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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 154, 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 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/154\hfil Robustness of a nonuniform trichotomy]
{Robustness of a nonuniform $(\mu,\nu)$ trichotomy in Banach spaces}

\author[Y. Jiang \hfil EJDE-2012/154\hfilneg]
{Yongxin Jiang} 

\address{Yongxin Jiang \newline
Department of Mathematics, College of Science \\
Hohai University, Nanjing 210098, China}
\email{yxinjiang@163.com}

\thanks{Submitted March 13, 2012. Published September 7, 2012.}

\subjclass[2000]{34D09, 34D10}
\keywords{Robustness; dichotomy; nonuniform $(\mu,\nu)$ trichotomy; perturbation}

\begin{abstract}
 In this article, we consider the robustness of a nonuniform $(\mu,\nu)$
 trichotomy in Banach spaces, in the sense that the existence of such
 a trichotomy for a given linear equation persists under sufficiently
 small linear perturbations.  The continuous dependence with the
 perturbation of  the constants in the notion of trichotomy is studied, and
 the related robustness of strong  $(\mu,\nu)$ trichotomy is also presented.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}

Exponential trichotomy is the most complex asymptotic property of dynamical 
systems arising from the central manifold theory. When asymptotic behavior 
around the equilibrium point of a dynamical system is controlled by either 
the attraction of the stable manifold or the repulsion of the unstable manifold,
 exponential dichotomy describes a rather idealistic situation where the solution 
is either exponentially stable on the stable subspaces or exponentially unstable
 on the unstable subspaces. When asymptotic behavior is described through the 
splitting of the main space into stable, unstable and central subspaces at each 
point from the flowĄŻs domain, exponential trichotomy reflects a deeper analysis 
of the behavior of solutions of dynamical systems. The conception of trichotomy
 was first introduced by Sacker and Sell \cite{SS}. They described SS-trichotomy 
for linear differential systems by linear skew-product flows. Later, Elaydi 
and H\'ajek \cite{EH88,EH90} gave the notions of exponential trichotomy
for differential systems and for nonlinear differential systems, respectively. 
These notions are stronger notions than SS-trichotomy. Recently, Barreira and 
Valls \cite{bv09j} considered a general concept of nonuniform exponential 
trichotomy, from which can see exponential trichotomy as a special case of 
the nonuniform exponential trichotomy. For more information about exponential 
trichotomy we refer the reader to \cite{pa}.

The notion of exponential trichotomy plays a central role in the study of 
center manifolds, it is one of the powerful tools in the analysis of the 
asymptotic behavior of dynamical systems. When a linear dynamics possesses 
no unstable directions, all solutions converge exponentially to the center 
manifold, and thus the stability of the zero solution under sufficiently small 
perturbations is completely determined by the behavior on any center manifold. 
The study of center manifolds can be traced back to the works of Pliss \cite{pv} 
and Kelley \cite{ka}. A very detailed exposition in the case of autonomous equations 
is given \cite{va1}, See also \cite{va3} for the case of infinite-dimensional systems.
 We refer the reader to \cite{cj,cwy1,va1} for more details and further references.

Inspired both in the classical notion of exponential dichotomy and in the notion 
of nonuniform hyperbolic trajectory introduced by Pesin in \cite{pesin76, pesin77},
 Barreira and Valls have introduced the notion of nonuniform exponential dichotomies 
and have developed the corresponding theory in a systematic way \cite{ bv08, bsv}. 
See also the book \cite{bv6} for details. In particular, the results proved by 
Barreira and Valls can be regarded as a nice contribution to the nonuniform 
hyperbolicity theory. We refer to \cite{bp} for a detailed exposition of the 
nonuniform hyperbolicity theory.

Furthermore, general nonuniform dichotomies have been studied, which extend 
the notion of nonuniform exponential dichotomy in various ways. 
In \cite{BS1,BS3}, Bento and Silva considered the nonuniform polynomial dichotomy, 
the existence of smooth stable manifolds in Banach spaces for sufficiently small 
perturbations of nonuniform polynomial dichotomy was obtained. 
Nonuniform $(\mu,\nu)$ dichotomy was studied in \cite{chu,BS11}, 
Barreira et al \cite{chu} have established robustness of this dichotomy, 
this general nonuniform dichotomies and local stable manifolds was given 
in \cite{BS11}. A similar dichotomy for the discrete case was  discussed 
in \cite{BS2,jie}

In \cite{bv09j}, Barreira and Valls have introduced the so-called nonuniform 
exponential trichotomy. Robustness of such a nonuniform exponential trichotomy 
was established, which means a nonuniform exponential trichotomy defined by 
a nonautonomous  linear equation 
\begin{equation} \label{le}
x'=A(t)x
\end{equation}
 in a Banach space, 
persists under sufficiently small linear perturbations in the equation
\begin{equation} \label{lpe}
x'=[A(t)+B(t)]x.
\end{equation} 
In \cite{bv10j}, they considered a linear equations \eqref{le} 
that may exhibit stable, unstable and central behaviors in different directions, 
with respect to arbitrary  asymptotic rates of the form $e^{c\rho(t)}$ 
determined by an arbitrary function $\rho(t)$ instead of the usual exponential 
behavior $e^{ct}$. They proposed a $\rho-$nonuniform exponential trichotomy and 
consider the Lyapunov functions for the trichotomy.

In the present paper, our main objective is to consider the general case of 
nonuniform $(\mu,\nu)$ trichotomy for an arbitrary
nonautonomous linear dynamics, and establish the robustness of the nonuniform 
$(\mu,\nu)$ trichotomy in Banach spaces, based on \cite{chu} for nonuniform 
$(\mu,\nu)$ dichotomy. This means that such a trichotomy persists under 
sufficiently small linear perturbations. Precisely, the perturbed equation 
\eqref{lpe} admits a nonuniform $(\mu,\nu)$ trichotomy if the same happens 
for \eqref{le} for any sufficiently small perturbations $B(t)$. 
We also establish the continuous dependence with the perturbation of 
the constants in the notion of trichotomy and robustness of strong 
nonuniform $(\mu,\nu)$ trichotomy. We note that the notion of nonuniform 
$(\mu,\nu)$ trichotomy is also an elaboration of the notion of 
nonuniform $(\mu,\nu)$ dichotomy.

We remark that the study of robustness in the case of uniform
exponential behavior has a long history. Early it was discussed by
Perron \cite{per}, Coppel \cite{coppel}.
 For more recent work, we refer to \cite{ h07, jw} and the references 
therein for uniform exponential behavior. We refer to 
\cite{bv08,bv6, bv09, cj} for the study of robustness in the setting
of a nonuniform exponential behavior. A trichotomy for the discrete
case was discussed in \cite{bvl,ccc}. We emphasize that the
trichotomy considered in this paper is more general, this may seem a
somewhat formal generalization of the notion of nonuniform exponential 
trichotomy in \cite{bv09j, bv10j}.  Particularly
in view of the applications, it is important to look for more general notions.
 Moreover, due to the central role played by the notion of trichotomy,
 most importantly in the theory of center manifolds which are crucial 
in the study of the
asymptotic behavior of trajectories, it is also helpful to understand
how trichotomies vary under perturbations.

 The remaining part of this paper is
organized as follows. Section 2 is a preliminary for our main results.
In Section 3, we establish the robustness of nonuniform $(\mu,\nu)$
trichotomies. The robustness of strong  $(\mu,\nu)$ trichotomies is
presented in Section 4.



\section{Preliminaries}

We say that an increasing function $\mu:\mathbb{R}^+\to[1,+\infty)$ is
a growth rate if
\[
\mu(0)=1\quad \text{and}\quad\lim_{t\to+\infty}\mu(t)=+\infty.
\]

Let $X$ be a Banach space and denote by $\mathcal {B}(X)$ the space
of bounded linear operators acting on $X$. Given a continuous
function $A:\mathbb{R}^+\to \mathcal{B}(X)$. We assume that each
solution of \eqref{le} is global and denote the evolution operator
associated with \eqref{le} by $T(t,s)$; i.e., the linear operator such
that 
\[
T(t,s)x(s)=x(t),\quad t,s>0,
\] 
where $x(t)$ is any solution
of \eqref{le}. Clearly, $T(t,t)=\operatorname{Id}$ and
\[
T(t,\tau)T(\tau,s)=T(t,s),\quad t,\tau,s>0.
\]

\begin{definition}\cite{chu} \rm  We say that equation \eqref{le}
 admits a nonuniform $(\mu,\nu)$ dichotomy in $\mathbb{R}^+$
if there exist projections $P(t):X\to X$ for each $t>0$
satisfying 
\begin{equation} \label{tppt}
T(t,s)P(s)=P(t)T(t,s),\quad t\geq s,
\end{equation} 
and there exist constants $\alpha,\beta,D>0$ $\varepsilon\geq0$ and two
continuously differentiable growth rates $\mu,\nu$ such that 
\begin{gather} \label{tpp}
\|T(t,s)P(s)\|\leq D\Big(\frac{\mu(t)}{\mu(s)}\Big)^{-\alpha}\nu^{\varepsilon}(s),
\quad t\geq s, \\
\label{tqq}
\|T(t,s)Q(s)\|\leq D\Big(\frac{\mu(s)}{\mu(t)}\Big)^{-\beta}\nu^{\varepsilon}(s),\quad
s\geq t,
\end{gather}
 where $Q(t)=\operatorname{Id}-P(t)$ for each $t>0$. When
$\varepsilon=0$, we say that \eqref{le} has a uniform $(\mu,\nu)$
dichotomy or simply a $(\mu,\nu)$ dichotomy.
\end{definition}

For the convenience of the reader, we recall the following result
about robustness of a nonuniform $(\mu,\nu)$ dichotomy obtained in
\cite{chu}.
Set 
\begin{gather} 
\begin{gathered}\label{con1} 
\tilde{\alpha}=\frac{(\alpha-\beta)+\sqrt{(\alpha+\beta)^{2}
-4\delta D(\alpha+\beta)}}{2}  \\ 
\tilde{D_{1}}=\frac{D}{1-\delta D/(\beta+\tilde{\alpha})}, \quad
\tilde{D_{2}}=\frac{D}{1-\delta D/(\alpha+\tilde{\alpha})},\quad
\tilde{D}=\max\{\tilde{D_{1}},\tilde{D_{2}}\},
\end{gathered}\\
\label{term}
\delta<\min\big\{\frac{\alpha+\beta}{4D},
\frac{\alpha\beta}{2D(\alpha+\beta)},\frac{\tilde{\alpha}+\beta}{D},
\frac{\tilde{\alpha}+\alpha}{D}, \frac{1}{4D\tilde{D}}\big\}.
\end{gather}
We  denote the evolution operator associated to equation \eqref{lpe}
by $\hat{T}(t,s)$

\begin{lemma}[\cite{chu}] \label{lemm1}
 Let $A,B:\mathbb{R}^+\to \mathcal{B}(X)$ be continuous functions such that equation
\eqref{le} admits a nonuniform $(\mu,\nu)$ dichotomy in $\mathbb{R}^+$ with
$\varepsilon<\min\{\alpha, \beta\}$. Moreover, assume that $B(t)$ satisfies
\begin{equation} \label{bc-1}
\|B(t)\|\leq \delta\nu^{-\varepsilon}(t)\frac{\mu'(t)}{\mu(t)},\quad t\geq0
\end{equation}
with \eqref{term}, then \eqref{lpe} admits a nonuniform
$(\mu,\nu)$ dichotomy in $\mathbb{R}^+$ with the projections
$\hat{P}(t),\hat{Q}(t)$ such that for each $t,s\in\mathbb{R}^+$: 
\begin{gather} \label{nop}
\hat{P}(t)=\hat{T}(t,0)\hat{P}(0)\hat{T}(0,t),\quad
\hat{Q}(t)=\hat{T}(t,0)\hat{Q}(0)\hat{T}(0,t),
\\
\label{imp} \|\hat{T}(t,s)|\operatorname{Im}\hat{P}(s)\|\leq\tilde{D}
\Big(\frac{\mu(t)}{\mu(s)}\Big)^{-\tilde{\alpha}}\nu^{\varepsilon}(s), \quad
t\geq s,\\
\label{imq} \|\hat{T}(t,s)|\operatorname{Im}\hat{Q}(s)\|\leq\tilde{D}
\Big(\frac{\mu(s)}{\mu(t)}\Big)^{-\tilde{\alpha}}\nu^{\varepsilon}(s),\quad s\geq t.
\end{gather} 
Suppose further that growth rates satisfy $\mu\geq\nu$ and
 \begin{equation} \label{bc2}
\|B(t)\|\leq \delta\nu^{-2\varepsilon}(t)
\big\{(\alpha+\tilde{\alpha})\frac{\mu'(t)}
{\mu(t)}+\varepsilon\frac{\nu'(t)}{\nu(t)}\big\},\quad t\geq0,
\end{equation}
then equation \eqref{lpe} admits a nonuniform $(\mu,\nu)$
dichotomy in $\mathbb{R}^+$ with the constants $\alpha,D,\varepsilon$
replaced respectively by $\tilde{\alpha},4D\tilde{D},2\varepsilon$
in \eqref{tpp}, \eqref{tqq} and 
\begin{equation} \label{pqpq}
\|\hat{P}(t)\|\leq 4D\nu^{\varepsilon}(t),\quad 
\|\hat{Q}(t)\|\leq 4D\nu^{\varepsilon}(t).
\end{equation}
\end{lemma}

\begin{definition}\rm  We say that \eqref{le}
 admits a nonuniform $(\mu,\nu)$ trichotomy in $I$ if there exist projections 
$P(t), Q (t), R(t): X\to X$ for each $t \in I$ such that 
\begin{equation} \label{PQR}
\begin{gathered}
T(t,s)P(s)=P(t)T(t,s),\quad T(t,s)Q(s)=Q(t)T(t,s),\quad
T(t,s)R(s)=R(t)T(t,s)
\end{gathered}
\end{equation}
and
\begin{equation} \label{PQRI}
P(t)+Q(t)+R(t)=\operatorname{Id}
\end{equation}
for every $t,s\in I$, and there exist
constants
\begin{equation} \label{abcd}
0\leq \eta<\alpha,\quad  0\leq \xi<\beta, \quad
\varepsilon\geq0, \quad D\geq1
\end{equation}
such that for every $t,s\in I$ with
$t\geq s$ we have:
 \begin{gather} \label{tp}
\|T(t,s)P(s)\|\leq D\Big(\frac{\mu(t)}{\mu(s)}\Big)^{-\alpha}
\nu^{\varepsilon}(s),\\
\|T(t,s)R(s)\|\leq D\Big(\frac{\mu(t)}{\mu(s)}\Big)^{\xi}\nu^{\varepsilon}(s),
\\ \label{tq}
\|T(t,s)^{-1}Q(t)\|\leq D\Big(\frac{\mu(t)}{\mu(s)}\Big)^{-\beta}\nu^{\varepsilon}(t),
\\
\|T(t,s)^{-1}R(t)\|\leq D\Big(\frac{\mu(t)}{\mu(s)}\Big)^{\eta}\nu^{\varepsilon}(t),
\end{gather}a\
We notice that setting $t = s$ in \eqref{tp} and \eqref{tq} we obtain
\begin{equation} \label{pqr}
P(t)\leq D \nu^{\varepsilon}(t), \quad
Q(t)\leq D \nu^{\varepsilon}(t), \quad
R(t)\leq D \nu^{\varepsilon}(t),
\end{equation}
for every $t\in I$. When $\varepsilon= 0$, we say that \eqref{le} admits
a uniform $(\mu,\nu)$ trichotomy.
\end{definition}


The following is an example with a nonuniform $(\mu,\nu)$ trichotomy which 
can not be uniform.

\begin{example} \label{examp2.4} \rm
 Given $\varepsilon>0$, $\alpha>0$, $\mu$ and $\nu$ are arbitrary differentiable 
growth rates, consider the differential equation in $\mathbb{R}^{3}$
given by
\begin{equation} \label{exp}
\begin{split}
\dot{x}&=\Big(\frac{-\alpha\mu'(t)}{\mu(t)}+\frac{\varepsilon\nu'(t)}{2\nu(t)}
(\cos t-1)-\frac{\varepsilon}2\log \nu(t)\sin t\Big)x\\
\dot{y}&=0\\
\dot{z}&=\Big(\frac{\alpha\mu'(t)}{\mu(t)}-\frac{\varepsilon\nu'(t)}{2\nu(t)}
(\cos t-1)+\frac{\varepsilon}2\log \nu(t)\sin t\Big)z.
\end{split}
\end{equation}
It is easy to verify that \eqref{exp} has the  evolution operator
\begin{align*}
 T(t,s)(x,y,z)
&= (X(t,s)x, Y(t,s)y, Z(t,s)z)\\
&= (T(t,s)P(s)x,T(t,s)R(s)y,T(t,s)Q(s)z),
\end{align*}
where
\begin{gather*}
X(t,s)=\Big(\frac{\mu(t)}{\mu(s)}\Big)^{-\alpha}
\exp\Big(\frac{\varepsilon}{2}\log\nu(t)(\cos t-1)
-\frac{\varepsilon}{2}\log\nu(s)(\cos s-1)\Big),\\
 Y(t,s)=1,
\\
Z(t,s)=X(s,t)=\Big(\frac{\mu(t)}{\mu(s)}\Big)^{\alpha}
\exp\Big(-\frac{\varepsilon}{2}\log\nu(t)(\cos t-1)
+\frac{\varepsilon}{2}\log\nu(s)(\cos s-1)\Big)
\end{gather*}
One can easily verify that
\begin{gather*}
\|T(t,s)P(s)\|=\|X(t,s)\|\leq \Big(\frac{\mu(t)}{\mu(s)}\Big)^{-\alpha}
 \nu^{\varepsilon}(s),\\
\|T(t,s)R(s)\|=\|Y(t,s)\|= 1\leq\nu^{\varepsilon}(s),\\
\|T(t,s)^{-1}Q(t)\|=\|Z(t,s)^{-1}\|\leq
\Big(\frac{\mu(t)}{\mu(s)}\Big)^{-\alpha}\nu^{\varepsilon}(t),\\
 \|T(t,s)^{-1}R(t)\|=\|Y(t,s)^{-1}\|=1\leq \nu^{\varepsilon}(t),
\end{gather*}
This shows that \eqref{exp} admits a nonuniform $(\mu,\nu)$
trichotomy in $\mathbb{R}^{+}$. Moreover,
if we take $t=2k\pi$ and $s=(2k-1)\pi,k\in\mathbb{N}$, then
\[
\|X(t,s)\|=\Big(\frac{\mu(t)}{\mu(s)}\Big)^{-\alpha}\nu^{\varepsilon}(s),
\]
which ensures us that the nonuniform part can not removed when $\varepsilon>0$.
\end{example}

\section{Robustness in semi-infinite intervals}

\begin{theorem} \label{main-a}
Let $A, B : I\to B(X)$ be continuous functions in an
interval $I = [0,+\infty)$ such that \eqref{le} admits a nonuniform 
$(\mu,\nu)$ trichotomy with $\mu\geq\nu$ in $I$ satisfying 
\begin{equation} \label{e}
\varepsilon< \min\{(\alpha-\eta)/2,(\beta-\xi)/2\},
\end{equation}
and assume that $B(t)$ satisfies \eqref{bc2} and \eqref{pqpq} with 
\eqref{term}, then \eqref{lpe} admits a nonuniform $(\mu,\nu)$  trichotomy 
in $[0,+\infty)$; i.e.

(i) there exist projections $\hat{P}(t),\hat{Q}(t)$ and $\hat{R}(t)$ for 
 $t\in I$ satisfying \eqref{PQR} and \eqref{PQRI} for every $t,s\in I$;

(ii) for every $t,s\in I$ with $t\geq s$, the corresponding estimates to the
ones in \eqref{tp}--\eqref{pqr} are valid
with constants $\alpha, \beta, \xi, \eta, \varepsilon, D$ replaced respectively by
\begin{gather*}
\hat{\alpha}=(\alpha+\eta)/2+L((\alpha-\eta)/2),\quad 
\hat{\beta}=(\beta+\xi)/2+L((\beta-\xi)/2), \\
\hat{\xi}=(\beta+\xi)/2-L((\beta-\xi)/2),\quad 
\hat{\eta}=(\alpha+\eta)/2-L((\alpha-\eta)/2),\\
\hat{\varepsilon}=3\varepsilon, \quad 
\hat{D}= \max\big\{\frac{D}{1-\delta D/(\hat{\alpha}-\eta)},\frac{D}{1-\delta
D/(\hat{\beta}-\xi)}\big\}, 
\end{gather*}
 where $L(x)=x\sqrt{1-2\delta D/x}$,
\end{theorem}

\begin{proof}  
Let $x(t)=T(t,s)x(s)$ be a solution of \eqref{le}. We
consider the change of variables $y(t)=x(t)\mu^{k}(t)$, where
$k=(\alpha+\eta)/2$. Then $y(t)$ satisfies the linear equation
\begin{equation} \label{n1}
y'= [A(t)+k\frac{\mu'(t)}{\mu(t)}]y.
\end{equation}
Denoting by $T_{k}(t,s)$ its evolution operator we have 
\begin{equation}
T_{k}(t,s)=T(t,s)\Big(\frac{\mu(t)}{\mu(s)}\Big)^{k}.
\end{equation}
Since \eqref{le} admits a nonuniform $(\mu,\nu)$ trichotomy in $I$ , we
conclude that \eqref{n1} admits a nonuniform $(\mu,\nu)$ dichotomy in
$I$ with $\alpha_{1}=\beta_{1}=(\alpha-\eta)/2$, and projections
$P_{1}(t)=P(t)$ and $Q_{1}(t)=Q(t)+R(t)$ for each $t\in I$. It
follows from Lemma \eqref{lemm1} that the equation 
\begin{equation} \label{n2}
 y'= [A(t)+k\frac{\mu'(t)}{\mu(t)}+B(t)]y
\end{equation}
 admits a nonuniform $(\mu,\nu)$ dichotomy, say with projections 
$\hat{P}_{1}(t)$ and $ \hat{Q}_{1}(t)$. 
In particular, the linear subspaces
$\hat{E}_{1}(t)=\hat{P}_{1}(t)(X)$ and
$\hat{F}_{1}(t)=\hat{Q}_{1}(t)(X)$ satisfy 
\begin{equation} \label{n3}
\hat{E}_{1}(t)\oplus\hat{F}_{1}(t)=X.
\end{equation}
Now we consider a second
change of variables $z(t)=x(t)\mu^{k'}(t)$, where
$k'=-(\beta+\xi)/2$. Then $z(t)$ satisfies the linear equation
\begin{equation}\label{n11} 
z'= [A(t)+k'\frac{\mu'(t)}{\mu(t)}]z,
\end{equation}
and denoting by $T_{k'}(t,s)$ its evolution operator we have 
\begin{equation}
T_{k'}(t,s)=T(t,s)\Big(\frac{\mu(t)}{\mu(s)}\Big)^{k'}.
\end{equation}
Since \eqref{le} admits a nonuniform $(\mu,\nu)$ trichotomy in $I$ , we
conclude that \eqref{n11} admits a nonuniform $(\mu,\nu)$ dichotomy
in $I$ with $\alpha_{2}=\beta_{2}=(\beta-\xi)/2$, and projections
$P_{2}(t)=P(t)$ and $Q_{2}(t)=Q(t)+R(t)$ for each $t\in I$. It
follows from Lemma \eqref{lemm1} that the equation 
\begin{equation} \label{n12}
z'= [A(t)+k'\frac{\mu'(t)}{\mu(t)}+B(t)]y,
\end{equation}
admits a nonuniform $(\mu,\nu)$ dichotomy, say with projections
 $\hat{P}_{2}(t)$ and $\hat{Q}_{2}(t)$. In particular, the linear subspaces
$\hat{E}_{2}(t)=\hat{P}_{2}(t)(X)$ and
$\hat{F}_{2}(t)=\hat{Q}_{2}(t)(X)$ satisfy 
\begin{equation} \label{n13}
\hat{E}_{2}(t)\oplus\hat{F}_{2}(t)=X.
\end{equation}
We also consider the evolution operators in \eqref{n2} and \eqref{n12}, 
namely
\begin{equation}\label{change}
\hat{T}_{k}(t,s)=\Big(\frac{\mu(t)}{\mu(s)}\Big)^{k}\hat{T}(t,s)\quad
\text{and}\quad
\hat{T}_{k'}(t,s)=\Big(\frac{\mu(t)}{\mu(s)}\Big)^{k'}\hat{T}(t,s).
\end{equation}
\end{proof}

In the following, we firstly consider the relationship of these
linear subspaces and projections.

\begin{lemma} \label{lem2} For every $t\in I$ we have
$$
\hat{E}_{1}(t)\subset\hat{E}_{2}(t)\quad
\text{and}\quad \hat{F}_{2}(t)\subset\hat{F}_{1}(t).
$$
\end{lemma}

\begin{proof}  Set 
\begin{equation}
U(x)=\limsup_{t\to+\infty}\frac{\ln\|\hat{T}_{k}(t,s)x\|}{\ln \mu(t)}.
\end{equation}
If there exists
$x\in\hat{E}_{1}(t)\backslash\hat{E}_{2}(t)$, then we write $x=y+z$
with $y\in \hat{E}_{2}(t)$ and $z\in\hat{F}_{2}(t)$. Since 
$x\in \hat{E}_{1}(t)$, by Lemma {\rm\eqref{lemm1}} we have
\begin{equation} \label{con}
\|\hat{T}_{k}(t,s)x\|\leq \tilde{D}\Big(\frac{\mu(t)}{\mu(s)}\Big)
^{-L(\alpha_{1})}\nu^{\varepsilon}(s)\|x\|.
\end{equation}
and hence 
\begin{equation} \label{md1}
U(x)\leq-L(\alpha_{1})<0
\end{equation}
where $L(x)=x\sqrt{1-2\delta D/x}$.
Moreover, since $x\in\hat{E}_{1}(t)\backslash\hat{E}_{2}(t)$, $y\in \hat{E}_{2}(t)$, 
we have $z\neq 0$, and hence
$$
U(x)=\max\{U(y),U(z)\}=U(z)
=\limsup_{ t\to+\infty}\frac{\ln\|\hat{T}_{k}(t,s)z\|}{\ln\mu(t)}.
$$
Since $z\in\hat{F}_{2}(t)$, for $t\geq s$ we have
\begin{align*}
 \|\hat{T}_{k}(t,s)z\|
&= \Big(\frac{\mu(t)}{\mu(s)}\Big)^{(k-k')}\|\hat{T}_{k'}(t,s)z\|\\
&\geq  \frac{1}{\tilde{D}}\|z\|\Big(\frac{\mu(t)}{\mu(s)}\Big)
 ^{(k-k'+L(\alpha_{2}))}\nu^{-\varepsilon}(t)\\
&\geq \frac{1}{\tilde{D}}\|z\|\Big(\frac{\mu(t)}{\mu(s)}\Big)
 ^{(k-k'+L(\alpha_{2}))}
\frac{1}{\mu^{\alpha_{2}+L(\alpha_{2})}(t)}.
\end{align*}
So 
\begin{equation}
U(x)=(k-k'-\alpha_{2})=\frac{\alpha+\eta+2\xi}{2}>0.
\end{equation}
This contradicts the inequality \eqref{md1}. Therefore, 
$\hat{E}_{1}(t)\subset\hat{E}_{2}(t)$.
In a similar manner,  we obtain that 
$\hat{F}_{2}(t)\subset\hat{F}_{1}(t)$ for each $t\in I$.
\end{proof}

By Lemma \eqref{lem2}, we can prove the following two Lemmas. 
These proofs are  similar to those in \cite{bv09j}.

\begin{lemma} \label{lemm3} 
 For every $t\in I$ we have 
\begin{equation} \label{zhihe}
(\hat{E}_{2}(t)\cap\hat{F}_{1}(t))\oplus\hat{E}_{1}(t)\oplus\hat{F}_{2}(t)=X.
\end{equation}
\end{lemma}

\begin{proof} 
Since $\hat{E}_{1}(t)\oplus\hat{F}_{1}(t)=X$, we have
$$
(\hat{E}_{2}(t)\cap\hat{E}_{1}(t))\oplus(\hat{E}_{2}(t)\cap\hat{F}_{1}(t))
=\hat{E}_{2}(t).
$$
According to Lemma \eqref{lem2}, $\hat{E}_{1}(t)\subset \hat{E}_{2}(t)$, we have 
$$
\hat{E}_{2}(t)\cap\hat{E}_{1}(t)=\hat{E}_{1}(t),
$$ 
and hence
$$
\hat{E}_{1}(t)\oplus(\hat{E}_{2}(t)\cap\hat{F}_{1}(t))=\hat{E}_{2}(t).
$$
Then we obtain 
$$
(\hat{E}_{2}(t)\cap\hat{F}_{1}(t))\oplus\hat{E}_{1}(t)\oplus\hat{F}_{2}(t)
=\hat{E}_{2}(t)\cap\hat{F}_{2}(t)=X
.$$
\end{proof}

\begin{lemma} \label{lemm4} 
 For every $t\in I$ we have 
\begin{equation} \label{zhiji}
\hat{P}_{1}(t)\hat{Q}_{2}(t)=\hat{Q}_{2}(t)\hat{P}_{1}(t)=0.
\end{equation}
\end{lemma}

\begin{proof} 
According to Lemma \eqref{lem2}, $\hat{E}_{1}(t)\subset \hat{E}_{2}(t)$, 
$\hat{F}_{2}(t)\subset \hat{F}_{1}(t)$,
for each $x\in X$ we have 
\begin{gather*}
\hat{Q}_{2}(t)x\in\hat{F}_{2}(t)\subset\hat{F}_{1}(t),\\
\hat{P}_{1}(t)x\in\hat{E}_{2}(t)\subset\hat{E}_{2}(t).
\end{gather*} 
Therefore,
\begin{gather*}
\hat{P}_{1}(t)\hat{Q}_{2}(t)x\in \hat{P}_{1}(t)\hat{F}_{1}(t)
 =\hat{P}_{1}(t)\operatorname{Im}\hat{Q}_{1}(t)=\{0\},\\
\hat{Q}_{2}(t)\hat{P}_{1}(t)x\in \hat{Q}_{2}(t)\hat{E}_{1}(2)
 =\hat{Q}_{2}(t)\operatorname{Im}\hat{P}_{2}(t)=\{0\}\,.
\end{gather*}
\end{proof}

We proceed with the proof of Theorem \eqref{main-a}. Set
\[
\hat{P}(t)=\hat{P}_{1}(t),\quad \hat{Q}(t)=\hat{Q}_{2}(t),\quad
\hat{R}(t)=\operatorname{Id}-\hat{P}_{1}(t)-\hat{Q}_{2}(t).
\]
In view of Lemma \eqref{lemm1}, we have 
\[
\hat{T}_{k}(t,s)\hat{P}(s)=\hat{P}(t)\hat{T}_{k}(t,s),\quad
\hat{T}_{k'}(t,s)\hat{Q}(s)=\hat{Q}(t)\hat{T}_{k'}(t,s), 
\]
according to \eqref{change}, we obtain 
\[
\hat{T}(t,s)\hat{P}(s)=\hat{P}(t)\hat{T}(t,s),\quad 
\hat{T}(t,s)\hat{Q}(s)=\hat{Q}(t)\hat{T}(t,s). 
\] 
This implies that 
\[
 \hat{T}(t,s)\hat{R}(s)=\hat{R}(t)\hat{T}(t,s).
\]
Since the operators $\hat{P}(t)$ and $\hat{Q}(t)$ are
projections, 
$$
\hat{P}(t)=\hat{P}(t)^{2},\quad
\hat{Q}(t)=\hat{Q}(t)^{2},
$$ 
and by Lemma \eqref{lemm4} we have 
\begin{align*}
\hat{R}(t)^{2}
&= (\operatorname{Id}-\hat{P}_{1}(t)-\hat{Q}_{2}(t))^{2}\\
&=\operatorname{Id}-2\hat{P}_{1}(t)-2\hat{Q}_{2}(t)+\hat{P}_{1}(t)^{2}+\hat{Q}_{2}(t)^{2}
+\hat{P}_{1}(t)\hat{Q}_{2}(t)+\hat{Q}_{2}(t)\hat{P}_{1}(t)\\
&=\operatorname{Id}-\hat{P}_{1}(t)-\hat{Q}_{2}(t)=\hat{R}(t).
\end{align*}
 Now we consider the subspaces 
\begin{equation} \label{k1}
\hat{E}(t)=\hat{P}(t)(X),\quad
\hat{F}(t)=\hat{Q}(t)(X) \quad
\hat{G}(t)=\hat{R}(t)(X).
\end{equation}
 since $\hat{E}_{1}(t)=\hat{P}(t)(X)$ and $\hat{F}_{2}(t)=\hat{Q}(t)(X)$,
We have 
\begin{equation} \label{k2}
\hat{E}(t)=\hat{E}_{1}(t),\quad \hat{F}(t)=\hat{F}_{2}(t).
\end{equation}
by Lemma \eqref{lemm3}, the image
of $\operatorname{Id}-\hat{P}(t)-\hat{Q}(t)=\hat{R}(t)$ is 
$\hat{E}_{2}(t)\cap\hat{F}_{1}(t)$.
So 
\begin{equation} \hat{G}(t)=\hat{R}(t)(X)=\hat{E}_{2}(t)\cap \hat{F}_{1}(t).
\end{equation}
Furthermore, according to \eqref{pqpq}
\begin{equation}\label{r1}
\|\hat{R}(t)\|=\|Id-\hat{P}_{1}(t)-\hat{Q}_{2}(t)\|\leq1+8D\nu^{\varepsilon}(t)
\leq(1+8D)\nu^{\varepsilon}(t).
\end{equation}
By Lemma \eqref{lemm1}, since $\hat{P}(t)=\hat{P}_{1}(t)$, for every
$t\geq s$ we have 
\begin{align*}
\|\hat{T}(t,s)|\hat{E}(s)\|
&= \|\hat{T}_{k}(t,s)\big( \frac{\mu(t)}{\mu(s)}\big)^{-k}|\hat{E}_{1}(s)\|\\
&\leq  D_{1} \big( \frac{\mu(t)}{\mu(s)}\big)^{-(k+L(\alpha_{1}))}
 \nu^{2\varepsilon}(s)
\end{align*}
for some constant $D_{1}>0$. Similarly, since
$\hat{Q}(t)=\hat{Q}_{2}(t)$ for every $t\geq s\geq 0$ we have 
\begin{align*}
\|\hat{T}(t,s)^{-1}|\hat{F}(t)\|
&= \|\hat{T}_{k'}(t,s)^{-1}\big( \frac{\mu(t)}{\mu(s)}\big)^{k'}|\hat{F}_{2}(t)\|\\
&\leq  D_{2} \Big(\frac{\mu(t)}{\mu(s)}\Big)^{(k'-L(\alpha_{2}))}\nu^{2\varepsilon}(t)
\end{align*}
for some constant $D_{2}>0$. Furthermore, by \eqref{k2}, for every
$t\geq s$ we have 
\begin{equation} \label{t1}
\begin{split}
\|\hat{T}(t,s)|\hat{R}(s)\|&\leq \|\hat{T}(t,s)|\hat{G}(s)\|\cdot\|\hat{R}(s)\|\\
&= \|\hat{T}(t,s)|(\hat{E}_{2}(s)
\cap\hat{F}_{1}(s))\|\cdot\|\hat{R}(s)\|\\
&\leq \|\hat{T}(t,s)|\hat{E}_{2}(s)\|\cdot\|\hat{R}(s)\|\\ &=
D_{2}\Big(\frac{\mu(t)}{\mu(s)}\Big)^{-k'}\|\hat{T}_{k'}(t,s)|\hat{E}_{2}(s)\|\cdot\|\hat{R}(s)\|\\
&\leq  (1+8D)D_{2} \Big( \frac{\mu(t)}{\mu(s)}\Big)^{(-k'-L(\alpha_{2}))}
\nu^{3\varepsilon}(s).
\end{split}
\end{equation}
The last inequality follows from  Lemma \eqref{lemm1} and \eqref{r1},
on the other hand, again by \eqref{k2}, for every  $t\geq s\geq 0$ we
have
\begin{equation} \label{t2}
\begin{split}
\|\hat{T}(t,s)|^{-1}\hat{R}(t)\|
&\leq \|\hat{T}(t,s)^{-1}|\hat{G}(t)\|\cdot\|\hat{R}(s)\|\\
&= \|\hat{T}(t,s)^{-1}|\hat{F}_{1}(t)\|\cdot\|\hat{R}(t)\|\\
&=\Big(\frac{\mu(t)}{\mu(s)}\Big)^{k}\|\hat{T}_{k'}^{-1}(t,s)|
 \hat{F}_{1}(t)\|\cdot\|\hat{R}(t)\|\\
&\leq (1+8D)D_{1} \Big(\frac{\mu(t)}{\mu(s)}\Big)^{(k-L(\alpha_{1}))}
\nu^{3\varepsilon}(t).
\end{split}
\end{equation}
 The last inequality also follows from  Lemma \eqref{lemm1} and \eqref{r1}.
This shows that  \eqref{lpe} admits a nonuniform $(\mu,\nu)$
trichotomy in $[0,+\infty)$.


\section{Robustness of strong $(\mu,\nu)$ trichotomy}

We can also consider a stronger version of $(\mu,\nu)$ trichotomy
and establish a corresponding robustness result. Namely, we say that
\eqref{le} admits a strong nonuniform $(\mu,\nu)$ trichotomy in $I$ if
it admits a nonuniform $(\mu,\nu)$ trichotomy in $I$ and there exist
constants $a\geq\alpha$ and $b\geq\beta$ such that for every 
$t,s\in I$ with $t\geq s$ we have 
\begin{gather} \label{tp1}
\|T(t,s)^{-1}P(t)\|\leq D\Big(\frac{\mu(t)}{\mu(s)}\Big)^{a}\nu^{\varepsilon}(t),\\
\label{tq1}
\|T(t,s)Q(s)\|\leq D\Big(\frac{\mu(t)}{\mu(s)}\Big)^{b}\nu^{\varepsilon}(s),
\end{gather}
We recall that \eqref{le} is said to admit a strong
nonuniform $(\mu,\nu)$  dichotomy if it admits a nonuniform
$(\mu,\nu)$  dichotomy and there exists $a>\max\{\alpha,\beta\}$ such that for
every $t,s\in I$ with $t\geq s$ we have 
\begin{gather} \label{tp2}
\|T(t,s)^{-1}P(t)\|\leq D\Big(\frac{\mu(t)}{\mu(s)}\Big)^{a}\nu^{\varepsilon}(t),\\
 \label{tq2}
\|T(t,s)Q(s)\|\leq D\Big(\frac{\mu(t)}{\mu(s)}\Big)^{a}\nu^{\varepsilon}(s),
\end{gather}


\begin{lemma}[\cite{chu}] \label{lemm5} 
Let $A, B : R^{+}\to B(X)$ be continuous functions such that  \eqref{le} admits a
nonuniform $(\mu,\nu)$ dichotomy in $R^{+}$ with $\mu\geq\nu$
satisfying $\epsilon<\min\{\alpha.\beta\}$
and assume that $B(t)$ satisfies \eqref{bc2} with
\eqref{term}, then \eqref{lpe} admits a strong nonuniform
$(\mu,\nu)$ dichotomy in $\mathbb{R}^{+}$.
\end{lemma}

\begin{theorem} \label{main-b} 
Let $A, B : \mathbb{R}^{+}\to B(X)$ be
continuous functions such that \eqref{le} admits a nonuniform
$(\mu,\nu)$ trichotomy in $R^{+}$ with $\mu\geq\nu$ satisfying \eqref{e} and assume
that $B(t)$ satisfies \eqref{bc2} with
\eqref{term}, then \eqref{lpe} admits a strong nonuniform
$(\mu,\nu)$ trichotomy in $R^{+}$.
\end{theorem}

\begin{proof} 
 Following the proof of Theorem \eqref{main-a}, We
consider the projections $\hat{P}(t)=\hat{P}_{1}(t)$ and
$\hat{Q}(t)=\hat{Q}_{2}(t)$. According to lemma \eqref{lemm5}, for every
$t\geq s$ we have 
\begin{gather*} %\label{tp3}
\|\hat{T}(t,s)^{-1}\hat{P}(t)\|
=\Big(\frac{\mu(t)}{\mu(s)}\Big)^{k}\|\hat{T}_{k}(t,s)^{-1}\hat{P}(t)\|
\leq 4D\tilde{D}\Big(\frac{\mu(t)}{\mu(s)}\Big)^{k+L(\alpha_{1})}
 \nu^{3\varepsilon}(t),
\\ % \label{tq3}
\|\hat{T}(t,s)\hat{Q}(s)\|=\Big(\frac{\mu(t)}{\mu(s)}\Big)^{-k'}
\|\hat{T}_{k'}(t,s)^{-1}\hat{Q}(s)\|
\leq 4D\tilde{D}\Big(\frac{\mu(t)}{\mu(s)}\Big)^{-k'+L(\alpha_{2})}
\nu^{3\varepsilon}(s),
\end{gather*}
where $k=\alpha+\eta$, $k'=-(\beta+\xi)$, so \eqref{lpe} admits
a strong nonuniform $(\mu,\nu)$ trichotomy in $\mathbb{R}^{+}$.
\end{proof}

\subsection*{Acknowledgments}
 This work is partly supported by grants 11171090 from the National Natural 
Science Foundation of China,  409281 from the Natural Science Foundation 
of Hohai University, and by the  Fundamental Research Funds for the 
Central Universities.

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