\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 13, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/13\hfil Limit behavior]
{Limit behavior of monotone and concave
skew-product semiflows with applications}

\author[B.-G. Wang \hfil EJDE-2015/13\hfilneg]
{Bin-Guo Wang}

\address{Bin-Guo Wang \newline
School of Mathematics and Statistics, Lanzhou University,
Lanzhou, Gansu 730000, China}
\email{wangbinguo@lzu.edu.cn}

\thanks{Submitted October 21, 2013. Published January 19, 2015.}
\subjclass[2000]{34C12, 34D08, 34D45}
\keywords{Monotone; skew-product semiflow;
attractor; \hfill\break\indent  almost periodic equation}

\begin{abstract}
 In this article, we study the long-time behavior of monotone and concave
 skew-product semiflows. We show that if there are two strongly
 ordered omega limit sets, then one of them is a copy of the base.
 Thus, we obtain a global attractor result. As an application,
 we consider a delay differential equation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Recently, monotone skew-product semiflows generated by
nonautonomous systems, in particular almost periodic systems,
have extensively investigated,  see \cite{hetzer, jiang,
novo2, novo1, novo3,shen11, zhao}. 
Hetzer and Shen \cite{hetzer} considered the convergence of positive solutions 
of almost periodic competitive diffusion systems. Jiang and Zhao \cite{jiang} 
established the $1$-covering property of the omega  limit set for monotone and 
uniformly stable skew-product semiflows with the componentwise
separating property of bounded and ordered full orbits, which is an important 
property for considering the long-time behavior of skew-product semiflows.
 Novo et al \cite{novo2, novo1, novo3} considered the skew-product semiflow 
generated by almost periodic systems. Under the assumption that there existed 
two strongly ordered minimal subsets or completely strongly ordered minimal 
subsets, a complete description of the long-time behavior of the trajectories 
was given and a global picture of the dynamics was provided for a class of 
monotone and convex skew-product semiflows.
Zhao \cite{zhao} proved a global attractivity theory for a class of skew-product
semiflows.

In conclusion, the properties of the
omega limit set of skew-product semiflows, especially its structure,
play an important role in considering the convergent behavior of the
orbit. Shen and Yi \cite{shen11} told us if the omega limit set
$\mathcal{O}$ is linearly stable, then there exists an
integral number $N$ such that $\mathcal{O}$ is the $(N-1)$-almost
periodic extension; i.e., there exists a subset $Y_0\subset Y$
(the definition of $Y$ see Section 2) such that for any $g_0\in
Y_0$, card$(\mathcal{O}\cap\pi^{-1}(g_0))=N$ ($\pi$ is the
natural projector). If it is uniformly stable, then it is the
extension of $Y$; i.e., card$(\mathcal{O}\cap\pi^{-1}(g))=N$ for any
$g\in Y$. This is not enough to understand the structure of the
omega limit set thoroughly. 
If we can obtain the conclusion that $\mathcal{O}$ is the  copy of the base $Y$; 
i.e.,  $\operatorname{card}(\mathcal{O}\cap\pi^{-1}(g))=1$ for any $g\in Y$, 
it would give a  complete description for the long-time behavior of the orbit. 
For this purpose, under the 
assumption of the existence of two completely strongly ordered omega limit sets
and motivated by \cite{novo2,novo1}, we deduce that one of them
is an equilibrium point set if monotonicity and concavity
are satisfied. Naturally, it is a copy of the base.
Furthermore, we establish the convergent results for skew-product semiflows.

This article is organized as follows.
In Section 2, we present some definitions and notation of skew-product semiflows.
In Section 3, we establish global attractor results and consider an almost
periodic delay differential equation.

\section{Preliminaries}

Let $(Y,d)$ be a compact metric space. A continuous flow
$(Y,\sigma,\mathbb{R})$ is defined by a continuous mapping
$\sigma$ : $Y\times \mathbb{R}\to Y$, $(g,t)\mapsto\sigma(g,t)$, which satisfies
(i) $\sigma_0= id$,
(ii) $\sigma_{t}\cdot\sigma_{s} =\sigma_{t+s}$,
for all $t,s\in \mathbb{R}$, where $\sigma_{t}(g):=\sigma (g,t)=g\cdot t$
for $g \in Y$ and $t \in\mathbb{R}$ with $g\cdot 0=g$ and $g\cdot
(s+t)=(g\cdot s)\cdot t$.  A continuous flow $(Y,\sigma,\mathbb{R})$
is \textbf{distal} if for any two distinct points $g_1$ and $g_2$
in $Y$, $\inf_{t\in \mathbb{R}}d(\sigma (g_1,t),\sigma (g_2,t))>0$.

A semiflow ($X$, $\Phi$, $\mathbb{R^{+}}$) on Banach space $X$
 is a continuous map $\Phi:X\times \mathbb{R^{+}} \to X$,
$(x,t)\mapsto\Phi(x,t)$, which satisfies
(i) $\Phi_0= id$,
(ii) $\Phi_{t}\cdot\Phi_{s} =\Phi_{t+s}$, where $\Phi_{t}(x):=\Phi (x,t)$
for $x \in X$ and $t \geq 0$.

A compact, positively invariant subset $S$ of a semiflow ($X$,
$\Phi$, $\mathbb{R^{+}}$) is \textbf{minimal} if it contains no
nonempty, closed and proper positively invariant subset. If $X$
itself is minimal, then ($X$, $\Phi$, $\mathbb{R^{+}}$) is called
minimal semiflow.

In this article, we assume that  $(X,X^{+})$ is an ordered Banach space
with $\operatorname{int}X^{+}\neq\emptyset$, where $\operatorname{int} X^{+}$
denotes the interior of the cone $X^{+}$. For $x$, $y\in X$, we write
$x\leq y$ if $y-x\in X^{+}$; $x< y$ if
$y-x\in X^{+}\backslash \{0\}$; $x\ll y$ if
$y-x\in \operatorname{int}X^{+}$. In addition, the norm of
Banach space $X$ is \textbf{monotone}, namely, if $0\leq x\leq y$, then
$\|x\|\leq \|y\|$ (see \cite{novo1}).

The ordering on $X$ induces the ordering on $Y\times X$ in the
following way:
\begin{gather*}
(g,x)\leq (g,y)\Leftrightarrow y-x \in X^{+},\quad  \forall g\in Y, \\
(g,x)<(g,y)\Leftrightarrow y-x \in X^{+},  \; x\neq y,\quad \forall g\in Y,\\
(g,x)\ll (g,y)\Leftrightarrow y-x\in \operatorname{int}X^{+},\quad \forall
g\in Y.
\end{gather*}


Consider a skew-product semiflow:
$\Pi:\mathbb{R^{+}}\times Y \times X\to Y \times X$,
\begin{equation}\label{skew}
(t,g,x)\mapsto (g\cdot t, u(t,g,x)).
\end{equation}
We assume that $(Y,\sigma,\mathbb{R})$ is a minimal flow defined by
$\sigma: Y\times \mathbb{R} \to Y$,
$(g,t)\mapsto g\cdot t$ and $u$ is locally  $C^{1}$ in $x\in X$;
that is, $u$ is $C^{1}$
in $x$, and $u_{x}$ is continuous in $g\in Y$, $t>0$ in a
neighborhood of each compact subset of $Y\times X$. Moreover, for
any $v\in X$, $\lim_{t\to 0^{+}}u_{x}(t,g,x)v=v$
uniformly in every compact subset of $Y\times X$. Sometimes, we also
use the notation $\Pi_{t}(g,x)\equiv\Pi(t,g,x)$. We denote
$\pi:Y\times X\to Y$ as the natural projection.

The forward orbit of $(g_0,x_0)$ is written as
\[
O(g_0,x_0)=\{\Pi(t,g_0,x_0):t\geq0\}.
\]
If $u(t,g_0,x_0)$ is convergent as $t\to\infty$, we can
define the omega limit set of $(g_0,x_0)$ as
\[
\mathcal {O}(g_0,x_0)=\{(g,x)\in Y\times X:
\exists t_n\to\infty\text{ such that }
g_0\cdot t_n\to g, \; u(t_n,g_0,x_0)\to x\}.
\]
Given a subset $K\subset Y\times X$, let us introduce the projection
set of $K$ into the fiber space
\[
 K_{Y}:=\{g\in Y: \text{ there exists }x\in X \text{ such that }
(g,x)\in K\}\subset Y.
\]

An \textbf{equilibrium} is a map $a: Y\to X$ such that
$a(g\cdot t)=u(t,g,a(g))$, for all $g\in Y$, $t\geq0$. A set
$E\subset Y\times X$ is called an \textbf{equilibrium point set} if
there exists a map $a$ such that $a(g)=x$, for all $(g,x)\in E$ and
$a(g\cdot t)=u(t,g,a(g))$, for all $g\in E_{Y}$, $t\geq0$.

We say that the skew-product semiflow \eqref{skew} is \textbf{monotone}
if
\begin{equation}
u(t,g,y)\geq u(t,g,x), \quad \forall y\geq x, \; t\geq0,
\end{equation}
and \textbf{strongly monotone} if
\[
u(t,g,y)\gg u(t,g,x),\quad \forall y\gg x, \; t\geq 0.
\]
The skew-product semiflow \eqref{skew} is said to be \textbf{eventually
strongly monotone} if there exists $t_0>0$ such that
\begin{equation}\label{2.3}
u(t,g,y)\gg u(t,g,x), \quad \forall y>x, \; t>t_0
\end{equation}
and it preserves the ordering; i.e.,
\[
u(t,g,y)>_ru(t,g,x), \quad \forall y>_rx,\; t>0,
\]
where $>_r$ denotes the relations $\geq$, $>$ or $\gg$.

The skew-product semiflow \eqref{skew} is called
\textbf{concave}, if, whenever $x\leq y$,
\begin{equation}\label{la2.2}
u(t,g,\lambda y+(1-\lambda)x)\geq \lambda u(t,g,y)+(1-\lambda)u(t,g,x)
\end{equation}
for  $g\in Y$, $\lambda \in[0,1]$\ and $t\in\mathbb{R}^{+}$;
\textbf{strongly} \textbf{concave}, if, whenever $x\ll y$,
\begin{equation}\label{la2.3}
u(t,g,\lambda y+(1-\lambda)x)\gg\lambda u(t,g,y)+(1-\lambda)u(t,g,x)
\end{equation}
for  $g\in Y$,  $\lambda \in(0,1)$ and $t\in\mathbb{R}^{+}$.

From the continuous hypothesis for $u$, \eqref{la2.2} is
equivalent to, whenever $y\geq x$,
\[
u_{x}(t,g,x)(y-x)\geq u_{x}(t,g,y)(y-x)
\]
for \ $g\in Y$  and $t\in\mathbb{R}^{+}$. Similarly, \eqref{la2.3}
is equivalent to, whenever $y\gg x$,
\[
u_{x}(t,g,x)(y-x)\gg u_{x}(t,g,y)(y-x)
\]
for  $g\in Y$  and $t\in\mathbb{R}^{+}$. Since $x\leq\lambda
y+(1-\lambda)x$ and $\lambda y+(1-\lambda)x\leq y$, we have
\begin{equation}\label{kconvex}
u_{x}(t,g,y)(y-x)\leq u(t,g,y)-u(t,g,x)\leq u_{x}(t,g,x)(y-x)
\end{equation}
for $g\in Y$ and $t\in\mathbb{R}^{+}$.

Let $y\geq x$, we have
\[
u(t,g,y)-u(t,g,x)=\int_0^{1} u_{x}(t,g,\lambda
y+(1-\lambda)x)(y-x)d\lambda.
\]

A forward orbit
$\{\Pi(t,g_0,x_0)|t\geq0\}$ of the skew-product semiflow
\eqref{skew} is said to be \textbf{uniformly stable} if for any
$\epsilon>0$ there is a $\delta=\delta(\epsilon)>0$, such that if
$s>0$ and $\|u(s,g_0,x_0)-u(s,g_0,x)\|\leq \delta(\epsilon)$,
we have
\[
\|u(t+s,g_0,x_0)-u(t+s,g_0,x)\|\leq \epsilon, \ \forall
t\geq0.
\]
A forward orbit $\{\Pi(t,g_0,x_0)|t\geq0\}$ of the skew-product
semiflow \eqref{skew} is said to be \textbf{uniformly  asymptotically
stable} if it is uniformly stable and there is $\delta_0>0$ with
the following property: for each $\epsilon>0$ there exists a
$t_0(\epsilon)>0$ such that if $s\geq0$ and
$\|u(s,g_0,x_0)-u(s,g_0,x)\|\leq \delta_0$, we get
\[
\|u(t+s,g_0,x_0)-u(t+s,g_0,x)\|\leq \epsilon,\ \forall t\geq
t_0(\epsilon).
\]


\section{Global attractor result}


In this section, we assume that the skew-product semiflow
\eqref{skew} satisfies eventually strong monotonicity and (strong)
concavity. Based on this, we establish the global attractor results.

\begin{definition} \label{def3.1} \rm
Two subsets $S_1$, $S_2$ of $Y\times X$ are ordered
$S_1\leq S_2$ if for each $(g,x_1)\in S_1$, there exists
$(g,x_2)\in S_2$ such that $x_1\leq x_2$. We say $S_1<
S_2$ if $S_1\leq S_2$ and they are different.
\end{definition}

\begin{definition}\label{def3.2} \rm
 We say the subset $S_1$, $S_2$ of $Y\times X$ to be ordered
$S_1\ll S_2$ if for each $(g,x_1)\in S_1$, there exists
$(g,x_2)\in S_2$ such that $x_1\ll x_2$.
\end{definition}

\begin{definition}\label{def3.3} \rm
Two subsets $S_1$, $S_2$ are said to be completely strongly
ordered $S_1\ll_{C} S_2$ if $x_1\ll x_2$ holds for all
$(g,x_1)\in S_1$ and $(g,x_2)\in S_2$.
\end{definition}

\begin{definition}\label{de3.3} \rm
Let $M\subset Y\times X$ be a compact, positively invariant subset of the skew-product semiflow
$\eqref{skew}$. For $(g,x)\in M$, we define the \textbf{Lyapunov exponent} $\lambda(g,x)$ as
\[
\lambda(g,x)=\limsup_{t\to
\infty}\frac{\ln \|u_{x}(t,g,x)\|}{t}.
\]
The number $\lambda_{M}=\sup_{(g,x)\in M}\lambda(g,x)$ is called the
 \textbf{upper Lyapunov exponent} on $M$. If $\lambda_{M}\leq0$, then $M$ is said to be \textbf{linearly stable}.
\end{definition}

In addition, the following assumptions are necessary.
\begin{itemize}
\item[(A1)] Every bounded forward orbit $\{\Pi(t,g,x):t\geq0\}$ is
precompact.
\item[(A2)] $u(t,g,0)=0$, for all $g\in Y$, $t\in\mathbb{R}^{+}$.
\end{itemize}


\begin{theorem}\label{th3.4}
Assume that {\rm (A2)} holds and $\mathcal{O}\subset
Y\times \operatorname{int}X^{+}$ with $\lambda_{\mathcal{O}}<0$. Then $\mathcal{O}$
is uniformly asymptotically stable, that is, for each $g\in Y$, the
forward orbit $\{\Pi(t,g,a(g)|t\geq0\}$ is uniformly asymptotically
stable. Moreover, $\mathcal{O}$ is the copy of the base $Y$, i.e.,
$\operatorname{card}(\mathcal{O}\cap\pi^{-1}(g))=1$, for all $g\in Y$.
\end{theorem}

\begin{proof}
The proof of the uniformly asymptotical stability is completely
similar to \cite[Theorem 8.1]{novo2}, we omit the details here.

In view of the theory of \cite{shen11} about the structure of omega
limit sets, we deduce that $\mathcal{O}$ is an $(N-1)$-extension of
$Y$ as $\lambda_{\mathcal{O}}<0$, that is,
$\operatorname{card}(\mathcal{O}\cap\pi^{-1}(g))=N$ for any $g\in Y$, where $N$
is an integral number, and hence, we denote
$\mathcal{O}\cap\pi^{-1}(g)=\{x_1(g),\dots ,x_{N}(g)\}$. Since
$X^{+}$ is a normal cone and $\operatorname{int}X^{+}\neq\emptyset$, it is easy
to deduce that, for each $g\in Y$, the finite set
$\{x_1(g),\dots ,x_{N}(g)\}$ is bounded with respect to the
ordering induced by $X^{+}$. Thus, there exists the supremum
\[
b(g)=\sup\{x_1(g),\dots ,x_{N}(g)\},
\]
which is a continuous map on $Y$. The positive invariance and
monotonicity of the semiflow imply that
\begin{equation}\label{la3.1}
b(g\cdot t)\leq u(t,g,b(g)),\quad \forall g\in Y,\; t\geq0.
\end{equation}
Furthermore, we claim that $b$ is invariant under
the flow $\sigma$, that is, $b(g\cdot t)= u(t,g,b(g))$ for each
$g\in Y$ and $t\geq0$.


On the contrary, we assume that there exist
$g\in Y$ and $s>0$ such that
\begin{equation}\label{la3.2}
b(g\cdot s)< u(s,g,b(g)).
\end{equation}
Our assumption implies that $x_{i}\gg0$, $i=1,\dots ,N$, from which we
deduce that $b(g)\gg0$. For $e\gg0$ we define $e$-norm by
\begin{equation}\label{enorm}
\|x\|_{e}=:\inf\{\gamma>0:-\gamma e\leq_{K}x\leq_{K}\gamma e\}.
\end{equation}
Let $e=b(g)\gg0$ and
\begin{equation}\label{la3.3}
\alpha=\inf\{\|b(g)-x_{i}(g)\|_{e}:i=1,\dots ,N\}.
\end{equation}
Obviously, $\alpha<1$ and there exists $j\in\{1,\dots ,N\}$ such that
$\alpha=\|b(g)-x_{j}(g)\|_{e}$. Hence, $b(g)-x_{j}(g)\leq\alpha
b(g)$, which is equivalent to
\[
x_{j}(g)\geq(1-\alpha) b(g).
\]
The monotonicity and concavity of the skew-product semiflow and
(A2) imply that
\[
u(s,g,x_{j}(g))\geq(1-\alpha)u(s,g,b(g))>(1-\alpha) b(g\cdot s).
\]
If $\alpha=0$, then we obtain
$b(g\cdot s)\geq x_{j}(g\cdot s)=u(s,g,x_{j}(g))\geq
u(s,g,b(g))$, which contradicts to \eqref{la3.2}, and hence,
$\alpha$ is strictly positive. Moreover, the eventually strong
monotonicity and strong concavity of the semiflow show that
\[
u(s+t_0,g,x_{j}(g))\gg(1-\alpha)u(t_0,g\cdot s,b(g\cdot s)).
\]
The property of cones implies that we can find $0<\alpha_0<\alpha$ such that
\[
u(s+t_0,g,x_{j}(g))\gg(1-\alpha_0)u(t_0,g\cdot s,b(g\cdot s)).
\]
Using the eventually strong monotonicity and strong concavity of the 
semiflow again, it then follows from
\eqref{la3.1} that
\[
u(t,g,x_{j}(g))\gg(1-\alpha_0)b(g\cdot t), \quad \forall t\geq s+t_0.
\]
Since the flow is minimal, there exists a
sequence $t_n\to\infty$ such that
\[
\lim_{n\to \infty}(g\cdot t_n,u(t_n,g,x_{j}(g))=(g,x_{k}(g))
\]
for some $k\in\{1,\dots ,N\}$. Thus, we have
\[
x_{k}(g)\geq(1-\alpha_0)b(g);
\]
i.e., $b(g)-x_{k}(g)\leq\alpha_0b(g)=\alpha_0 e$, which
contradicts to \eqref{la3.3}. Hence, $b$ is invariant under the
flow $\sigma$.


Define
\[
\mathcal{O}_{b}=\{(g,b(g)):g\in Y\}.
\]
Finally, we verify that $\mathcal{O}_{b}=\mathcal{O}$. On the
contrary, assume that there exist $g\in Y$ and $j\in\{1,\dots ,N\}$
such that $b(g)>x_{j}(g)$. The eventually strong monotonicity of the
semiflow implies that $b(g)\gg x_{j}(g)$, for all $g\in Y$,
$j\in\{1,\dots ,N\}$, which contradicts that $b$ is the supremum. 
Hence, we get $\mathcal{O}_{b}=\mathcal{O}$. Furthermore, the conclusion 
that $\mathcal{O}$ is a copy of the base $Y$ can be obtained straight.
\end{proof}

\begin{corollary}\label{co3.6}
Let the assumptions of Theorem \ref{th3.4} hold. Then
$\mathcal{O}$ is an equilibrium point set.
\end{corollary}

\begin{proof}
By Theorem \ref{th3.4}, we have
\[
\mathcal{O}=\{(g,b(g)):g\in Y\},
\]
and the map $g\mapsto b(g)$ is a bijection with $b(g\cdot t)=
u(t,g,b(g))$, $\forall g\in Y$, $t\geq0$. Hence, $\mathcal{O}$ is
the equilibrium point set.
\end{proof}

\begin{lemma}\label{le3.6}
Assume that two omega limit sets satisfy $\mathcal{O}_1\ll_{C}
\mathcal{O}_2$. Then there exists a positive constant $c_1$ such
that
\[
\|u_{x}(t,g,x_2)\|\leq c_1, \quad \forall (g,x_2)\in \mathcal{O}_2, \; t\geq0.
\]
\end{lemma}

\begin{proof}
In view of the proof of \cite[Lemma 5.6]{novo2}, we know
that, for $e\gg0$ there exists a constant $\bar{c}$ (depending on
$e$) such that
\begin{equation}\label{enorm1}
\|u_{x}(t,g,x)\|\leq \bar{c}\|u_{x}(t,g,x)e\|, \quad 
\forall (g,x)\in Y\times X,\; t\geq0.
\end{equation}
The conclusion of \cite[Lemma 5.3]{novo2} implies that
there exists a positive constant $\beta>0$ such that
$x_2-x_1\geq\beta e$,
for all $(g,x_1)\in\mathcal{O}_1$, $(g,x_2)\in\mathcal{O}_2$. 
The positiveness of the linear operator
$u_{x}(t,g,x_2)$ shows that
\[
u_{x}(t,g,x_2)(x_2-x_1)\geq\beta u_{x}(t,g,x_2)e.
\]
The monotonicity and concavity of the semiflow and \eqref{kconvex}
show that
\[
\|u_{x}(t,g,x_2)\|\leq\frac{\bar{c}}{\beta}
\|u_{x}(t,g,x_2)-u_{x}(t,g,x_1)\|,\quad \forall t\geq0.
\]
From the above and the compact positive invariance
of $\mathcal{O}_1$ and $\mathcal{O}_2$ we can conclude that
there exists a positive constant $c_1$ such that
\[
\|u_{x}(t,g,x_2)\|\leq c_1, \quad \forall (g,x_2)\in \mathcal{O}_2, \; t\geq0.
\]
The proof is complete.
\end{proof}

\begin{proposition}\label{pro3.8}
If $\mathcal{O}_1\ll_{C} \mathcal{O}_2$ holds, then
$\mathcal{O}_2$ is a linearly stable set, i.e.,
$\lambda_{\mathcal{O}_2}\leq0$.
\end{proposition}

\begin{proof}
By Definition \ref{de3.3} and Lemma \ref{le3.6}, the conclusion
can be obtained immediately.
\end{proof}


\begin{proposition}\label{pro3.9}
There exists the function $g\mapsto a(g)$ such that the set
\[
Y_0=\{g\in Y:(g,a(g))\in\mathcal{O}\}
\]
is the continuous point set of the mapping $g\mapsto a(g)$.
\end{proposition}

\begin{proof}
It is sufficient to prove that for any
$g_{k}\to g$ there exists $g\mapsto a(g)$ such that
$a(g_{k})\to a(g)$. Because of the minimality of the flow,
we only to prove $a(g\cdot t_{k})\to a(g\cdot t_0)$ for
any $t_{k}\to t_0$. Let $(g,x)\in\mathcal{O}$,
from the definition of the omega limit set, there exists a
sequence $t_n\to\infty$ such that $g_0\cdot
t_n\to g$, $u(t_n,g_0,x_0)\to x$. Let
\[
a(g):=\lim_{n\to \infty}u(t_n,g_0,x_0)=x.
\]
Then
\begin{align*}
a(g\cdot t_0)
&=\lim_{n\to \infty}u(t_n,g_0\cdot t_0,u(t_0,g_0,x_0))\\
&=\lim_{n\to \infty}u(t_n+t_0,g_0,x_0)\\
&=\lim_{n\to \infty}u(t_0,g_0\cdot t_n,u(t_n,g_0,x_0))\\
&=u(t_0,g,x),
\end{align*}
and for any $k\in\mathbb{N}$,
\begin{align*}
\lim_{k\to \infty}a(g\cdot t_{k})
&=\lim_{k\to \infty}\lim_{n\to \infty}u(t_n,g_0\cdot
t_{k},u(t_{k},g_0,x_0))\\
&=\lim_{k\to \infty}\lim_{n\to
\infty}u(t_{k},g_0\cdot
t_n,u(t_n,g_0,x_0))\\
&=\lim_{k\to \infty}u(t_{k},g,x)\\
&=u(t_0,g,x)
=a(g\cdot t_0).
\end{align*}
The proof is complete.
\end{proof}

From \cite[Proposition 6.1]{novo2}, we have the following result
.
\begin{proposition}\label{p6.1}
Suppose that $\mathcal{O}_1\ll_{C} \mathcal{O}_2$. If
$\lambda_{\mathcal{O}_2}=0$, there exist positive constant
$\hat{c}$ and $c$ such that
\begin{equation}\label{ne1}
\hat{c}\leq\|u_{x}(t,g,x_2)\|\leq c, \quad \forall (g,x_2)\in 
\mathcal{O}_2, \; t\geq0.
\end{equation}
\end{proposition}

\begin{proposition}\label{pro3.13}
Assume that $\mathcal{O}_1\ll_{C} \mathcal{O}_2$ holds
and $\lambda_{\mathcal{O}_2}=0$. Then there exists a minimal
subset $\mathcal{O}^{*}$ of $Y\times X$ such that
$\mathcal{O}_1\ll\mathcal{O}^{*}<\mathcal{O}_2$.
\end{proposition}

\begin{proof}
As in Proposition \ref{pro3.9}, define
$Y_0=\{g\in Y:(g,a(g))\in\mathcal{O}_2\}$. Let $g_0\in Y_0$, from the
definition of $Y_0$, we have
$(g_0,a(g_0))\in\mathcal{O}_2$. Since
$\mathcal{O}_1\ll_{C} \mathcal{O}_2$, for each
$(g_0,x_1)\in\mathcal{O}_1$, we have $x_1\ll a(g_0)$.
Fixed $0<\alpha<1$, define
\[
y_{\alpha}=\alpha x_1+(1-\alpha)a(g_0).
\]
Obviously, $x_1\ll y_{\alpha}< a(g_0)$. The precompactness of
the forward orbit $\{\pi(t,g_0,y_{\alpha}):t\geq\delta, \
\delta>0\}$ implies that its closure contains  a minimal subset,
denoted by $\mathcal{O}_{\alpha}$, i.e.,
\[
\mathcal{O}_{\alpha}\subset\operatorname{cls}\{(g_0\cdot
t,u(t,g_0,y_{\alpha})):t\geq\delta\}.
\]
The monotonicity of the skew-product semiflow implies
$\mathcal{O}_1\leq\mathcal{O}_{\alpha}\leq\mathcal{O}_2$. In
the following, we prove that $\mathcal{O}_{\alpha}$ is required.

First we check $\mathcal{O}_1\ll\mathcal{O}_{\alpha}$. For
$(g,z)\in\mathcal{O}_{\alpha}$, there exist a sequence
$t_n\to\infty$ such that
\[
\lim_{n\to \infty}\Pi(t_n,g_0,y_{\alpha})=(g,z).
\]
The concavity implies that
\[
u(t_n,g_0,y_{\alpha})\geq \alpha
u(t_n,g_0,x_1)+(1-\alpha)u(t_n,g_0,a(g_0)).
\]
In addition, there exists a subsequence (assume the whole sequence),
$(g,z_1)\in \mathcal{O}_1$ and $(g,z_2)\in \mathcal{O}_2$
such that
\[
\lim_{n\to \infty}\Pi(t_n,g_0,x_1)=(g,z_1),\quad
\lim_{n\to \infty}\Pi(t_n,g_0,a(g_0))=(g,z_2).
\]
Hence, we have
\[
z\geq\alpha z_1 +(1-\alpha)z_2.
\]
Since $\mathcal{O}_1\ll_{C} \mathcal{O}_2$, $z_1\ll z_2$
holds, from which we have $z\gg z_1$, Definition \ref{def3.2}
tells us $\mathcal{O}_1\ll \mathcal{O}_{\alpha}$.

In the following we prove $\mathcal{O}_2\neq \mathcal{O}_{\alpha}$. 
On the contrary, we assume that
$\mathcal{O}_2=\mathcal{O}_{\alpha}$ with
$(g_0,a(g_0))\in \mathcal{O}_2\cap \mathcal{O}_{\alpha}$. Thus, there exists a
sequence ${t_{k}}\to\infty$ such that
$\lim_{n\to \infty}\Pi(t_{k},g_0,y_{\alpha})=(g_0,a(g_0))$. Proposition
\ref{p6.1} implies that there exist a positive constant $\hat{c}>0$
such that $\hat{c}\leq\|u_{x}(t,g_0,a(g_0))\|$,
$\forall t\geq0$. From the inequality \eqref{kconvex} we deduce
that for all $k\in \mathbb{N}$,
\begin{align*}
u(t_{k},g_0,a(g_0))-u(t_{k},g_0,y_{\alpha})
&\geq u_{x}(t_{k},g_0,a(g_0))(a(g_0)-y_{\alpha})\\
&=\alpha u_{x}(t_{k},g_0,a(g_0))(a(g_0)-x_1).
\end{align*}
It then follows from \eqref{enorm1} and the
monotonicity of the skew-product semiflow that for $e=(a(g_0)-x_1)$, 
we can find $l$ (which only depends on $a(g_0)$ and $x_1$) such that
\[
\|a(g_0\cdot t_{k})-u(t_{k},g_0,y_{\alpha})\|\geq l>0, \quad \forall
k\in \mathbb{N}.
\]
This contradicts that $g_0$ is a point of continuity of $a(g_0)$, 
which implies $\lim_{n\to \infty}(g_0\cdot t_{k},u(t_{k},g_0,y_{\alpha}))
=(g_0,a(g_0))$. The proof is complete.
\end{proof}

\begin{theorem}\label{th4.16}
If $\mathcal{O}_1\ll_{C} \mathcal{O}_2$, then
$\lambda_{\mathcal{O}_2}<0$.
\end{theorem}

\begin{proof}
Proposition \ref{pro3.8} implies that $\lambda_{\mathcal{O}_2}\leq0$,
hence, it is sufficient to prove $\lambda_{\mathcal{O}_2}\neq0$. On the
contrary, we assume that $\lambda_{\mathcal{O}_2}=0$. It follows from
Proposition \ref{pro3.13} that there exists the subset
$\mathcal{O}^{*}$ of $Y\times X$ such that
$\mathcal{O}_1\ll\mathcal{O}^{*}<\mathcal{O}_2$. Let $g_0\in
Y_0$, then $(g_0,a(g_0))\in\mathcal{O}_2$ and there exist
$(g_0,z)\in\mathcal{O}^{*}$ and $(g_0,x_1)\in\mathcal{O}_1$
such that
\[
x_1\ll z< a(g_0).
\]
Let $e=a(g_0)-x_1\gg0$ in \eqref{enorm} and define
\[
\gamma=\inf\{\|a(g_0)-x\|_{e}:(g_0,x)\in \mathcal{O}^{*}\}.
\]
It is easy to see that there exists $(g_0,x)\in \mathcal{O}^{*}$ 
such that $\gamma=\|a(g_0)-x\|_{e}$ with
$0<\gamma<1$, which implies that $a(g_0)-x\leq\gamma (a(g_0)-x_1)$; i.e.,
\[
x\geq(1-\gamma)a(g_0)+\gamma x_1.
\]
Since $a(g_0)\gg x_1$, the monotonicity and strong concavity of
the skew-product semiflow implies that
\begin{equation}\label{4.7}
u(t,g_0,x)\gg(1-\gamma)u(t,g_0,a(g_0))+\gamma u(t,g_0,x_1).
\end{equation}
In view of the property of the cone, there exists $\gamma_0$ with
$0<\gamma_0<\gamma$ such that
\[
u(t,g_0,x)\gg(1-\gamma_0)a(g_0\cdot t)+\gamma_0 u(t,g_0,x_1),
\]
Hence, there exists $(g_0,y)\in \mathcal{O}^{*}$ such that
\[
y\geq(1-\gamma_0)a(g_0)+\gamma_0x_1;
\]
i.e., $a(g_0)-y\leq\gamma_0(a(g_0)-x_1)=\gamma_0 e$, which
implies that $\|a(g_0)-y\|_{e}\leq\gamma_0<\gamma$. This
contradicts  the definition of $\gamma$.
\end{proof}

\begin{theorem}\label{th4.17}
If $\mathcal{O}_1\ll_{C} \mathcal{O}_2$, then $\mathcal{O}_2$
is the copy of the base $Y$, i.e., for each $g\in Y$,
$\operatorname{card}(\mathcal{O}_2\cap\pi^{-1}(g))=1$.
\end{theorem}

\begin{proof}
Since $\mathcal{O}_1\ll_{C} \mathcal{O}_2$, Theorem
\ref{th4.16} tells us $\lambda_{\mathcal{O}_2}<0$, the remaining
is concluded by Theorem \ref{th3.4}.
\end{proof}

Next, we introduce the main result of this article.

\begin{theorem}\label{th4.18}
If {\rm (A1)} and {\rm (A2)} hold, then for any $(g,x)\in Y\times
X^{+}\setminus\{0\}$ either
\begin{itemize}
\item[(i)] $\lim_{t\to \infty}\|u(t,g,x)\|=+\infty$, or
\item[(ii)] there exists an equilibrium point set
 $\mathcal{O}^{*}\subset Y\times \operatorname{int}X^{+}$ 
such that $\mathcal{O}(g,x)=\mathcal{O}^{*}$ and
$\lim_{t\to \infty}\|u(t,g,x)-u(t,g,x^{*})\|=0$,
where $(g,x^{*})=\mathcal{O}^{*}\cap \pi^{-1}(g)$.
\end{itemize}
\end{theorem}

\begin{proof}
On the contrary, we assume that (i) does not hold; i.e.,
the forward orbit of the skew-product semiflow is bounded, From
(A1) we know $\{\Pi(t,g,x)|t\geq0\}$ is precompact. The
eventually strong monotonicity implies that if
$(g,x)\in Y\times (X^{+}\setminus\{0\})$, then 
$\mathcal{O}(g,x)=:\mathcal{O}^{*}\subset
Y\times \operatorname{int}X^{+}$.
It then follows from  (A2) that
$\mathcal{O}(g,0)=:\mathcal{O}^{0}\subset Y\times \{0\}$.
Hence, $\mathcal{O}^{0}\ll_{C} \mathcal{O}^{*}$. Thus, Theorem
\ref{th4.16} implies that $\lambda_{\mathcal{O}^{*}}<0$. Furthermore, Theorem
\ref{th4.17} and Corollary \ref{co3.6} show that
$\mathcal{O}^{*}$ is a copy of the base $Y$ and an equilibrium set, i.e.,
$\operatorname{card}(\mathcal{O}^{*}\cap\pi^{-1}(g))=1$, for all $g\in Y$.

Next we prove that $\lim_{t\to \infty}\|u(t,g,x)-u(t,g,x^{*})\|=0$.
On the contrary, we assume there exists a sequence $t_n\to\infty$
and a positive constant $\epsilon>0$ such that
$\|u(t_n,g,x)-u(t_n,g,x^{*})\|>\epsilon$ for all $n\geq1$.
Denote $\lim_{n\to \infty}\Pi(t_n,g,x)=(\bar{g},\bar{x}_1)$ and
$\lim_{n\to \infty}\Pi(t_n,g,x^{*})=(\bar{g},\bar{x}_2)$, where
$(g,x^{*})=\mathcal{O}^{*}\cap\pi^{-1}(g)$. Since
$\operatorname{card}(\mathcal{O}^{*}\cap \pi^{-1}(\bar{g}))=1$, we have
$\bar{x}_1=\bar{x}_2$. Thus,
$0=\|\bar{x}_1-\bar{x}_2\|=\lim_{n\to \infty}\|u(t_n,g,x)-u(t_n,g,x^{*})\|
\geq\epsilon$, a
contradiction holds. Hence, 
$\lim_{t\to \infty}\|u(t,g,x)-u(t,g,x^{*})\|=0$.
\end{proof}

Consider the almost periodic delay differential equation
\begin{equation}\label{4.4equa}
\begin{gathered}
y'(t)=f(t,y(t),y(t-1)), \quad \forall t\in \mathbb{R}^{+}, \\
y(s)=\phi(s),  \quad \forall s\in[-1,0], 
\end{gathered}
\end{equation}
where $\phi\in C^{+}:=C([-1,0],\mathbb{R}_{+}^{n})$, the function
$f=(f_1,f_2,\dots ,f_n):\mathbb{R}^{+}\times\mathbb{R}^{n}\times\mathbb{R}^{n}$
is almost periodic ( Let $(X,d)$ be metric space, a function 
$f\in C(\mathbb{R},X)$ is said to be \textbf{almost periodic} if for any
$\epsilon>0$, there exists $l=l(\epsilon)>0$ such that every
interval of $\mathbb{R}$ of length $l$ contains at least one point
of the set
$T(\epsilon)=\{\tau\in\mathbb{R}:d(f(t+\tau),f(t))<\epsilon,\forall
t\in\mathbb{R}\}$). In addition, we propose the following properties:
\begin{itemize}
\item[(i)] for each $y,z\in\mathbb{R}^{n}$, $t\in\mathbb{R}$ and
$i\neq j$, $\frac{\partial f_{i}}{\partial y_{j}}(t,y,z)\geq0$; If
$\tilde{I}$ and $\tilde{J}$\ form a partition of
$N=\{1,2,\dots ,n\}$, then there exist $\delta>0$, $i\in \tilde{I}$
and $j\in \tilde{J}$, such that
\[
\big|\frac{\partial f_{i}}{\partial
y_{j}}(t,y,z)\big|\geq\delta, \quad \forall y,z\in\mathbb{R}^{n}, t\in\mathbb{R};
\]

\item[(ii)] for $y,z\in\mathbb{R}^{n}$,
$t\in\mathbb{R}$ and $i,j\in\{1,2,\dots ,n\}$, $\frac{\partial
f_{i}}{\partial z_{j}}(t,y,z)\geq0$. Furthermore, There exists $\delta>0$ such
that
\[
\big|\frac{\partial f_{i}}{\partial z_{j}}(t,y,z)\big|\geq\delta;
\]

\item[(iii)] there exists $g_0\in Y$ such that $f$
\begin{itemize}
\item[(a)] is concave with respect to $(y,z)$, i.e., whenever $y^{1}\leq y^{2}$,\
$z^{1}\leq z^{2}$,
\[
f(t,\lambda(y^{1},z^{1})+(1-\lambda)(y^{2},z^{2}))\geq \lambda
f(t,(y^{1},z^{1}))+(1-\lambda)f(t,(y^{2},z^{2}))
\]
for $\lambda \in[0,1]$ and $t\in\mathbb{R}^{+}$;

\item[(b)] is strongly concave with respect to $(y,z)$;
 i.e., whenever $y^{1}\ll y^{2}$, $z^{1}\ll z^{2}$,
\[
f(t,\lambda(y^{1},z^{1})+(1-\lambda)(y^{2},z^{2}))\gg \lambda
f(t,(y^{1},z^{1}))+(1-\lambda)f(t,(y^{2},z^{2}));
\]
for $\lambda \in(0,1)$ and $t\in[0,1]$;
\end{itemize}

\item[(iv)] $f(\cdot,0,0)\equiv0$.

\end{itemize}
We embed \eqref{4.4equa} into the skew-product semiflow
$\Pi:\mathbb{R^{+}}\times Y \times C^{+}\to Y \times C^{+}$
\begin{equation}\label{4.3h}
\Pi(t,g,\phi)\mapsto (\sigma_{t}(g),u(t,g,\phi))\emph{жа},
\end{equation}
where for $\theta\in[-1,0]$,
$u(t,g,\phi)(\theta)=y(t+\theta,g,\phi)$, and
$\sigma_{t}(g(s,\cdot,\cdot))=g(s,\cdot,\cdot)\cdot
t=g(t+s,\cdot,\cdot)$. $y(t,g,\phi)$ is the solution of the
 equation
\begin{equation}\label{4.mequa}
y'(t)=g(t,y(t),y(t-1)),
\end{equation}
and for $\theta\in[-1,0]$ and $g=(g_1,g_2,\dots ,g_n)\in Y$,
$y(\theta,g,\phi)=\phi(\theta)$, where
\[
Y:=\operatorname{cls}\{f_{t}|t\geq0,\quad
f_{t}(s,\cdot,\cdot)=f(t+s,\cdot,\cdot)\},
\]
the closure is defined in the topology of uniform convergence on
compact set. From the above we deduce that $Y$ is compact metric
space and $(Y,\sigma,\mathbb{R^{+}})$ is minimal. By the standard
theory of delay differential equations (refer to
\cite{hale,hino}), we know that for all $g\in Y $  and initial
value $\phi\in C$, \eqref{4.4equa} admit a unique solution
$y(t,g,\phi)$, i.e., for $\theta\in[-1,0]$,
$y(\theta,g,\phi)=\phi(\theta)$. If $y(t,g,\phi)$ is the unique
solution of \eqref{4.4equa} in the existence interval of $t$, then
$u(t,g,\phi)$ exists for all $t>0$, and the forward orbit
$\{u(t,g,\phi)|t\geq1+\delta\}$ is precompact for $\delta>0$.

\begin{theorem}\label{monotonecave}
The skew-product semiflow \eqref{4.3h} is eventually strongly
monotone and satisfies concavity and strongly concavity,
respectively; i.e., there exists $g_0\in Y$ such that
\[
 \lambda
u(t,g,v)+(1-\lambda)u(t,g,w)\leq u(t,g,\lambda v+(1-\lambda)w)
\]
whenever $w\geq v$, $t\geq0$, $\lambda\in[0,1]$ and $g\in Y$, and
\[
\lambda u(t,g_0,v)+(1-\lambda)u(t,g_0,w)\ll u(t,g_0,\lambda
v+(1-\lambda)w)
\]
whenever $w\gg v$, $t\geq1$ and $\lambda\in(0,1)$.
\end{theorem}

\begin{proof}
The eventually strong monotonicity can be obtained from
\cite{novo2,novo1}. Let $\lambda\in(0,1)$ and 
$Z_g(t)=\lambda y(t,g,v)+(1-\lambda)y(t,g,w)$, so
\[
Z'_g=\lambda g(t,y(t,g,v),v(t-1))+(1-\lambda)
g(t,y(t,g,w),w(t-1)),\ \forall t\in[0,1].
\]
By the monotonicity of the skew-product semiflow, if $v\leq w$, 
then $y(t,g,v)\leq y(t,g,w)$. It then follows from (iii)(a)  that
\[
Z'_g(t)\leq g(t,Z_g(t),\lambda v(t-1)+(1-\lambda)w(t-1)), \quad
\forall t\in[0,1].
\]

From (i), (ii)  and comparison theorems for this kind of
ordinary differential equation (see \cite{guzman}), we have
\[
\lambda y(t,g,v)+(1-\lambda)y(t,g,w)\leq y(t,g,\lambda
v+(1-\lambda)w), \quad \forall t\in[0,1]
\]
An inductive argument shows that for each $n\in\mathbb{N}$,
\[
\lambda y(t,g,v)+(1-\lambda)y(t,g,w)\leq y(t,g,\lambda
v+(1-\lambda)w), \quad \forall t\in[n,n+1].
\]
Hence,
\[
\lambda u(t,g,v)+(1-\lambda)u(t,g,w))\leq u(t,g,\lambda
v+(1-\lambda)w), \quad \forall t\geq0.
\]

If $v\ll w$, the strong monotonicity implies 
$y(t,g_0,v)\ll y(t,g_0,w)$. From (iii)(b), for each $t\in[1,2]$,
\[
z'_{g_0}(t)\ll g_0(t,z_{g_0}(t),\lambda v(t-1)+(1-\lambda)w(t-1)).
\]
Using a same process, comparison theorems provide 
$Z_{g_0}(t)\ll y(t,g_0,\lambda v+(1-\lambda)w)$. Hence,
\[
\lambda y(t,g_0,v)+(1-\lambda)y(t,g_0,w))\ll y(t,g_0,\lambda
v+(1-\lambda)w), \quad  \forall t>0.
\]
That is,
\[
\lambda u(t,g_0,v)+(1-\lambda)u(t,g_0,w))\ll u(t,g_0,\lambda
v+(1-\lambda)w), \quad \forall t>1.
\]
The proof is complete.
\end{proof}


\begin{theorem}
If  \eqref{4.4equa} admits a bounded solution $y(t,\phi)$, then there 
exists an almost periodic solution $y^{*}(t)$, 
$\lim_{t\to \infty}\|y(t,\phi)-y^{*}(t)\|=0$ for 
$\phi\in C^{+}$ with $\phi(0)>0$.
\end{theorem}

\begin{proof}
Theorem \ref{monotonecave} tells us that the skew-product semiflow
\eqref{4.3h} is eventually strongly monotone and (strongly)
concave. For any $(g,\phi)\in Y\times C^{+}$ with $\phi(0)>0$, we
conclude $\mathcal{O}^{*}:=\mathcal{O}(g,\phi)\subset Y\times
\operatorname{int}C^{+}$. It then follows from Theorem \ref{th4.18} 
that $\lim_{t\to \infty}\|y(t,\phi)-y^{*}(t)\|=0$, where
$(g,y^{*}(t))=\mathcal{O}^{*}\cap \pi^{-1}(g)$.
\end{proof}

\subsection*{Acknowledgments}
This research was supported by the NSF of China under grant 10926091,
and by the Fundamental Research Funds for the Central Universities
lzujbky-2010-166.

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\end{document}