t\Vert A(\tau +r)-A(\tau )\Vert \Delta \tau \\&=C^2(1+\alpha \mathcal {K})e_{\ominus \alpha }(t,\sigma (s))\sum \limits _{j=1}^{n_{ts}}\int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}\Vert A(\tau +r)-A(\tau )\Vert \Delta \tau \\&\leqslant C^2(1+\alpha \mathcal {K})n_{ts}\mathcal {K} e_{\ominus \alpha }(t,\sigma (s))\Vert A(\cdot +r)-A(\cdot )\Vert _{S^p}\\&\leqslant C^2(1+\alpha \mathcal {K})N_{\alpha }\varepsilon /M=\varepsilon . \end{aligned}$$

This implies that $T(A,\varepsilon /M)\subset \varUpsilon (\varepsilon )$, and $\varUpsilon (\varepsilon )$ is relatively dense in $\Pi$. $\square$

3 Main results

Let $y(t)=x^{\Delta }(t)+\delta _1(t)x(t)$, Eq. (5) transforms into the following system:

$$\begin{aligned} \left\{ \begin{aligned}&x^{\Delta }(t)=-\delta _1(t)x(t)+y(t),\\&y^{\Delta }(t)=-\delta _2(t)y(t)+\beta (t)x(t)-b(t)x^m(t-\tau (t))+p(t), \end{aligned} \right. \end{aligned}$$

(10)

where $\delta _2(t)=c(t)-\delta _1(\sigma (t)),\ \beta (t)=a(t)+\delta ^{\Delta }_1(t)+\delta _1(t)\delta _2(t)$. To study (10), we first consider the following abstract linear equation:

$$\begin{aligned} x^{\Delta }(t)=A(t)x(t)+f(t), t\in \mathbb {T}, \end{aligned}$$

(11)

where $f=g+\phi \in S^pPAP(\mathbb {T};\mathbb {E}^n)\cap C(\mathbb {T};\mathbb {E}^n)$.

Lemma 3.1

(Tang and Li (2018)) Assume that $A\in \mathcal {R}(\mathbb {T};\mathbb {R}^{n\times n})$ with (9) satisfied. Then (11) admits a unique bounded continuous solution u(t) given by

$$\begin{aligned} u(t)=\int _{-\infty }^te_{A}(t,\sigma (s))f(s)\Delta s, t\in \mathbb {T}, \end{aligned}$$

(12)

and $\vert u(t)\vert \leqslant C\lambda _{\alpha }\mathcal {K}\Vert f\Vert _{S^p}$, where $\lambda _{\alpha }$ is given in Lemma 2.14 (iv).

Theorem 3.1

Assume that all conditions in Lemma 2.15 are satisfied. Then (11) admits a unique pseudo almost periodic solution given by (12).

Proof

By Lemma 3.1, it suffices to prove that $u\in PAP(\mathbb {T};\mathbb {E}^n)$. For $t\in \mathbb {T}$, let

$$\begin{aligned} u(t)=\int _{-\infty }^te_{A}(t,\sigma (s))f(s)\Delta s=\sum \limits _{j=1}^{\infty }u_j(t), \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} u_j(t)&=\int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}e_A(t,\sigma (s))f(s)\Delta s\\&=\int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}e_A(t,\sigma (s))g(s)\Delta s+\int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}e_A(t,\sigma (s))\phi (s)\Delta s\\&:=g_j(t)+\phi _j(t),\qquad j\in \mathbb {N}. \end{aligned} \end{aligned}$$

Now we prove $u_j\in PAP(\mathbb {T};\mathbb {E}^n)$. For $\varepsilon >0$, it follows from Lemma 2.15 that $\varUpsilon (\varepsilon )\cap T(g,\varepsilon )$ is relatively dense in $\Pi$. For $r\in \varUpsilon (\varepsilon )\cap T(g,\varepsilon )$, by Lemma 2.1 and 2.11,

$$\begin{aligned}{} & {} \vert g_j(t+r)-g_j(t)\vert \\{} & {} \quad =\left| \int _{t+r-j\mathcal {K}}^{t+r-(j-1)\mathcal {K}}e_A(t+r,\sigma (s))g(s)\Delta s-\int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}e_A(t,\sigma (s))g(s)\Delta s\right| \\{} & {} \quad =\left| \int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}e_A(t+r,\sigma (s)+r)g(s+r)-e_A(t,\sigma (s))g(s)\Delta s\right| \\{} & {} \quad \leqslant \int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}\Vert e_A(t+r,\sigma (s)+r)-e_A(t,\sigma (s))\Vert \cdot \vert g(s)\vert \Delta s\\{} & {} \qquad +\int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}\Vert e_A(t,\sigma (s))\Vert \cdot \vert g(s+r)-g(s)\vert \Delta s\\{} & {} \quad \leqslant \varepsilon \int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}\vert g(s)\vert \Delta s+Ce_{\ominus \alpha }(t,\sigma (t)-(j-1)\mathcal {K})\int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}\vert g(s+r)-g(s)\vert \Delta s\\{} & {} \quad \leqslant \mathcal {K}\Vert g\Vert _{S^p}\varepsilon +C(1+\alpha \bar{\mu })\mathcal {K}\Vert g(\cdot +r)-g(\cdot )\Vert _{S^p}\\{} & {} \quad \leqslant \mathcal {K}\Vert g\Vert _{S^p}\varepsilon +C(1+\alpha \bar{\mu })\mathcal {K}\varepsilon =(\mathcal {K}\Vert g\Vert _{S^p}+C(1+\alpha \bar{\mu })\mathcal {K})\varepsilon ,\\ \end{aligned}$$

which means that $g_j(t)$ is almost periodic for $j\in \mathbb {N}$.

Next, we prove that $\phi _j(t)\in PAP_0(\mathbb {T};\mathbb {E}^n)$.

$$\begin{aligned} \begin{aligned} \vert \phi _j(t)\vert&\leqslant \int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}\Vert e_A(t,\sigma (s))\Vert \cdot \vert \phi (s)\vert \Delta s\leqslant C\int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}} e_{\ominus \alpha }(t,\sigma (s))\cdot \vert \phi (s)\vert \Delta s\\&\leqslant Ce_{\ominus \alpha }(t,\sigma (t)-(j-1)\mathcal {K})\int _{t-j\mathcal {K}}^{t-(j-1)\mathcal {K}}\vert \phi (s)\vert \Delta s\leqslant C(1+\alpha \bar{\mu })\mathcal {K}\mathcal {N}_p(\phi )(t-j\mathcal {K}). \end{aligned} \end{aligned}$$

Notice that $\phi \in S^pPAP_0(\mathbb {T};\mathbb {E}^n)$. Thus, for a fixed $t_0\in \mathbb {T}$,

$$\begin{aligned} \lim \limits _{r\rightarrow \infty }\frac{1}{2r}\int _{t_0-r}^{t_0+r}\vert \phi _j(t)\vert \Delta t\leqslant C(1+\alpha \bar{\mu })\mathcal {K}\lim \limits _{r\rightarrow \infty }\frac{1}{2r}\int _{t_0-r}^{t_0+r}\mathcal {N}_p(\phi )(t-j\mathcal {K})\Delta t=0. \end{aligned}$$

This implies that $\phi _j\in PAP_0(\mathbb {T};\mathbb {E}^n)$, and then $u_j(t)\in PAP(\mathbb {T};\mathbb {E}^n)$. This together with the boundedness of u(t) yields that $u(t)\in PAP(\mathbb {T};\mathbb {E}^n)$. $\square$

The following conditions will be useful in the proof of our main results.

($\textrm{H}_1$):

$\delta _1,\delta _2\in C(\mathbb {T};\mathbb {R}^+)\cap S^pAP(\mathbb {T};\mathbb {R}^+)$ and $-\delta _1,-\delta _2\in \mathcal {R}^+$. We denote $\delta ^-_i=\inf \limits _{t\in \mathbb {T}}\delta _i(t),i=1,2$, $\bar{\delta }=\textrm{min}(\delta _1^-,\delta _2^-)$.

($\textrm{H}_2$):

$\beta ,\ b,\ p\in S^pPAP(\mathbb {T};\mathbb {R}),\ \tau \in PAP(\mathbb {T};\Pi )$.

($\textrm{H}_3$):

$\displaystyle \theta _1=\text{ max }\left\{ \frac{1}{\delta _1^{-}},\lambda _{\delta _2^-}\mathcal {K}(\Vert \beta \Vert _{S^p}+m\Vert b\Vert _{S^p})\right\} <1$.

We note that, in this work, $z(t) =(z_1(t), z_2(t))$ is assumed to be a column vector function without any further comments. In the rest of this work, we will use the following norm for $PAP(\mathbb {T};\mathbb {R}^2)$, which is equivalent to the one mentioned in Lemma 2.3 (ii):

$$\begin{aligned} \Vert \varphi \Vert _{*}=\text{ max }\left\{ \sup \limits _{t\in \mathbb {T}}\vert \varphi _1(t)\vert ,\ \sup \limits _{t\in \mathbb {T}}\vert \varphi _2(t)\vert \right\} , \end{aligned}$$

for $\varphi =(\varphi _1,\varphi _2)\in PAP(\mathbb {T};\mathbb {R}^2)$. For $\mathbb {T}\ne \mathbb {R}$, let

$$\begin{aligned} E_1^*=\left\{ \varphi \in PAP(\mathbb {T};\mathbb {R}^2):\Vert \varphi -\varphi ^0\Vert _*\leqslant \frac{\theta _1\lambda }{1-\theta _1}\right\} , \end{aligned}$$

where

$$\begin{aligned} \varphi ^0(t)=\left( 0,\varphi _2^0(t)\right) ,\ \varphi ^0_2(t)=\int _{-\infty }^{t}e_{-\delta _2}(t,\sigma (s))p(s)\Delta s,\ \lambda =\lambda _{\delta _2^-}\mathcal {K}\Vert p\Vert _{S^p}, \end{aligned}$$

and by Lemma 3.1, we can get that $\Vert \varphi ^0\Vert _{*}\leqslant \lambda$.

Now we are in a position to give our main result.

Theorem 3.2

Suppose that $\mathbb {T}\ne \mathbb {R}$ and the assumptions $\mathrm {(H_1)}$-$\mathrm{(H_3)}$ hold, and $\displaystyle \frac{\lambda }{1-\theta _1}<1$, then (10) has a unique pseudo almost periodic solution in $E_1^*$ and (5) has a unique pseudo almost periodic solution u satisfying that $(u,u^{\Delta }+\delta _1u)\in E_1^*$.

Proof

It is easy to see that u(t) is a solution of (5) if and only if $(u(t), u^\Delta (t)+\delta _1(t)u(t))$ is a solution of (10). Consider the following system:

$$\begin{aligned} \left\{ \begin{aligned}&x^{\Delta }(t)=-\delta _1(t)x(t)+{\varphi _2(t),}\\&y^{\Delta }(t)=-\delta _2(t)y(t)+\beta (t)\varphi _1(t)-b(t)\varphi _1^m(t-\tau (t))+p(t), \end{aligned} \right. \end{aligned}$$

(13)

for $t\in \mathbb {T},\varphi =(\varphi _1,\varphi _2)\in PAP(\mathbb {T};\mathbb {R}^2)$.

Let

$$\begin{aligned} A(t)=\left( \begin{array}{cc} -\delta _1(t) &{} 0\\ 0 &{}-\delta _2(t) \end{array} \right) , \end{aligned}$$

(14)

then the homogeneous equation of (13) is

$$\begin{aligned} \displaystyle z^\Delta (t)=A(t)z(t),\ t\in \mathbb {T}, \end{aligned}$$

and we can get that

$$\begin{aligned} \Vert e_A(t,s)\Vert \leqslant 2e_{-\bar{\delta }}(t,s)\leqslant 2e_{\ominus \bar{\delta }}(t,s),\ t\geqslant s, \end{aligned}$$

since

$$\begin{aligned} e_{-\bar{\delta }}(t,s)=\exp \left( \int _{s}^t\frac{\mathrm{{Log}}(1-\bar{\delta }\mu (\tau ))}{\mu (\tau )}\Delta \tau \right) \leqslant \exp \left( \int _{s}^t\frac{\mathrm{{Log}}\frac{1}{1+\bar{\delta }\mu (\tau )}}{\mu (\tau )}\Delta \tau \right) =e_{\ominus \bar{\delta }}(t,s). \end{aligned}$$

By Lemma 2.8, we have $\varphi _1(t-\tau (t))$ is pseudo almost periodic, and by Lemma 2.11 and 2.12, we derive that

$$\begin{aligned} \varphi _2,\ \beta \varphi _1-b\varphi _1^m(\cdot -\tau (\cdot ))+p\in S^1PAP(\mathbb {T};\mathbb {R}). \end{aligned}$$

Denote

$$\begin{aligned} z(t)=(x(t),y(t)),\ F(t)=(\varphi _2(t),\beta (t)\varphi _1(t)-b(t)\varphi _1^m(t-\tau (t))+p(t)). \end{aligned}$$

We can rewrite (13) as

$$\begin{aligned} z^{\Delta }(t)=A(t)z(t)+F(t), t\in \mathbb {T}, \end{aligned}$$

(15)

where A(t) is given by (14). By $(\mathrm {H_1})$ and $(\mathrm {H_2})$, it is easy to see that all conditions in Theorem 3.1 are satisfied with (15) instead of (11). Thus, we obtain that (15) has a unique pseudo almost periodic solution $z^{\varphi }(t)=(x^{\varphi }(t),y^{\varphi }(t))$, which is expressed as follows:

$$\begin{aligned} \left\{ \begin{aligned} x^{\varphi }(t)&=\int _{-\infty }^te_{-\delta _1}(t,\sigma (s))\varphi _2(s)\Delta s,\\ y^{\varphi }(t)&=\int _{-\infty }^te_{-\delta _2}(t,\sigma (s))\left( \beta (s)\varphi _1(s)-b(s)\varphi _1^m(s-\tau (s))+p(s)\right) \Delta s. \end{aligned} \right. \end{aligned}$$

(16)

For $\varphi \in E_1^*$, we have

$$\begin{aligned} \Vert \varphi \Vert _*\leqslant \Vert \varphi -\varphi ^0\Vert _*+\Vert \varphi ^0\Vert _*\leqslant \displaystyle \frac{\theta _1\lambda }{1-\theta _1}+\lambda =\displaystyle \frac{\lambda }{1-\theta _1}<1. \end{aligned}$$

Define a nonlinear operator:

$$\begin{aligned} T:E_1^*\mapsto PAP(\mathbb {T};\mathbb {R}^2),\ \varphi =(\varphi _1,\varphi _2)\mapsto z^{\varphi }=(x^{\varphi },y^{\varphi }). \end{aligned}$$

Then (10) has a unique pseudo almost periodic solution in $E_1^*$ if and only if T has a fixed point in $E_1^*$. So we only need to prove that T has a fixed point in $E_1^*$.

First, we show that for any $\varphi \in E_1^*$, $T\varphi \in E_1^*$, we have

$$\begin{aligned} \begin{aligned} \Vert T\varphi -\varphi _0\Vert _*&=\text{ max }\left\{ \sup \limits _{t\in \mathbb {T}}\left| \int _{-\infty }^te_{-\delta _1}(t,\sigma (s))\varphi _2(s)\Delta s\right| \right. ,\\&\qquad \qquad \sup \limits _{t\in \mathbb {T}}\left. \left| \int _{-\infty }^te_{-\delta _2}(t,\sigma (s))\left( \beta (s)\varphi _1(s)-b(s)\varphi _1^m(s-\tau (s))\right) \Delta s\right| \right\} \\&\leqslant \text{ max }\left\{ \sup \limits _{t\in \mathbb {T}}\int _{-\infty }^te_{-\delta _1}(t,\sigma (s))\Delta s\Vert \varphi \Vert _*\right. ,\\&\qquad \qquad \sup \limits _{t\in \mathbb {T}}\left. \int _{-\infty }^te_{-\delta _2}(t,\sigma (s))\left( \vert \beta (s)\vert +\vert b(s)\vert \right) \Delta s\Vert \varphi \Vert _*\right\} .\\ \end{aligned} \end{aligned}$$

By Lemma 2.11 (iii) and 2.14 (iv), we can get that

$$\begin{aligned}&\int _{\infty }^t e_{-\delta _2}(t,\sigma (s))(\vert \beta (s)\vert +\vert b(s)\vert )\Delta s \end{aligned}$$

(17)

$$\begin{aligned}&\quad =\sum \limits _{n=1}^{\infty }\int _{t-n\mathcal {K}}^{t-(n-1)\mathcal {K}}e_{-\delta _2}(t,\sigma (s))(\vert \beta (s)\vert +\vert b(s)\vert )\Delta s \nonumber \\&\quad \leqslant \sum \limits _{n=1}^{\infty }\int _{t-n\mathcal {K}}^{t-(n-1)\mathcal {K}}e_{-\delta _2^-}(t,\sigma (s))(\vert \beta (s)\vert +\vert b(s)\vert )\Delta s\nonumber \\&\quad \leqslant \sum \limits _{n=1}^{\infty }e_{-\delta _2^-}(t,\sigma (t)-(n-1)\mathcal {K})\int _{t-n\mathcal {K}}^{t-(n-1)\mathcal {K}}(\vert \beta (s)\vert +\vert b(s)\vert )\Delta s\nonumber \\&\quad \leqslant \lambda _{\delta _2^-}\mathcal {K}(\Vert \beta (s)\Vert _{S^p}+\Vert b\Vert _{S^p})\leqslant \lambda _{\delta _2^-}\mathcal {K}(\Vert \beta \Vert _{S^p}+m\Vert b\Vert _{S^p}). \end{aligned}$$

(18)

Then we obtain that

$$\begin{aligned} \Vert T\varphi -\varphi _0\Vert _*\leqslant \text{ max }\left\{ \frac{1}{\delta _1^{-}},\lambda _{\delta _2^-}\mathcal {K}(\Vert \beta \Vert _{S^p}+m\Vert b\Vert _{S^p})\right\} \Vert \varphi \Vert _*=\theta _1\Vert \varphi \Vert _*\leqslant \frac{\theta _1\lambda }{1-\theta _1}, \end{aligned}$$

that is $T\varphi \in E_1^*$. Next, we will prove that T is a contraction. In fact, for any $\varphi =(\varphi _1,\varphi _2),\psi =(\psi _1,\psi _2)\in E_1^*$, by Lemma 2.11 (iii), 2.14 (iv) and the same calculation in (18), we can get

$$\begin{aligned}{} & {} \Vert T\varphi -T\psi \Vert _*\\{} & {} \quad =\text{ max }\Bigg \{\sup \limits _{t\in \mathbb {T}}\left| \int _{-\infty }^te_{-\delta _1}(t,\sigma (s))(\varphi _2(s)-\psi _2(s))\Delta s\right| , \ \sup \limits _{t\in \mathbb {T}}\Big \vert \int _{-\infty }^t e_{-\delta _2}(t,\sigma (s))\\{} & {} \qquad \qquad \cdot \Big (\beta (s)(\varphi _1(s)-\psi _1(s))-b(s)\big (\varphi _1^m(s-\tau (s))-\psi _1^m(s-\tau (s))\big )\Big )\Delta s\Big \vert \Bigg \}\\{} & {} \quad \leqslant \text{ max }\Bigg \{\sup \limits _{t\in \mathbb {T}}\int _{-\infty }^te_{-\delta _1}(t,\sigma (s))\Delta s\Vert \varphi -\psi \Vert _*, \ \sup \limits _{t\in \mathbb {T}}\int _{-\infty }^t e_{-\delta _2}(t,\sigma (s))\\{} & {} \qquad \qquad \cdot \Bigg (\vert \beta (s)\vert +\vert b(s)\vert \sum _{i+j=m-1}\Vert \varphi \Vert ^i\Vert \psi \Vert ^j\Bigg )\Delta s\Vert \varphi -\psi \Vert _*\Bigg \}\\{} & {} \quad \leqslant \text{ max }\left\{ \frac{1}{\delta _1^-},\lambda _{\delta _2^-}\mathcal {K}(\Vert \beta \Vert _{S^p}+m\Vert b\Vert _{S^p})\right\} \Vert \varphi -\psi \Vert _*=\theta _1\Vert \varphi -\psi \Vert _*. \end{aligned}$$

Thus, T is a contraction mapping, and by the Banach fixed point theorem, T has a unique fixed point in $E_1^*$. $\square$

For $\mathbb {T}=\mathbb {R}$, the following conditions will be useful.

($\textrm{H}_4$):

$\delta _i\in BC(\mathbb {R};\mathbb {R})\cap S^pAP(\mathbb {R};\mathbb {R}), i= 1,2$ and denote $\delta _i^+=\sup \limits _{t\in \mathbb {R}}\delta _i(t),\ \delta _i^-=\inf \limits _{t\in \mathbb {R}}\delta _i(t), i =1,2,\ \bar{\delta }=\min \{\delta _1^-,\delta _2^-\}>0$;

($\textrm{H}_5$):

$\beta ,\ b,\ p\in S^pPAP(\mathbb {R};\mathbb {R})\cap BC(\mathbb {R};\mathbb {R}),\ \tau \in PAP(\mathbb {R};\mathbb {R})$;

($\textrm{H}_6$):

$\displaystyle \theta _2=\text{ max }\left\{ \frac{1}{\delta _1^{-}},\frac{\Vert \beta \Vert +m\Vert b\Vert }{\delta _2^-}\right\} <1$.

Let

$$\begin{aligned} E_2^*=\left\{ \varphi \in PAP(\mathbb {R};\mathbb {R}^2):\Vert \varphi -\varphi _0\Vert _*\leqslant \frac{\theta _2\lambda }{1-\theta _2} \text{ and } \varphi \text{ is } \text{ uniformly } \text{ continuous }\right\} , \end{aligned}$$

where $\displaystyle \varphi ^0(t)=(0,\varphi _2^0(t)),\ \varphi _2^0(t)=\int _{-\infty }^te_{-\delta _2}(t,s)p(s)ds,\ \lambda =\frac{\Vert p\Vert }{\delta _2^-}$. It is easy to verify that $\Vert \varphi _0\Vert _{*}\leqslant \lambda$.

Lemma 3.2

(Liu and Tunç (2015)) $E_2^*$ is a closed subset of $PAP(\mathbb {R};\mathbb {R}^n)$.

Theorem 3.3

Suppose that $\mathbb {T}=\mathbb {R}$ and assumptions $\mathrm {(H_4)}$-$\mathrm {(H_6)}$ hold, and $\displaystyle \frac{\lambda }{1-\theta _2}<1$, then (10) has a unique pseudo almost periodic solution in $E_2^*$ and (5) has a unique pseudo almost periodic solution u satisfying that $(u,u'+\delta _1u)\in E_2^*$.

Proof

We replace $\varphi =(\varphi _1,\varphi _2)\in PAP(\mathbb {R};\mathbb {R}^2)\cap BUC(\mathbb {R};\mathbb {R}^2)$ in (3.4), then we get the following system:

$$\begin{aligned} \left\{ \begin{aligned}&x'(t)=-\delta _1(t)x(t)+\varphi _2(t),\\&y'(t)=-\delta _2(t)y(t)+\beta (t)\varphi _1(t)-b(t)\varphi _1^m(t-\tau (t))+p(t). \end{aligned} \right. \end{aligned}$$

(19)

Let

$$\begin{aligned} A(t)=\left( \begin{array}{cc} -\delta _1(t) &{} 0\\ 0 &{}-\delta _2(t) \end{array} \right) , \end{aligned}$$

(20)

and the homogeneous equation of (19) is

$$\begin{aligned} z'(t)=A(t)z(t),\ t\in \mathbb {R}. \end{aligned}$$

We can check that $\Vert e_{A}(t,s)\Vert \leqslant 2e_{-\bar{\delta }}(t,s)$ for $t\geqslant s$. By Lemma 2.7, we have $\varphi _1(t-\tau (t))$ is pseudo almost periodic, and by Lemma 2.11 (i), we derive that

$$\begin{aligned} \varphi _2,\ \beta \varphi _1-b\varphi _1^m(\cdot -\tau (\cdot ))+p\in S^1PAP(\mathbb {T};\mathbb {R}). \end{aligned}$$

Denote

$$\begin{aligned} z(t)=(x(t),y(t)),\ F(t)=(\varphi _2(t),\beta (t)\varphi _1(t)-b(t)\varphi _1^m(t-\tau (t))+p(t)). \end{aligned}$$

We can rewrite (19) as

$$\begin{aligned} z'(t)=A(t)z(t)+F(t),\ t\in \mathbb {R}, \end{aligned}$$

(21)

where A(t) is given by (20). By $(\mathrm {H_4})$ and $(\mathrm {H_5})$, it is easy to see that all conditions in Theorem 3.1 are satisfied with (21) instead of (11). Thus, we obtain that (21) has a unique pseudo almost periodic solution $z^{\varphi }(t)=(x^{\varphi }(t),y^{\varphi }(t))$, which is expressed as (16).

For $\varphi \in E_2^*$, we have

$$\begin{aligned} \Vert \varphi \Vert _*\leqslant \Vert \varphi -\varphi ^0\Vert _*+\Vert \varphi ^0\Vert _*\leqslant \displaystyle \frac{\theta _2\lambda }{1-\theta _2}+\lambda =\displaystyle \frac{\lambda }{1-\theta _2}<1. \end{aligned}$$

Define a nonlinear operator

$$\begin{aligned} T:E_2^*\mapsto PAP(\mathbb {T};\mathbb {R}^2),\ \varphi =(\varphi _1,\varphi _2)\mapsto z^{\varphi }=(x^{\varphi },y^{\varphi }). \end{aligned}$$

Then (10) has a unique pseudo almost periodic solution in $E_2^*$ if and only if T has a fixed point in $E_2^*$. So we only need to prove that T has a fixed point in $E_2^*$.

We first prove that $x^{\varphi },y^{\varphi }$ are uniformly continuous. For $\varepsilon >0$, let $\displaystyle 0<\eta <\min \left\{ \frac{-\ln (1-\varepsilon )}{\delta _1^+},\ \varepsilon \right\}$. For $t_1,t_2\in \mathbb {R},\vert t_1-t_2\vert <\eta$, without loss generality we assume that $t_1>t_2$, we have

$$\begin{aligned}&\vert x^{\varphi }(t_1)-x^{\varphi }(t_2)\vert \\&\quad =\left| \int _{-\infty }^{t_1}e_{-\delta _1}(t_1,s)\varphi _2(s)ds-\int _{-\infty }^{t_2}e_{-\delta _1}(t_2,s)\varphi _2(s)ds\right| \\&\quad =\left| \int _{-\infty }^{t_2}(e_{-\delta _1}(t_1,s)-e_{-\delta _1}(t_2,s))\varphi _2(s)ds+\int _{t_2}^{t_1}e_{-\delta _1}(t_1,s)\varphi _2(s)ds\right| \\&\quad \leqslant \left| \int _{-\infty }^{t_2}(e_{-\delta _1}(t_1,t_2)-1)e_{-\delta _1}(t_2,s)\varphi _2(s)ds\right| +\left| \int _{t_2}^{t_1}{e_{-\delta _1}(t_1,s)}\varphi _2(s)ds\right| \\&\quad \leqslant (1-e_{-\delta _1}(t_1,t_2))\int _{-\infty }^{t_2}e^{-\delta _1^-(t_2-s)}\vert \varphi _2(s)\vert ds+\varepsilon \Vert \varphi _2\Vert \\&\quad \leqslant \varepsilon \cdot \frac{\Vert \varphi _2\Vert }{\delta _1^-}+\varepsilon \Vert \varphi _2\Vert =C\varepsilon , \end{aligned}$$

where $\displaystyle C=\frac{1+\delta _1^-}{\delta _1^-}\Vert \varphi _2\Vert$. Hence, $x^{\varphi }$ is uniformly continuous. Similarly, we can prove that $y^{\varphi }$ is uniformly continuous and we omit the details here. Next, we show that for any $\varphi \in E_2^*$, $T\varphi \in E_2^*$.

$$\begin{aligned} \Vert T\varphi -\varphi _0\Vert _*&=\text{ max }\Big \{\sup \limits _{t\in \mathbb {R}}\Big \vert \int _{-\infty }^te_{-\delta _1}(t,s)\varphi _2(s) ds\Big \vert ,\ \sup \limits _{t\in \mathbb {R}}\Big \vert \int _{-\infty }^t e_{-\delta _2}(t,s)\big (\beta (s)\varphi _1(s)\\&\qquad \qquad -b(s)\varphi _1^m(s-\tau (s))\big ) d s\Big \vert \Big \}\\&\leqslant \text{ max }\Big \{\sup \limits _{t\in \mathbb {R}}\int _{-\infty }^t e_{-\delta _1}(t,s) d s\cdot \Vert \varphi \Vert _*,\ \sup \limits _{t\in \mathbb {R}}\int _{-\infty }^te_{-\delta _2}(t,s)\\&\qquad \qquad \cdot \big (\vert \beta (s)\vert +\vert b(s)\vert \big ) d s\cdot \Vert \varphi \Vert _*\Big \}\\&\leqslant \text{ max }\left\{ \frac{1}{\delta _1^{-}},\frac{\Vert \beta \Vert +\Vert b\Vert }{\delta _2^-}\right\} \cdot \Vert \varphi \Vert _*\\&\leqslant \text{ max }\left\{ \frac{1}{\delta _1^{-}},\frac{\Vert \beta \Vert +m\Vert b\Vert }{\delta _2^-}\right\} \cdot \Vert \varphi \Vert _*\\&=\theta _2\Vert \varphi \Vert _*\leqslant \frac{\theta _2\lambda }{1-\theta _2}, \end{aligned}$$

that is $T\varphi \in E_2^*$. At the last, we will prove that T is a contraction. In fact, for any $\varphi =(\varphi _1,\varphi _2),\psi =(\psi _1,\psi _2)\in E_2^*$, we can get

$$\begin{aligned}&\Vert T\varphi -T\psi \Vert _*\\&\quad =\text{ max }\Big \{\sup \limits _{t\in \mathbb {R}}\Big \vert \int _{-\infty }^te_{-\delta _1}(t,s)(\varphi _2(s)-\psi _2(s)) ds\Big \vert ,\ \sup \limits _{t\in \mathbb {R}}\Big \vert \int _{-\infty }^te_{-\delta _2}(t,s)\big (\beta (s)(\varphi _1(s)\\&\qquad \qquad -\psi _1(s))-b(s)(\varphi _1^m(s-\tau (s))-\psi _1^m(s-\tau (s)))\big ) d s\Big \vert \Big \}\\&\quad \leqslant \text{ max }\Bigg \{\sup \limits _{t\in \mathbb {R}}\int _{-\infty }^t e_{-\delta _1}(t,s) d s\cdot \Vert \varphi -\psi \Vert _*,\ \sup \limits _{t\in \mathbb {R}}\int _{-\infty }^t e_{-\delta _2}(t,s)\Bigg (\vert \beta (s)\vert +\vert b(s)\vert \\&\qquad \qquad \cdot \sum _{i+j=m-1}\Vert \varphi \Vert ^i\Vert \psi \Vert ^j\Bigg ) d s\cdot \Vert \varphi -\psi \Vert _*\Bigg \}\\&\quad \leqslant \text{ max }\left\{ \frac{1}{\delta _1^-},\frac{\Vert \beta \Vert +m\Vert b\Vert )}{\delta _2^-}\right\} \cdot \Vert \varphi -\psi \Vert _*\\&\quad \leqslant \theta _2\cdot \Vert \varphi -\psi \Vert _*. \end{aligned}$$

Thus, T is a contraction mapping, and by the Banach fixed point theorem, T has a unique fixed point in $E_2^*$. $\square$

Remark 3.1

We note that $\delta _i, i=1,2, \beta , b$ and p are not assumed to be bounded in $\mathrm{({H}_1)}$ and $\mathrm{({H}_2)}$, but in $\mathrm{({H}_4)}$ and $\mathrm{({H}_5)}$, the boundedness is needed. In fact, for $\mathbb {T}\ne \mathbb {R}$, under the conditions $\mathrm{({H}_1)}$ and $\mathrm{({H}_2)}$, we can get the pseudo almost periodicity of $\varphi _1(\cdot -\tau (\cdot )),\ x^{\varphi }$ and $y^{\varphi }$ in Theorem 3.2 by Lemma 2.8 without the uniform continuity of $\varphi$. On the other hand, for $\mathbb {T}=\mathbb {R}$, to ensure $\varphi _1(\cdot -\tau (\cdot ))\in PAP(\mathbb {R};\mathbb {R})$, we have to prove that the uniform continuity of $\varphi$, where the boundedness of the parameters is essential. There exists counterexamples showing that $\varphi (\cdot -\tau (\cdot ))\notin PAP(\mathbb {R};\mathbb {R}^2)$ if $\varphi$ is not uniformly continuous. For more details of this problem, readers may refer to Zhang (2003). Moreover, assumption $\mathrm{({H}_3)}$ is replaced by a simple form $\mathrm{({H}_6)}$.

Let us end this work with two examples.

Example 3.1

Let

$$\begin{aligned} \displaystyle \mathbb {T}=\bigcup \limits _{k\in \mathbb {Z}}\left( \left[ 2k,2k+\frac{1}{2}\right] \cup \bigcup \limits _{n=2}^{\infty }\left\{ 2k+1-\frac{1}{2^n}\right\} \cup \left\{ 2k+1\right\} \cup \left\{ 2k+\frac{3}{2}\right\} \right) , \end{aligned}$$

denote $\displaystyle g_{nl}=10^n\cdot l+1-\frac{1}{2^{10^n-1}},n\geqslant 2, l \text{ is } \text{ odd }.$ Consider the following Duffing equation on $\mathbb {T}$.

$$\begin{aligned} (x^{\Delta })^{\Delta }(t)+c(t)x^{\Delta }(t)-a(t)x(t)+b(t)x^3(t-\tau (t))=p(t), \end{aligned}$$

(22)

where

$$\begin{aligned}{} & {} c(t)=\left\{ \begin{array}{ll} 0.1 \sin 2\pi t+3,&{}\ t\in [2k,2k+\frac{1}{2}];\\ 0.1 n+1.5,&{}\ t=g_{nl}, n\geqslant 2, l \text{ is } \text{ odd };\\ 3,&{}\ \text{ otherwise }, \end{array} \right. \\{} & {} a(t)=\left\{ \begin{array}{ll} -0.2\pi \cos 2\pi t-1.5(0.1\sin 2\pi t+1.5)+\frac{1}{60}(\sin 2\pi t+h(t)),&{}\ t\in [2k,2k+\frac{1}{2});\\ -1.5n+\frac{h(t)}{60},&{}\ t=g_{nl}, n\geqslant 2, l \text{ is } \text{ odd };\\ -2.25+\frac{h(t)}{60},&{}\ \text{ otherwise } , \end{array} \right. \\{} & {} b(t)=\left\{ \begin{array}{ll} \frac{1}{60}(\sin 2\pi t+h(t)),&{}\ t\in [2k,2k+\frac{1}{2}];\\ \frac{h(t)}{60}, &{} \text{ otherwise }, \end{array} \right. \\{} & {} h(t)=\left\{ \begin{array}{ll} 1,&{}\ t\in [-1,1]_{\mathbb {T}};\\ \frac{1}{\sqrt{\vert t\vert }}, &{} \text{ otherwise }, \end{array} \right. \\{} & {} p(t)=\left\{ \begin{array}{ll} \frac{12}{49}\sin 2\pi t,&{}\ t\in \Big [2k,2k+\frac{1}{2}\Big ];\\ 0,&{} \text{ otherwise }, \end{array} \right. \\{} & {} \tau (t)=\left\{ \begin{array}{ll} 8+\tau _0(t),&{}\ t\in \Big [4k,4k+\frac{1}{2}\Big ];\\ 2+\tau _0(t),&{}\ t\in \Big [4k+2,4k+\frac{5}{2}\Big ];\\ 0,&{}\ \text{ otherwise }, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} \tau _0(t)=\left\{ \begin{array}{ll} 4,&{}\ t\in \Big [2^n,2^n+\frac{1}{2}\Big ],\ n\in \mathbb {N}^+;\\ 0, &{} \text{ otherwise }. \end{array} \right. \end{aligned}$$

It is easy to see that $\tau \in PAP(\mathbb {T};\Pi )$. Let $y(t)=x^{\Delta }(t)+\delta _1(t)x(t)$ where

$$\begin{aligned} \delta _1(t)=\left\{ \begin{array}{ll} 0.1\sin 2\pi t+1.5,&{}\ t\in \Big [2k,2k+\frac{1}{2}\Big ];\\ 1.5,&{} \text{ otherwise }. \end{array}\right. \end{aligned}$$

Then we transform (22) into the following system:

$$\begin{aligned} \left\{ \begin{aligned}&x^{\Delta }(t)=-\delta _1(t)x(t)+y(t),\\&y^{\Delta }(t)=-\delta _2(t)y(t)+\beta (t)x(t)-b(t)x^3(t-\tau (t))+p(t), \end{aligned}\right. \end{aligned}$$

where

$$\begin{aligned}{} & {} \delta _2(t)=\left\{ \begin{array}{ll} n,&{}\ t=g_{nl},\ n\geqslant 2,\ l \text{ is } \text{ odd };\\ 1.5, &{} \text{ otherwise }, \end{array}\right. \\{} & {} \beta (t)=\left\{ \begin{array}{ll} \frac{1}{60}(\sin 2\pi t+h(t)),&{}\ t\in \Big [2k,2k+\frac{1}{2}\Big ];\\ \frac{h(t)}{60},&{} \text{ otherwise }. \end{array} \right. \end{aligned}$$

Obviously, $\delta _1\in S^1AP(\mathbb {T};\mathbb {R}),\ b,\ \beta \in PAP(\mathbb {T};\mathbb {R})$, and use the same calculation in Example 3.2 in Yang and Li (2022), we can get that $\delta _2\in S^1AP(\mathbb {T};\mathbb {R})$. From Lemma 2.14 (iv), we have $\displaystyle \lambda _{\delta _2^-}=\frac{49}{12},$ and we derive that $\displaystyle \Vert \beta \Vert _{S^1}=\Vert b\Vert _{S^1}=\frac{1}{120}\left( 2+\frac{1}{\pi }\right) ,\ \lambda =\lambda _{\delta _2^-}\mathcal {K}\Vert p\Vert _{S^1}=\frac{1}{\pi }$, $\displaystyle \theta =\text{ max }(\delta _1^-,\lambda _{\delta _2^-}\mathcal {K}(\Vert \beta \Vert _{S^1}+3\Vert b\Vert _{S^1}))=\frac{2}{3}$, $\displaystyle \frac{\lambda }{1-\theta }=\frac{3}{\pi }<1$. Let

$$\begin{aligned} \displaystyle E_1^*=\left\{ \phi \in PAP(\mathbb {T};\mathbb {R}^2):\Vert \phi -\phi _0\Vert \leqslant \frac{2}{\pi }\right\} , \end{aligned}$$

where $\phi _0(t)=(0,\phi _2^0(t)),\ \phi _2^0(t)=\int _{-\infty }^te_{-\delta _2}(t,s)p(s)ds.$ Thus, all conditions in Theorem 3.2 are satisfied, (22) has a unique pseudo almost periodic solution u satisfying that $(u,u^{\Delta }+\delta _1u)\in E_1^*$.

Example 3.2

Consider the following Duffing equation on $\mathbb {R}$ with time-varying coefficients:

$$\begin{aligned} \begin{aligned}&x''(t)+(6+\sin t+\cos g(t))x'(t)+(9+\cos t+3 \cos g(t)+3\sin t+\cos g(t)\sin t\\&-\frac{1}{24}(\sin t+\sin \sqrt{2}t))x(t)+\frac{1}{24}(\cos t+\cos \sqrt{2}t)x^3(t-\cos t)=\frac{1}{10}(3\sin t+h(t)), \end{aligned} \end{aligned}$$

(23)

where

$$\begin{aligned} g(t)=\frac{1}{2+\cos t+\cos \sqrt{2}t},\ h(t)=\left\{ \begin{array}{ll} 1,&{}-1<t<1;\\ \frac{1}{\sqrt{\vert t\vert }},&{} \text{ otherwise }. \end{array} \right. \end{aligned}$$

From the result in Levitan (1959) (see page 212–213), we know that $\cos g\in S^1AP(\mathbb {R};\mathbb {R})$, and it is easy to see that $h\in PAP_0(\mathbb {R};\mathbb {R})$. Let

$$\begin{aligned} y=x'(t)+(3+\sin t)x(t), \end{aligned}$$

then we can transform (23) into the following system:

$$\begin{aligned}\left\{ \begin{aligned}&x'(t)=-(3+\sin t)x(t)+y(t)\\&y'(t)=-(3+\cos g(t))y(t)+\frac{1}{24}(\sin t+\sin \sqrt{2}t)x(t)-\frac{1}{24}(\cos t+\cos \sqrt{2}t)\\&\qquad \qquad \qquad \cdot x^3(t-\cos t)+\frac{1}{10}(2\sin t+\sin \sqrt{3}t+h(t)). \end{aligned} \right. \end{aligned}$$

Denote $\delta _1(t)=3+\sin t,\ \delta _2(t)=3+\cos g(t),\ \displaystyle \beta (t)=\frac{1}{24}(\sin t+\sin \sqrt{2}t),\ \displaystyle b(t)=\frac{1}{24}(\cos t+\cos \sqrt{2}t),\ \displaystyle p(t)=\frac{1}{10}(2\sin t+\sin \sqrt{3}t+h(t))$. Let

$$\begin{aligned} E^*=\left\{ \varphi \in PAP(\mathbb {R};\mathbb {R}^2):\Vert \varphi -\varphi ^0\Vert _*\leqslant \frac{1}{5} \text{ and } \varphi \text{ is } \text{ uniformly } \text{ continuous }\right\} , \end{aligned}$$

where

$$\begin{aligned} \varphi ^0(t)=(0,\varphi _2^0(t)),\ \varphi _2^0(t)=\int _{-\infty }^te_{-\delta _2}(t,s)p(s)ds. \end{aligned}$$

It is easy to see that $\delta _1^-=\delta _2^-=2$, $\displaystyle \Vert \beta \Vert =\Vert b\Vert =\frac{1}{12}$, and $p\in PAP(\mathbb {R};\mathbb {R})$, and consequently $\displaystyle p(t)\in S^1PAP(\mathbb {R};\mathbb {R})$ $\displaystyle \Vert p \Vert \leqslant \frac{2}{5}$. Moreover,

$$\begin{aligned} \begin{aligned}&\theta = \text{ max }\left\{ \frac{1}{\delta _1^-},\ \frac{\Vert \beta \Vert +3\Vert b\Vert }{\delta _2^-}\right\} = \text{ max }\left\{ \frac{1}{2},\ \frac{1}{6}\right\} = \frac{1}{2}<1;\ {}&\lambda =\frac{\Vert p \Vert }{\delta _2^-}\leqslant \frac{1}{5};\ \frac{\lambda }{1-\theta }\leqslant \frac{2}{5}<1. \end{aligned} \end{aligned}$$

Now it is easy to check that all conditions in Theorem 3.3 are satisfied. Hence, (23) has a unique pseudo almost periodic solution u satisfying that $(u,u'+\delta _1u)\in E^*$.