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\markboth{ A Lyapunov function for pullback attractors }
{ Peter E. Kloeden }
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Nonlinear Differential Equations, \newline
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 91--102\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 A Lyapunov function for pullback attractors of
 nonautonomous differential equations
% 
\thanks{ {\em Mathematics Subject Classifications:}  34D20, 34D45.
 \hfil\break\indent 
{\em Key words:} 
Lyapunov function, nonautonomous system, pullback attractor.
 \hfil\break\indent
\copyright 2000 Southwest Texas State University. 
\hfil\break\indent Published October 25, 2000.  \hfil\break\indent
Partly supported by the DFG Forschungschwerpunkt 
``Ergodentheorie, Analysis und \hfil\break\indent
effiziente Simulation dynamischer Systeme".} } 

\date{}
\author{ Peter E. Kloeden \\[12pt]
{\em Dedicated to Alan Lazer} \\ {\em on  his 60th birthday }}
\maketitle

\begin{abstract} 
 The contruction of a Lyapunov function  characterizing the pullback
 attractor of a cocycle dynamical system  is presented.
 This system is the state space component of a skew-product flow   
 generated by a nonautonomous differential equation that
 is driven by an autonomous dynamical system on  a metric space.
\end{abstract}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}


\section{Introduction}

Lyapunov functions are an effective practical as well as theoretical tool for
the investigation of stability properties of dynamical systems, see e.g.
\cite{KMS,KL,KS,S,Y}.  Converse results ensuring the existence of a Lyapunov
function that characterizes a particular type of stability property, such as the
uniform asymptotic stability of a global attractor of an autonomous dynamical
system, have been particularly useful in numerical dynamics \cite{KL,KS}.  Such
a result is presented here for the pullback attractor of a cocycle dynamical
system generated by a nonautonomous differential equation.  The idea of pullback
attraction, which has been used for a long time in other contexts (see e.g.
\cite{MAK}) provides a means of constructing limiting objects in nonautonomous
systems that exist in actual time rather than asymptotically in the future
\cite{A,CKS,CF,FS,KS}.

The cocycle formalism and pullback attractors are briefly recalled in the next
two sections, the reader being referred to the literature (e.g.
\cite{A,CKS,CF,FS,KS}) for more details, examples and motivation.  The main
result is then formulated in Section 4 and the proof is presented in Section 6
following the proof of a lemma on the existence of a pullback absorbing
neighbourhood system in Section 5.  In order to focus on the idea behind the
construction of the Lyapunov function rather than on technical details, the
differential equations considered here are assumed to be globally defined and to
satisfy a global Lipschitz condition.  The paper is concluded with a comment on
the properties of the Lyapunov function and an example in Section 7.

The following notation and definitions will be used. $H^{*}(A,B)$ denotes
the Hausdorff
separation or semi--metric between nonempty compact subsets $A$ and $B$ of
${\mathbb R}^d$,
and is defined by
$$
H^{*}(A,B) := \max_{a\in A} \mathop{\rm dist}(a,B)
$$
where $\mathop{\rm dist}(a,B):=\min_{b\in B} \|a-b\|$. For a nonempty compact
subset $A$ of ${\mathbb R}^d$ and $r>0$, the open  and closed balls
about $A$ of radius $r$ are
defined, respectively, by
$$
B(A; r) := \{x \in {\mathbb R}^d : \ \mathop{\rm dist}(x,A) < r\}, \quad
B[A; r] := \{x \in {\mathbb R}^d : \ \mathop{\rm dist}(x,A) \leq r\}.
$$


\section{The cocycle formalism}

Consider a  parametrized  differential equation
$$
\dot{x} = f(p,x)
$$
on ${\mathbb R}^d$, where  $p$ is a  parameter that  is allowed to vary with
time in a certain way.
In particular, let $P$ be  a topological space and  consider a group
$\Theta = \{\theta_t\}_{t\in {\mathbb R}}$ of
mappings $\theta_t : P \to P$ for each $t \in {\mathbb R}$ such
that $(t,p) \mapsto \theta_t p$ is continuous.
The autonomous dynamical system  $\Theta$ on $P$ acts as a driving
mechanism that generates the time variation in the parameter
$p$ in the parametrized  differential equation above to form a
nonautonomous differential equation
\begin{equation}\label{PDE}
\dot{x} = f(\theta_t p,x)
\end{equation}
on ${\mathbb R}^d$ for each $p \in P$.

It will be assumed amongst other things (see later) that 
$f:P\times{\mathbb R}^d \to {\mathbb R}^d$ is
continuous,  that $f(p,\cdot)$ is  globally Lipschitz continuous on ${\mathbb
R}^d$ for each $p \in P$
and that    the global  forwards existence and uniqueness of solutions of
(\ref{PDE}) holds (e.g.,
due to an additional  dissipativity structural assumption).  The solution
mapping $\Phi :{\mathbb R}^{+}\times P\times {\mathbb R}^d \to {\mathbb R}^d$ 
of (\ref{PDE}), for which
\begin{equation} \label{cprop1}
\frac{d}{dt} \Phi(t,p,x_0) =  f\left(\theta_t p, \Phi(t,p,x_0) \right), \quad
x_0 \in {\mathbb R}^d, p \in P, t \in {\mathbb R}^{+},
\end{equation}
with the {\em initial condition property\/}
\begin{equation}\label{cprop2}
\Phi(0,p,x_0)  =  x_0, \quad  x_0 \in {\mathbb R}^d, p \in P,
\end{equation}
then satisfies  the {\em cocycle property\/}
\begin{equation}\label{cprop3}
\Phi(s+t,p,x_0) =  \Phi(s,\theta_t p,\Phi(t,p,x_0) ), \quad
x_0 \in {\mathbb R}^d,\  p \in P, \  s, t \in {\mathbb R}^{+}.
\end{equation}
That is,  $\Phi$ is a {\em  cocycle mapping\/} on ${\mathbb R}^d$ with respect
to the autonomous
dynamical system $\Theta$ on $P$. In fact, the product mapping
$(\Phi,\Theta)$ then forms an
autonomous semi--dynamical system, or skew--product flow,  on the product
space ${\mathbb R}^d \times P$.

Note that the $t$ variable in $\Phi$ is now the time that has elapsed since
starting rather than
absolute time. Although solutions  of initial value problems may also be
(at least partially)
extendable backwards in time, interest  in this paper is on what happens
forwards in time since
starting,  as is typical in investigations of  systems with some kind of
dissipative behaviour.

A simple example is a conventionally written nonautonomous differential
equation
\begin{equation}\label{NDE}
\dot{x} = g(t,x), \quad t \in {\mathbb R}, x \in {\mathbb R}^d,
\end{equation}
with $p = t_0$, the initial time instant, and shift mappings 
$\theta_t t_0:= t_0 +t$ on $P = {\mathbb R}$. Thus, 
$f(\theta_t p,x) :=g(t_0+t,x)$ here and the solution
mapping $\Phi(t,t_0,x_0) := x(t+t_0;t_0,x_0)$ in terms of the  solution
of the corresponding
initial value problem as it is usually written.  A less trivial example of
the above skew--product
formalism is given by Sell's investigations  of  almost periodic
differential equations \cite{S},
in which $P$ is a compact metric space of admissible vector field functions
and  $\theta_t$ is a
temporal shift  operator acting on these vector field functions. Random
dynamical systems
\cite{A,CF,FS} also provide examples with a measure space rather than a
topological space as the
parameter  space.



\section{Pullback attractors}

The most obvious way to formulate  asymptotic behaviour for a nonautonomous
dynamical system is consider the limit set of the forwards trajectory
$\{\Phi(t,p’,x_0\}_{t\geq 0}$ as $t \to \infty$ for each fixed
initial value $(p,x_0)$. The resulting (omega) limit set
$\omega^{+}(p,x_0)$ now depends on
both the starting parameter $p$ and the starting state $x_0$. In general,
the limit sets
$\omega^{+}(p,x_0)$ are  not invariant under $\Phi$  in the sense that
$\Phi(t,p, \omega^{+}(p,x_0))= \omega^{+}(p,x_0)$ for all 
$t\in {\mathbb R}^{+}$. In fact, it is
too restrictive to define
invariance   like this in terms of just a single set. Instead, it is more
useful to say that  a family
$\widehat{A} = \left\{A_{p}; \, p \in P \right\}$   of nonempty compact
subsets  of ${\mathbb R}^d$  is {\em invariant  under} $\Phi$, or $\Phi$-{\em
invariant,} if
$$
\Phi(t,p, A_{p}) = A_{\theta_t p}, \quad p \in P, t \in {\mathbb R}^{+}.
$$
For example, with $P = {\mathbb R}$, the family of singleton sets defined
by $A_{t_0} =\{\bar{\phi}(t_0)\}$,  where $\bar{\phi}$ is a solution of the
nonautonomous differential equations
(\ref{NDE}) that exists for all $t \in {\mathbb R}$, is $\Phi$-invariant.

The natural generalization of convergence  seems to be the  {\em forwards
running
convergence\/} defined by
$$
H^{*}(\Phi(t,p,x_0),A_{\theta_t p}) \to 0 \quad \mbox{as} \ \ t \to
\infty.
$$
However,  this does not ensure convergence to a specific component set
$A_{p}$ for a fixed
$p$.  For  that one needs to  start ``progressively earlier''  at
$\theta_{-t} p$ in order to
``finish''  at $p$.  (Think of $P = {\mathbb R}$ with $p$  being the final
time $t_0$ and $\theta_{-t} t_0= t_0-t$ the  new starting time). This leads 
to the concept of {\em pullback  convergence\/}  defined
by
$$
H^{*}(\Phi(t,\theta_{-t} p,x_0),A_{p}) \to 0 \quad \mbox{as} \ \ t \to
\infty.
$$
The invariant family $\widehat{A}$ is then called a {\em pullback
attractor.\/}


The concepts of forwards and pullback attraction are independent of each
other. For example, consider the
scalar differential equations  $\dot{x} = g(t,x) = \pm 2t x$ with
the parameter set $P= {\mathbb R}$ and the shift mappings as above.   
In both cases the
invariant families have components
sets $A_{t_0} = \{0\}$ for all $t_0 \in {\mathbb R}$. The zero
solution is forwards attracting only
for the ``$-$" system and pullback attracting only for the  ``$+$" system.
(See the example at the end
of the paper for some additional  details).

The purpose of  this paper is to  construct a Lyapunov function  that
characterizes
such pullback attraction and  pullback  attractors. This will be done in
terms of a more general  definition
of a pullback  attractor that encompasses local attraction as well as
parametrically dependent  regions of
pullback attraction.

A $\Phi$-invariant family of compact subsets $\widehat{A} = \{A_{p} ;
\, p \in P\}$ will be called a
{\em pullback attractor with respect to a  basin of attraction system}
${\cal D}_{att}$  if it
satisfies  the pullback attraction property
\begin{equation}\label{pb}
\lim_{t \to \infty} H^*\left(\Phi(t,\theta_{-t} p,D_{\theta_{-t}
p}),A_{p}\right) = 0
\end{equation}
for all $p \in P$ and all $\widehat{D} = \left\{D_{p};\ p \in P
\right\}$
belonging to a basin of attraction system ${\cal D}_{att}$, that is,   a
collection   of families of nonempty sets $\widehat{D} = \left\{D_{p}
;\  p \in P\right\}$
where $D_{p}$ is  bounded in ${\mathbb R}^d$ for each  $p \in P$ with the
property that $\widehat{D}^{(1)}= \left\{D^{(1)}_{p}; 
p \in P \right\} \in {\cal D}_{att}$ if
$\widehat{D}^{(2)} =\left\{D^{(2)}_{p}\  ;\  p \in P\right\} \in {\cal D}_{att}$
and $D^{(1)}_{p} \subseteq D^{(2)}_{p}$ for all $p \in P$.


Note that the mapping $t \mapsto A_{\theta_t p}$  is continuous  for
each fixed $p \in\Phi$ due to the continuity of  $\Phi$ in $t$ and the 
$\Phi$-invarianceof $\widehat{A}$.
(However, the mapping $p \mapsto A_{p}$  is usually only upper semi
continuous, see
\cite{CKS}).   Obviously  $\widehat{A} \in {\cal D}_{att}$. In fact,
$A_p \subset \mathop{\rm int}{\cal D}_{att}(p)$,  where  
${\cal D}_{att}(p) :=\bigcup_{\widehat{D} = \{D_{p} ;\  p \in
P\}  \in {\cal D}_{att}} D_p$,  for each  $p \in P$.

\section{Lyapunov Functions for Pullback Attractors}

The  main result is to establish the existence  of a Lyapunov function that
characterizes  pullback attraction and pullback attractors.

\begin{theorem}\label{th1}

Let $\widehat{A}$ be a pullback attractor with a basin of attraction system
${\cal D}_{att}$ for the cocycle dynamical system $(\Phi,\Theta)$ generated
by the
differential equation (\ref{PDE}), where
\begin{itemize}

\item $(p,x) \mapsto f(p,x)$ is continuous in $(p,x) \in P\times
{\mathbb R}^d$;

\item $x \mapsto f(p,x)$ is globally Lipschitz continuous on ${\mathbb R}^d$
with  Lipschitz constant $L(p)$ for each $p \in P$;
\item $p \mapsto L( p)$ is continuous;

\item $(t,p) \mapsto \theta_t p$ is continuous.
\end{itemize}
Then there exists a function $V : \bigcup_{p\in P} \left(\{p\}\times
{\cal D}_{att}(p)\right) \to {\mathbb R}^{+}$ such that
\vspace*{2mm}

\noindent {\bf Property 1 (upper bound):  } For all $p \in P$
and $x_0 \in {\cal D}_{att}(p)$
\begin{equation}\label{ub}
V(p,x_0) \leq \mathop{\rm dist}(x_0, A_{p}) ;
\end{equation}

\vspace*{2mm}

\noindent {\bf Property 2 (lower bound):  } For each  $p \in P$ there exists 
a function $a(p, \cdot) : {\mathbb R}^{+} \to {\mathbb R}^{+}$ with 
$a(p,0) = 0$ and $a(p,r) > 0$ for  $r > 0$ which is  increasing in $r$ such 
that
\begin{equation}\label{lb}
a(p,\mathop{\rm dist}(x_0, A_{p})) \leq V(p,x_0)
\end{equation}
for all  $x_0 \in {\cal D}_{att}(p)$;

\vspace*{2mm}

\noindent {\bf Property 3 (Lipschitz condition):  } For all $p \in P$ and 
$x_0$, $y_0 \in {\cal D}_{att}(p)$
\begin{equation}\label{lip}
\left|V(p,x_0)- V(p,y_0)\right| \leq \|x_0-y_0\| ;
\end{equation}

\vspace*{3  mm}

\noindent {\bf Property 4 (pullback convergence):  } For all 
$p \in P$ and any $\widehat{D} \in {\cal D}_{att}$
\begin{equation}\label{pbc}
\limsup_{t \to \infty} \sup_{z \in D_{\theta_{-t} p}}
V(p,\Phi(t,\theta_{-t} p,z)) = 0.
\end{equation}

\vspace*{3  mm}

\noindent In addition,\\
\noindent {\bf Property 5 (forwards convergence):   }   There exists
$\widehat{N} \in {\cal D}_{att}$  consisting of nonempty compact sets
$N_{p}$  which are $\Phi$-positively invariant  in the sense that
$\Phi(t,p,N_{p}) \subseteq N_{\theta_t p}$ for all $t \geq 0$, 
$p\in P$, and satisfy $A_{p} \subset {\rm int} N_{p}$ for each 
$p\in P$ such that
\begin{equation}\label{for}
V(\theta_t p,\Phi(t,p,x_0)) \leq e^{-t} V(p,x_0)
\end{equation}
for all $x_0 \in N_{p}$ and $t \geq 0$.
\end{theorem}

The proof will be given in Section 6, but first  it will be shown in  the
next section  that the assumed
pullback  attractor has a pullback absorbing neighbourhood system.

\section{Pullback Absorbing Neighbourhood Systems}

A  family $\widehat{B} = \left\{B_{p}\  ;\  p\in P \right\} \in
{\cal D}_{att}$ of
nonempty compact subsets $B_{p}$  of ${\mathbb R}^d$ with nonempty interior is
called a {\em pullback
absorbing neighbourhood system\/} for a $\Phi$-pullback attractor
$\widehat{A}$ if it is
$\Phi$-positively invariant   and  if it {\em pullback absorbs\/} all
$\widehat{D} \in {\cal D}_{att}$,
that is for each $\widehat{D} \in {\cal D}_{att}$ and $p \in P$ 
there exists a $T(\widehat{D},p) \in {\mathbb R}^{+}$ such that
$$
\Phi(t,\theta_{-t} p,D_{\theta_{-t} p}) \subset \mbox{\rm int}  B_{p} \quad
\mbox{for all} \quad t \geq T(\widehat{D},p).
$$
Obviously,  $\widehat{A} \subset \widehat{B} \in {\cal D}_{att}$.
 Moreover, by  positive invariance  and the cocycle property
$$
\Phi(s+t,\theta_{-s-t}p,B_{\theta_{-s-t}p}) \subset \Phi(t,\theta_{-t}p,
B_{\theta_{-t}p})
$$
for all $s$, $t \geq 0$ and $p \in P$, from which it follows that
\begin{equation}\label{pba}
A_{p} = \bigcap_{t \geq 0} \Phi(t,\theta_{-t}p,B_{\theta_{-t}p})
\quad \mbox{for all} \quad p \in P.
\end{equation}

The following lemma shows that there always exists such a pullback
absorbing neighbourhood system for any given cocycle attractor. This
will be required for the construction of the Lyapunov function for the
proof of Theorem \ref{th1}.

\begin{lemma}\label{lem}
If $\widehat{A}$ is a cocycle attractor with a basin of attraction
system ${\cal D}_{att}$ for a cocycle dynamical system $(\Phi,\Theta)$ for
which $(t,p,x) \mapsto \Phi(t, \theta_{-t} p,x)$ is continuous, 
then there exists a pullback absorbing
neighbourhood system $\widehat{B} \subset {\cal D}_{att}$ of $\widehat{A}$
with respect to $\Phi$.
\end{lemma}

\noindent {\bf Proof:  } Since $A_p \subset {\rm int}{\cal D}_{att}(p)$,
there is a  $\delta_{p} \in (0,1]$  such that 
$B[A_{p}; 2\delta_{p}] \subset {\rm int}{\cal
D}_{att}(p)$  for each $p \in P$.   Define

$$
B_{p} := \overline{\bigcup_{t\geq 0}
\Phi(t,\theta_{-t}p,B[A_{\theta_{-t}p};\delta_{\theta_{-t}p}])}.
$$
Obviously, $A_{p} \subset B[A_{p};\delta_{p}] 
\subset B_{p} \subset {\cal D}_{att}(p)$
for  each $p \in P$.

By the cocycle property
\begin{eqnarray*}
\Phi(t,p,B_{p}) & \subseteq & \overline{\bigcup_{s\geq 0} \Phi(t,p,
\Phi(s,\theta_{-s}p,B[A_{\theta_{-s}p};  \delta_{\theta_{-s}p}]))}
\\
& = &   \overline{\bigcup_{s\geq 0}
\Phi(s+t,\theta_{-s}p,B[A_{\theta_{-s}p};  \delta_{\theta_{-s}p}])}
\\
& = &   \overline{\bigcup_{r\geq t}
\Phi(r,\theta_{-r+t}p,B[A_{\theta_{-r+t}p};  \delta_{\theta_{-r+t}p}])}
\\
& \subseteq &   \overline{\bigcup_{r\geq 0}
\Phi(r,\theta_{-r}\theta_{t}p,B[A_{\theta_{-s}\theta_{t}p};
\delta_{\theta_{-r}
\theta_{t}p}])} \ = \ B_{\theta_t p}
\end{eqnarray*}
for all $t \geq 0$, so $\Phi(t,p,B_{p}) \subseteq B_{\theta_t p}$,
that is,
$\widehat{B} = \left\{B_{p}\  ;\  p\in P \right\}$ is $\Phi$-positively 
invariant.


Now by pullback convergence, there exists a $T = T(p,\delta_{p}) 
\in {\mathbb R}^{+}$ such that
$$
\Phi(t,\theta_{-t}p,B[A_{\theta_{-t}p};\delta_{\theta_{-t}p}])  \subseteq
B[A_{p};\delta_{p}] \subset B_{p}
$$
for all $t \geq T$. Hence
\begin{eqnarray*}
B_{p} & = & \overline{\bigcup_{t\geq 0}
\Phi(t,\theta_{-t}p,B[A_{\theta_{-t}p};  \delta_{\theta_{-t}p}])}
\\
& \subseteq &  B[A_{p}; \delta_{p}]  \cup
\overline{\bigcup_{t\in [0,T]}\Phi(t,\theta_{-t}p,B[A_{\theta_{-t}p};
\delta_{\theta_{-t}p}])}
\\
& = & \overline{\bigcup_{t\in [0,T]}\Phi(t,\theta_{-t}p,B[A_{\theta_{-t}p};
\delta_{\theta_{-t}p}])},
\\
& \subseteq & \overline{\bigcup_{t\in
[0,T]}\Phi(t,\theta_{-t}p,B[A_{\theta_{-t}p};1])}
\\
& \subseteq & \overline{\bigcup_{t\in [0,T]}\Phi(t,\theta_{-t}p,B^{*})}
=: U_{p,T}.
\end{eqnarray*}
Here $B^{*} := \overline{\bigcup_{t\in [0,T]}B[A_{\theta_{-t}p};1]}$ is
compact by the continuity of
the mapping $t \mapsto A_{\theta_{-t}p}$ and the compactness of the
sets $B[A_{\theta_{-t}p};1]$.
The compactness of the set  $U_{p,T}$ then follows   by the continuity of
the setvalued mapping  $t \mapsto \Phi(t,\theta_{-t}p,B^{*})$. 
Hence $B_{p}$ is compact for each $p \in P$.

To see that $\widehat{B}$ so constructed is pullback absorbing from ${\cal
D}_{att}$, let $\widehat{D} \in {\cal D}_{att}$ and  fix $p \in P$. 
Since $\widehat{A}$ is pullback attracting, there exists
a  $T(\widehat{D},\delta_{p},p) \in {\mathbb R}^{+}$ such  that
$$
H^{*}\left(\Phi(t,\theta_{-t}p,D_{\theta_{-t}p}), A_{p}\right) < \delta_{p},
$$
that is, $\Phi(t,\theta_{-t}p,D_{\theta_{-t}p}) \subset B(A_{p};\delta_{p})$, 
 for all $t \geq T(\widehat{D},\delta_{p},p)$. But  
 $B(A_{p};\delta_{p}) \subset \mathop{\rm int} B_{p}$, so
$\Phi(t,\theta_{-t}p,D_{\theta_{-t}p}) \subset {\rm int}  B_{p}$    for all
$t \geq T(\widehat{D},\delta_{p}, p)$. Hence
$\widehat{B}$  is pullback absorbing as required. \hfill $\Box$

\section{Proof of Theorem \protect{\ref{th1}}}

A Lyapunov function $V$ that characterizes a  pullback attractor
$\widehat{A}$ and satisfies properties
1--5 of Theorem \ref{th1} will be constructed by  modifying to the
ordinary differential equation
setting under consideration the construction used in  \cite{K} for
nonautonomous  difference
equations.

Define $V(p,x_0)$ for all $p \in P$ and $x_0 \in {\cal D}_{att}(p)$
by
$$
V(p,x_0) := \sup_{t\geq 0} e^{-T_{p,t}} \mathop{\rm dist}\left(x_0,
\Phi(t,\theta_{-t}p,B_{\theta_{-t}p})\right),
$$
where
$$
T_{p,t} = t + \int_{0}^{t} L(\theta_{-s}p) \, ds \quad \mbox{with} \quad
T_{p,0} = 0.
$$
The integral here exists due to the continuity assumptions. Note that
$T_{p,t} \geq t$  and
that
$$
T_{\theta_{t} p,s+t} = T_{p,s} + t + \int_0^t L(\theta_r p)\, dr
$$
for all $s$, $t \geq 0$ and $p \in P$, the latter holding  because
\begin{eqnarray*}
T_{\theta_{t} p,s+t} & = & s+t + \int_0^{s+t} L(\theta_{-r}\theta_t p)\, dr
\\
& = & s+ \int_t^{s+t} L(\theta_{-r+t} p)\, dr +t + \int_0^{t} L(\theta_{-r+t}
p)\, dr
\\
& = & s+ \underbrace{\int_0^{s} L(\theta_{-u}p)\, du}_{u=r-t}  + t -
\underbrace{\int_{t}^{0} L(\theta_{v} p)\, dv}_{v=t-r}
\\
& = & T_{p,s}  + t +  \int_{0}^{t} L(\theta_{v}  p)\, dv.
\end{eqnarray*}



\subsection{Proof of property 1}

Since $e^{-T_{p,t}} \leq 1$ for all $t \geq 0$ and  since
$\mathop{\rm dist}\left(x_0, \Phi(t,\theta_{-t}p,B_{\theta_{-t}p})\right)$ is
monotonically increasing from $0 \leq \mathop{\rm dist}\left(x_0,
\Phi(0,p,B_{p})\right) = \mathop{\rm dist}\left(x_0, B_{p}\right)$  
at $t = 0$ to
$\mathop{\rm dist}\left(x_0,A_{p}\right)$  as $t \to \infty$, it follows that
$$
V(p,x_0)  =  \sup_{t\geq 0} e^{-T_{p,t}} \mathop{\rm dist}\left(x_0,
\Phi(t,\theta_{-t}p,B_{\theta_{-t}p})\right)  \leq  1 \cdot  {\rm
dist}\left(x_0,A_{p}\right) .
$$

\subsection{Proof of property 2}

By Property 1, $V(p,x_0) = 0$ for $x_0 \in A_{p}$. Assume instead
that  $x_0 \in {\cal
D}_{att}(p)\setminus A_{p}$.  Now the supremum in
$$
V(p,x_0)   =   \sup_{t\geq 0} e^{-T_{p,t}} \mathop{\rm dist}\left(x_0,
\Phi(t,\theta_{-t}p,B_{\theta_{-t}p})\right)
$$
involves the product of an exponentially decreasing quantity bounded below
by zero and a bounded
increasing function, since the $\Phi(t,\theta_{-t}p,B_{\theta_{-t}p})$ are
a nested family of
compact sets decreasing to $A_{p}$with increasing $t$.  Hence there exists
a $T^{*} = T^{*}(p,x_0) \in {\mathbb R}^{+}$ such that
$$
\frac{1}{2}\mathop{\rm dist}(x_0, A_{p})\leq  \mathop{\rm dist}\left(x_0,
\Phi(t,\theta_{-t}p,
B_{\theta_{-t}p})\right)
$$
for all $t \geq T^{*}$, but not for $t < T^{*}$. Thus,  from above,
\begin{eqnarray*}
V(p,x_0) &  \geq  &  e^{-T_{p,T^{*}}} \mathop{\rm dist}\left(x_0,
\Phi(T^{*},\theta_{-T^{*}}p,B_{\theta_{-T^{*}}p})\right)
\\
& \geq &  \frac{1}{2} e^{-T_{p,T^{*}}} \mathop{\rm dist}\left(x_0, A_{p}\right).
\end{eqnarray*}
Define
$$
\hat{T}(p,r) := \sup \{ T^{*}(p,x_0) :  x_0 \in {\cal D}_{att}(p), \  \
\mathop{\rm dist}\left(x_0, A_{p}\right)=r \}.
$$
Then   $\hat{T}(p,r) < \infty$.  To see this note that  by the triangle
rule
$$
\mathop{\rm dist}(x_0,A_{p}) \leq {\rm
dist}(x_0,\Phi(t,\theta_{-t}p,B_{\theta_{-t}p}))  +
H^{*}(\Phi(t,\theta_{-t}p,B_{\theta_{-t}p}),A_{p}).
$$
Also, by pullback convergence, there exists a finite $T(p,r/2)$ such that
$$
H^{*}(\Phi(t,\theta_{-t}p,B_{\theta_{-t}p}),A_{p}) < \frac{1}{2} r
$$
for all $t \geq T(p,r/2)$. Hence
$$
r \leq \mathop{\rm dist}(x_0,\Phi(t,\theta_{-t}p,B_{\theta_{-t}p})) + \frac{1}{2} r
$$
for $\mathop{\rm dist}(x_0,A_{p}) = r$ and $t \geq T(p,r/2)$,  that is
$$
\frac{1}{2} r \leq \mathop{\rm dist}(x_0,\Phi(t,\theta_{-t}p,B_{\theta_{-t}p})).
$$
Thus ,  $\hat{T}(p,r) \leq T(p,r/2) < \infty$. In addition,
$\hat{T}(p,r)$ is obviously nondecreasing
in $r$ as  $r \to 0$.

Finally,  define
\begin{equation}\label{lbf}
a(p,r) := \frac{1}{2}r \  e^{-T_{p,\hat{T}(p,r)}},
\end{equation}
which satisfies the stated properties.


\subsection{Proof of property 3}

 From the definition 
\begin{eqnarray*}
\lefteqn{ \left|V(p,x_0) - V(p,y_0)\right| }\\
& = & \big|\sup_{t\geq 0} e^{-T_{p,t}} \mathop{\rm dist}\left(x_0,
\Phi(t,\theta_{-t}p,B_{\theta_{-t}p})\right)  \\
&&\quad- \sup_{t\geq 0} e^{-T_{p,t}}
\mathop{\rm dist}\left(y_0, \Phi(t,\theta_{-t}p,B_{\theta_{-t}p})\right) \big|
\\
& \leq &
\sup_{t\geq 0} e^{-T_{p,t}} \left|\mathop{\rm dist}\left(x_0, \Phi(t,\theta_{-t}p,
B_{\theta_{-t}p})\right) - \mathop{\rm dist}\left(y_0, \Phi(t,\theta_{-t}p,
B_{\theta_{-t}p})\right) \right|
\\
& \leq &
\sup_{t\geq 0} e^{-T_{p,t}} \|x_0-y_0\| \leq \|x_0-y_0\|.
\end{eqnarray*}

\subsection{Proof of property 4}


Assume the opposite. Then there exists an $\varepsilon_0 > 0$, a sequence
$t_j \to \infty$ in ${\mathbb R}^{+}$ and points $x_j \in \Phi(t_j,\theta_{-t_j}p,
D_{\theta_{-t_j}p})$ such that $V(p,x_j) \geq \varepsilon_0$ for all 
$j \in {\mathbb N}$. Since $\widehat{D} \in {\cal D}_{att}$ and $\widehat{B}$
is pullback absorbing, there exists a 
$T = T(\widehat{D},p) \in {\mathbb R}^{+}$ such that
$$
\Phi(t_j,\theta_{-t_j}p,D_{\theta_{-t_j}p}) \subset B_{p}, \quad \mbox{for
all}
\quad t_j \geq T.
$$
Hence $x_j \in B_{p}$ for all $j$ such that $t_j \geq T$. Then,
since  $B_{p}$
is a compact set, there exists a convergent subsequence 
$x_{j'} \to x^{*} \in B_{p}$. But
$$
x_{j'} \in \overline{\bigcup_{t \geq t_{j'}}
\Phi(t,\theta_{-t}p,D_{\theta_{-t}p}) }
$$ and
$$
\bigcap_{t_{j'}} \overline{\bigcup_{t \geq t_{j'}} \Phi(t,\theta_{-t}p,
D_{\theta_{-t}p}) }  \subseteq A_{p}
$$
by (\ref{pba}) and the definition (and existence) of a pullback absorbing
system. Hence  $x^{*} \in A_{p}$ and $V(p,x^{*}) = 0$ must hold.  
But $V$ is  Lipschitz continuous in
its second variable by property 3, so
$$
\varepsilon_0  \leq V(p,x_{j'}) = \|V(p,x_{j'})-V(p,x^{*})\|  \leq
\|x_{j'}- x^{*}\|,
$$
which contradicts the convergence $x_{j'} \to x^{*}$. Hence property 4
must hold.


\subsection{Proof of property 5}

Take $N_{p} \equiv B_{p}$ for each $p \in P$. Thus  
$\widehat{N} = \left\{N_{p}\ ; p\in P \right\}$
is positively invariant.

It remains to establish the exponential decay inequality (\ref{for}). Note
that the
cocycle mapping $\Phi$,  considered as the solution mapping of the
nonautonomous differential
equation (\ref{PDE}),   satisfies the Lipschitz  condition
$$
\left\|\Phi(t,p,x_0) - \Phi(t,p,y_0) \right\| \leq e^{\int_0^t
L(\theta_s p)\, ds}  \|x_0-y_0\|
$$
for all $x_0$, $y_0 \in {\mathbb R}^d$, from which it follows  that
$$
\mathop{\rm dist}( \Phi(t,p,x_0), \Phi(t,p,C_{p}))
\leq  e^{\int_0^t L(\theta_s p)\, ds} \mathop{\rm dist}(x_0, C_{p})
$$
for any nonempty compact subset $C_{p}$ of ${\mathbb R}^d$.

Now $\Phi(t,p,x_0) \in N_{\theta_t p}$ when $x_0 \in N_{p}$.
Re-indexing and then using the cocycle property  and the above Lipschitz
condition  thus gives
\begin{eqnarray*}
V(\theta_t p,\Phi(t,p,x_0)) & = & \sup_{s\geq 0} e^{-T_{\theta_t p,s+t}}
\mathop{\rm dist}(
\Phi(t,p,x_0), \Phi(s+t,\theta_{-s} p,B_{\theta_{-s} p}))
\\
& = &  \sup_{s\geq 0}  e^{-T_{\theta_t p,s+t}}  \mathop{\rm dist}(
\Phi(t,p,x_0), \Phi(t,p, \Phi(s,\theta_{-s} p ,B_{\theta_{-s} p})))
\\
& \leq & \sup_{s\geq 0} e^{-T_{\theta_{t} p,s+t}} e^{\int_0^t L(\theta_r
p)\, dr}
\mathop{\rm dist}( x_0, \Phi(s,\theta_{-s} p,B_{\theta_{-s} p} ))
\end{eqnarray*}
However,  $T_{\theta_{t} p,s+t} = T_{p,s}+t + \int_0^t L(\theta_r p)\,
dr$, so
\begin{eqnarray*}
\lefteqn{ V(\theta_t p,\Phi(t,p,x_0))}\\
& \leq & \sup_{s\geq 0} e^{-T_{p,s}-t-\int_0^t L(\theta_r p)\, dr
+ \int_0^t L(\theta_r p)\, dr}
\mathop{\rm dist}( x_0, \Phi(s,\theta_{-s} p,B_{\theta_{-s}p} ))
\\
& = & \sup_{s\geq 0}e^{-T_{p,s}-t} \mathop{\rm dist}(
x_0, \Phi(s,\theta_{-s}p,B_{\theta_{-s}p} ))
\\
& =  & e^{-t} \sup_{s\geq 0} e^{-T_{p,s}} \mathop{\rm dist}(
x_0, \Phi(s,\theta_{-s} p,B_{\theta_{-s} p} ))  =   e^{-t} V(p,x_0),
\end{eqnarray*}
which is the desired inequality.

\vspace*{2mm}

This completes the proof of Theorem \ref{th1}. \hfill $\Box$

\section{Example}

The forwards convergence inequality (\ref{for}) of the pullback Lyapunov
function
 does not imply the usual forwards  Lyapunov stability  or asymptotic
stability. Athough
the inequality
$$
a(\theta_{t} p,\mathop{\rm dist}(\Phi(t,p,x_0),A_{\theta_{t} p})) \leq e^{-t}
V(p,x_0)
$$
then holds, $\mathop{\rm dist}(\Phi(t,p,x_0),A_{\theta_{t} p})$ need not become
small as $t \to \infty$. The reason
for this
is that, without additional assumptions on the dynamical bahaviour,   it is
possible  that
$$
\inf_{t\geq 0} a(\theta_{t} p,r) = 0
$$
for some $r > 0$ and $p \in P$.

In fact, this is  what happens with the differential equation
$$
\dot{x} = 2tx
$$
with  the solution $x(t;t_0,x_0) = x_0 e^{t^2-t_0^2}$, where 
$t \geq t_0$, and the cocycle mapping
$$
\Phi(t;t_0,x_0) = x_0 e^{(t+t_0)^2-t_0^2},  \quad t \geq 0.
$$
Here the parameter $p = t_0 \in P = {\mathbb R}$ and $\theta_t
t_0 = t+t_0$. The pullback attractor here has components
$A_{t_0} = \{0\}$ for each $t_0 \in {\mathbb R}$ and the pullback
attraction is global, i.e. there is no restriction on the
 bounded subsets that are considered in the  basin of attraction system.  A
Lyapunov function satisfying  the  properties
of Theorem \ref{th1}  is given by
$$
V(t_0,x_0) = |x_0| e^{-t_0-t_0^2-\frac{1}{4}}.
$$
Property 1 with $a(t_0,r) = r e^{-|t_0|-t_0^2-\frac{1}{4}}$ and
property 2  are immediate,
while  property 3  follows from
\begin{eqnarray*}
V(t_0,\Phi(t;t_0-t,x_0)) & = &
\left| x_0e^{(t+t_0-t)^2-(t_0-t)^2} \right| e^{-t_0-t_0^2-\frac{1}{4}} \\
& = & e^{-(t_0-t)^2 - t_0 -\frac{1}{4}}|x_0| \to 0 \quad \mbox{as} \quad
t \to
\infty.
\end{eqnarray*}
In addition, $V$ satisfies inequality (\ref{for}), since
\begin{eqnarray*}
V(t_0+t,\Phi(t;t_0,x_0))  & = & \left|x_0 e^{(t+t_0)^2-t_0^2}\right|
e^{-(t_0+t)-(t_0+t)^2-\frac{1}{4}}  \\
& = & e^{-t} V(t_0,x_0) \to 0 \quad \mbox{as} \quad  t \to \infty.
\end{eqnarray*}
However,  the zero solution is obviously not forwards Lyapunov stable.



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}\end{thebibliography} \medskip
 
 \noindent{\sc Peter E.~Kloeden}\\
 FB Mathematik, Johann Wolfgang Goethe Universit\"at\\
 D-60054 Frankfurt am Main, Germany\\
 email: kloeden@math.uni-frankfurt.de



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