\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 107, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/107\hfil Attractors of asymptotically periodic systems]
{Attractors of asymptotically periodic multivalued dynamical
systems governed by time-dependent subdifferentials}

\author[N. Yamazaki\hfil EJDE-2004/107\hfilneg]
{Noriaki Yamazaki}

\address{Noriaki Yamazaki \hfill\break
Department of Mathematical Science\\
Common Subject Division\\
Muroran Institute of Technology\\
27-1 Mizumoto-cho, Muroran, 050-8585, Japan }
\email{noriaki@mmm.muroran-it.ac.jp}

\date{}
\thanks{Submitted April 17, 2004. Published September 10, 2004.}
\thanks{Partially supported by the Ministry of Education, Science, Sports
and Culture, \hfill\break\indent
Grant-in-Aid for Young Scientists (B), No. 14740109.}
\subjclass[2000]{35B35, 35B40, 35B41, 35K55, 35K90}
\keywords{Subdifferentials; multivalued dynamical systems;
attractors; stability}

\begin{abstract}
 We study a nonlinear evolution equation associated with time-dependent
 subdifferential in a separable Hilbert space. In particular,
 we consider an asymptotically periodic system, which means that
 time-dependent terms converge to time-periodic terms as time approaches
 infinity. Then we consider the large-time behavior of solutions
 without uniqueness. In such a situation the corresponding dynamical
 systems are multivalued. In fact, we discuss the stability of
 multivalued semiflows from the view-point of attractors.
 Namely, the main object of this paper is to construct a global attractor
 for asymptotically periodic multivalued dynamical systems,
 and to discuss the relationship to one for the limiting periodic systems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

We consider non-autonomous systems, in a real separable Hilbert
space $H$, of the form
\begin{equation}
 v'(t)+ \partial \varphi^t (v(t)) + G(t, v(t)) \ni f(t) \quad \mbox{ in } H,
 \quad t > s \; ( \geq 0), \label{e1.1}
\end{equation}
where $v'= \frac{dv}{dt}$, $ \partial \varphi^t $ is a subdifferential of
time-dependent proper lower semicontinuous (l.s.c.) convex function
$\varphi^t $ on $ H $, $G (t , \cdot )$ is a multivalued perturbation
small relative to $\varphi ^t $, and $f$ is a forcing term.

In the case when $ G(t , \cdot ) \equiv 0 $, many mathematicians studied the existence-uniqueness and the asymptotic behavior
 of solutions, the time periodic problem and the almost periodic case for \eqref{e1.1} (cf. \cite{D}, \cite{FMK}, \cite{K}, \cite{KO},
 \cite{KO2}, \cite{KO3}, \cite{Ku}, \cite{O3}, \cite{O4}).

For the multivalued nonmonotone perturbation $ G(t , \cdot ) $, \^Otani has
already shown the existence of solution for \eqref{e1.1} in \cite{O}.
The large-time behavior of solutions for \eqref{e1.1} was discussed in
\cite{Y} from the view-point of attractors.
For the time periodic case, assuming the periodicity conditions with same
period $ T_0 $, $ 0 < T_0 < + \infty $, i.e.
 $$ \varphi^t = \varphi^{ t+T_0} , \quad G ( t , \cdot ) =
 G ( t + T_0 , \cdot ) , \quad f ( t ) = f ( t + T_0 ) ,\quad
 \quad \forall t \in R_+ := [ 0 , \infty ) ,
$$
 the existence of periodic solution for \eqref{e1.1} was proved in \cite{O2}.
 Moreover, the periodic stability was discussed in \cite{Yp}.
In fact, the author showed the existence and characterization of time-periodic
global attractors for \eqref{e1.1} in \cite{Yp}.

In this paper, for a given positive number $ T_0 >0 $, we treat the case when
$ \varphi^t $, $ G (t, \cdot )$ and $ f (t)$ are asymptotically $ T_0 $-periodic
in time. Namely we assume that
\begin{equation}
\varphi^t - \varphi _p ^t \to 0 , \quad G(t, \cdot ) - G_p (t , \cdot ) \to 0, \quad
 f(t) - f_p (t) \to 0 \label{e1.2}
\end{equation}
in appropriate senses as $ t \to +\infty $, where
$ \varphi^t _p= \varphi^{ t+T_0} _p $,
$ G_p ( t , \cdot ) = G_p ( t + T_0 , \cdot ) $ and
$ f_p ( t ) = f_p ( t + T_0 ) $ for any $ t \in R_+ $.
By the asymptotically $ T_0 $-periodic condition \eqref{e1.2}, we have the
limiting $ T_0 $-periodic system for \eqref{e1.1} of the form:
\begin{equation}
u'(t)+ \partial \varphi^t _p (u(t))+ G_p (t,u(t)) \ni f_p (t)
\quad \mbox{in }H, \quad t > s \quad ( \geq 0) . \label{e1.3}
\end{equation}
In the case when $ G ( t , \cdot ) $ and $ G_p ( t , \cdot ) $ are single-valued,
 the asymptotically $ T_0 $-periodic problem has already been discussed in
 \cite{IKYp}. To guarantee the uniqueness of solutions for the
 Cauchy problem of \eqref{e1.1} and \eqref{e1.3}, they assumed some conditions on
$ \varphi^t $, $\varphi^t_p $, $ G(t , \cdot ) $ and $ G_p ( t , \cdot ) $.
Then, they discussed the asymptotically $ T_0 $-periodic stability for \eqref{e1.1}
 from the view-point of attractors (cf. \cite{IKYp}).
 The main object of this paper is to develop the result obtained in \cite{IKYp}
in order to consider the large-time behavior of solution for \eqref{e1.1}
 without uniqueness. Namely, we would like to construct the attractor for
the asymptotically $ T_0 $-periodic multivalued flows associated with \eqref{e1.1}.
Moreover we shall discuss the relationship to the $ T_0 $-periodic attractor
for \eqref{e1.3} obtained in \cite{Yp}.

In the next Section 2, we recall the known results for the Cauchy problem
of \eqref{e1.1}.
In Section 3 we consider the limiting $ T_0 $-periodic problem \eqref{e1.3}
and recall
the abstract results obtained in \cite{Yp}. In Section 4, we introduce the notion of
a metric topology on the family $\{ \varphi ^t ; t \ge 0\}$ which was constructed
in \cite{KO3}. And we present and prove the main results in this paper.
In proving main results, we generalize the results obtained in \cite{IKYp}
and \cite{Y3}. In the final section we apply our abstract results to the
parabolic variational inequality with asymptotically $ T_0 $-periodic double
obstacles. Then we can discuss the asymptotic stability for the asymptotically
$ T_0 $-periodic double obstacle problem without uniqueness of solutions.

\noindent{\bf Notation.}
Throughout this paper, let $ H $ be a (real) separable Hilbert space with norm
 $ | \cdot |_H $ and inner product $ ( \cdot , \cdot )_H $. For a proper l.s.c.
convex function
 $ \varphi $ on $ H $ we use the notation $ D( \varphi ) $, $ \partial \varphi$ and
 $ D( \partial \varphi )$ to indicate the effective domain, subdifferential and
its domain of $ \varphi $, respectively; for their precise definitions and
 basic properties see \cite{Bre}.

For two non-empty sets $ A $ and $ B $ in $H$, we define the so-called Hausdorff
semi-distance
$$
\mathop{\rm dist}{}_H ( A , B ):= \sup_{ x\in A} \inf_{y \in B} | x-y|_H .
$$

\section{Preliminaries}

In this section, we recall the known results for a nonlinear evolution equation
in $ H $ of the form:
\begin{equation}
u' (t) + \partial \varphi^t ( u(t)) + G (t, u (t)) \ni f(t) \quad \mbox{ in } H ,
\quad t \in J , \label{e2.1}
\end{equation}
where $ J $ is an interval in $ R_+ $, $ \partial \varphi^t $ is the
subdifferential of a time-dependent proper l.s.c.
 and convex function $ \varphi ^t $ on $ H $, $ G( t, \cdot ) $ is a
 multivalued operator from a subset $ D( G ( t , \cdot )) \subset H $
 into $ H $ for each $ t \in R_+ $ and $ f $ is a given function in
 $ L^2_{\rm loc} ( J ; H)$. 
 
We begin by defining a solution for \eqref{e2.1}.

\begin{definition} \label{def2.1}\rm
 (i) For a compact interval $J := [ t_0 , t_1 ] \subset R_+ $ and $ f \in L^2 (J;H)$,
a function $u: J \to H$ is called a solution of \eqref{e2.1} on $J$, if
$ u \in C(J;H) \cap W^{1,2}_{\rm loc}(( t_0, t_1 ];H)$,
$ \varphi^{( \cdot )}(u( \cdot )) \in L^1(J)$, $u(t) \in D( \partial \varphi^t )$
for a.e. $ t \in J$, and if there exists a function $ g \in L^2_{\rm loc}(J;H) $
such that $ g( t) \in G( t, u(t)) $ for a.e. $ t \in J $ and
$$
f(t)-g(t)-u'(t) \in \partial \varphi^t ( u(t)), \quad \mbox{a.e. } t \in J.
$$
(ii) For any interval $J$ in $ R_+ $ and $ f \in L^2_{\rm loc}(J;H)$, a function
$ u: J \to H$ is called a solution of \eqref{e2.1} on $J$, if it is a solution of \eqref{e2.1}
 on every compact subinterval of $J$ in the sense of (i).

\noindent
(iii) Let $ J $ be any interval in $ R_+ $ with initial time $ s \in R_+$.
 For $ f \in L^2_{\rm loc}(J;H)$, a function $ u : J \to H $ is called a solution of the Cauchy
problem for \eqref{e2.1} on $J$ with given initial value $ u_0 \in H $, if it
 is a solution of \eqref{e2.1} on $ J $ satisfying $ u( s ) = u_0 $.
\end{definition}

For the rest of this paper, let $\{a_r \}:= \{ a_r ; r \geq 0 \}$ and
$\{ b_r \}:= \{b_r ; r \geq 0 \}$ be families of real functions in
$ W^{1,2}_{\rm loc} ( R_+ )$ and $ W^{1,1}_{\rm loc} ( R_+ )$, respectively, such that
$$
\sup_{ t \in R_+ } | a' _r |_{ L^2 ( t, t+1 )}
+ \sup_{ t \in R_+ } | b' _r |_{ L^1 ( t, t+1 )} < + \infty \quad \mbox{ for each }
r \geq 0.
$$

Now we define the class $ \Phi (\{ a_r \}, \{ b_r \})$ of time-dependent
convex function $ \varphi^t $.

\begin{definition} \label{def2.2}\rm
A function $ \{ \varphi^t \}$ belongs to $\Phi (\{ a_r \}, \{ b_r \})$ if
$ \varphi^t $ is a proper l.s.c. convex function on $H$ and satisfies
the following three properties:
\begin{itemize}
\item[($ \Phi $1)]
For each $r > 0$, $s, t \in R_+ $ and $z \in D( \varphi^s)$ with $ |z|_H \le r$,
there exists $ \tilde{z} \in D( \varphi^t)$ such that
\begin{gather*}
| \tilde{z}-z |_H \le | a_r (t)- a_r (s)|(1+| \varphi^s (z)|^{ \frac12 }),\\
\varphi^t( \tilde{z} )- \varphi^s (z) \le | b_r (t)- b_r (s)|(1+| \varphi^s(z)|) .
\end{gather*}

\item[($ \Phi $2)] There exists a positive constant $ C_1$ such that
 $$ \varphi^t (z) \geq C_1 | z | ^2 _H , \quad \forall t \in R_+,
 \ \forall z \in D ( \varphi^t ) .$$

\item[($ \Phi $3)] For each $ k > 0 $ and $ t \in R_+ $,
 the level set $ \left \{ z \in H ; \varphi^t (z) \leq k \right \}$
 is compact in $H$.
\end{itemize}
\end{definition}

Next, we introduce the class $ \mathcal{G} ( \{ \varphi^t \}) $ of time-dependent
multivalued perturbation $ G(t, \cdot )$ associated with
$ \{ \varphi^t \} \in \Phi ( \{ a_r \} , \{ b_r \} )$.

\begin{definition} \label{def2.3}\rm
An operator $ \{ G(t, \cdot )\}$ belongs to $\mathcal{G} ( \{ \varphi^t \}) $ if
$ G (t, \cdot )$ is a multivalued operator from $ D(G(t, \cdot )) \subset H$ into
$H$ which fulfills the following five conditions:
\begin{itemize}
 \item[(G1)] $D( \varphi^t ) \subset D( G(t, \cdot )) \subset H $ for
 any $ t \in R_+ $. And for any interval $ J \subset R_+ $
 and $ v \in L^2_{\rm loc}
 (J;H )$ with $ v(t) \in D ( \varphi^t )$ for a.e. $ t \in J $, there exists
 a strongly measurable function $ g( \cdot ) $ on $ J $ such that
 $g( t) \in G( t, v(t))$ for a.e. $t \in J $.

\item[(G2)] $ G(t , z) $ is a convex subset of $ H $ for any
$ z \in D( \varphi^t ) $ and $ t \in R_+ $.

\item[(G3)] There are positive constants $ C_2 $, $ C_3 $ such that
$$
 |g |_H^2 \leq C_2
\varphi^t (z) + C_3 , \quad \forall t \in R_+ ,\
 \forall z \in D( \varphi^t ), \forall g \in G( t, z ).
$$

\item[(G4)](demi-closedness) If $ z_n \in D( \varphi^{ t_n }) $, $ g_n \in G ( t_n , z_n ) $,
$ \{ t_n \} \subset
 R_+ $, $ \{ \varphi^{ t_n }( z_n ) \} $ is bounded, $ z_n \to z $
 in $ H $, $ t_n \to t$ and $ g_n \to g $
 weakly in $ H $ as $ n \to +\infty $, then $ g \in G(t , z)$.

\item[(G5)] For each bounded subset $ B $ of $ H $, there exist positive constants
 $ C_4 (B) $ and $ C_5 (B)$ such that
$$
\varphi^t (z) + (g,z-b)_H \geq C_4 (B) |z|_H^2 - C_5 (B),
$$
for all $t \in R_+$, all $g \in G(t,z)$,
all $z \in D( \varphi^t )$, and all $b \in B$.
\end{itemize}
\end{definition}

For a given $ \{ \varphi^t \}$ in $\Phi ( \{ a_r \}, \{ b_r \} )$,
$\{ G(t, \cdot ) \}$ in $\mathcal{G} ( \{ \varphi^t \})$ and a forcing term $f$
in $L^2_{\rm loc} ( R_+ ; H)$, we consider the evolution equation
\begin{equation}
u'(t)+ \partial \varphi^t(u(t))+G(t,u(t)) \ni f(t) \quad \mbox{in }H, \quad t>s
\label{Es}
\end{equation}
for each $ s \in R_+ $.

Now we recall the known results on the existence and global estimates
of solutions for the Cauchy problem of \eqref{Es}:
\begin{itemize}
 \item[(A)] [Existence of solution for \eqref{Es}] (cf. \cite[Theorem II, III]{O})
 The Cauchy problem for \eqref{Es} has at least one
 solution $ u $ on $ J= [s, +\infty) $ such that
$( \cdot -s)^{ \frac12} u' \in L^2_{\rm loc}(J;H)$,
 $( \cdot -s) \varphi^{( \cdot )}( u( \cdot )) \in L^\infty_{\rm loc}(J)$ and
 $ \varphi^{( \cdot )}(u( \cdot ))$ is absolutely continuous on any compact
subinterval of $( s, +\infty )$, provided that given initial value 
$ u_0 \in \overline{ D( \varphi^s)}$. In particular,
if $ u_0 \in D( \varphi^s)$, then the solution $ u $ satisfies that
$ u' \in L^2_{\rm loc}(J;H)$ and $ \varphi^{( \cdot )}(u( \cdot ))$
is absolutely continuous on any compact interval in $J$.

 \item[(B)] [Global boundedness of solutions for \eqref{Es}]
 (cf. \cite[Theorem 2.2]{SIKY})
 Suppose that
$$ S_f := \sup_{ t \in R_+ } |f|_{ L^2 (t, t+1 ;H)}< +\infty . $$
Then, the solution $ u $ of the Cauchy problem for \eqref{Es} on $ [ s , +\infty )$
satisfies the global estimate
$$ \sup_{t \geq s}|u(t)|_H^2 + \sup_{ t \geq s} \int_t^{t+1} \varphi^\tau
(u( \tau )) d \tau \leq N_1 (1+ S_f^2 + | u_0 |_H^2 ), $$
where $ N_1 $ is a positive constant independent of $ f $, $ s \in R_+ $ and a given initial value
$ u_0 \in \overline{ D( \varphi^s )}$. Moreover, for each $ \delta >0 $ and each
bounded subset $ B $ of $ H $, there is a constant $ N_2 ( \delta , B )>0 $, depending
only on $ \delta >0$ and $ B $, such that
$$ \sup_{ t \geq s + \delta }| u' |^2_{ L^2 (t,t+1;H)} + \sup_{ t \geq s + \delta }
 \varphi^t (u(t)) \leq N_2 ( \delta , B )
$$
for the solution $ u $ of the Cauchy problem for \eqref{Es} on $ [ s , +\infty )$ with $ s \in R_+ $
 and $ u_0 \in \overline{D( \varphi^s )} \cap B$.
\end{itemize}


Next, we remember a notion of convergence for convex functions.

\begin{definition}[cf. \cite{M}] \label{def2.4} \rm
Let $ \psi $, $ \psi_n $  ($ n \in N $) be proper l.s.c. and convex functions
on $ H $. Then we say that $ \psi_n $ converges to $ \psi $ on $ H $ as
$ n \to +\infty $
 in the sense of Mosco \cite{M}, if the following two conditions are satisfied:
\begin{itemize}
\item[(i)] For any subsequence $ \{ \psi_{ n_k } \} \subset \{ \psi_n \} $,
if $ z_k \to z $ weakly in $ H $ as $ k \to +\infty $, then
$$
\liminf_{ k \to +\infty } \psi_{ n_k } ( z_k ) \geq \psi (z).
$$
\item[(ii)] For any $ z \in D( \psi ) $, there is a sequence $ \{ z_n \} $ in $ H $
such that
$$
z_n \to z \ {\rm in} \ H \ {\rm as} \ n \to +\infty , \quad
\lim_{ n \to +\infty } \psi_n ( z_n ) = \psi (z).
$$
\end{itemize}
\end{definition}

Now, we recall a convergence result (cf. \cite[Lemma 4.1]{SIKY}) as follows.
\begin{itemize}
 \item[(C)] Let $\{ \varphi^t_n \} \in \Phi( \{a_r \},
 \{ b_r \})$, $\{ G_n ( t, \cdot ) \} \in \mathcal{G} ( \{ \varphi^t _n \} )$
 with common positive constants $ C_1 , C_2 , C_3 , C_4 (B),C_5 (B)$,
$\{ f_n \} \subset L^2 (J;H)$, $J=[ s , t_1 ] \subset R_+ $, and
$ u_{0,n} \in \overline{D( \varphi^{ s } _n )} $ for $n = 1, 2 , \dots $.
Assume that
\begin{itemize}
\item[(i)] $ \varphi^t _n $ converges to $ \varphi^t $ on $H$ in the sense
 of Mosco \cite{M} for each $t \in J $ (as $ n \to +\infty $) and
 $ \bigcup_{ n =1 }^ { +\infty } \{ z \in H ; 
\varphi^t _n (z) \le k \} $ is relatively compact in $H$ for every real $k > 0 $ and $ t \in J$, where $\{ \varphi^t \} \in \Phi( \{a_r \}, \{ b_r \}) $
 and $ \varphi^t _n = \varphi^t $ if $ n = + \infty $.

\item[(ii)] If $ z_n \in D( \varphi^{ t_n } _n ) $, $ g_n \in G_n ( t_n , z_n ) $,
$ \{ t_n \} \subset R_+ $, $ \{ \varphi^{ t_n } _n ( z_n ) \} $ is bounded,
$ z_n \to z $ in $ H $, $ t_n \to t$ and $ g_n \to g $
 weakly in $ H $ as $ n \to +\infty $, then $ g \in G(t , z)$, where
 $ \{ G ( t, \cdot ) \} \in \mathcal{G} ( \{ \varphi^t \} )$.

\item[(iii)] $ f_n \to f$ weakly in $ L^2 (J;H) $ for some $ f \in L^2 (J;H) $ and
$ u_{0,n} \to u_0$ in $H$ for some $ u_0 \in \overline{ D( \varphi ^{ s } )} $.
\end{itemize}

Denote by $u$ the solution of the Cauchy problem for \eqref{Es} on $J$ with
$u(s)= u_0 $ and by $ u_n $ the solution of the Cauchy problem for
\eqref{Es} with $ \varphi^t , G , f $ replaced by $ \varphi^t_n , G_n , f_n $,
and with $ u_n (s) = u_{ 0, n } $. Then $ u_n $ converges to $u$ on $J$ in
the sense that
\begin{gather*}
 u_n \to u \mbox{ in } C(J;H),\quad
 (\cdot -s)^{\frac12} u' _n \to ( \cdot -s)^{ \frac12 } u' \mbox{ weakly in }
 L^2(J;H), \\
\int_{J} \varphi^t_n ( u_n (t))dt \to \int_{J} \varphi^t (u(t))dt \quad \mbox{ as }
n \to +\infty .
\end{gather*}
\end{itemize}

\section{Attractor for periodic multivalued dynamical system }

In this section we recall the known results obtained in \cite{Yp} for a
 $ T_0 $-periodic system in $ H $, of the form:
\begin{equation}
 u'(t)+ \partial \varphi^t_p (u(t))+ G_p (t,u(t)) \ni f_p (t)
 \quad \mbox{in }H, \quad t>s \label{Ps}
\end{equation}
for each $s \in R_+ $, where $ \varphi^t_p $, $ G_p (t, \cdot ) $ and
$ f_p (t) $ are $ T_0 $-periodic, namely periodic in time with the same
period $ T_0 $, $ 0 < T_0 < + \infty $.

\begin{definition} \label{def3.1}\rm
Let $ T_0 $ be a positive number. Then\\
(i) $ \Phi _p ( \{ a_r \}, \{ b_r \} ; T_0 )$ is the set of all
$ \{ \varphi ^t _p \} \in \Phi ( \{ a_r \}, \{ b_r \} )$ satisfying the
$ T_0 $-periodicity condition
\[
\varphi^{t+ T_0 } _p ( \cdot )= \varphi^t _p ( \cdot ) \quad \mbox{ on }H,
\quad \forall t \in R_+ . %\label{e3.1}
\]
(ii) $ \mathcal{G} _p ( \{ \varphi^t_p \} ; T_0 )$ is the set of all
$ \{ G_p ( t, \cdot ) \} \in \mathcal{G} ( \{ \varphi^t _p \} )$ satisfying the
 $ T_0 $-periodicity condition
\[
G_p ( t+ T_0 , \cdot )= G_p ( t, \cdot ) \quad \mbox{ in }H, \quad
\forall t \in R_+ . %\label{e3.2}
\]
\end{definition}

For the rest of this section we assume that
$ \{ \varphi^t_p \} \in \Phi_p ( \{ a_r \}, \{ b_r \} ; T_0 )$,
$ \{ G_p (t, \cdot ) \} \in \mathcal{G}_p( \{ \varphi^t_p \} ; T_0 )$ and
$ f_p \in L^2 _{\rm loc} ( R_+ ; H) $ is $ T_0 $-periodic in time, namely
\begin{equation}
f_p ( t + T_0 ) = f_p (t) \quad \mbox{ in }H, \quad \forall t \in R_+ . \label{e3.3}
\end{equation}
Here we note that \eqref{Ps} can be considered as \eqref{Es} in Section 2.
So, by the result (A) in Section 2, the Cauchy problem for \eqref{Ps}
has at least one solution $u$ on $ [ s , + \infty ) $. Hence we can
define the multivalued dynamical process associated with \eqref{Ps}
as follows:

\begin{definition} \label{def3.2}\rm
For every $ 0 \le s \le t< +\infty$ we denote by $U(t,s)$ the mapping from
$ \overline{D( \varphi^s _p )}$ into $ \overline{D( \varphi^t _p )}$ which
assigns to each $u_0 \in \overline{D( \varphi^s _p )}$ the set
\begin{equation} \label{e3.4}
\begin{aligned}
U(t,s) u_0 := \big\{& z \in H ; \mbox{There is a solution $u$ of \eqref{Ps} on }
 [s, + \infty ) \\
 & \mbox{such that $u(s)= u_0$ and $u(t)=z$}. \big\}
\end{aligned}
\end{equation}
\end{definition}
Then we deduce easily the following properties of $ \{ U(t,s) \} := \{ U(t,s) ;
0 \le s \le t< +\infty \}$:
\begin{itemize}
\item[(U1)] $U(s,s)=I $ on $ \overline{D( \varphi^s _p )} \quad $ for any $s \in R_+ $.
\item[(U2)] $U(t_2,s)z=U(t_2,t_1) U(t_1,s) z $ for any $ 0 \le s \le t_1 \le t_2 < +\infty$
 and $ z \in \overline{D( \varphi^s _p )}$.
\item[(U3)] $U(t+T_0, s+ T_0 ) z =U(t,s)z $ for any $ 0 \le s \le t< +\infty $ and $ z \in \overline{D( \varphi^s _p )}$, that is, $U$ is $T_0$-periodic.
\item[(U4)] $ \{ U(t,s)\} $ has the following demi-closedness:\\
 If $ 0 \le s_n \le t_n < + \infty $, $ s_n \to s $, $ t_n \to t$, $ z_n \in \overline{D(\varphi _p ^{s_n} )}$,
 $ z \in \overline{D(\varphi _p ^{s } )}$, $ z_n \to z $ in $H$ and
an element $ w_n \in U(t_n , s_n ) z_n $ converges to some element $ w \in H $ as $n \to + \infty $, then $ w \in U(t,s) z $.
\end{itemize}

Next we define the discrete dynamical system in order to construct a global
attractor for \eqref{Ps}.

\begin{definition} \label{def3.3} \rm
Let $ U ( \cdot , \cdot ) $ be the solution operator for
\eqref{Ps} defined in Definition \ref{def3.2}. Then \\
(i) For each $ \tau \in R_+ $, we denote by $ U_\tau $ the $T_0$-step mapping from
$ \overline{ D( \varphi^\tau _p )}$ into
$ \overline{ D( \varphi ^{ \tau +T_0} _p )} = \overline{ D( \varphi^\tau _p )}$,
namely, $U_\tau := U( \tau +T_0 ,\tau )$.\\
(ii) For any $k \in Z_+ := N \cup \{ 0 \} $, we define
$$U_\tau^k :=\underbrace{U_\tau \circ U_\tau \circ \dots \circ U_\tau }
_{k \mbox{ iterations}}. $$
Clearly we have $ U^k _\tau = U( \tau +kT_0 , \tau ) $ for any
$ \tau \in R_+ $ and $ k \in Z_+ $.
\end{definition}

Now, we recall the known result on the existence of global attractors
for discrete multivalued dynamical systems $ U_\tau $ associated with
\eqref{Ps}.

\begin{theorem}[{\cite[Theorem 3.1]{Yp}}] \label{thm3.1}
Assume that $\{ \varphi^t_p \} \in \Phi_p ( \{ a_r \}, \{ b_r \} ; T_0 )$,\break
$\{ G_p (t, \cdot ) \} \in \mathcal{G}_p( \{ \varphi^t_p \} ; T_0 )$, and
$ f_p \in L^2_{\rm loc}( R_+ ;H)$ satisfies the $ T_0 $-periodicity condition
\eqref{e3.3}. Then, for each $ \tau \in R_+ $, there exists a subset
$\mathcal{A}_\tau $ of $ D( \varphi^\tau _p )$ such that
\begin{itemize}
 \item[(i)] $\mathcal{A}_\tau $ is non-empty and compact in $H$;
 \item[(ii)] for each bounded set $B$ in $H$ and each number
 $ \epsilon >0$ there exists $N_{B, \epsilon } \in N$ such that
$$
\mathop{\rm dist}{}_H( U _\tau ^k z, \mathcal{A}_\tau )< \epsilon
$$
for all $z \in \overline{D( \varphi^\tau _p )} \cap B$ and all $k \geq N_{B, \epsilon }$;
 \item[(iii)] $ U_\tau^k\mathcal{A}_\tau =\mathcal{A}_\tau $ for any $k \in N$.
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk3.1} \rm
By \cite[Lemma 3.1]{Yp} we can get the compact absorbing set
$B_{0, \tau }$ of $ \overline{ D ( \varphi _p ^\tau )} $ for $ U_\tau $
 such that for each bounded subset $B$ of $H$ there is a positive integer $ n_B$
 (independent of $\tau \in R_+ $) satisfying
$$
U^n_\tau \left( \overline{ D(\varphi^\tau _p )} \cap B \right)
\subset B_{0, \tau } \quad \mbox{for all }n \geq n_B .
$$
Then we observe that the global attractor $\mathcal{A}_\tau $ is given by
the $ \omega $-limit set of the absorbing set $ B_{0, \tau }$ for $ U_\tau $, i.e.
$$\mathcal{A}_\tau = \bigcap_{n \in Z_+ } \overline{ \bigcup_{ k \geq n}
U _\tau ^k B_{0, \tau }}. $$

The next theorem concerns a relationship between global attractors
$ \mathcal{A} _s $ and $\mathcal{A}_\tau $. For detail proof, see \cite{Yp}.
\end{remark}

\begin{theorem}[{\cite[Theorem 3.2]{Yp}}] \label{thm3.2}
Suppose the same assumptions are made as in Theorem \ref{thm3.1}.
Let $\mathcal{A}_s $ and $\mathcal{A}_\tau $ be global attractors for
 $ U_s $ and $ U_\tau $, with $ 0 \le s \le \tau \le T_0 $, respectively.
Then, we have
$$\mathcal{A}_\tau =U(\tau , s)\mathcal{A}_s ,$$
where $U(\tau , s)$ is the $ T_0 $-periodic process given in Definition
\ref{def3.2}.
\end{theorem}

\begin{remark} \label{rmk3.2} \rm
 By Theorem \ref{thm3.1} (iii) and Theorem \ref{thm3.2}, we see that the global attractor
 $\mathcal{A}_\tau $ for $ U_\tau $ is $ T_0 $-periodic in $ \tau $.
In fact, for each $ \tau \in R_+ $
 choose $ m_\tau \in Z_+ $ and $ \sigma_\tau \in [0, T_0 )$ so that
$ \tau = \sigma_\tau + m_\tau T_0 $. Then, we have
$\mathcal{A}_\tau = \mathcal{A}_{ \sigma_\tau }$.
\end{remark}

The third known result is the existence of a global attractor for the
$ T_0 $-periodic multivalued dynamical system \eqref{Ps}.

\begin{theorem}[{cf. \cite[Theorem 3.3]{Yp}}] \label{thm3.3}
 Under the assumptions of Theorem \ref{thm3.1}, put
$$ \mathcal{A}:= \bigcup_{0 \le \tau \le T_0 } \mathcal{A}_\tau ,
$$
where $\mathcal{A}_\tau $ is as obtained in Theorem \ref{thm3.1}. Then, $\mathcal{A}$ has the following properties:
\begin{itemize}
 \item[(i)] $\mathcal{A} $ is non-empty and compact in $H$;
 \item[(ii)] for each bounded set $B$ in $H$ and each number $ \epsilon >0$ there exists a finite time $ T_{B, \epsilon } >0 $ such that
$$
\mathop{\rm dist}{}_H ( U( t+ \tau , \tau ) z, \mathcal{A} )< \epsilon
$$
for all $ \tau \in R_+ $, all $z \in \overline{D( \varphi^{ \tau } _p )} \cap B$
and all $t \geq T_{B, \epsilon }$.
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk3.3} \rm
 In \cite[Section 4]{Yp} the characterization of the
$ T_0 $-periodic global attractor was discussed. The author proved that for
each time $ \tau \in R_+ $ the global attractor $ \mathcal{A} _\tau $
for the discrete multivalued dynamical system $ U_\tau $ coincides with
the cross-section of the family of all global bounded complete trajectories
for the $ T_0 $-periodic system \eqref{Ps}.
\end{remark}

\section{Attractor for asymptotically periodic multivalued dynamical system}

Throughout this section, let $ M >0 $ be a fixed (sufficiently) large positive
number. Now we put
\begin{align*}
 \Psi _M := \big\{&\psi ; \psi \mbox{ is proper, l.s.c. and convex on }H, \\
&\exists z \in D( \psi ) \mbox{ s.t. } |z|_H \le M, \; \psi (z) \le M \big\}
\end{align*}
Then we state the notion of a metric topology on $ \Psi _M $ introduced
in \cite{KO3}.

Given $ \varphi, \psi \in \Psi _M , $ we define
$ \rho ( \varphi, \psi ; \cdot ): D( \varphi) \to R $ by putting
$$
\rho ( \varphi, \psi ;z)= \inf\{ \max ( |y-z| _H , \psi (y)- \varphi (z));
y \in D( \psi )\}
$$
for each $z \in D( \varphi)$, and for each $ r \geq M $
$$
\rho_r( \varphi, \psi ):= \sup _{ z \in L _ \varphi (r) } \rho ( \varphi, \psi ;z) ,
$$
where $ L_\varphi (r) := \{ z \in D( \varphi) ; |z|_H \le r, \varphi (z) \le r \}$.
 Moreover, for each $ r \geq M, $ we define the functional
$ \pi_r ( \cdot , \cdot ) $ on $ \Psi _M \times \Psi _M $ by
$$
\pi_r ( \varphi , \psi ) := \rho_r ( \varphi , \psi )
+ \rho_r ( \psi, \varphi ) \quad \mbox{ for } \varphi , \psi \in \Psi _M .
$$
Then, according to \cite[Proposition 3.1]{KO3}, we can define a complete metric topology on $ \Psi _M $ so that the convergence $ \psi _n \to \psi $ in $ \Psi _M $ (as $ n \to +\infty $) if and only if
$$ \pi _r ( \psi _n , \psi ) \to 0 \quad \mbox{for every } r \geq M.
$$


Now by using the above topology on $ \Psi _M $, we consider an asymptotically
 $ T_0 $-periodic system as follows.


\begin{definition} \label{def4.1} \rm
 Assume $ \{ \varphi^t \} \in \Phi ( \{ a_r \} , \{ b_r \} ) \cap \Psi _M $,
$ \{ G (t, \cdot ) \} \in \mathcal{G} ( \{ \varphi^t \} ) $ and
$f \in L^2_{\rm loc} (R_+ ; H)$. Then the system
\begin{equation} \label{APs}
v'(t)+ \partial \varphi^t(v(t))+G(t, v(t)) \ni f(t) \quad \mbox{ in }H, \; t>s
\;( \geq 0)
\end{equation}
is asymptotically $ T_0 $-periodic, if there are
$ \{ \varphi^t _p \} \in \Phi _p ( \{ a_r \}$,
$\{ b_r \} ; T_0 ) \cap \Psi _M $, \break
$ \{ G _p (t, \cdot ) \} \in \mathcal{G} _p ( \{ \varphi^t _p \} ; T_0 ) $
and a $ T_0 $-periodic function $ f_p \in L^2_{\rm loc}( R_+ ; H) $ such that
\begin{itemize}
 \item[(A1)] (Convergence of $ \varphi^t - \varphi^t _p \to 0$ as $t \to +\infty $)
 For each $ r \geq M$,
$$
J_m^{(r)} := \sup_{ \sigma \in [0, T_0 ] } \pi_r( \varphi^{mT_0+ \sigma },
\varphi^{ \sigma } _p ) \to 0 \quad \mbox{ as }m \to +\infty .$$

\item[(A2)] (Convergence of $ G(t, \cdot )- G_p (t, \cdot ) \to 0$ as
$t \to +\infty $) If $\{ \tau _n \} \subset [0, T_0 ]$,
$ \{ m_n \} \subset Z_+$, $ m_n \to +\infty$,
$ z_n \in D( \varphi^{ m_n T_0 + \tau _n } ) $,
$ g_n \in G ( m_n T_0 + \tau _n , z_n ) $,
$ \{ \varphi^{ m_n T_0 + \tau _n } (z_n) \}$ is bounded,
$ z_n \to z $ in $H$, $ \tau _n \to \tau $ and $ g_n \to g $ weakly in
$ H $ (as $n \to +\infty $), then
$g \in G_p ( \tau ,z)$.

 \item[(A3)] (Convergence of $ f(t)- f_p (t) \to 0 \mbox{ as }t \to +\infty$)
$$| f( mT_0 + \cdot )- f_p |_{L^2(0, T_0 ;H)} \to 0 \quad
\mbox{as }m \to +\infty . $$
\end{itemize}
\end{definition}

By Definition \ref{def4.1} we easily see that a limiting system for \eqref{APs}
 is a $ T_0 $-periodic one \eqref{Ps} of the form:
$$ %( \mbox{P} )_s
u'(t)+ \partial \varphi^t _p (u(t))+ G_p (t, u(t)) \ni f_p (t)
\quad \mbox{ in }H, \; t>s \;( \geq 0) .
$$
Here we note that \eqref{APs} is also considered as \eqref{Es}.
So, by the result (A) in Section 2, the Cauchy problem for
\eqref{APs} has at least one solution $v$ on $ [ s , + \infty ) $. Hence we can
define the multivalued dynamical system associated with \eqref{APs}
as follows:

\begin{definition} \label{def4.2} \rm
For every $ 0 \le s \le t< +\infty$ we denote by $E(t,s)$ the mapping from
$ \overline{D( \varphi^s )}$ into $ \overline{D( \varphi^t )}$ which
assigns to each $v_0 \in \overline{D( \varphi^s )}$ the set
\begin{align*}
 E(t,s) v_0 := \big\{&z \in H ; \mbox{ There is a solution $v$ of \eqref{APs}
  on $[s, + \infty )$} \\
 & \mbox{such that } v(s)= v_0 \mbox{ and } v(t)=z.\big\}
\end{align*}
\end{definition}

Then we easily see that $ \{ E(t,s) \} := \{ E(t,s) ;
0 \le s \le t< +\infty \}$ has the following evolution properties:
\begin{itemize}
 \item[(E1)] $E(s,s)=I$ on $ \overline{D( \varphi^s )}$ for any $s \in R_+ $.

 \item[(E2)] $E(t_2,s)z=E(t_2,t_1) E(t_1,s) z $ for any
 $ 0 \le s \le t_1 \le t_2 < +\infty$ and $ z \in \overline{D( \varphi^s )}$.

 \item[(E3)] $ \{ E(t,s)\} $ has the following demi-closedness:\\
  If $ 0 \le s_n \le t_n < + \infty $, $ s_n \to s $, $ t_n \to t$,
  $ z_n \in \overline{D(\varphi ^{s_n} )}$,
 $ z \in \overline{D(\varphi ^{s } )}$, $ z_n \to z $ in $H$ and
an element $ w_n \in E(t_n , s_n ) z_n $ converges to some element
$ w \in H $ as $n \to + \infty $, then $ w \in E(t,s) z $.
\end{itemize}

Now we give  the definition of a discrete $ \omega $-limit set for
$ E ( \cdot , \cdot)$.


\begin{definition}[Discrete $ \omega $-limit set for $ E ( \cdot , \cdot)$]
 \label{def4.3.}\rm
Let $ \tau \in R_+ $ be fixed. Let $\mathcal{B} ( H ) $ be a family of bounded
subsets of $ H $.
Then for each $ B \in \mathcal{B} (H) $, the set
$$
\omega _\tau (B) := \bigcap_{n \in Z_+ } \overline{ \bigcup_{k \geq n, m \in Z_+ }
E(kT_0+mT_0+ \tau , mT_0+ \tau )( \overline{D( \varphi^{mT_0+ \tau })} \cap B)}
 $$
is called the discrete $ \omega $-limit set of $ B $ under
$ E ( \cdot , \cdot )$.
\end{definition}

\begin{remark} \label{rmk4.1}\rm
By the definition of the discrete
$ \omega $-limit set $ \omega _\tau (B) $, it is easy to see that $ x \in \omega _\tau (B) $
if and only if there exist sequences $ \{ k_n \} \subset Z_+ $ with $ k_n \uparrow +\infty $, $\{ m_n \} \subset Z_+ $, $ \{ z_n \} \subset B $ with $ z_n \in \overline{ D( \varphi^{ m_n T_0 + \tau })} $ and $ \{ x_n \} \subset H $ with $ x_n \in
E( k_n T_0 + m_n T_0 + \tau , m_n T_0 + \tau ) z_n $ such that
$$ x_n \to x \mbox{ in }H \mbox{ as } n \to + \infty.
 $$
\end{remark}

Now we state the main theorems in this paper.

\begin{theorem}[Discrete attractors of \eqref{APs}] \label{thm4.1}
For each $ \tau \in R_+ $, let $\mathcal{A}_\tau $ be the global attractor of
$ T_0 $-periodic dynamical systems $ U_\tau $, which is obtained in Section 3.
For $ \{ \varphi^t \} \in \Phi ( \{ a_r \}$, $\{ b_r \} ) \cap \Psi_M $,
 $ \{ G (t, \cdot ) \} \in \mathcal{G} ( \{ \varphi^t \} ) $ and
 $ f \in L^2 _{\rm loc}( R_+ ; H )$, we assume that the system \eqref{APs} is
 asymptotically $ T_0 $-periodic. Here we put
\begin{equation}
\mathcal{A}_{ \tau }^* := \overline{ \bigcup_{ B \in \mathcal{B} (H) }
\omega _\tau (B)} . \label{e4.1}
\end{equation}
Then, we have
\begin{itemize}
\item[(i)] $\mathcal{A}_{ \tau }^* ( \subset D( \varphi ^\tau _p ) )$
is non-empty and compact in $H$;

\item[(ii)] for each bounded set $ B \in \mathcal{B} (H)$ and each number
$ \epsilon >0 $ there exists $ N_{B, \epsilon } \in N $ such that
$$
\mathop{\rm dist}{}_H ( E ( k T_0 + \tau , \tau ) z,\mathcal{A}_{ \tau }^*)
< \epsilon
$$
for all $z \in \overline{D( \varphi^\tau )} \cap B$ and all
$ k \geq N_{ B, \epsilon }$;

\item[(iii)] $ \mathcal{A}_{ \tau }^* \subset U_\tau ^l \mathcal{A}_{ \tau }^*
 \subset \mathcal{A}_{ \tau } $ for any $ l \in N $, where $ U _\tau $ is
 the discrete dynamical system for \eqref{Ps} given in Definition \ref{def3.3}.
\end{itemize}
\end{theorem}


\begin{remark} \label{rmk4.2}\rm
 By the definition of the discrete $ \omega $-limit set $ \omega _\tau (B) $ and $\mathcal{A}_{ \tau }^*$, we easily see that
$$\mathcal{A}_{ \tau }^*= \mathcal{A}_{ \tau + n T_0 } ^* , \quad \forall n \in N .$$
Hence $ \mathcal{A}_{ \tau }^* $ is $ T_0 $-periodic in time in the above sense.
\end{remark}

The second main theorem concerns a relationship between attractors
$ \mathcal{A}_{ s }^* $ and $\mathcal{A}_{ \tau }^* $.


\begin{theorem} \label{thm4.2}
Suppose the same assumptions are made as in Theorem \ref{thm4.1}. Let $\mathcal{A}_{ s }^*$
and $\mathcal{A}_{ \tau }^* $ be discrete attractors for
 $ E ( \cdot , s ) $ and $ E( \cdot , \tau ) $ with $ 0 \le s \le \tau < + \infty $,
respectively. Then,
$$\mathcal{A}_{ \tau }^* \subset U( \tau , s) \mathcal{A}_{ s }^* ,$$
where $ U(\tau , s)$ is the $ T_0 $-periodic process for \eqref{Ps}
which is given in Definition \ref{def3.2}.
\end{theorem}

By Theorems \ref{thm4.1}-\ref{thm4.2}, we can get the attractor for the asymptotic
$ T_0 $-periodic system \eqref{APs}.


\begin{theorem}[Global attractor for \eqref{APs}] \label{thm4.3}
Suppose the same assumptions are made as in Theorem \ref{thm4.1}. For any
$ \tau \in R_+ $, let $ \mathcal{A}_{ \tau }^*$ be the discrete attractor
for $ E( \cdot , \tau ) $ obtained in Theorem \ref{thm4.1}. Here we put
\begin{equation}
\mathcal{A} ^* := \bigcup_{ \tau \in [ 0 , T_0 ] }\mathcal{A}_{ \tau }^*.
\label{e4.2}
\end{equation}
Then, for any bounded set $ B \in \mathcal{B} (H)$,
\begin{equation}
\bigcap_{ s \geq 0 } \overline{ \bigcup_{ t \geq s , \tau \in R_+ }
E( t+ \tau , \tau )(\overline{D( \varphi ^\tau )} \cap B)} \subset \mathcal{A} ^* .
 \label{e4.3}
\end{equation}
\end{theorem}

By Theorem \ref{thm4.3}, the set $ \mathcal{A} ^* $ can be called the global attractor
of \eqref{APs}.

Here we give some key lemmas.

\begin{lemma} \label{lm4.1}
If $ \{ s_n \} \subset R_+ $, $ \{ \tau _n \} \subset R_+ $, $ s \in R_+ $,
$ \tau \in R_+ $, $ s_n \to s $, $ \tau _n \to \tau $, $ \{ m_n \} \subset Z_+ $
with $ m_n \to + \infty $, $ z_n \in \overline{D( \varphi ^{ m_n T_0 + s_n } )}$,
 $ z \in \overline{ D( \varphi _p ^{ s } )}$, $ z_n \to z $ in $H$ and
an element $ w_n \in E ( m_n T_0 + \tau_n + s_n , m_n T_0 + s_n ) z_n $
converges to some element $ w \in H $ as $n \to + \infty $, then
$ w \in U( \tau + s ,s) z $.
\end{lemma}

\begin{proof}
Since $ \tau _n \to \tau $, without loss of generality we may assume that
there exists a finite time $ T>0 $ such that $ \{ \tau_n \}\subset [0, T] $
and $ \tau \in [ 0, T] $. By
$ w_n \in E ( m_n T_0 + \tau_n + s_n , m_n T_0 + s_n ) z_n $, there is a solution
$ v_n $ of \eqref{APs}
 on $ [ m_n T_0 + s_n , + \infty ) $ such that
$$
v_n ( m_n T_0 + \tau_n + s_n ) = w_n \mbox{ and } v_n ( m_n T_0 + s_n ) = z_n .
$$

Now we put $ u_n ( t ):= v_n ( t + m_n T_0 + s_n )$, then we
easily see that $ u_n $ is the solution for
\begin{gather*}
 u_n '( t )+ \partial \varphi^{ t + m_n T_0 + s_n } ( u_n ( t ))
 + G ( t+ m_n T_0 + s_n , u_n ( t )) \\ 
\quad  \quad \quad \ni f ( t + m_n T_0 + s_n ),  \quad t >0 , \\
 u_n (0)= z_n.
 \end{gather*}
 
Let $ \delta \in ( 0 , 1 )$ be fixed. Since $ z_n \to z $ in $ H $ as
$ n \to + \infty $, $ \{ z_n \} $ is bounded in $ H$.
Hence, from global estimates of solutions (cf. (B) in Section 2) it follows
that there is a positive constant $ M_\delta >0 $ (independent of $ n $)
satisfying
\begin{equation}
\sup_{ t \ge \delta }| u_n ( t ) |_H ^2 + \sup_{ t \ge \delta }
| u' _n |^2_{ L^2 (t,t+1;H)}
+ \sup_{ t \ge \delta } \varphi^{ t + m_n T_0 + s_n } ( u_n ( t )) \le M_\delta .
\label{e4.4}
\end{equation}

By \cite[Lemma 4.1]{KO3} we note that the convergence assumption (A1) implies
\begin{equation} \label{e4.5}
 \varphi^{ t+ m_n T_0 + s_n } \to \varphi^{ t+ s }_p
\end{equation}
in the sense of Mosco \cite{M} for each $ t \geq 0 $ as $ n \to + \infty $.
Moreover by the same argument in \cite[Lemma 3.1]{IKY} we can prove that
\begin{equation}
\bigcup_{ n =1 }^ { +\infty } \{ z \in H ;
\varphi^{ t+ m_n T_0 + s_n } (z) \le k \}
\label{e4.6}
\end{equation}
is relatively compact in $H$ for every real $k > 0 $ and $ t \geq 0 $, where
$ \varphi^{ t+ m_n T_0 + s_n } = \varphi^{ t + s } _p $ if $ n = + \infty $.
Therefore, by \eqref{e4.4}-\eqref{e4.6}, (A2), (A3) and the convergence result
(C) in Section 2, (by taking a subsequence of $\{ n \}$, if necessary)
we see that there is a function $ u_\delta $ such that
$$
u_\delta '( t )+ \partial \varphi _p ^{ t + s } ( u_\delta ( t ))
+ G_p ( t+ s , u_\delta ( t )) \ni f_p ( t + s ), \quad t > \delta .
$$

By the standard diagonal process and the same argument in
\cite[Lemma 3.10]{O}, we can construct the solution $ u $ on
$[ 0, + \infty ) $ satisfying
\begin{gather*}
 u '( t )+ \partial \varphi _p ^{ t + s } ( u ( t ))
 + G_p ( t+ s , u ( t )) \ni f_p ( t + s ), \quad t >0 , \\
 u (0)= z
 \end{gather*}
and
\begin{equation}
u_n \to u \mbox{ in } C( [0, T]; H) \quad \mbox{as } n \to + \infty . \label{e4.7}
\end{equation}
Then, by \eqref{e4.7} and $ u_n ( \tau _n ) = w_n $ we have $ u ( \tau ) = w$,
which implies $ w \in U( \tau + s ,s) z $.
\end{proof}

By (B) in Section 2, for each $ B \in \mathcal{B} (H)$ we can choose constants
$ r_B >0$ and $ M_B >0$ so that
\begin{equation}
|v |_H \le r_B \quad \mbox{ and } \quad \varphi^{t+s}(v) \le M_B , \label{e4.8}
\end{equation}
for any $ s \in R_+ ,$ $ t \geq T_0 $, $z \in \overline{ D( \varphi^s )} \cap B$
 and $ v \in E(t+s,s)z .$
Hence it follows from condition (A1) that for each $m \in Z_+ $,
$ \tau \in [ 0 , T_0 ]$, $n \in N$ and
$z \in \overline{D( \varphi^{mT_0+ \tau })} \cap B$ there is
$ \tilde{z}:= \tilde{z}_{mT_0+ \tau ,z, n T_0} \in D( \varphi^{ \tau } _p )$
such that
\begin{gather*}
| \tilde{z} - v |_H \le J_{m+ n }^{( r_B + M_B +M )} , \\
 \Big( \mbox{hence }
 | \tilde{z}|_H \le r_B + J_{m+n}^{( r_B + M_B +M)} \Big)
 \end{gather*}
and
\begin{gather*}
\varphi^{ \tau } _p ( \tilde{z} ) - \varphi^{nT_0+mT_0+ \tau }( v )
\le J_{m+n}^{( r_B + M_B + M )} , \\
 \Big( \mbox{hence }
 \varphi^{ \tau } _p ( \tilde{z}) \le M_B + J_{m+n}^{( r_B + M_B +M )} \Big),
\end{gather*}
where $ v \in E(n T_0+mT_0+ \tau ,mT_0+ \tau )z $.

Since $ J_{ k }^{( r_B + M_B +M )} \to 0 $ as $ k \to + \infty $, there is a number $ N_0 \in N $ such that
$$ J_{ k }^{( r_B + M_B +M )} \le 1 , \quad \forall k > N_0 .
$$
Now, put $ J_0 := 1 + \sup_{ 1 \le k \le N_0 } J_{ k }^{( r_B + M_B +M )} <+\infty $.
 Then, we define the bounded set $ \widetilde{B _\tau }$ by
$$ \widetilde{B _{ \tau } }:= \{ z \in H ; | z |_H \le r_B + J_0 \}
\cap \overline{ D ( \varphi ^\tau _p)} .
$$

Let $ B_{0 , \tau } $ be the compact absorbing set for $ U_\tau $
introduced by Remark \ref{rmk3.1}. Then, we see that there exists a number
$ \widetilde{ N } \in N $ so that
\begin{equation}
U _\tau ^l \widetilde{ B _\tau } \subset B_{0 , \tau }, \quad
\forall l \geq \widetilde{ N }. \label{e4.9}
\end{equation}

The next lemma is very important for proving Theorem \ref{thm4.1} (iii).

\begin{lemma} \label{lm4.2}
Let $ \tau \in R_+ $ and $ B_{0, \tau } $ be the compact absorbing set for
$ U_\tau $. Then we have
$$ \omega _\tau ( B) \subset B_{0 , \tau }, \quad \forall B \in \mathcal{B} ( H ) .
$$
\end{lemma}

\begin{proof} At first we assume $ \tau \in [ 0 , T_0]$.
For each $ B \in \mathcal{B} (H)$, let $ x $ be any element
of $ \omega_\tau ( B ) $.
Then, it follows from Remark \ref{rmk4.1} that there exist sequences $ \{ k_n \} \subset Z_+ $
with $ k_n \to + \infty $, $ \{ m_n \} \subset Z_+ $, $ \{ z_n \} \subset B $
with $ z_n \in \overline{ D ( \varphi^{ m _n T_0 + \tau })} $ and $ \{ x_n \} \subset H $
 with $ x_n \in E( k_n T_0 + m_n T_0 + \tau , m_n T_0 + \tau ) z_n $ such that
\begin{equation}
 x_n \to x \mbox{ in }H \quad \mbox{as } n \to + \infty .
\label{e4.10}
\end{equation}

Let $ \widetilde{ N } $ be the positive integer obtained in \eqref{e4.9}.
 Then by (E2) we have
\begin{equation}
\begin{aligned}
 x_n \in &E( k_n T_0 + m_n T_0 + \tau , k_n T_0 - \widetilde{ N } T_0 + m_n T_0
 + \tau )\\
& \circ  E( k_n T_0 - \widetilde{ N } T_0 + m_n T_0 + \tau , m_n T_0
+ \tau ) z_n
\end{aligned}\label{e4.11}
\end{equation}
for any $ n $ with $k_n \geq \widetilde{ N } + 1 $.
Hence, there exists an element
$ y_n \in E( k_n T_0 - \widetilde{ N } T_0 + m_n T_0 + \tau , m_n T_0 + \tau ) z_n $
such that
\begin{equation}
x_n \in E( k_n T_0 + m_n T_0 + \tau , k_n T_0 - \widetilde{ N } T_0 + m_n T_0
+ \tau ) y_n . \label{e4.12}
\end{equation}

Since $ \{ z_n \} \subset B $, we see that
$| y_n |_H \le r_B$  and
$$
\varphi^{ k_n T_0 - \widetilde{ N } T_0 + m_n T_0 + \tau } ( y_n ) \le M_B
$$
for any $ n $ with  $k_n\geq \widetilde{ N } +1 $,
where $ r_B $ and $ M_B $ are same positive constants in \eqref{e4.8}.

 From the convergence condition (A1) it follows that for
 $ y_n \in E( k_n T_0 - \widetilde{ N } T_0 + m_n T_0 + \tau , m_n T_0 + \tau ) z_n $
  there is $ \widetilde{ z } _n \in D( \varphi _p ^\tau )$
such that
\begin{gather*}
 | \widetilde{ z }_n - y_n |_H \le J_{ k_n - \widetilde{ N }+ m_n }
 ^{ ( r_B + M_B + M )}, \\
 \Big(\mbox{hence }
 | \widetilde{ z }_n |_H \le r_B + J_{ k_n - \widetilde{ N }+ m_n }
 ^{ ( r_B + M_B + M )}\Big)
 \end{gather*}
and
$$
 \varphi _p ^\tau ( \widetilde{ z }_n )  \le M_B
 + J_{ k_n - \widetilde{ N }+ m_n } ^{ ( r_B + M_B + M )} .
$$

Since $ \{ \widetilde{ z }_n \in D( \varphi _p^\tau ) \ ; \ n \in N
\mbox{ with } k_n \geq \widetilde{ N } +1 \} ( \subset \widetilde{ B _\tau }) $ is relatively compact in $ H $,
we may assume that
$$ \widetilde{ z }_n \to \widetilde{ z }_\infty \mbox{in } H
\quad \mbox{ as } n \to + \infty
$$
for some $ \widetilde{ z }_\infty \in H$. Then we easily see that $ \widetilde{ z }_\infty
 \in \widetilde{ B _\tau }$ and
\begin{equation}
 y_n \to \widetilde{ z }_\infty \mbox{ in } H \quad \mbox{as } n
\to + \infty . \label{e4.13}
\end{equation}

By Lemma \ref{lm4.1} and \eqref{e4.10}-\eqref{e4.13}, we observe that
$x \in U ( \widetilde{ N } T_0 + \tau , \tau ) \widetilde{ z }_\infty $,
which implies that
$$
x \in U ( \widetilde{ N } T_0 + \tau , \tau ) \widetilde{ B _\tau }
= U ^{ \widetilde{ N } } _\tau \widetilde{ B _\tau } \subset B_{ 0 , \tau }.
$$
Hence we have
$\omega_\tau ( B) \subset B_{0 , \tau } $.

For the general case of $ \tau \in R_+ $, choose positive numbers $ i_\tau \in N $ and $ \tau _0 \in [ 0 , T_0 ] $ so that
$ \tau = \tau_0 + i_\tau T_0 $. Then, we can
 show $ \omega_\tau ( B) \subset B_{0 , \tau } $ by the same argument as
 above.  \end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.1}]
 On account of Lemma \ref{lm4.2} we can get $ \mathcal{A}_{ \tau }^*
 \subset B_{ 0 , \tau }$. Hence, Theorem \ref{thm4.1} (i) holds. Also, by \eqref{e4.1}
 and Remark \ref{rmk4.1} we observe that Theorem \ref{thm4.1} (ii) holds.

 Now, we prove Theorem \ref{thm4.1} (iii). At first, let us prove that $ \mathcal{A}_{ \tau }^*
 \subset U _\tau ^l \mathcal{A}_{ \tau }^* $ for any $ l \in N $.
Let $ x $ be any element of $ \mathcal{A}_{ \tau }^* $. By the definition of
$ \mathcal{A}_{ \tau } ^* $, there are sequences $ \{ B_n \}
\subset \mathcal{B} ( H ) $ and $ \{ x_n \} \subset H $ with $ x_n \in
\omega _\tau ( B_n ) $ such that
\begin{equation}
 x_n \to x \quad \mbox{ in } H \quad
\mbox{as } n \to + \infty . \label{e4.14}
\end{equation}
Then, for each $ n $ it follows from Remark \ref{rmk4.1} that there exist sequences
$ \{ k_{n, j} \} \subset Z_+ $ with $ k_{n, j} \to + \infty $,
$ \{ m_{n, j} \} \subset Z_+ $,
 $ \{ z_{n, j} \} \subset B_n $ with $ z_{n, j} \in \overline{ D ( \varphi^{ m_{n, j} T_0 + \tau })} $
 and $ \{ v_{n,j} \} \subset H $ with $ v_{n, j} \in E( k_{n, j} T_0 + m_{n, j} T_0 + \tau , \
 m_{n, j} T_0 + \tau ) z_{n, j} $ such that
\begin{equation}
 v_{n, j} \to x_n \mbox{ in }H \quad \mbox{as } j \to + \infty .
\label{e4.15}
\end{equation}

Let $ l $ be any number in $ N $, then we see that
\begin{align*}
 v_{n, j} &\in E( k_{n, j} T_0 + m_{n, j} T_0 + \tau ,
 \ k_{n, j} T_0 - l T_0+ m_{n, j} T_0 + \tau ) \\
&\circ E( k_{n, j} T_0 - l T_0+ m_{n, j} T_0 + \tau , \ m_{n, j} T_0
+ \tau ) z_{n, j}
\end{align*}
for $ j $ with $ k_{n, j} \geq l +1 $. So, there exists an element
$ w_{n, j} \in E( k_{n, j} T_0 - l T_0+ m_{n, j} T_0 + \tau ,
 m_{n, j} T_0 + \tau ) z_{n, j} $ such that
\begin{equation}
 v_{n, j} \in E( k_{n, j} T_0 + m_{n, j} T_0 + \tau ,
  k_{n, j} T_0 - l T_0+ m_{n, j} T_0 + \tau ) w_{n, j} .
\label{e4.16}
\end{equation}

By the global estimates (B) in Section 2,
$ \{ w_{n,j} \in H \ ; \ j \in N
 \mbox{ with } k_{n,j} \geq l +1 \} $ is relatively compact in $ H $ for each $ n $. Therefore we may
 assume that the element $ w_{n, j} $ converges to some element
 $ \widetilde{ w }_{ n, \infty } \in H $ as $ j \to + \infty $. Clearly,
 $ \widetilde{ w }_{ n , \infty } \in \omega _\tau ( B_n ) $. Moreover, it
 follows from Lemma \ref{lm4.1}
 and \eqref{e4.15}-\eqref{e4.16} that
$$
x_n \in U ( l T_0 + \tau , \tau ) \widetilde{ w }_{ n , \infty } \subset
U ( l T_0 + \tau , \tau ) \omega _\tau ( B_n ) ,
$$
hence, we have
\begin{equation}
x_n \in \bigcup_{ n \geq 1 } U^l _\tau \omega _\tau ( B_n ) , \quad \forall n \geq 1 .
\label{e4.17}
\end{equation}

Here, by the closedness of $ U ( \cdot , \cdot )$ we note that for each subset $ X $ of $ B_{ 0, \tau } $,
\begin{equation}
\overline{ U^l _\tau X} \subset U_\tau ^l \overline{ X } , \quad \forall l \in N .
 \label{e4.18}
\end{equation}
Taking into account Lemma \ref{lm4.2}, \eqref{e4.14}, \eqref{e4.17} and \eqref{e4.18},
we observe that
\[
x  \in \overline{ \bigcup_{ n \geq 1 } U_\tau ^l \omega _\tau ( B_n ) }
 = \overline{ U_\tau ^l \bigcup_{ n \geq 1 } \omega _\tau ( B_n ) } 
 \subset U_\tau ^l \overline{ \bigcup_{ n \geq 1 } \omega _\tau ( B_n ) }
 \subset U_\tau ^l \mathcal{A}_{ \tau }^* ,
\]
which implies that $ \mathcal{A}_{ \tau }^* $ is semi-invariant under the
$ T_0 $-periodic dynamical systems $ U_\tau $, i.e.
\begin{equation}
\mathcal{A}_{ \tau }^* \subset U _\tau ^l \mathcal{A}_{ \tau }^* , \quad
\forall l \in N. \label{e4.19}
\end{equation}

Next we shall prove that
$ U _\tau ^l \mathcal{A}_{ \tau }^* \subset \mathcal{A}_{ \tau } $
for any $ l \in N $. By \eqref{e4.19}, for each $ l \in N $
\begin{equation}
U _\tau ^l \mathcal{A}_{ \tau }^* \subset U _\tau ^l U _\tau ^n
\mathcal{A}_{ \tau }^* = U _\tau ^{l + n } \mathcal{A}_{ \tau }^* ,
\quad \forall n \in N . \label{e4.20}
\end{equation}
By $ \mathcal{A}_{ \tau }^* \subset B_{ 0 , \tau }$, \eqref{e4.20} and the
attractive property of $ \mathcal{A}_{ \tau } $, we have
$$
U _\tau ^l \mathcal{A}_{ \tau }^* \subset \mathcal{A}_{ \tau } ,
\quad \forall l \in N.
$$
Therefore, we conclude that
$\mathcal{A}_{ \tau }^* \subset U _\tau ^l \mathcal{A}_{ \tau }^* \subset \mathcal{A}
_\tau $ for all $l \in N $.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.2}]
Let $ x $ be any element of $ \mathcal{A}_{ \tau }^* $.
Then by the definition of $ \mathcal{A}_{ \tau }^* $, there exist sequences
$ \{ B_n \} \subset \mathcal{B} ( H ) $ and $ \{ x_n \} \subset H $ with $ x_n \in
\omega _\tau ( B_n ) $ such that
\begin{equation}
 x_n \to x \mbox{ in } H \quad\mbox{as } n \to + \infty . \label{e4.21}
\end{equation}
 From Remark \ref{rmk4.1} it follows that for each $ n $, there are sequences
$ \{ k_{n, j} \} \subset Z_+ $ with $ k_{n, j} \to + \infty $,
$ \{ m_{n, j} \} \subset Z_+ $,
 $ \{ z_{n, j} \} \subset B_n $ with $ z_{n, j} \in \overline{ D ( \varphi^{ m_{n, j} T_0 + \tau })} $
 and $ \{ v_{n,j} \} \subset H $ with $ v_{n, j} \in E( k_{n, j} T_0
 + m_{n, j} T_0 + \tau ,  m_{n, j} T_0 + \tau ) z_{n, j} $ such that
\begin{equation}
v_{n, j} \to x_n \mbox{ in }H \quad \mbox{ as } j \to + \infty .
\label{e4.22}
\end{equation}

Note that for given $ s , \tau \in R_+ $ with $ s \le \tau $ there is a
positive number $ l_s \in N $ satisfying
$$ s \le \tau \le l_s T_0 +s .$$
Using the property (E2) we see that
\begin{align*}
v_{n, j} &\in E( k_{n, j} T_0 + m_{n, j} T_0 + \tau , k_{n, j} T_0
+ m_{n, j} T_0 + s ) \\
&\circ E( k_{n, j} T_0 + m_{n, j} T_0 + s , T_0 +m_{n, j} T_0 +l_s T_0 + s ) \\
&\circ E( T_0+m_{n, j} T_0 + l_s T_0 + s , m_{n, j} T_0 + \tau ) z_{n, j}
\end{align*}
for any $ j \in Z_+ $ with $ k_{ n , j } \geq l_s + 2 $.
Here we can take elements $ w_{n, j} \in H $ and $ y_{n, j} \in H $ so that
\begin{gather}
v_{n, j} \in E( k_{n, j} T_0 + m_{n, j} T_0 + \tau , k_{n, j} T_0
+ m_{n, j} T_0 + s ) w_{n, j} , \label{e4.23}\\
 w_{n, j} \in E( k_{n, j} T_0 + m_{n, j} T_0 + s , T_0 +m_{n, j} T_0
 +l_s T_0 + s ) y_{n, j}, \label{e4.24} \\
y_{n, j} \in E( T_0 + m_{n, j} T_0 + l_s T_0 +s , m_{n, j} T_0 + \tau ) z_{n, j}
. \label{e4.25}
\end{gather}

By $ \{ z_{n, j} \} \subset B_n $ and the global boundedness result (B)
in Section 2, we can get a positive
 constant $ C_n := C_n ( B_n ) >0 $ satisfying
\begin{equation}
| y_{n, j} |_H \le C_n , \quad \forall y_{n, j} \in E( T_0 + m_{n, j} T_0
+l_s T_0 + s ,  m_{n, j} T_0 + \tau ) z_{n, j}. \label{e4.26}
\end{equation}
Here we define the bounded set $ B_{ C_n } $ by
$$ B_{ C_n } :=\{ b \in H : | b |_H \le C_n \}.
$$
 From \eqref{e4.26} and the result (B) in Section 2 it follows that the set
\begin{align*}
\big\{& w_{n, j} \in H ;
w_{n, j} \in E( k_{n, j} T_0 + m_{n, j} T_0 + s , T_0 + m_{n, j} T_0
+ l_s T_0 +s ) y_{n, j} \\
&\mbox{ for any $j \in Z_+ $ with $ k_{ n , j } \geq l_s + 2 $}\big\}
\end{align*}
is relatively compact in $ H $. Hence, we may
 assume that the element $ w_{n, j} $ converges to some element
 $ \widetilde{ w }_{ n, \infty } \in H $ as $ j \to + \infty $. Clearly,
 $ \widetilde{ w }_{ n , \infty } \in \omega _s ( B_{ C_n } ) $, and
 it follows from Lemma \ref{lm4.2} that
$$
\omega _s ( B_{ C_n } ) \subset B_{ 0 , s} \subset \overline{ D ( \varphi^s _p)}\,.
$$
Moreover, by Lemma \ref{lm4.1} and \eqref{e4.22}-\eqref{e4.23} we have
$$
x_n \in U ( \tau , s ) \widetilde{ w }_{ n , \infty } \subset U ( \tau , s )
\omega _s ( B_{ C_n } ) , \quad \forall n \geq 1 ,
$$
hence, we see that
\begin{equation}
x_n \in \bigcup_{ n \geq 1 } U ( \tau , s ) \omega _s ( B_{ C_n } ) ,
\quad \forall n \geq 1\, . \label{e4.27}
\end{equation}

Here, by the closedness of $ U ( \cdot , \cdot )$, we note that for each subset
$ X $ of $ B_{ 0, s} $,
\begin{equation}
 \overline{ U ( \tau , s ) X} \subset U ( \tau , s ) \overline{ X } . \label{e4.28}
\end{equation}

On account of Lemma \ref{lm4.2}, \eqref{e4.21}, \eqref{e4.27} and \eqref{e4.28},
we observe that
\[
x  \in \overline{ \bigcup_{ n \geq 1 } U ( \tau , s ) \omega _s ( B_{ C_n }) }
  = \overline{ U( \tau , s ) \bigcup_{ n \geq 1 } \omega _s ( B_{ C_n } ) }
 \subset U( \tau , s ) \overline{ \bigcup_{ n \geq 1 } \omega _s ( B_{ C_n } ) }
  \subset U ( \tau , s ) \mathcal{A}_{ s }^* ,
\]
which implies that $ \mathcal{A}_{ \tau }^* $ is a subset of
$ U ( \tau , s ) \mathcal{A}_{ s }^* $, namely
$\mathcal{A}_{ \tau }^* \subset U ( \tau , s ) \mathcal{A}_{ s }^*$.
 \end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.3}]
For any $ B \in \mathcal{B}(H)$, let $ z_0 $ be any element of the $ \omega $-limit
set $ \omega _E (B) $ which is define by
$$
\omega _E (B):= \bigcap _{ s \geq 0 } \overline{ \bigcup_{ t \geq s ,
\tau \in R_+ }E( t+ \tau , \tau )( \overline{D( \varphi ^\tau )} \cap B)}.
$$
Then we easily see that there exist sequences $ \{ t_n \} \subset R_+ $ with
$ t_n \to +\infty$,
$\{ \tau _n \} \subset R_+ $, $ \{ y_n \} \subset B $ with
$ y_n \in \overline{D( \varphi ^{ \tau _n } ) } $ and
$ \{ z_n \} \subset H $ with $ z_n \in E( t_n + \tau _n , \tau _n ) y_n $
such that
\begin{equation}
\begin{gathered}
t _n := k_n T_0 + t ' _n ,\quad k _n \in Z_+ , \; k_n \to + \infty ,\quad
 t ' _n \in [ T_0 , 2 T_0 ] ,\; t ' _n \to t ' _0 , \\
\tau _n := l_n T_0 + \tau ' _n ,\quad
l _n \in Z_+ ,\; \tau ' _n \in [ 0 , T_0 ] ,\; \tau ' _n \to \tau ' _0,\\
z_n \to z_0 \quad \mbox{ in }H
\end{gathered} \label{e4.29}
\end{equation}
as $ n \to +\infty $. Without loss of generality, we may assume that
\begin{itemize}
\item[(a)] $t ' _n + \tau ' _n \nearrow t ' _0 + \tau ' _0$
 or
\item[(b)] $t ' _n + \tau ' _n \searrow t ' _0 + \tau ' _0 $.
\end{itemize}

Now, assume that (a) holds. Then let us consider the multivalued semiflow
\begin{equation}
v_n \in E( 1+ k_n T_0 + l_n T_0 + t ' _0 + \tau_0 ' , \ k_n T_0 + l_n T_0 + t_n ' + \tau_n ' ) z_n .\label{e4.30}
\end{equation}
Then, there is a solution $ u _n $ on
$ [ k_n T_0 + l_n T_0 + t_n ' + \tau_n ' , + \infty ) $ for
\begin{gather*}
u_n '( t )+ \partial \varphi^{ t + k_n T_0 + l_n T_0 + t_n '
+ \tau_n ' } ( u_n ( t ))+ G ( t+ k_n T_0 + l_n T_0 + t_n ' + \tau_n ' ,
 u_n ( t )) \\
\ni f ( t + k_n T_0 + l_n T_0 + t_n ' + \tau_n ' ), \quad t >0 , \\
 u_n (0)= z_n \quad \mbox{ and } \quad u_n ( 1 + t_0 ' + \tau_0 ' - t_n ' - \tau_n ') = v_n.
\end{gather*}
Since $ z_n \to z_0 $ in $ H$, $ \{ z_n \} $ is bounded in $ H $. Therefore,
by the global estimate (B) in Section 2, we see that the set
\begin{align*}
&\big\{ v_n \in H ; v_n \in E( 1+ k_n T_0 + l_n T_0 + t ' _0 + \tau_0 ' ,
 k_n T_0 + l_n T_0 + t_n ' + \tau_n ' ) z_n \\
&\mbox{ for any } n \in N\big\}
\end{align*}
is relatively compact in $H$. Hence we may assume that
\begin{equation}
 v_n \to v \mbox{ in } H \mbox{ for some } v \in H . \label{e4.31}
\end{equation}

Now applying Lemma \ref{lm4.1} with \eqref{e4.29}-\eqref{e4.31}, we obtain
$$
v \in U( 1+ t ' _0 + \tau_0 ' , \ t_0 ' + \tau_0 ' ) z_0 ,
$$
more precisely, (taking the subsequence of $ \{ n \} $ if necessary)
we observe that
\begin{equation}
u_n \to u \quad \mbox{ in } C( [ 0, 1 ] ; H) \quad
\mbox{ as } n \to + \infty , \label{e4.32}
\end{equation}
where $ u $ is the solution on $ [ t_0 ' + \tau_0 ' , + \infty ) $ satisfying
\begin{gather*}
u '( t )+ \partial \varphi^{ t + t_0 ' + \tau_0 ' } _p ( u ( t ))
+ G_p ( t+ t_0 ' + \tau_0 ' , u ( t )) \ni f_p ( t + t_0 ' + \tau _0 ' ),
\quad t >0 , \\
 u (0)= z_0 \quad \mbox{ and } \quad u ( 1 ) = v .
\end{gather*}
By \eqref{e4.32} we easily see that
\begin{equation}
u_n ( t_0 ' + \tau _0 ' - t_n ' - \tau_n ' ) \to z_0 \quad \mbox{ as }
 n \to + \infty. \label{e4.33}
\end{equation}
Note that
\begin{align*}
&u_n ( t_0 ' + \tau _0 ' - t_n ' - \tau_n ' ) \\
&\in E( k_n T_0 + l_n T_0 + t ' _0 + \tau_0 ' , \ k_n T_0 + l_n T_0 + t_n '
 + \tau_n ' ) z_n \\
&= E( k_n T_0 + l_n T_0 + t ' _0 + \tau_0 ' , \ l_n T_0 + \tau _n ' ) y_n \\
&= E( k_n T_0 + l_n T_0 + t ' _0 + \tau_0 ' , \ l_n T_0 + t_0 ' + \tau_0 ' )
E( l_n T_0 + t_0 ' + \tau_0 ', \ l_n T_0 + \tau _n ' ) y_n .
\end{align*}
So, we can take an element $ x_n \in E( l_n T_0 + t_0 ' + \tau_0 ', \ l_n T_0
+ \tau _n ' ) y_n $ such that
\begin{equation}
u_n ( t_0 ' + \tau _0 ' - t_n ' - \tau_n ' ) \in
E( k_n T_0 + l_n T_0 + t ' _0 + \tau_0 ' , \ l_n T_0 + t_0 ' + \tau_0 ' ) x_n .
\label{e4.34}
\end{equation}
By $ \{ y_n \} \subset B $ and the global estimate (B) in Section 2,
we easily see that $ \{ x_n \} $ is bounded, i.e.
\begin{equation}
\{ x_n \} \subset \widetilde{ B } \mbox{ for some } \widetilde{ B }
 \in \mathcal{B} (H ) . \label{e4.35}
\end{equation}
Therefore, from Remarks \ref{rmk4.1}-\ref{rmk4.2} and \eqref{e4.33}-\eqref{e4.35} we observe that
$$ z_0 \in \omega _{ t ' _0 + \tau_0 ' } ( \widetilde{B}) \subset \mathcal{A} ^* _{ t_0 ' + \tau_0 ' } \subset \mathcal{A} ^* .$$
Thus \eqref{e4.3} holds.

In the case (b) when $ t ' _n + \tau ' _n \searrow t ' _0 + \tau ' _0 $, we can
prove \eqref{e4.3} by a slight modification of the proof as above.
\end{proof}

Note that Theorem \ref{thm4.1} implies that the attracting set
$ \mathcal{A}_{ \tau }^* $ for \eqref{APs} is
semi-invariant under $ U_\tau $ associated with the limiting $ T_0 $-periodic
system \eqref{Ps}, in general. Moreover, from Theorem \ref{thm4.2} we observe that
$$
\mathcal{A}_{ \tau }^* \subset U( \tau , s) \mathcal{A}_{ s }^* \quad
\mbox{ for any } 0 \le s \le \tau < + \infty .
$$

To get the invariance of $ \mathcal{A}_{ \tau }^* $ under $ U_\tau $ and
$ \mathcal{A}_{ \tau }^* = U( \tau , s) \mathcal{A}_{ s }^* $,
let us use a concept of a regular approximation, which was introduced
in \cite{KY}.


\begin{definition}[Regular approximation] \label{def4.4}\rm
 Let $ s \in R_+ $ be fixed. Let $ z \in D( \varphi _p ^s ) $. Then, we
say that $ U( t + s , s ) z $ is regularly approximated by
$ E( t + k T_0 + s , \ k T_0 + s ) $ as
 $ k \to + \infty$, if for each finite $ T > 0 $ there are sequences
 $ \{ k_n \} \subset Z_+ $ with $ k_n \to + \infty $ and $ \{ z_n \} \subset H
 $ with $ z_n \in D( \varphi^{ k_n T_0 + s } ) $ and $ z_n \to z $ in $H$
satisfying the following property:
for any function $ u \in W^{1,2} ( 0, T ; H) $ satisfying
$ u(t) \in U(t + s , s ) z $
for all $ t \in [ 0 , T ] $ there is a sequence $ \{ u_n \} \subset W^{1,2}
(0, T ; H) $ such that $ u_n ( t ) \in E(t+ k_n T_0 + s , \ k_n T_0 + s ) z_n $
for all $ t \in
[ 0 , T] $ and $ u_n \to u $ in $ C([0, T ] ; H) $ as $ n \to + \infty$.
\end{definition}

Using the above concept, we can show the invariance of $ \mathcal{A}_{ \tau }^* $
under $ U_\tau $. Moreover we can get
$$\mathcal{A}_{ \tau }^* = U( \tau , s) \mathcal{A}_{ s }^*.$$


\begin{theorem} \label{thm4.4}
Suppose all assumptions in Theorem \ref{thm4.1}. Let $\mathcal{A}_{ s }^*$ and
$\mathcal{A}_{ \tau }^* $ be discrete attractors for
 $ E ( \cdot , s ) $ and $ E( \cdot , \tau ) $, with $ 0 \le s \le \tau < + \infty $,
respectively. Assume that for any point $ z $ of $ \mathcal{A}_{ s } ^*$,
$ U(t+ s , s ) z $ is regularly approximated by $ E(t + k T_0 + s , k T_0 + s ) $
 as $ k \to + \infty $. Then we have
$$
\mathcal{A}_{ \tau }^* = U( \tau , s) \mathcal{A}_{ s }^*.$$
\end{theorem}

\begin{proof}
By Theorem \ref{thm4.2}, we have only to show that
$$ U( \tau , s) \mathcal{A}_{ s }^* \subset \mathcal{A}_{ \tau }^* .$$
To do so, let $ x $ be any element of $ U( \tau , s) \mathcal{A}_{ s }^* $.

At first, taking into account Definitions \ref{def3.2}-\ref{def3.3}
 and Theorem \ref{thm4.1} (iii),
we see that for each $ n \in N$
\begin{equation}
\begin{aligned}
\ & U_\tau ^n U ( \tau , s) \mathcal{A}_{ s }^* \\
& =  U( n T_0 + \tau , \tau ) U ( \tau , s) \mathcal{A}_{ s }^* 
=  U( n T_0 + \tau , n T_0 + s ) U ( n T_0 +s , s) \mathcal{A}_{ s }^* \\
& =  U( \tau , s ) U_s ^n \mathcal{A}_{ s }^* \supset U ( \tau , s) \mathcal{A}_{ s }^* . 
 \end{aligned} \label{e4.36}
\end{equation}
Hence, there exists an element $ y_n \in \mathcal{A}_{ s }^* $ such that
$$ x \in U_\tau ^n U ( \tau , s) y_n = U( n T_0 + \tau -s +s, s) y_n. $$

Using our assumption as $ t = n T_0 + \tau -s $, we observe that for each $ n $,
there are sequences
$ \{ k_{n,j} \} \subset Z_+ $, $ \{ x_{n,j} \} \subset H $ and $ \{ y_{n,j} \}
\subset H $ such that
 $$
 k_{n,j} \to +\infty, \quad
 y_{n,j} \in D( \varphi^{ k_{n,j} T_0 + s }), \quad
 y_{n,j} \to y_n \quad \mbox{in } H
$$
and
\begin{equation}
 x_{n,j} \in E( n T_0 + \tau -s + k_{n,j} T_0 + s , \ 
k_{n,j} T_0 + s ) y_{n,j}, \quad
x_{n,j} \to x \quad \mbox{in } H \label{e4.37}
\end{equation}
as $j \to + \infty $. Therefore, by the usual diagonal argument,
we can find a subsequence $ \{ j_n \}$ of $ \{ j \} $ such that
$ \widetilde x_n := x_{n,j_n} $, $ \widetilde y_n:=y_{n,j_n} $ and
$ \widetilde k_n:= k_{n,j_n} $ satisfy
\begin{equation}
\begin{gathered}
| \widetilde x_n- x |_H <\frac 1n, \quad
\widetilde x_n \in E( n T_0 + \tau -s +\widetilde k_n T_0 + s , 
\widetilde k_n T_0 + s ) \widetilde y_n, \\ \quad
|\widetilde y_n - y_n |_H < \frac 1n
\end{gathered} \label{e4.38}
\end{equation}
for  $n=1,2,\dots$.
Since $\{ \widetilde y_n \}$ is bounded in $H$, there is a bounded set
$ B \in \mathcal{B} (H) $ so that $\{ \widetilde y_n \} \subset B $.
By (E2), we see that
\begin{align*}
\widetilde x_n & \in E( n T_0 + \tau -s +\widetilde k_n T_0 + s ,
 \widetilde k_n T_0 + s ) \widetilde y_n \\
& = E( n T_0 + \widetilde k_n T_0 + \tau , T_0 + \widetilde k_n T_0 + \tau )
E( T_0 + \widetilde k_n T_0 + \tau , \widetilde k_n T_0 + s ) \widetilde y_n ,
\end{align*}
hence there is an element
$ \widetilde z_n \in E( T_0 + \widetilde k_n T_0 + \tau , \widetilde k_n T_0 + s )
 \widetilde y_n $ such that
\begin{equation}
 \widetilde x_n \in E( n T_0 + \widetilde k_n T_0 + \tau ,
 T_0 + \widetilde k_n T_0 + \tau ) \widetilde z_n . \label{e4.39}
\end{equation}
Since $\{ \widetilde y_n \}\subset B $ and the global estimate (B) in Section 2,
we see that  $\{ \widetilde z_n \}$ is also bounded in $H$. Hence,
there is a bounded set $ \widetilde B \in \mathcal{B} (H) $ so that
$\{ \widetilde z_n \} \subset \widetilde B $.
The above fact \eqref{e4.37}-\eqref{e4.39}
 implies (cf. Remark \ref{rmk4.1}) that
 $ x \in \omega _\tau ( \widetilde B ) \subset \mathcal{A}_{ \tau }^*$.
Thus we have $ U ( \tau , s ) \mathcal{A}_{ s }^* \subset \mathcal{A}_{ \tau }^*$.
\end{proof}

By Remark \ref{rmk4.2} and the same argument in Theorem \ref{thm4.4}, we can get the following
corollary.


\begin{corollary} \label{coro4.1}
(i) Suppose the same assumptions of Theorem \ref{thm4.4}. Then, $ \mathcal{A}_{ s }^* $
is invariant under the $ T_0 $-periodic dynamical system
$ U_s (:= U ( T_0 + s , s) )$. Namely,
$$
\mathcal{A}_s ^* = U_s^l \mathcal{A}_s ^* \quad \mbox{ for any } l \in N.
$$
(ii) Assume that for any point $ z $ of $ \mathcal{A}_{ \tau } $,
$ U(t+ \tau , \tau ) z $ is regularly
approximated by $ E(t + k T_0 + \tau , k T_0 + \tau ) $ as
$ k \to + \infty $. Then, $ \mathcal{A} _\tau ^* \supset \mathcal{A}_{ \tau }$.
 Hence we have $\mathcal{A}_{ \tau }^* = \mathcal{A} _{ \tau } $
 (cf. Theorem \ref{thm4.1} (iii)).
\end{corollary}

\begin{remark} \label{rmk4.3}\rm
 If the solution operator $ U(t,s) $ is single valued, namely
the solution for the Cauchy problem of \eqref{Ps}
 is unique, the assumptions of
Theorem \ref{thm4.4} always hold. Thus, Theorem \ref{thm4.4} implies the
 abstract results obtained in
\cite{IKYp} which was concerned with the asymptotically $ T_0 $-periodic
stability for the single valued dynamical system associated with
 time-dependent subdifferentials.
\end{remark}

\section{Applications to obstacle problems for PDE's}

Let $ \Omega $ be a bounded domain in $ R^N $ ($ 1 \le N< + \infty $) with
smooth boundary $ \Gamma = \partial \Omega $,
$q$ be a  fixed number with
$ 2 \le q < + \infty $ and $ T_0 $ be a fixed positive number. We use the
notation
$$
 a_q ( v , z ) := \int_{ \Omega } | \nabla v |^{ q-2 } \nabla v \cdot
 \nabla z dx, \quad \forall v,
\  z \in W^{1,q} ( \Omega )
$$
and denote by $ ( \cdot , \cdot )$ the usual inner product in $ L^2 ( \Omega )$.

For prescribed obstacle functions $ \sigma_{0 } \le \sigma_{1 } $ and each
$ t \in R_+ $ we define the set
$$
K (t) := \big\{ z \in W^{1,q} ( \Omega ) ;  \sigma _{0 } ( t , \cdot )
\le z \le \sigma _{1} ( t , \cdot )  \mbox{ a.e. on } \Omega \big\} .
$$

Let $ f $ be a function in $ L^2_{\rm loc} ( R_+ ; L^2 ( \Omega ))$ and
$ h $ be a non-negative function on $ R_+ \times R $.

Then for given ${\bf b} \in L^\infty ( \Omega ) ^N $
 we consider an interior asymptotically $ T_0 $-periodic double obstacle
 problem for each initial time $s \in R_+$:
 \\
  Find functions $ v \in C([s , + \infty ) ; L^2 ( \Omega)) $ and
 $ \theta \in L^2_{\rm loc} (( s , + \infty ) ; L^2 ( \Omega )) $ such that
\begin{equation} \label{OPap}
\begin{gathered}
 v \in L^{q}_{\rm loc}
( ( s , + \infty ) ; W^{1,q} ( \Omega ))
\cap W^{1,2}_{\rm loc}( ( s, + \infty ) ;
 L^2 ( \Omega )); \\
 v(t)\in K (t) \quad  \mbox{ for a.e. } t \geq s ; \\
 0 \le \theta (t , x ) \le h (t , v(t,x)) \quad \mbox{ a.e. on } ( s , + \infty )
 \times \Omega ;\\
(v'(t)+ \theta (t) + {\bf b} \cdot \nabla v(t) - f (t), v(t) -z)+
a_q ( v(t), v(t) -z ) \le 0 \\
\mbox{for } z \in K (t) \mbox{ and a.e. }t \geq s.
\end{gathered}
\end{equation}

The main object of this section is to consider the large-time behavior of solution
for \eqref{OPap} under  asymptotically $ T_0 $-periodicity assumptions
$$
 \sigma _i (t) - \sigma _{ i,p } (t) \to 0 \; (i=0,1),\quad
 h (t, \cdot ) - h_p (t , \cdot ) \to 0, \quad  f(t) - f_p (t) \to 0
$$
 as $ t \to \infty $
in the sense specified below, where $ \sigma _{ i,p } (t) $, $ h_p (t , \cdot ) $,
$ f_p (t) $ are periodic in time with the same period $ T_0 $. By the above
assumptions, the limiting system of \eqref{OPap} is a $ T_0 $-periodic one
as follows: %\mbox{ (OP)$_s ^P $ }
\\
 Find functions $ u \in C([s , + \infty ) ; L^2 ( \Omega)) $ and
 $ \theta \in L^2_{\rm loc} (( s , + \infty ) ; L^2 ( \Omega )) $ such that
\begin{equation} \label{OPp}
\begin{gathered}
 u \in L^{q}_{\rm loc} ( ( s , + \infty ) ; W^{1,q} ( \Omega ))
\cap W^{1,2}_{\rm loc}( ( s, + \infty ) ;  L^2 ( \Omega )); \\
  u(t)\in K_p (t) \quad \mbox{ for a.e. } t \geq s ; \\
 0 \le \theta (t , x ) \le h_p (t , u(t,x)) \quad \mbox{ a.e. on } ( s , + \infty ) \times \Omega ;\\
(u'(t)+ \theta (t) + {\bf b} \cdot \nabla u(t) - f_p (t), u(t) -z)+ a_q ( u(t),
 u(t) -z ) \le 0 \\
\mbox{for any } z \in K_p (t) \mbox{ and a.e. }t \geq s,
\end{gathered}
\end{equation}
where $ K_p (t) := \big\{ z \in W^{1,q} ( \Omega ) : \sigma _{0, p } ( t , \cdot )
\le z \le \sigma _{1 , p} ( t , \cdot )\mbox{ a.e. on } \Omega\big\}$.

Now we suppose the following conditions:
\begin{itemize}
\item $ \sigma_{i } $ and $ \sigma_{ i,p} $ are functions on $ R_+ \times \Omega $
such that
$$ \sup_{ t \in R_+ } \Big| \frac{ d \sigma _{i } }{ dt } \Big|_{ L ^2 ( t, t+1 ;
W^{1,q} ( \Omega ))} + \sup_{ t \in R_+ } \Big| \frac{ d \sigma _{i } }{ dt }
\Big|_{ L ^2 ( t, t+1 ; L^{ \infty } ( \Omega ))} < + \infty ,$$
$$
 \sup_{ t \in R_+ } \Big| \frac{ d \sigma _{i,p} }{ dt } \Big|_{ L ^2 ( t, t+1 ;
  W^{1,q} ( \Omega ))} + \sup_{ t \in R_+ } \Big| \frac{ d \sigma _{i,p} }{ dt }
\Big|_{ L ^2 ( t, t+1 ; L^{ \infty } ( \Omega ))} < + \infty
$$
and $ \sigma_{i ,p} $ is a $ T_0 $-periodic obstacle function, i.e.
$$
\sigma_{i,p} ( t + T_0 ,x ) = \sigma _{i,p} (t,x) \quad \mbox{ for a.e. }
x \in \Omega \mbox{ and any } t \in R_+
$$
for $ i= 0 , 1$. Moreover, there are positive constants $ k_1 >0 $ and
$ k_2 >0$  such that
$$
\sigma_{1 } - \sigma_{0 } \geq k_1 \quad \mbox{ and } \quad
\sigma_{1,p} - \sigma_{0,p} \geq k_1 \quad \mbox{ a.e. on } R_+ \times \Omega $$
and
$$ | \sigma_{i } | _{L^\infty ( R_+ ; W^{1,q} ( \Omega ))} + | \sigma
 _{i} |_{ L^\infty
( R_+ \times \Omega )} +
| \sigma_{i,p} | _{L^\infty ( R_+ ; W^{1,q} ( \Omega ))} + | \sigma
 _{i,p} |_{ L^\infty
( R_+ \times \Omega )} \le k_2 $$
for $ i = 0 ,1 $.



\item $ h $ and $ h_p $ are non-negative continuous functions on $ R_+ \times R $.
There is a positive constant $ L $ such that
\begin{gather*}
|h ( t , z_1 ) - h ( t , z_2 )| \leq L | z_1 - z_2 | \\
|h_p ( t , z_1 ) - h_p ( t , z_2 )| \leq L | z_1 - z_2 |
\end{gather*}
for all $ t \in R_+ $, $ z_i \in R $ and $ i=1,2 $.
Moreover, $ h_p $ is a $ T_0 $-periodic function, i.e.
 for any $ z \in R $, $ h_p ( t + T_0 , z) = h_p (t , z) $ for
 any $ t \in R_+ $.

 \item $ f $, $ f_p \in L^2_{\rm loc} ( R_+ ; L^2 ( \Omega ))$, and $ f_p $ is a $ T_0 $-periodic function, i.e.
$$ f_p ( t + T_0 ) = f_p (t) \quad \mbox{ in } L^2 ( \Omega ) , \quad
\forall t \in R_+ . $$
 \end{itemize}

Moreover, we suppose the following convergence conditions:
\begin{itemize}
\item (Convergence of $ \sigma _i (t) - \sigma _{ i,p } (t) \to 0 $ as
$ t \to + \infty $) Put
\begin{align*}
I_m &:= \sup_{ t \in [ 0 , T_0]} |
\sigma_0 ( m T_0 + t ) - \sigma _{ 0, p} (t)|_{ W^{ 1,q } ( \Omega )}\\
&+\sup_{ t \in [ 0 , T_0]} |
\sigma_1 ( m T_0 + t ) - \sigma _{ 1, p} (t)|_{ W^{ 1,q } ( \Omega )}\\
&+\sup_{ t \in [ 0 , T_0]} |
\sigma_0 ( m T_0 + t ) - \sigma _{ 0, p} (t)|_{ L^\infty ( \Omega )}\\
&+\sup_{ t \in [ 0 , T_0]} |
\sigma_1 ( m T_0 + t ) - \sigma _{ 1, p} (t)|_{ L^\infty ( \Omega )} .
\end{align*}
Then,
$I_m \to 0$  as $m \to + \infty$.

\item (Convergence of $ h (t, \cdot ) - h_p (t , \cdot ) \to 0 $ as
$ t \to + \infty $) For any $ z \in R $,
\begin{equation}
 \sup_{ t \in [ 0 , T_0]} |
h ( m T_0 + t , z ) - h_p (t,z) | \to 0 \quad \mbox{ as } m \to +\infty . 
 \label{e5.1}
\end{equation}

\item (Convergence of $ f(t) - f_p (t) \to 0 $ as $ t \to + \infty $)
\begin{equation}
| f ( m T_0 + \cdot ) - f_p |_{ L^2 ( 0 , T_0 ; L^2 ( \Omega ))} \to 0
\quad \mbox{ as } m \to +\infty . \label{e5.2}
\end{equation}
\end{itemize}

Under the above assumptions, let us consider problems \eqref{OPap} and
\eqref{OPp}.
To apply the abstract results in Sections 2-4, we choose
$ L^2 ( \Omega ) $ as a real separable Hilbert space $ H $. And we define a
proper l.s.c. convex function $ \varphi^t $ on $ L^2( \Omega )$ by
\begin{equation}
\varphi^t (z) = \begin{cases}
\displaystyle \frac{1}{q} \int_\Omega | \nabla z|^q dx &\mbox{if } z \in K (t) , \\[0.2cm]
+ \infty &\mbox{if } z \in L^2 ( \Omega ) \setminus K (t) ,
\end{cases} \label{e5.3}
\end{equation}
and define $ \varphi^t_p $ by replacing $ K(t) $ by $ K_p (t) $ in
\eqref{e5.3}.


Also, we define a multivalued operator
$ G ( \cdot , \cdot ) $ from  $ R_+ \times H^1 ( \Omega ) $ into
$ L^2 ( \Omega ) $ by
\begin{equation}
\begin{aligned}
 G (t,z) := \big\{ g \in L^2 ( \Omega ) ;
g= l + {\bf b} \cdot \nabla z \quad \mbox{in } L^2 ( \Omega ) \\
 0 \le l( x ) \le h (t , z(x)) \quad \mbox{a.e. on } \Omega\big\}
\end{aligned}\label{e5.4}
\end{equation}
for all $ t \in R_+ $ and $ z \in H^1 ( \Omega ) $. And we define
$ G_p ( \cdot , \cdot ) $ by replacing $ h(t , \cdot ) $ by $ h_p (t , \cdot ) $
in \eqref{e5.4}.

By the same argument as in \cite[Lemma 5.1]{YIK}, we can obtain the following
lemmas.

\begin{lemma}[{cf. \cite[Lemma 5.1]{YIK}}] \label{lm5.1}
For any $ r > 0 $ and $ t \in R_+ $, put
\begin{align*}
 a_r (t) &= b_r (t) \\
 &:= k_3 \int_0^t \left\{ | \sigma_{0,p}'|_{ L^\infty ( \Omega )}+
| \sigma_{0,p} ' |_{ W^{1,q} ( \Omega )}+| \sigma_{1,p} ' |_{ L^\infty ( \Omega )}+
| \sigma_{1,p} '|_{ W^{1,q} ( \Omega )} \right\} d \tau \\
&\quad + k_3 \int_0^t \left\{ | \sigma_{0}'|_{ L^\infty ( \Omega )}+
| \sigma_{0} ' |_{ W^{1,q} ( \Omega )}+| \sigma_{1} ' |_{ L^\infty ( \Omega )}+
| \sigma_{1} '|_{ W^{1,q} ( \Omega )} \right\} d \tau ,
\end{align*}
where $ k_3 $ is a (sufficiently large) positive constant.
Then, $\{ \varphi^t \} \in \Phi ( \{ a_r \}, \{ b_r \} ) $ and
$\{ \varphi^t_p \} \in \Phi_p ( \{ a_r \}, \{ b_r \} ; T_0 ) $.
Moreover we have $ \{ G (t, \cdot ) \} \in \mathcal{G} ( \{ \varphi^t \} )$ and
$ \{ G_p (t, \cdot ) \} \in \mathcal{G}_p( \{ \varphi^t_p \} ; T_0 )$.
\end{lemma}


\begin{lemma} \label{lm5.2}
The convergence assumptions (A1)-(A3) hold.
\end{lemma}

\begin{proof} We see easily that (A2) and (A3) hold by assumptions
 \eqref{e5.1} and \eqref{e5.2}.
Now let us show (A1). For each $ t \in R_+ $ there are $ m \in Z_+ $ and
$ \tau \in [ 0 , T_0 ] $ so that $ t = m T_0 + \tau $.
For each $ z_p \in D( \varphi^t _p ) = K_p (t) $, we put
$$
z := ( z_p - \sigma_{0,p}(t)) \frac{\sigma_1(t)-\sigma_0(t)}{\sigma_{1,p}(t)
-\sigma_{0,p}(t)}+\sigma_0(t) .
$$
Then we see that $ z \in D( \varphi^t ) = K (t) $. Moreover, by the same
argument in \cite[Lemma 5.1]{YIK}, we see that
\begin{equation}
| z - z_p |_{ L^2 ( \Omega)} \le k_4 I_m \quad \mbox{and} \quad
| \nabla z - \nabla z_p |_{ L^q ( \Omega)}
\le k_4 I_m ( 1 + | \nabla z_p |_{L^q ( \Omega) }) \label{e5.5}
\end{equation}
for some constant $ k_4 >0 $. Hence we have
\begin{equation}
\varphi^t ( z) - \varphi^t_p ( z_p ) \le k_5 I_m ( 1 + \varphi^t _p ( z_p )) 
\label{e5.6}
\end{equation}
for a sufficiently large $ k_5 >0$.

Conversely, let $ z \in D( \varphi^t ) = K (t) $ and we put
$$
z_p := ( z - \sigma_{0}(t)) \frac{\sigma_{1,p} (t)
-\sigma_{0,p} (t)}{\sigma_{1}(t)-\sigma_{0}(t)}+\sigma_{0,p}(t) .
$$
Then, we observe that $ z_p \in D( \varphi^t _p ) = K_p (t) $ and
\begin{equation}
 | z_p - z |_{ L^2 ( \Omega)} \le k_4 I_m \quad \mbox{ and } \quad
\varphi^t _p ( z_p ) - \varphi^t ( z ) \le k_5 I_m ( 1 + \varphi^t ( z )) .
\label{e5.7}
\end{equation}
Therefore, by \eqref{e5.5}-\eqref{e5.7} we see that the convergence assumption
(A1) holds. \end{proof}

Clearly, the obstacle problem \eqref{OPap} can be reformulated as an evolution
equation \eqref{APs} involving the subdifferential of $ \varphi ^t $ given
by \eqref{e5.3} and the multivalued operator $ G (t , \cdot )$ defined by
\eqref{e5.4}. Also, the limiting $ T_0 $-periodic problem \eqref{OPp}
can be reformulated as an evolution equation \eqref{Ps}.
Therefore, by Lemmas \ref{lm5.1}-\ref{lm5.2}  we can apply abstract results in Section 2-4.
Namely, we can obtain an attractor $ \mathcal{A}_s ^* $ for
\eqref{OPap}, a $ T_0 $-periodic attractor $ \mathcal{A}_s $ for \eqref{OPp}
and the relationships between \eqref{OPap} and \eqref{OPp}

Additionally, we assume that $ f(t) \equiv f_p (t) $ for any $ t \in R_+ $ and
$$
\sigma _{0 } (t, z ) \equiv \sigma _{0, p } ( t, z) , \quad
 \sigma _{1, p } (t, z ) \equiv \sigma _{1 } ( t, z) , \quad
 h_p ( t, z ) \le h ( t, z)
$$
for any $ 0 \le t < + \infty $ and $ z \in R$.
Then we easily see that the assumptions of Theorem \ref{thm4.4} and its Corollary hold.
Hence we can get $\mathcal{A}_s ^* = \mathcal{A}_s $ by the same argument in
\cite[Theorem 5.4]{Y3}.

Unfortunately, we do not give assumptions for $ \sigma _i ( t , \cdot )$,
$ h ( t , \cdot ) $ and $ f (t)$ in order to get
\begin{equation}
U ( \tau , s )\mathcal{A}_s ^* = \mathcal{A}_\tau ^* \subset \mathcal{A}_\tau
\mbox{ for any } 0 \le s \le \tau < + \infty .
\label{e5.8}
\end{equation}
It seems difficult to show \eqref{e5.8}, so we leave it as an open problem.

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