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.
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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:
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
where \(\delta >1\) is a constant, (1) transforms into the following system:
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:
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
where \(\xi >1\) is a constant and \(Q_1(t)\) is continuous and differentiable, (2) transforms into the following system:
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:
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:
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
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
and define
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:
Denote
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
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.
Using \(m_1,\) we can generate the outer measure \(m_1^{*}\) on the power set \(\mathcal {P}(\mathbb {T})\) of \(\mathbb {T}\): for \(E\in \mathcal {P}(\mathbb {T})\)
where \(\beta =\sup \mathbb {T}\) and
A set \(A\subset \mathbb {T}\) is called \(\Delta -\)measurable if for \(E\subset \mathbb {T},\) we have
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
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
Definition 2.10
A function \(g\in C(\mathbb {T};\Pi )\) is almost periodic on \(\mathbb {T}\) if for \(\varepsilon >0,\)
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
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
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
We can derive that
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
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 \(t\in \mathbb {T}\),
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 \(r>T\), \(t_0\in \mathbb {T}\),
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
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 \(r>T_1\),
Then we can get
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
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:
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 \(f,g\in BS^p(\mathbb {T};\mathbb {E}^n),\ t\in \mathbb {T}\),
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 \(t\in \mathbb {T}\),
In addition, if f is bounded and continuous, we have
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
If f is bounded and continuous, then we have
\(\square \)
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
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
for a fixed \(t_0\in \mathbb {T}\) and \(r\in \Pi \). Let \(r\rightarrow \infty \) in (8) we derive that
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
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
for \(t,s\in \mathbb {T}\) with the cylinder transformation
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
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
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 \(\varepsilon >0\),
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
Note that \(U(\sigma (s),\sigma (s))=0\), then by the variation of constants formula,
For \(\mathbb {T}=\mathbb {R}\),
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 \(t,s\in \mathbb {T},t\geqslant \sigma (s)\),
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:
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:
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
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
where
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,
which means that \(g_j(t)\) is almost periodic for \(j\in \mathbb {N}\).
Next, we prove that \(\phi _j(t)\in PAP_0(\mathbb {T};\mathbb {E}^n)\).
Notice that \(\phi \in S^pPAP_0(\mathbb {T};\mathbb {E}^n)\). Thus, for a fixed \(t_0\in \mathbb {T}\),
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):
for \(\varphi =(\varphi _1,\varphi _2)\in PAP(\mathbb {T};\mathbb {R}^2)\). For \(\mathbb {T}\ne \mathbb {R}\), let
where
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:
for \(t\in \mathbb {T},\varphi =(\varphi _1,\varphi _2)\in PAP(\mathbb {T};\mathbb {R}^2)\).
Let
then the homogeneous equation of (13) is
and we can get that
since
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
Denote
We can rewrite (13) as
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:
For \(\varphi \in E_1^*\), we have
Define a nonlinear operator:
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
By Lemma 2.11 (iii) and 2.14 (iv), we can get that
Then we obtain that
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
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
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:
Let
and the homogeneous equation of (19) is
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
Denote
We can rewrite (19) as
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
Define a nonlinear operator
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
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 \(\varphi \in E_2^*\), \(T\varphi \in E_2^*\).
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
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
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 \(\mathbb {T}\).
where
where
It is easy to see that \(\tau \in PAP(\mathbb {T};\Pi )\). Let \(y(t)=x^{\Delta }(t)+\delta _1(t)x(t)\) where
Then we transform (22) into the following system:
where
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
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:
where
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
then we can transform (23) into the following system:
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
where
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,
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 \((u,u'+\delta _1u)\in E^*\).
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This paper is supported by the National Natural Science Foundation of China (No.11971329).
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Yang, H., Li, HX. Pseudo almost periodic solutions for a class of nonlinear Duffing equations on time scales. Comp. Appl. Math. 43, 258 (2024). https://doi.org/10.1007/s40314-024-02742-2
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DOI: https://doi.org/10.1007/s40314-024-02742-2