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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 65, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/65\hfil Existence and convergence theorems]
{Existence and convergence theorems for evolutionary
hemivariational inequalities of second order}

\author[Z. Peng, C. Xiao \hfil EJDE-2015/65\hfilneg]
{Zijia Peng, Cuie Xiao}

\address{Zijia Peng (corresponding author)\newline
Guangxi Key Laboratory of Universities Optimization Control
and Engineering Calculation, and School of Sciences,
Guangxi University for Nationalities,
Nanning, Guangxi 530006, China}
\email{pengzijia@126.com}

\address{Cuie Xiao \newline
Department of Mathematics and Computation Sciences,
Hunan City University,
Yiyang, Hunan 413000, China}
\email{xiaocuie@163.com}

\thanks{Submitted September 23, 2014. Published March 18, 2015.}
\subjclass[2000]{35K15, 35K86}
\keywords{Hemivariational inequality; nonlinear evolution inclusion; 
\hfill\break\indent Rothe method; pseudomonotone operator; Clarke's generalized gradient}

\begin{abstract}
 This article concerns with a class of evolutionary hemivariational
 inequalities in the framework of evolution triple.
 Based on the Rothe method, monotonicity-compactness technique and the
 properties of Clarke's generalized derivative and gradient, the existence
 and convergence theorems to these problems are established.
 The main idea in the proof is using the time difference to construct
 the approximate problems. The work generalizes the existence results
 on evolution inclusions and hemivariational inequalities of second order.
\end{abstract}

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


\section{Introduction}

Let $V$ be a separable and reflexive Banach space and $V^*$ be its dual space.
Let $H$ be a separable Hilbert space which is identified with its dual $H^*$.
We assume that $V,H$ and $V^*$ form an evolution triple, i.e.,
$ V\subset H\subset V^*$ with the embeddings being dense and continuous.

Setting $T>0$ and $I=[0,T]$, the evolutionary hemivariational inequalities (EHI)
considered in this article is stated as follows: find a pair $(z,v)$
with $z(0)=z_0,v(0)=v_0$ such that for a.e. $t\in I$,
 $v(t)\in B(z'(t))$ and
\begin{equation}\label{S-HVI}
\begin{split}
 & \big\langle v'(t)+A(t,z'(t))+E(z(t))-f(t),w\big\rangle \\
 &+\int_{\Omega}j^\circ(x,z(x,t),z'(x,t);w(x),w(x))dx\geq 0
 \end{split}
 \end{equation}
holds for all $w\in V$. Here $A(t,\cdot):V\mapsto V^*$ is a nonlinear 
and pseudomonotone operator, $B:H\to 2^{H}$ is the subdifferential of 
convex functional $\Psi$ defined on $H$, $E(\cdot):V\mapsto V^*$ is
 a linear and bounded operator, $\Omega\in\mathbb{R}^{N}$ and 
$j^\circ(x,z(x,t),z'(x,t); w(x),w(x))$ denotes the Clarke's generalized 
direction derivative of a locally Lipschitz function $j(x,\cdot,\cdot) $.

The notation of hemivariational inequality was introduced by Panagiotopoulos 
in the 1980s and 1990s as the variational formulations of the problems in
 Mechanics and Engineering Science (cf. \cite{pana1985,pana1993}). 
These variational forms involve nonconvex and nonsmooth energy functionals 
and express the principle of virtual work in
their inequality forms. In last several decades, plenty of monographs are 
concerned with the study in this field, see, e.g., Naniewicz and 
Panagiotopoulos \cite{Naniewicz1995}, Motreanu and 
Panagiotopoulos \cite{Motreanubook}, Motreanu and 
Radulescu \cite{ Motreanu-Radulescu2}, Carl and Heikkila \cite{Carl-heikkila},
 Migorski, Ochal and Sofonea \cite{Migorskibook2013}. 
As to related papers, Motreanu and Radulescu \cite{Motreanu-Radulescu}, Radulescu
 and Repovs \cite{Radulescu-Repovs} have studied the existence results for 
elliptic inequality problems lack of convexity; Goeleven, Motreanu 
and Panagiotopoulos \cite{GoelevenMotreanuPana1997}, Ciulcu, Motreanu 
and Radulescu \cite{Ciulcu-Motreanu-Radulescu} have dealt with existence and 
multiplicity of solutions to elliptic hemivariational inequalities. 
For recent studies on the first-order evolution inclusions and parabolic
hemivariational inequalities, we refer readers to Papageorgiou et al 
\cite{papag1999}, Migorski and  Ochal \cite{Migorski2004}, 
Liu \cite{Liu2008,Liu2005},
Carl and Motreanu \cite{CarlMo2003}, Carl, Le and 
Motreanu \cite{Carl-Le-Motreanubook}, Peng and Liu \cite{PengJGO,Peng Math2013},
 Kalita \cite{Kalita-JMAA,Kalita-IJNAM}. Besides, the evolution
inclusions and hemivariational inequalities of second order have been 
considered by Papageorgiou and Yannakakis \cite{papag2005}, Gasinski and 
Smolka\cite{Gasinski2002}, Migorski \cite{Migorski-AA}, Migorski 
and Ochal \cite{Migorski2006}, Li and Liu \cite{Yunxiang Li JMAA2011} and so on.

The study of hemivariational problem (EHI) is connected with the 
nonlinear evolution inclusion
\begin{equation}\label{S-Inclusion}
\begin{gathered}
 \big(B(z'(t))\big)'+{A}(t,z'(t))+E(z(t))+{G}(z(t),z'(t))\ni f(t)\\
  v_0\in{B}(z'(0)),\quad  z(0)=z_0,
\end{gathered}
 \end{equation}
where $B=\partial \Psi$ and $G: H\times H\mapsto 2^{H}$ is a multivalued 
operator with nonempty,
closed and convex values. Particularly, if 
$\Psi(u)=\frac{1}{2}\| u\|_H^2,$ $ B(u)=u$.
As a result, \eqref{S-Inclusion} reduces to the following second order
evolution inclusion:
\begin{equation}
\begin{gathered}
 z''(t)+{A}(t,z'(t))+E(z(t))+{G}(z(t),z'(t))\ni f(t),\\
  z'(0)=v_0,\quad  z(0)=z_0.
\end{gathered}
\end{equation}
This kind of evolution inclusions and their applications to dynamic viscoelastic 
contact problems were studied in such
references as \cite{Yunxiang Li JMAA2011, Migorski-AA, Migorski2006,papag2005}. 
Usually, by introducing an integration operator, this inclusion was transformed 
into a first order integro-differential inclusion to which the existence theorem 
is based on the classical surjectivity result for parabolic inclusions; 
see, e.g., \cite[Theorem 2.1]{papag1999}. Other methods such as upper 
and lower method (see, e.g., \cite {Carl-Le-Motreanubook, CarlMo2003}) and 
Rothe method (see \cite{Kalita-IJNAM}) are also used to parabolic
 hemivariational inequalities and inclusions. In this paper, Rothe method is 
adopted and the reason is two folds: one is this method is more effective to 
\eqref{S-Inclusion} than others because of the nonlinearity of $B$. 
Another reason lies that it is more constructive in numerical sense 
(cf. \cite{Kalita-JMAA, Kalita-IJNAM}). The main idea in this paper is more 
or less similar to \cite{Kalita-IJNAM,PengJGO, Peng Math2013} due to the same 
structure of Rothe method. However, the difference and difficulties usually 
lie in the concrete estimates and limit procedure. As the same strategy 
in \cite{Yunxiang Li JMAA2011, Migorski-AA, Migorski2006, papag2005}, 
\eqref{S-Inclusion} will be studied by being transformed into the first 
order integral and differential inclusion.

The rest of this paper is organized as follows. Section 2 is concerned 
with some definitions and Lemmas. The hypotheses and the existence theorem 
to \eqref{S-Inclusion} (Theorem \ref{Thm2}) are presented in Section 3; 
Section 4 and 5 are concerned with the estimates and convergence of Rothe 
sequences and the proof the main results. Finally, we solve the 
hemivariational inequality problem (EHI) in Section 6 by using the existence 
theorem in Section 3 and the properties of Clarke's generalized directional 
derivative and gradient.

\section{Preliminaries}

In this section, we introduce the necessary preliminary material on convex, 
nonsmooth analysis and monotone operators
in Banach spaces. We refer the reader to
monographs and textbooks, e.g.,
 \cite{Carl-Le-Motreanubook, Clarkebook, Eklandbook,Naniewicz1995}
for the proofs of the results presented in this section.

 Let $\Gamma_0(X)$ denote the set of proper, convex and
lower semicontinuous functionals defined on a Banach space $X$.
 For $\phi\in\Gamma_0(X)$, the subdifferential $\partial \phi$ is 
the subset of $X^*$, defined by
\begin{equation*}
\partial \phi (x) := \{ x^*\in X^* :
\langle x^*,v -x \rangle_{ X}\le \phi(v)-\phi(x)
 \text{ for all }  v \in X  \}.
\end{equation*}
The convex conjugate of $\phi$ is defined as
\begin{equation*}\phi^*(v):=\sup_{u\in X}\{\langle v,u \rangle_X-\phi(u)\},
\quad \text{for all } v\in X^*.
 \end{equation*}
It is well known that $\phi^*$ belongs to $\Gamma_0(X^*)$, and for each pair 
$(u,w)\in X\times X^*$, the following three conditions are equivalent to 
each other:
 \begin{equation}
 w\in\partial \phi(u); \quad \phi^*(w)=\langle w,u\rangle_X-\phi(u);
\quad u\in\partial \phi^*(w).
 \end{equation}
Next, we recall some basic tools from  nonsmooth analysis.

\begin{definition} \label{def2.1} \rm
Let $X$ be a Banach space and let $\varphi \colon X \to \mathbb{R}$
be a locally Lip\-schitz function.
The generalized (Clarke) directional derivative of $\varphi$ at
$x \in X$ in the direction $v \in X$, denoted by $\varphi^{0}(x; v)$, is
defined by
\begin{equation*}
 \varphi^{0}(x; v) := \limsup_{y \to x,  \lambda \downarrow 0}
 \frac{\varphi(y + \lambda v) - \varphi(y)}{\lambda}
\end{equation*}
and the generalized gradient (subdifferential) of $\varphi$ at $x$,
denoted by $\partial \varphi (x)$,
is a subset of a dual space $X^*$ given by
\begin{equation*}
 \partial \varphi (x) := \{ x^* \in X^* :
 {\langle x^*, v \rangle}_{ X} \le \varphi^{0}(x; v)
  \text{ for all }  v \in X \}.
\end{equation*}
\end{definition}

It is well known, that for every $x \in X$, the set $\partial \varphi (x)$
is nonempty, convex and $w^*$-compact in $X^*$.
If $\varphi \colon X \to \mathbb{R}$ is a convex and continuous function,
then $\varphi$ is locally Lipschitz and the generalized subdifferential of 
$\varphi$ coincides with the subdifferential in the sense of convex analysis;


\begin{definition} \label{def2.2} \rm
Let $A: X\mapsto X^*$ be a bounded operator. We say it is pseudomonotone 
if for any sequence $\{u_n\}_{n\ge 1}$ weakly
convergent to $u$ in X , from 
$\limsup_{n\to\infty}\langle A(u_n), u_n-u\rangle_X\leq 0$, 
it follows that
 \begin{equation}\label{eqv-pseudo}
\liminf_{n\to \infty}\langle A(u_n),u_n-v\rangle_X\geq \langle A(u),u-v\rangle_X,
\quad \forall v\in X.
 \end{equation}
\end{definition}

Recall that in the above definition the inequality \eqref{eqv-pseudo} 
could be equivalently replaced by
$\lim_{n\to\infty}\langle A(u_n),u_n-u\rangle_X=0$ and 
$A(u_n)\to A(u)$ weakly in $ X^*$ as $n\to \infty$. 
Pseudomonotone operator, introduced by Br\'{e}zis \cite{Brezis1968}, 
is a significant generalization of monotone operator. 
It contains a variety of variational-type operators as special cases and 
play an important role in the studies of nonlinear
problems (see, for example, \cite{Lionsbook,Carl-Le-Motreanubook}). 
Moreover, this notation was generalized by Browder and Hess \cite{Browder1972} 
to multi-valued case and it has been
widely applied to dealing with hemivariational or variational-hemivariational 
inequality problems.

\begin{definition} \label{D-mpseu} \rm
A multi-valued operator $G:X\to 2^{X^*}$ is said to be pseudomonotone
if the following are satisfied:
\begin{itemize}
\item [$(i)$] for each $u\in X$, $G(u)\subset X^*$ is nonempty, bounded, 
closed and convex;
\item [$(ii)$] the restriction of $G$ to each finite-dimensional subspace 
$F$ of $X$ is weakly upper semicontinuous as an operator from $F$ to $X^*$ ;
\item [$(iii)$]if for any sequence $\{u_n\}_{n\ge 1}$ weakly convergent to 
$u$ in X, from $\limsup_{n\to\infty}\\
 \langle u_n^*,u_n-u\rangle_X\leq 0$ with $u_n^*\in G(u_n)$, 
it follows: to each element $v\in X$, there exists $u^*(v)\in G(u)$ such that
 \[
\liminf_{n\to \infty}\langle u_n^*,u_n-v\rangle_X\geq \langle u^*(v),u-v\rangle_X.\]
\end{itemize}
\end{definition}

\begin{lemma}[{\cite[Proposition 2]{Migorski-Ochal-SIAM}}]\label{Con-Lemma}
 Let $X$ and $Y$ be Banach spaces, and let $F: (0, T )\times X \to 2^Y$ be
a multifunction such that
\begin{itemize}
\item[(1)] the values of $F$ are nonempty, closed and convex subsets of $Y $;
\item[(2)] for each $x \in X$, $F(\cdot, x)$ is measurable;
\item[(3)] for a.e. $t \in (0, T ), F(t,\cdot)$ is upper semicontinuous 
from $X$ to $\text{w-}Y$ .
\end{itemize}
Let $x_n: (0, T )\to X$, $y_n: (0, T )\to Y , n\in N$, be measurable 
functions such that $x_n$
converges almost everywhere on $(0, T )$ to a function 
$x: (0, T ) \to X $ and $y_n$ converges
weakly in $L^1(0, T ; Y )$ to $y: (0, T ) \to Y$. 
If $y_n(t) \in F(t, x_n(t))$ for all $n \in N$ and almost
all $t \in(0, T )$, then $y(t) \in F(t, x(t))$ for a.e. $t \in (0, T )$.
\end{lemma}

Following is an existence lemma to elliptic inclusions governed by the sum 
of a maximal monotone operator and a multi-valued
pseudomonotone operator \cite[Theorem 2.11]{Naniewicz1995}.
This lemma shall be used to show the existence
of solutions to the approximate problems in Section 4.

\begin{lemma}\label{sum mono}
Let $V$ be a reflexive Banach Space and $\tilde{T}:V\to 2^{V^*}$ a maximal 
monotone mapping with $u^0\in D(\tilde{T})$.
Let $T$ be a bounded and pseudomonotone mapping from $V$ to $2^{V^*}$. 
Suppose that there exists a function $c: \mathcal {R}^+\mapsto \mathcal{R}$ 
with $c(r)\to+\infty$ as $r\to+\infty$ such that for $(u^*,u)\in$Graph$(T)$, 
$\langle u^*,u-u^0\rangle_V\geq c(\| u\|_V)\| u\|_V$.
Then $R(\tilde{T}+T)=V^*$.
\end{lemma}

\section{Hypotheses and main theorems}

Now we are in a position to present the hypotheses of the operators 
involving in our problems.

 $A:[0,T]\times V$ to $ V^*$ is such that
\begin{itemize}
\item[(A1)] for all $u\in V$, $t\to A(t,u)$ is measurable;

\item[(A2)] $A(t,\cdot): V\to V^*$ is
demicontinuous and pseudomonotone for a.e. $t\in I$ and there exists a 
constant $c_1>0$ such that
 \begin{equation*}
\| A(t,u)\|_{V^*} \leq c_1(1+\| u\|_V), \quad \text{a.e. } t\in I;
 \end{equation*}

\item [(A3)] there exists a constant $c_2>0$ such that
\begin{equation*}
 \langle A(t,u),u\rangle \geq c_2\|u\|^2_V-1,\quad \text{for all }
 u\in V \text{ and a.e. } t\in I.
\end{equation*}
\end{itemize}


 $B=\partial \Psi$ where $\Psi\in\Gamma_0(H)$ and it is finite and continuous 
at $0$. We assume that there exist $c_3,c_4>0$ such that
\begin{itemize}
\item[(B1)] $ \| \xi \|_{H} \leq c_3\left(1+\| u\|_{H}\right)$ for all 
$u\in H$, $\xi\in\partial \Psi(u)$.

\item[(B2)] for all $u_1,\,u_2\in H$ and $v_1\in \partial \Psi(u_1)$,
$v_2\in \partial \Psi(u_2)$, one has
\begin{equation*}
\langle v_1-v_2,u_1-u_2\rangle \geq c_4 \| u_1-u_2\|^2_{H}.
 \end{equation*}
\end{itemize}
Without loss of generality, we assume $\Psi(0)=0$ and thus $\Psi^*(v)\geq 0$
for all $v\in H$. Let $i$ and $i^*$ denote the injection from $V$ to $H$ 
and $H$ to $V^*$, respectively. From the chain rule (
\cite[Proposition 5.7]{Eklandbook} it follows that
 $\partial(\Psi\circ i)=i^*\circ\partial \Psi \circ i$. So if we restrict 
the domain of $B$ on $V$, then we have $B=\partial(\Psi\circ i)$. 
For simplicity, we sometimes omit notations $i$ and $i^*$.

Let $E: V\to V^*$ be a linear, bounded,
symmetric and monotone operator, namely, one has
\begin{itemize}
\item [(E1)] $E\in \mathcal {L}(V,V^*)$, 
$\| E(u)\|_{V^*}\leq c_5\| u\|_V$ for all $u\in V$ for some constant $c_5>0$.

\item [(E2)] $\langle E(u),v\rangle=\langle E(v),u\rangle$, 
$\langle E(u),u\rangle\geq 0$  for all $u,v \in V$.
\end{itemize}

$G:H\times H\to 2^{H}$ is a multi-valued operator
with nonempty, convex and closed values and satisfies
\begin{itemize}
\item [(G1)]$ \| \eta\|_{H} \leq c_6(1+\| u\|_{H}+\| v\|_{H})$ for all 
$u,v\in H$, $\eta\in G(u,v)$ with $c_6>0$;

\item [(G2)]the graph of $G$, i.e., $\left(u,v, G(u,v)\right)$, is sequentially 
closed in $H\times H\times H_w$ (here by $H_w$ we denote the Hilbert 
space $H$ equipped with the weak topology).
\end{itemize}
Also we assume that
\begin{itemize}
\item[(H0)] $f\in L^2(I;V^*)$ and $v_0\in B(u_0)$ for some $u_0\in V$.


\item[(H1)] the embedding operator $i$ is compact and 
$ c_6c_0^2 < c_2$ where $c_0$ is the embedding constant from $V$ to $H$.
\end{itemize}

\begin{definition} \label{D-solution1} \rm
Given $f\in L^2(I;V^*), z_0\in V,v_0\in V^*$, a triple $(z,v,g)$ of functions 
from $ I $ to $V\times V^*\times V^*$ is said to be a solution to 
\eqref{S-Inclusion} if:
\begin{itemize}
\item[(a)] $z\in H^1(I;V)$, $v\in L^{\infty}(I;H)\cap H^{1}(I;V^*)$
 and $g\in L^2(I;H)$;
\item[(b)] $v'(t)+{A}(t,z'(t))+E(z(t))+g(t)= f(t)$ in $V^*$ a.e. $t\in I $; 
\item[(c)] $z(0)=z_0$, $v(0)=v_0$, $v(t)\in B(z'(t))$, $g(t)\in G(z(t)$,
$z'(t))\text{ a.e. }t\in I $.
\end{itemize}
\end{definition}

\begin{theorem}\label{Thm2}
Under the hypotheses {\rm (A1)--(A3), (B1)--(B2), (E1)--(E2), 
(G1)--(G2), (H0)--(H1)}, the nonlinear evolution inclusion \eqref{S-Inclusion}
 admits at least one solution.
\end{theorem}

To prove this theorem, we set $u(t)=z'(t), K(u)(t)=z_0+\int_0^t u(s)ds$ 
and transform \eqref{S-Inclusion} into
the  first-order evolution inclusion
\begin{equation}\label{F-Inclusion}
\begin{gathered}
{B'}(u(t))+{A}(t,u(t))+E(K(u)(t))+{G}(K(u)(t),u(t))\ni f(t),\\
 v_0\in{B}(u(0)).
\end{gathered}
\end{equation}
As we can see, \eqref{F-Inclusion} is an integro-differential inclusion
 where the integration term involves in both $E$ and $G$, which 
complicates the study.

\begin{theorem}\label{Thm1}
Under assumptions {\rm (A1)--(A3),(B1)--(B2),(E1)--(E2), (G1)--(G2), (H0)--(H1)},
 there exist $u\in L^2(I;V)$, $v\in L^{\infty}(I;H)\cap H^{1}(I;V^*)$
and $g\in L^2(I;H)$ such that
\begin{equation}\label{F-Inclusion2}
\begin{gathered}
 v'(t)+A(t,u(t))+E(K(u)(t))+g(t)=f(t) \quad\text{in } V^*
 \text{ a.e. }t\in I ,\\
  v(0)=v_0,\quad  v(t)\in B(u(t)), \quad
g(t)\in G(K(u)(t), u(t))\quad \text{a.e. }t\in I .
\end{gathered}
\end{equation}
\end{theorem}

 Note that $v(0)=v_0$ makes sense, because $v, v'\in L^2(I; V^*)$ 
implies $v\in C( I ; V^*)$ by possibly modifying the values on a
 set of null measure.

\section{Rothe method and a priori estimates}

\subsection{Rothe method and approximate problems}
In the sequel, the Rothe method, also known as implicit time-discritization 
method, is applied for proving Theorem \ref{Thm1}.

 Let $m\in \mathbb{Z}^+$ and $(t_i)_{0\leq i \leq m}$ be a subdivision of 
$ I $ whose step is $\mu=T/m$.
Setting $u_{\mu}^0=u_0, v_{\mu}^0=v_0$ and $g_{\mu}^0=0$; for $n=0,1,\ldots,m-1$,
 we aim to find
$(u_{\mu}^{n+1},v_{\mu}^{n+1},g_{\mu}^{n+1})\in V\times V^*\times V^*$ such that
\begin{equation} \label{AP} %(D_\mu)\ \
\begin{gathered}
 \frac{v_{\mu}^{n+1}-v_{\mu}^n}{\mu}
+A_\mu^n(u_{\mu}^{n+1})+E\big(\mu  \sum_{j=0}^{n+1}u_\mu^j\big)+
 g_\mu^{n+1} = f_\mu^n\quad \text{in }V^*,\\
 v_{\mu}^{n+1}\in B(u_{\mu}^{n+1}),\quad
g_\mu^{n+1}\in G\Big(\mu\sum_{j=0}^{n+1}u_\mu^j,u_{\mu}^{n+1}\Big).
 \end{gathered}
 \end{equation}
where we set
\begin{equation*}
t_\mu^n=n\mu; \quad
A_\mu^n(u)=\frac{1}{\mu}\int_{t_\mu^n}^{t_\mu^{n+1}}A(t,u)dt; \quad
f_\mu^n=\frac{1}{\mu}\int_{t_\mu^n}^{t_\mu^{n+1}}f(t)dt.
 \end{equation*}
As is seen, the integration term $K(u)(t)$ in \eqref{F-Inclusion} 
is approximated by $\mu  \sum_{j=0}^{n+1}u_\mu^j$ for
 $t\in (t_\mu^n, t_\mu^{n+1}]$. Before proceeding further, we present a 
proposition which is useful to limit procedure of evolutionary problems 
with pseudomonotone operator. We refer the reader to
\cite[Lemma 4.1]{Peng Math2013} for its proof.

\begin{proposition}\label{pseu} 
Suppose that {\rm (A1)--(A3)} hold, $u_n\to u $ weakly in $L^2(I;V)$ as 
$n\to \infty$ and $\limsup_{n\to \infty}\langle{A}u_n,u_n-u\rangle_{L^2(I;V)}\leq 0$.
 If we further have $u_n\to u$ in $L^2(I;H)$, then
$\lim_{n\to\infty}\langle{A}u_n,u_n\rangle =\langle{A}u,u\rangle$ and
 ${A}u_n\to{A}u$ weakly in $L^2(I;V^*)$ as $n\to \infty$.
\end{proposition}


\begin{theorem}\label{Apm} 
Provided that {\rm (A1)--(A3)} are satisfied, then operator $A_\mu^n$ 
is bounded and pseudomonotone from $V$ to $V^*$
for each $n$, $n=0,1,\ldots,m-1$.
 \end{theorem}

This theorem can be proved by using Proposition \ref{pseu}, we also refer 
reader to \cite{Peng Math2013} for its proof and omit it here.

\begin{theorem}\label{Pseudo} 
Let us define $F_\mu^n:V\to 2^{V^*}$ by $F_\mu^n(u)=A_\mu^n(u)+\mu E( u)+G\big(\mu
\sum_{j=0}^nu_\mu^j$ $+\mu u,u\big),\forall u\in V$. Then $F_\mu^n$
is pseudomonotone and bounded.
\end{theorem}


\begin{proof}
We shall verify the items of Definition \ref{D-mpseu} one by one. 
As (i) is obviously true, we focus on the second and third ones.
To prove (ii), it suffice to show 
$(F_\mu^n)^{-1}(D):=\{u\in V:F_\mu^n(u)\cap D\neq \emptyset \}$ 
is closed in $V$ for any weakly closed subset $D$ of $V^*$. Indeed,
letting $\{u_k\}\subset(F_\mu^n)^{-1}(D)$ and $u_k \to u$ in $V$ as 
$k\to \infty$, there exist sequences $\eta_k\in V^*$ and 
$w_k\in G\big(\mu  \sum_{j=0}^nu_\mu^j+\mu u_k,u_k\big)$ such that
$\eta_k=A_\mu^n(u_k)+\mu E( u_k)+w_k$. From (A2) and (G1), it follows 
$A_\mu^n(u_k)$ and $w_k$ are bounded in $V^*$ and $H$, respectively, and thus
 we may assume $A_\mu^n(u_k)\to \xi$ weakly in $V^*$, $w_k\to w $ weakly in 
$H$ as $k\to \infty$. By
Theorem \ref{Apm}, one has $A_\mu^n(u)=\xi$. Meanwhile, 
$w\in G\big(\mu  \sum_{j=0}^nu_\mu^j+\mu u,u\big)$ follows from (G2).
 From $E\in \mathcal{L}(V,V^*)$, $E(u_k)\to E(u)$. Therefore, 
$\xi+\mu E(u)+w\in F_\mu^n(u)$. Since $\eta_k\in D, \eta_k\to \xi+\mu E(u)+w $ 
weakly in $V^*$ as
$k\to \infty$ and $D$ is a weakly closed set in $V^*$, we have 
$\xi+\mu E(u)+w\in D$. This, together with $\xi+\mu E(u)+w\in F_\mu^n(u)$, 
implies that $(F_\mu^n)^{-1}(D)$ is closed in $V$.

We proceed to check condition (iii). To this
end, we assume that $u_k\to u$ weakly in $V$, $\eta_k\to \eta$
weakly in $V^*$ as $k\to \infty$ such that 
$\eta_k= A_\mu^n(u_k)+\mu E( u_k)+w_k$ with 
$w_k \in G\big(\mu \sum_{j=0}^nu_\mu^j+\mu u_k,u_k\big)$ and 
$\limsup_{k\to \infty}\langle \eta_k,u_k-u\rangle_V\leq 0$. 
Owing to $V\subset H$ compactly, one has $ u_k\to u$ in $H$. Similar as before,
 we assume that $A_\mu^n(u_k)\to \xi$ weakly in $V^*$, $w_k\to w $ weakly in 
$H$ as $k\to \infty$. 
By (G2), we have $w\in G\big(\mu  \sum_{j=0}^nu_\mu^j+\mu
u,u\big)$. On the other hand, due to 
$\langle E(u_k-u),u_k-u\rangle\geq 0$ and $u_k\to u$ weakly in $V$ as 
$k\to \infty$, one has
\begin{equation*}
\liminf_{k\to\infty}\mu\langle E(u_k),u_k-u\rangle
=\liminf_{k\to\infty} \big(\mu\langle
E(u_k-u),u_k-u\rangle+\mu\langle E(u),u_k-u\rangle\big)
\geq 0.
\end{equation*}
 Moreover, taking $\langle w_k, u_k-u\rangle_V=\langle w_k,
u_k-u\rangle_{H}\to 0$ as $k\to \infty$ into account, one has
$\limsup_{k\to \infty}\langle A_\mu^n(u_k),u_k-u\rangle_V\leq 0$.
Again by Theorem \ref{Apm}, one has $A_\mu^n(u)=\xi$ and
$\lim_{k\to\infty}\langle A_\mu^n(u_k),u_k-u\rangle=0$. Besides, from
$E\in \mathcal{L}(V,V^*)$, $u_k\to u$ weakly in $V$ and the fact that 
a bounded and linear operator is weakly continuous, we have
$E(u_k)\to E(u)$ weakly in $V^*$. Consequently, $\eta=\xi+\mu
E(u)+w\in F_\mu^n(u)$. Moreover, using the above results, it is easy
to check that
\begin{equation*}
\liminf_{k\to\infty}\langle \eta_k,u_k-v\rangle_{V}\geq \langle \eta,u-v
\rangle_{V}, \quad \forall v\in V.
 \end{equation*}
Therefore, $F_\mu^n:V\to 2^{V^*}$ is pseudomonotone. On the other
hand, by virtue of (A2), (E1) and (G1), we can obtain that
$F_\mu^n$ is bounded. The proof is complete.
\end{proof}

Now we are in a position to show the existence of solutions to the
discrete problem \eqref{AP} for $ n=1,2,\ldots,m-1$. Firstly,
\eqref{AP} can be equivalently written in the following form: find
$u_\mu^{n+1}$ such that
\begin{equation}B(u_\mu^{n+1})+\mu F_\mu^n(u_\mu^{n+1})\ni \mu
 f_\mu^n+\mu^2 \sum_{j=0}^nE( u_\mu^j)
 +v_{\mu}^n.
 \end{equation}
We have $F_\mu^n$ is a bounded and pseudomonotone
mapping from $V$ to $2^{V^*}$ by Theorem \ref{Pseudo}. So, $\mu F_\mu^n$ is also
bounded and pseudomonotone for given $ \mu>0$. Since the subdifferential 
operator is maximal monotone, $B$ is a maximal monotone operator from 
$V$ to $2^{V^*}$. Next, we check that $F_\mu^n$, so as to $\mu F_\mu$, is coercive
 for small and fixed $\mu$. Let $u\in V$ and $u^*\in F_\mu^n(u)$. 
So, there exists $g\in G\big(\mu \sum_{j=0}^nu_\mu^j+\mu u,u\big)$ 
such that $u^*= A_\mu^n(u)+\mu E(u) + g$. By (G1), we have
$$
\| g\|_H\leq c_6\big(1+\mu
\sum_{j=0}^n\| u_\mu^j\|_H+\mu \| u\|_H+\| u\|_H\big).
$$
Furthermore, by using (A3) and (E1) we have
\begin{equation} \label{coer:1}
\begin{split}
&\langle u^*,u\rangle_V\\
&=\langle A_\mu^n(u)+\mu E(u),u\rangle_V+\langle g,u\rangle_H\\
&\geq c_2 \| u\|^2_V-1-\mu c_5\| u\|_V^2- c_6(1+\mu)\| u\|^2_H
- c_6\Big( \mu\sum_{j=0}^n\| u_\mu^j\|_H+1\Big)\| u\|_H \\
 & \geq( c_2-\mu c_5-c_0^2c_6-\mu c_0^2c_6) \| u\|^2_V
  - c_6c_0\Big( \mu \sum_{j=0}^n\| u_\mu^j\|_H +1\Big)\| u\|_V-1
\end{split}
 \end{equation}
In view of $c_6c_0^2 < c_2$, let us choose 
$\mu < \mu_0=(c_5+c_6c_0^2)^{-1}(c_2-c_6c_0^2) $. By taking 
$u^0=0\in D(B)$, $T=\mu F_\mu^n$ and $\tilde{T}=B$, it is easy to 
see the coercive condition in Lemma \ref{sum mono}
is satisfied for $\mu<\mu_0$.
So, by Lemma \ref{sum mono}, we have the following conclusion:

\begin{theorem}\label{sub prbm} 
Under the hypotheses of Theorem \ref{Thm1}, there exists at least one 
solution to the discrete problem \eqref{AP} for $\mu<\mu_0$ and $n=0,1,\ldots m-1$.
 \end{theorem}

In what follows, we always assume $\mu<\mu_0$.

\subsection{A priori estimates}

Let the functions $u_{\mu}$ and $v_{\mu}$ be defined as
follows:
\begin{gather*} 
u_{\mu}(t)=u_{\mu}^{n+1}, \quad t\in (t_\mu^n,t_\mu^{n+1}],\quad
 u_{\mu}(0)=u_0 ; \\
 v_{\mu}(t)= v_{\mu}^{n+1}, \quad t\in (t_\mu^n,t_\mu^{n+1}], v_{\mu}(0)=v_0.
\end{gather*}
Let $\widehat{v}_{\mu}$ denote the linear time-interpolate
function of ${v}_{\mu}$, i.e.,
\begin{equation*}
\widehat{v}_{\mu}(t)= v_{\mu}^n+\frac{t-n\mu}{\mu}(v_{\mu}^{n+1}
-v_{\mu}^n)\quad \text{for } t\in
(t_\mu^n,t_\mu^{n+1}];\quad \widehat{v}_{\mu}(0)= v_0.
 \end{equation*}
Clearly, the time derivative of $\widehat{v}_{\mu}$ on
$(t_\mu^n,t_\mu^{n+1})$ is
$\frac{1}{\mu}(v_{\mu}^{n+1}-v_{\mu}^n)$. It follows from Theorem
\ref{sub prbm} that
 \begin{equation}\label{Appro-equation}
 \frac{d}{dt}\widehat{v}_{\mu}(t)+{A_\mu}(t)+{E_\mu}(t)+g_{\mu}(t)=f_\mu(t)
 \text{ in }V^*,\quad \text{for a.e. } t\in I ,
 \end{equation}
where ${A}_\mu, {E_\mu}(t)$, $g_\mu$ and $f_\mu: I \mapsto V^*$ take values
$A_\mu^n(u_\mu^{n+1})$, $E\big(\mu \sum_{k=0}^{n+1}u_\mu^k\big)$, 
$g_\mu^{n+1}$ and $f_\mu^n$, respectively for
$t\in(t_\mu^n,t_\mu^{n+1})$. We remark that $f_\mu$ is a
possible approximate choice that converges to $f$ in $L^2(I; V^*)$
and satisfies 
$\| f_\mu\|_{L^2(I; V^*)} \leq \| f\|_{L^2(I; V^*)}$.
 Multiplying the equation in \eqref{AP} by $u_{\mu}^{n+1}$ , we have
\begin{equation*}\label{eq:13}
\frac{1}{\mu}\langle v_{\mu}^{n+1}-v_{\mu}^n,u_{\mu}^{n+1}\rangle+\big\langle
A_\mu^n(u_{\mu}^{n+1})+E\big(\mu
\sum_{k=0}^{n+1}u_\mu^k\big)+g_{\mu}^{n+1},u_{\mu}^{n+1}\big\rangle
= \langle f_\mu^n,u_{\mu}^{n+1}\rangle.
 \end{equation*}
Noting that $(\Psi\circ i)^*\in \Gamma_0 (V^*)$ and
$u_{\mu}^{n+1}\in\partial (\Psi\circ i)^*(v_{\mu}^{n+1})$, we have
\begin{equation*}
(\Psi\circ i)^*(v_{\mu}^{n+1})-(\Psi\circ i)^*(v_{\mu}^n)\leq\langle v_{\mu}^{n+1}-v_{\mu}^n,u_{\mu}^{n+1}\rangle.
 \end{equation*}
Applying this inequality to the above equation and summing it from $n=0$
to $j, 0< j \leq m-1$, we deduce that
\begin{equation} \label{Ineq-1}
\begin{aligned}
&(\Psi\circ i)^*(v_{\mu}^{j+1})+\mu\sum_{n=0}^{j}
\big\langle A_\mu^n(u_{\mu}^{n+1})+E\big(\mu
\sum_{k=0}^{n+1}u_\mu^k\big)+g_{\mu}^{n+1},u_{\mu}^{n+1}
\big\rangle \\
&\leq  \mu\sum_{n=0}^{j}\langle
f_\mu^n,u_{\mu}^{n+1}\rangle+(\Psi\circ i)^*(v_0).
 \end{aligned}
\end{equation}
By (A2), (E1), (G1) and $\|f_\mu\|_{L^2(I; V^*)} \leq \| f\|_{L^2(I; V^*)}$,
 we  deduce that
\begin{align*}
 &(\Psi\circ i)^*(v_{\mu}^{j+1})+\mu ( c_2-c_0^2c_6-\mu c_5-\mu c_0^2c_6)
\sum_{n=0}^{j}\| u_{\mu}^{n+1}\|_V^2\\
&\leq \mu \sum_{n=0}^{j}\| u_\mu^{n+1}\|_V\big(\mu c_5 \sum_{k=0}^n\|
u_\mu^k\|_V+\mu  c_0c_6 \sum_{k=0}^n\| u_\mu^k\|_H+c_0c_6\big)\\
&\quad +\| f\|\Big(\mu
\sum_{n=0}^{j}\| u_{\mu}^{n+1}
 \|_V^2\Big)^{1/2}+C\\
&\leq \mu \sum_{n=0}^{j}\|
 u_\mu^{n+1}\|_V(c_5+c_0^2c_6)((n+1)\mu)^{1/2}\big(\mu \sum_{k=0}^n\|
u_\mu^k\|^2_V\big)^{1/2}\\
&\quad +c_0c_6\mu \sum_{n=0}^{j}\|
u_{\mu}^{n+1}\|_V+\| f\|\Big(\mu
\sum_{n=0}^{j}\| u_{\mu}^{n+1} \|_V^2\Big)^{1/2}+C\\
&\leq (c_5+c_0^2c_6)\sqrt{T}\Big(\mu \sum_{n=0}^{j}\|
 u_\mu^{n+1}\|_V^2\Big)^{1/2}\Big(\mu \sum_{n=0}^{j}\mu \sum_{k=0}^n\|
u_\mu^k\|^2_V\Big)^{1/2}\\
&\quad +\big( c_0c_6\sqrt{T}+\|
f\|\big)\Big(\mu \sum_{n=0}^{j}\|
 u_\mu^{n+1}\|_V^2\Big)^{1/2}+C,
\end{align*}
where the H\"{o}lder inequality and similar estimates to \eqref{coer:1} are used.
Here and in what follows, $C$ denotes a positive constant that only
 depends on $c_0,c_1,\ldots,c_7 $, $\| f \|_{L^2(I;V^*)}$,
$(\Psi\circ i)^*(v_0)$, and may change from line to line.
 Using the Young inequality, we have
\begin{equation}
(\Psi\circ i)^*(v_{\mu}^{j+1})+\mu \sum_{n=0}^{j}\|
u_{\mu}^{n+1}\|_V^2 \leq
C\Big(1+\mu\sum_{n=0}^{j}\mu\sum_{k=0}^n\|
u_\mu^k\|^2_V\Big).
\end{equation}
Recall that
 $(\Psi\circ i)^*(v)\geq 0, \forall v\in V^*$. By applying the
 discrete Gronwall lemma, we deduce that
\begin{equation}\label{Bu}
\| u_{\mu}\|^2_{L^2(I;V)}=\mu\sum_{n=0}^{m-1}\|
 u_{\mu}^{n+1} \|_V^2\leq C.
 \end{equation}
Taking into account (A2), we can deduce that
\begin{gather}\label{Bound-A}
\| A u_{\mu} \|^2_{L^2(I;V^*)}= \sum_{n=0}^{m-1}\int_{t_\mu^n}^{t_\mu^{n+1}}\|
 A(t,u_{\mu}^{n+1})\|^2_{V^*}dt\leq C.
\\
\label{Bound-A2}
\| A_\mu \|^2_{L^2(I;V^*)}=\mu\sum_{n=0}^{m-1}\|
A_\mu^n(u_{\mu}^{n+1})\|_{V^*}^2 \leq  C.
\end{gather}
Moreover, from (E1) and (G1), we compute
\begin{equation}
\begin{split}
 \| E_\mu\|^2_{L^2(I;V^*)}
&=\int_0^T\| E_\mu(t)\|^2_{V^*}dt=\mu\sum_{n=0}^{m-1}
 \| E(\mu\sum_{k=0}^{n+1}u_\mu^k)\|^2_{V^*}\\
 &\leq\mu\sum_{n=0}^{m-1} (n+1)\mu c_5^2\mu\sum_{k=0}^{n+1}\| u_\mu^k\|_V^2\\
 &\leq Tc_5^2 \mu\sum_{n=0}^{m-1} \mu\sum_{k=0}^{n+1}\| u_\mu^k\|_V^2
 \leq C,
 \end{split}
\end{equation}
and
\begin{equation}
\begin{split}
 \| g_\mu\|^2_{L^2(I;H)}
&=\int_0^T\| g_\mu(t)\|^2_{H}dt=\mu\sum_{n=0}^{m-1}
 \| G(\mu\sum_{k=0}^{n+1}u_\mu^k,u_\mu^{n+1})\|^2_{H}\\
 &\leq2\mu c_6^2\sum_{n=0}^{m-1}\big(1+T\mu
 \sum_{k=0}^{n+1}\| u_\mu^k\|_V^2+\| u_\mu^{n+1}\|_V^2\big)
 \leq C.
 \end{split}
\end{equation}
Thus, from the above four inequalities  and \eqref{Appro-equation}, it
follows that
\begin{equation}\label{Bound-v'}
\|\widehat{v}'_{\mu}\|_{L^2(I;V^*)}\leq C.
\end{equation}
It follows from \eqref{Bu}, \eqref{Bound-A} and $v_\mu^0=v_0$ that fo rall
$\mu>0$,
\begin{equation*}(\Psi\circ i)^*(v_\mu^j)\leq C, \,\, j\in\{0,1,\ldots,m-1\}.
\end{equation*}
Furthermore, for any $w\in B(u)=i^*\circ\partial \Psi \circ i (u)$
with $u\in V$, we have
\begin{equation*}
(\Psi\circ i)^*(w)=\langle w,u\rangle_{V}-(\Psi \circ i) (u)= \langle
w,u\rangle_{H}-\Psi(u)=\Psi^*(w).
 \end{equation*}
Hence, we have $\Psi^*(v_\mu(t))=(\Psi\circ i)^*(v_\mu(t))\leq
C,\,t\in I $. By the definition of the conjugate functional, we
have
\begin{equation*} \Psi^*(v_\mu(t))\geq \langle
v_\mu(t),u\rangle-\Psi(u)\quad \forall u\in H,\; t\in I .
\end{equation*}
Choosing $u=\| v_{\mu}(t)\|_{H}^{-1}v_\mu(t)$, we
have $\| u\|_{H}=1$ and
\begin{equation*}
 \Psi^*(v_\mu(t))\geq \| v_{\mu}(t)\|_{H}-\Psi(u).
\end{equation*}
Consequently,
\begin{align*}
\| v_{\mu}(t)\|_{H}
 &\leq \Psi^*(v_\mu(t))+ \Psi(u)\\
 &\leq \Psi^*(v_\mu(t))+ \langle\xi,u\rangle\quad (\xi\in \partial \Psi(u), \Psi(0)=0)\\
 &\leq \Psi^*(v_\mu(t))+ c_3\| u\|_{H}(1+\| u\|_{H}),
\end{align*}
which implies
\begin{equation}\label{Bound-v}
\| v_{\mu}\|_{L^\infty(I; H) }
\leq C, \quad\| \widehat{v}_{\mu}\|_{C(0,T; H) }
\leq C.
\end{equation}

\section{Convergence and limit procedure}

By the a priori estimates\eqref{Bu}-\eqref{Bound-v'}, \eqref{Bound-v}, 
there exist $u\in L^2(I;V)$, $v\in L^{\infty}(I;H)\cap H^{1}(I;V^*)$ and 
$g\in L^2(I;H)$ such that
\begin{gather} 
\label{Conver-u} u_{\mu}\to u \quad\text{weakly in }L^2(I;V) \\
\label{Conver-A} A_{\mu},Au_{\mu}\to\xi\quad\text{weakly in } L^2(I;V^*)\\
\label{Conver-E} E_\mu\to \eta\quad \text{weakly in } L^2(I;V^*)\\
\label{Conver-G} g _{\mu}\to g\quad \text{weakly in }L^2(I;H)\\
\label{Conver-v'} \widehat{v}'_{\mu}\to v'\quad\text{weakly in }L^2(I;V^*)\\
\label{Conver-vv} v_{\mu},\, \widehat{v}_{\mu}\to v \quad\text{weakly star in }
 L^\infty(I;H),
\end{gather}
by possibly taking subsequences. We remark that \eqref{Conver-A} follows from
\eqref{Bound-A}, \eqref{Bound-A2} and the fact that 
$\lim_{\mu\to 0}\langle A_{\mu}-A{u_\mu},v \rangle=0$  for all
$ v\in L^2(I;V)$; \eqref{Conver-vv} is
followed by \eqref{Bound-v}, the density embedding of $V$ into $H$ and the
fact that
\begin{equation}\label{Difference-vv}
\| \widehat{v}_{\mu}-v_{\mu}\|_{L^2(I;V^*)}^2
 \leq \frac{1}{3}\mu^2\big\| \widehat{v}'_{\mu}\big\|_{
 L^2(I;V^*)}^2 \longrightarrow 0, \quad\text{as } \mu\to 0.
 \end{equation}
In view of \eqref{Conver-u}-\eqref{Conver-v'}, passing to the limit in 
\eqref{Appro-equation} we have
\begin{equation}\label{Limit-equation}
v'(t)+ \xi(t)+\eta(t)+g(t)= f(t) \quad \text{in $V^*$ a.e. } t\in I .
 \end{equation}
Note that $\widehat{v}_{\mu}\to v$ in $L^2(I;V^*)$ due to
\eqref{Conver-vv}, \eqref{Conver-vv} and the compact embedding of 
$H\subset V^*$. On the
other hand, we shall show that $u_\mu$ converges to $u$ in
$L^2(I;H)$ in the
subsequent Theorem \ref{relativecompact}.

 It follows from \eqref{Difference-vv} that $v_{\mu}\to  v$ in $L^2(I;V^*)$ 
as $\mu\to 0$. Observing that $B$ is a maximal monotone operator from
 $L^2(I;V)$ to $L^2(I;V^*)$, from \eqref{Conver-u} and
 $v_\mu\in B(u_\mu)$, we conclude that $v\in B(u)$, i.e., $v(t)\in B(u(t))$ 
a.e. $t\in I $.

By its construction, $\widehat{v}_\mu \in C(0,T;V^*)$ for any
$\mu>0$. Owing to $v\in W^{1,q}(0,T;V^*)$, we may assume $v\in
C(0,T;V^*)$ by modifying the values on a set of null measure. Since
$\widehat{v}_\mu\to v$ in $L^2(I;V^*)$, we obtain that passing to a
subsequence, $\widehat{v}_\mu(t)\to v(t)$ in $V^*$ for all $t\in I $.
Therefore,
$\widehat{v}_\mu(0)= v_0$ leads to $v(0)=v_0$.

Next, we aim to show that $\eta(t)=E(K(u)(t))$ a.e. $t\in I $.
For convenience, we define 
$\tilde{E}_\mu, \tilde{E}_0\in L^2(I; V^*)$ as $\tilde{E}_\mu(t)=E(K(u_\mu)(t))$ 
and $\tilde{E}_0(t)=E(K(u)(t))$ respectively, a.e. $t\in I $. We
check that
\begin{equation}\label{Difference-E}
\begin{split}
 &\| E_\mu- \tilde{E}_\mu\|_{L^2(I; V^*)}^2\\
 &=\sum_{n=0}^{m-1}\int_{t_\mu^n}^{t_\mu^{n+1}}
 \big\| E(\mu\sum_{k=0}^{n+1}u_\mu^k)-E
 (\mu\sum_{k=0}^nu_\mu^k+(t-n\mu)u_\mu^{n+1})\big\|_{V^*}^2dt\\
 &\leq \sum_{n=0}^{m-1}\int_{t_\mu^n}^{t_\mu^{n+1}}\mu^2
 \| E(u_\mu^{n+1})\|_{V^*}^2dt\\
 &\leq  \mu^2c_5^2\mu\sum_{n=0}^{m-1}\| u_\mu^{n+1})\|_V^2
 \leq \mu^2C.
\end{split}
\end{equation}
Since $u_\mu\to u$ weakly in $L^2(I;V)$, we have
$\tilde{E}_\mu(t)=K(u_\mu)(t) \to \tilde{E}_0(t)=K(u)(t) $ weakly in
$V^*$ for all $t\in I $. On the other hand,
\begin{equation*}
\int_0^T\|\tilde{E}_\mu(t)\|_{V^*}^2dt=\int_0^T\|
E(K(u_\mu)(t))\|_{V^*}^2dt\leq 2c_5^2T\big( \| u_0
\|^2_V +\| u_{\mu}\|^2_{L^2(I;V)}\big).
\end{equation*}
Thus, we can apply the dominated convergence result to get
$\tilde{E}_\mu \to \tilde{E}_0$ weakly in $L^2(I; V^*)$, which,
together with \eqref{Conver-E}, \eqref{Difference-E}, implies 
$\eta=\tilde{E}_0$, i.e.,
$\eta(t)=E(K(u)(t))$ a.e. $t\in I $.

In the sequel, a compact argument with respect to $u_\mu$ in
$L^2(I;H)$ is developed with the aid of compactness conclusion in
Simon \cite{Simon1987}.

\begin{theorem}\label{relativecompact} 
Suppose that $\{u_\mu\}_{\mu>0}$ is generated by \eqref{AP} under 
the assumptions of Theorem \ref{Thm1}. Then it
is relatively compact in $L^2(I;H)$.
\end{theorem}

\begin{proof}
 First of all, $\{u_\mu\}_{\mu>0}$ is bounded in $L^2(I;V)$
from \eqref{Bu}. Since $V\subset H$ compactly, according to
\cite[Theorem 3]{Simon1987}, to prove this theorem it suffices to
show that
\begin{equation}\label{relative-u}
\int_{0}^{T-h} \| u_{\mu}(t+h)-u_{\mu}(t)\|_{H}^2dt\to 0,\quad
 \text{ as $h\to 0$, uniformly for } \mu>0.
 \end{equation}
We first show
\begin{equation}\label{relative-v}
 \int_{0}^{T-h} \langle v_{\mu}(t+h)-v_{\mu}(t),u_{\mu}(t+h)-u_{\mu}(t)\rangle dt
\leq  Ch^{1/2}.
 \end{equation}
Actually, for all $h>0$, we can assume that
$h=k\mu+\tau$, $k\in\{0,1,\ldots,m-1\}$, $0\leq\tau<\mu$. We compute
\begin{align*}
&\int_{0}^{T-h} \left\langle
v_{\mu}(t+h)-v_{\mu}(t),u_{\mu}(t+h)-u_{\mu}(t)\right\rangle dt\\
&=\sum_{n=0}^{m-k-1}\int_{n\mu}^{(n+1)\mu-\tau}\left\langle
v_{\mu}(t+h)-v_{\mu}(t),u_{\mu}(t+h)-u_{\mu}(t)\right\rangle dt\\
&\quad +\sum_{n=1}^{m-k-1}\int_{n\mu-\tau}^{n\mu}\left\langle
v_{\mu}(t+h)-v_{\mu}(t),u_{\mu}(t+h)-u_{\mu}(t)\right\rangle dt\\
&= (\mu-\tau)\sum_{n=0}^{m-k-1}\left\langle
v_{\mu}^{n+k+1}-v_{\mu}^{n+1},u_{\mu}^{n+k+1}-u_{\mu}^{n+1}\right\rangle\\
&\quad +\tau\sum_{n=1}^{m-k-1}\left\langle
v_{\mu}^{n+k+1}-v_{\mu}^n,u_{\mu}^{n+k+1}-u_{\mu}^n\right\rangle.
 \end{align*}
On the other hand, for $k\in\{0,1,\ldots, m-1\}$ and
$n\in\{0,1,\ldots,m-k-1\}$,
\begin{align*}
\|v_{\mu}^{n+k+1}-v_{\mu}^{n+1}\|_{V^*}
&\leq  \mu\sum_{l=1}^{k}\|\frac{v_{\mu}^{l+n+1}-v_{\mu}^{l+n}}{\mu}\|_{V^*} \\
&= \mu\sum_{l=1}^{k}\| f_{\mu}^{l+n}-A_{\mu}^{l+n}(u_{\mu}^{l+n+1})-E\big(\mu  \sum_{j=0}^{l+n+1}u_\mu^j\big)-g_\mu^{l+n+1}\|_{V^*} \\
&\leq  \Big(\mu\sum_{l=1}^{k} 1^2\Big)^{1/2}\left(\|
f_\mu-A_\mu-E_\mu -g_\mu\|_{L^2(I;V^*)}\right) \\
&\leq C(k\mu)^{1/2}.
 \end{align*}
Similar computation gives
\begin{equation*}
 \|v_{\mu}^{n+k+1}-v_{\mu}^n\|_{V^*}\leq
C\big(k\mu\big)^{1/2}\leq Ch^{1/2}.
 \end{equation*}
Consequently, we deduce that
\begin{align*}
&\int_{0}^{T-h} \left\langle
v_{\mu}(t+h)-v_{\mu}(t),u_{\mu}(t+h)-u_{\mu}(t)\right\rangle dt\\
&\leq Ch^{1/2}\big((\mu-\tau)\sum_{n=0}^{m-k-1}\|
u_{\mu}^{n+k+1}-u_{\mu}^{n+1}\|_{V}+\tau\sum_{n=1}^{m-k-1}\|
u_{\mu}^{n+k+1}-u_{\mu}^n\|_{V} \big)\\
&\leq  Ch^{1/2}\big(2\mu\sum_{n=0}^{m-1}\|
 u_{\mu}^{n+1}\|_{V}\big).
\end{align*}
Thus we obtain \eqref{relative-v} from \eqref{Bu} and H\"{o}lder inequality. 
Recalling that $v_{\mu}(s)\in B (u_{\mu}(s)),\,s\in I $, we conclude from 
\eqref{relative-v} and (B2) that
\begin{equation*}
\int_{0}^{T-h} \| u_{\mu}(t+h)-u_{\mu}(t)\|_{H}^2dt
\leq Ch^\frac{1}{2},\quad 
\forall \mu>0,\; \forall h\in I ,
 \end{equation*}
which implies \eqref{relative-u}. This completes the proof.
\end{proof}

We proceed to prove Theorem \ref{Thm1}. Taking $j=m-1$ in \eqref{Ineq-1},
 and noticing
\begin{align*}
 \langle A_{\mu},u_{\mu}\rangle
&=\sum_{n=0}^{m-1}\int_{t_\mu^n}^{t_\mu^{n+1}}\langle
 A_\mu^n(u_{\mu}^{n+1}),u_{\mu}^{n+1}\rangle_Vdt\\
 &=\mu\sum_{n=0}^{m-1}\langle A_\mu^n(u_{\mu}^{n+1}),u_{\mu}^{n+1}\rangle_V\\
 &=\sum_{n=0}^{m-1}\int_{t_\mu^n}^{t_\mu^{n+1}}
\langle A(t,u_{\mu}^{n+1}),u_{\mu}^{n+1}\rangle_Vdt\\
 &=\langle Au_\mu, u_\mu\rangle,
\end{align*}
we have
\begin{equation*}
(\Psi\circ i)^*(v_{\mu}(T))+\langle Au_{\mu}+E_\mu +g_{\mu},u_{\mu}\rangle\leq
 \langle f_\mu,u_{\mu}\rangle+(\Psi\circ i)^*(v_0).
 \end{equation*}
In view of \eqref{Conver-u}-\eqref{Conver-v'} and $f_\mu\to f$ in $L^q(0,T;V^*)$,
we further deduce that
\begin{equation}\label{Au-limit-1}
\begin{split}
 &\limsup_{\mu\to 0}\langle Au_{\mu}+E_\mu+g_{\mu},u_{\mu}-u\rangle\\
&=\limsup_{\mu\to 0}\big(\langle f_{\mu},u_{\mu}\rangle
 -(\Psi\circ i)^*(v_{\mu}(T))-\langle Au_{\mu}+E_\mu 
 +g_{\mu},u\rangle\big)+(\Psi\circ i)^*(v_0) \\
&=\langle f-\xi-\eta-g,u\rangle+(\Psi\circ i)^*(v_0)
 -\liminf_{\mu \to 0}(\Psi\circ i)^*(v_{\mu}(T)).
 \end{split}
 \end{equation}
On the other hand, multiplying by $u(t)$ in \eqref{Limit-equation} and
integrating over $ I $, we have
\begin{equation}\label{Au-limit-2}
\int_0^T\langle\frac{dv(t)}{dt},u(t)\rangle dt
=\langle f-\xi-\eta-  g,u\rangle.
\end{equation}
Since $u(t)\in\partial(\Psi\circ i)^*(v(t))$, a.e. $t\in I $, we
conclude from the chain rule (see e.g., \cite[Lemma 1]{PengJGO} ) that
\begin{equation*}
\frac{d}{dt}(\Psi\circ i)^*(v(t))=\langle\frac{dv(t)}{dt},u(t)\rangle,
\quad\text{a.e. }t\in I .
 \end{equation*}
Integrating over $I$ and using \eqref{Au-limit-2}, we have
\begin{equation}\label{Au-limit-3}
(\Psi\circ i)^*(v(T))-(\Psi\circ i)^*(v_0)
=\langle f-\xi-  g,u\rangle.
 \end{equation}
Observing that $v_{\mu}(T)\to v(T)$ weakly in $V^*$, we have
\begin{equation}\label{Au-limit-4}
\liminf_{\mu\to 0}(\Psi\circ i)^*(v_{\mu}(T)) \geq (\Psi\circ i)^*(v(T)).
 \end{equation}
Finally, from \eqref{Au-limit-1}, \eqref{Au-limit-3} and \eqref{Au-limit-4}, 
we deduce that
\begin{equation}\label{Au-limit-5}
\limsup_{\mu\to 0}\langle Au_{\mu}+E_\mu +g_{\mu},u_{\mu}- u\rangle\leq 0.
\end{equation}
On the other hand, we have
\begin{equation*}
 \langle E_\mu, u_{\mu}-u\rangle= \langle E_\mu-\tilde{E}_\mu,
 u_{\mu}-u\rangle+\langle \tilde{E}_\mu-\eta, u_{\mu}-u\rangle+\langle \eta,
 u_{\mu}-u\rangle.
\end{equation*}
Note that
\begin{align*}
 &\langle \tilde{E}_\mu-\eta, u_{\mu}-u\rangle\\
 &=\int_0^T\langle E(K(u_\mu)(t))-E(K(u)(t)), u_\mu(t)-u(t)\rangle dt\\
 &\geq \int_0^T\langle E(K(u_\mu)(t)-K(u)(t)),
 \frac{d}{dt}(K(u_\mu)(t)-K(u)(t))\rangle dt\\
 &= \frac{1}{2}\int_0^T \frac{d}{dt}\langle E(K(u_\mu)(t))
 - E(K(u)(t)), K(u_\mu)(t)- K(u)(t)\rangle dt\\
 &=\langle E(K(u_\mu)(T))-E(K(u)(T)),
 K(u_\mu)(T)-K(u)(T)\rangle
 \geq 0,
\end{align*}
So, this inequality, together with \eqref{Difference-E} and $u_\mu \to u$ 
weakly in $L^2(I;V)$ implies
\begin{equation}\label{Au-limit-6}
 \limsup_{\mu\to 0}\langle E_\mu, u_{\mu}-u\rangle\geq 0.
\end{equation}
It follows from \eqref{Conver-G} and $u_\mu\to u$ in
$L^2(I;H) $ that
\begin{equation}\label{Au-limit-7}
\limsup_{\mu\to 0}\langle g_\mu,u_{\mu}-u\rangle
=\lim_{\mu\to 0}\langle g_\mu,u_{\mu}-u\rangle_{L^2(I;H)}=0.
\end{equation}
Hence, we conclude from \eqref{Au-limit-5}, \eqref{Au-limit-6} 
and \eqref{Au-limit-7} that
\begin{equation}\label{Au-limit-8}
 \limsup_{\mu\to 0}\langle Au_{\mu},u_{\mu}-u\rangle\leq 0.
\end{equation}
Recalling that \eqref{Conver-A} and $u_\mu\to u$ in $L^2(I;H)$, we
have $Au=\xi$ and $\lim_{\mu\to 0}\langle Au_{\mu},u_{\mu}-u\rangle=
0$ by applying Proposition \ref{pseu}.

To finish the proof, we still need to show that 
$g(t)\in G(K(u)(t),u(t))$ a.e. $t\in I $. Define 
$\tilde{K}_\mu\in L^2(I;V)$ as
$\tilde{K}_\mu(t)=\mu\sum_{k=0}^{n+1}u_\mu^k$,
$t\in (t_\mu^n, t_\mu^{n+1}]$; $\tilde{K}_\mu(0)=z_0$. Then, we have
\begin{align*}
  \| \tilde{K}_\mu-K(u)\|^2_{L^2(I;H)}&\leq 2\big(\|\tilde{K}_\mu-K(u_\mu)
 \|^2_{L^2(I;H)}+\| K(u_\mu)-K(u)\|^2_{L^2(I;H)}\big)\\
 &\leq 2\sum_{n=0}^{m-1}\int_{t_\mu^n}^{t_\mu^{n+1}}\big\| \mu\sum_{k=0}^{n+1}
 u_\mu^k-\big(\mu\sum_{k=0}^nu_\mu^k+(t-n\mu)u_\mu^{n+1}\big)\big\|_{H}^2dt\\
 &\quad +2\int_0^T\big\|\int_0^t(u_\mu(s)-u(s))ds\big\|_H^2dt\\
 &\leq 2\mu\sum_{n=0}^{m-1}\mu^2\|
 u_\mu^{n+1}\|_{H}^2+2T\big(\int_0^T\| u_\mu(s)-u(s)\|_Hds\big)^2\\
 &\leq 2\mu^2c_0^2\mu\sum_{n=0}^{m-1}\|
 u_\mu^{n+1}\|_{V}^2+2T^3\int_0^T\| u_\mu(s)-u(s)\|_H^2  ds\\
 &=2\mu^2c_0^2\| u_\mu\|_{L^2(I;V)}^2+2T^3\| u_\mu-u\|_{L^2(I;H)}^2.
\end{align*}
It implies $\tilde{K}_\mu\to K(u)$ in $L^2(I;H)$. Thus we have
$\tilde{K}_\mu(t)\to K(u)(t)$ in $H$ a.e. $t\in I$ by possibly taking
a subsequence. Recall that $g_\mu\to g$ weakly in $L^2(I;H)$,
$g_\mu(t)\in G(\tilde{K}_\mu(t),u_\mu(t))$, and $u_\mu(t)\to u(t)$ in $H$
a.e. $t\in I$ by possibly taking a subsequence. Besides, by (G1)--(G2), 
it is easy to show that $G$ is upper semicontinuous from $H\times H$ to $H_w$. 
Consequently, by Lemma \ref{Con-Lemma}, we have
$g(t)\in G(K(u)(t),u(t))$ a.e. $t\in I$. So, we finally get
\begin{equation}
\begin{gathered}
\frac{d}{dt} v(t)+A(t,u(t))+E(u(t))+g(t)=f(t) \quad 
\text{in $V^*$ a.e. }t\in I ,\\
  v(0)=v_0,\quad v(t)\in B(u(t)), \quad
g(t)\in  G(K(u)(t), u(t))\text{ a.e. }t\in I .
\end{gathered}
\end{equation}
This completes the proof of Theorem \ref{Thm1}.

\begin{remark}\rm
 Suppose that $u,v,g $ are any weak accumulation points of ${u_\mu}, v_\mu, g_\mu$
 in $L^2(I;V)$, $L^{\infty}(0,T;H)$ and $L^2(I;H)$,
 respectively, then, the triple $(u,v,g)$ is a solution to \eqref{F-Inclusion}.
\end{remark}

\section{Hemivariational inequality (EHI)}

We turn our attention to the second order nonlinear evolution
inclusion \eqref{S-Inclusion} and the evolutionary hemivariational 
inequality problem (EHI).

 In view of Theorem \ref{Thm1}, the nonlinear evolution inclusion
\eqref{F-Inclusion} admits at least one solution $(u,v,g)$, where 
$u\in L^2(I;V), v\in L^{\infty}(0,T;H)\cap H^{1}(I;V^*)$,
$g\in L^2(I;H)$ and \eqref{F-Inclusion2} holds. 
Since $u(t)=z'(t), K(u)(t)=z_0+\int_0^t u(s)ds$, it is
easy to deduce that the triple $(z,v,g)$ satisfies the conditions in 
definition \ref{D-solution1} by taking $z(t)=K(u)(t)$ for every $t\in I$. 
Therefore, $(z,v,g)$ is a solution to \eqref{S-Inclusion} which completes 
the proof of Theorem \ref{Thm2}.

Assume that $\Omega$ is an open and bounded subset of $\mathbb{R}^N$ and 
$V=H_0^1(\Omega), H=L^2(\Omega), V^*=H^{-1}(\Omega)$. In the following, 
the hypotheses on $j$ are given to investigate the weak solution of the 
evolutionary hemivariational inequality problems (EHI).

Let $j:\Omega\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ be a
function such that
\begin{itemize}
\item[(J1)] $j(\cdot,\xi,\eta)$ is measurable for all $\xi,\eta\in\mathbb{R}$, 
and $j(\cdot,0,0)\in L^1(\Omega)$;

\item[(J2)] $ j(x,\cdot,\cdot)$ is locally Lipschitz continuous on 
$\mathbb{R}\times\mathbb{R}$ for all $x\in \Omega$;

\item[(J3)] there exists a constant $c_7$ such that 
$|\partial j(x,\xi_1,\xi_2)|_{\mathbb{R}\times\mathbb{R}}
\leq c_7(1+|\xi_1|+|\xi_2|)$ for all $x\in \Omega$.
\end{itemize}
Note that the function $j$, in particular, can take the form of 
$j(x,\xi,\eta)=j_1(x,\xi)+j_2(x,\eta)$, where both 
$j_1,j_2:\Omega\times\mathbb{R}\to\mathbb{R}$ are locally Lipschitz 
continuous functions. Let us introduce integral function 
$J:H\times H \mapsto \mathbb{R}$, defined by
\begin{equation*} 
J(u,v)=\int_\Omega j(x,u(x),v(x))dx, \quad \text{for } u,v\in H.
\end{equation*}
 Observing that $J$ is well defined and finite at
$(0,0)$ by $(j1)$ and $(j2)$. Thus, by conditions (J1)-(J3) and 
\cite[Theorem 2.7.5]{Clarkebook}, one has
$J$ is uniformly Lipschitz continuous on each bounded subset of $H\times H$. 
The Clarke's generalized gradient $\partial J:
H\times H\to 2^{H\times H}$ is well defined and satisfies
\begin{equation*} 
\partial J(u,v)\subset\int_\Omega \partial j(x,u(x),v(x))dx, 
\quad \text{for } u,v\in H.
\end{equation*}
Moreover, the Clarke's generalized direction derivative of
$J$ satisfies
\begin{equation}\label{6-inequality1}
J^\circ(u,v;z,w)\leq \int_\Omega j^\circ
(x,u(x),v(x);z(x),w(x))dx, \quad \forall z,w\in H.
 \end{equation}
We define the multivalued operator $G: H\times H\to 2^{H}$ as
\begin{equation}\label{Def-G}
G(u,v):=\big\{\eta_1+\eta_2: (\eta_1,\eta_2)\in\partial J( u,v)\big\},\quad
\forall u,v\in H.
 \end{equation}
Thus, by (J3), (G1) is satisfied with $c_6$ depending on $|\Omega|$ and $c_7$.

\begin{theorem}\label{Thm-sec6}
Under hypotheses {\rm (A1)--(A3), (B1)--(B2), (E1)--(E2), (J1)--(J3)}
and {\rm (H0)--(H1)}, 
there exist $z\in H^1(I;V), v\in L^{\infty}(I;H)\cap H^{1}(I;V^*)$ with 
$v(0)=v_0$, $z(0)=z_0$ such that the inequality \eqref{S-HVI} holds and 
$ v(t)\in B(z'(t))$ for a.e. $t\in I$.
\end{theorem}

Note that since $B$ is a nonlinear and multi-valued operator, this theorem 
generalizes existence results in \cite{Migorski2006}.

\begin{proof}
 Let the multivalued function $G$ be defined as \eqref{Def-G}. 
We check the assumption (G1)--(G2). First of all, $G$ takes nonempty, convex 
and closed values since the Clarke's generalized gradient is nonempty 
convex and closed. Since (G1) has been proved by (J3), it remains to show (G2). 
It suffices to show that $\eta\in G(u,v)$ whenever the sequences $u_n\to u$, 
$v_n\to v$ in $H$ and $\eta_n\to \eta$ weakly in
$H$ as $n\to \infty$ with $\eta_n\in G(u_n,v_n)$. 
In fact, by $\eta_n\in G(u_n,v_n)$, there exist sequences $\eta_{1n}$ 
and $\eta_{2n}$ with $\eta_n=\eta_{1n}+\eta_{2n}$ and 
$(\eta_{1n},\eta_{2n})\in \partial J(u_n,v_n)$ for each $n$. 
Since $(\eta_{1n},\eta_{2n})$ is bounded, up to a subsequence, we may 
assume $\eta_{1n}\to \eta_1$, $\eta_{2n}\to \eta_2$ weakly in $H$. 
Consequently, $\eta_{1n}+\eta_{2n}\to \eta_1+\eta_2$ weakly in $H$. 
On the other hand, it follows from the weak closeness properties of the 
Clarke's generalized gradient  \cite[Proposition 2.1.5 (b)]{Clarkebook}, 
we have $(\eta_1,\eta_2)\in \partial J(u,v)$. Thus, we have 
$\eta=\eta_1+\eta_2\in G(u,v)$, i.e., the graph of $G$ is sequentially 
closed in $H\times H\times H_w$.

Therefore, by Theorem \ref{Thm2}, there exists a triple $(z,v,g)$ with 
$z\in H^1(I;V)$, $ v\in L^{\infty}(I;H)\cap H^{1}(I;V^*)$  and 
$g\in L^2(I;H)$ such that
 \begin{equation}\label{6-equation1}
v'(t)+{A}(t,z'(t))+E(z(t))+g(t)= f(t) \quad\text{in $V^*$ a.e. } t\in I,
 \end{equation}
where $g(t)\in G(z(t),z'(t))$, $v(t)\in B(z'(t))$, a.e. 
$t\in I $ and $ z(0)=z_0$, $v(0)=v_0$.
It follows from the definition of $G(\cdot,\cdot)$, there exist 
$g_1,g_2\in L^2(I;H)$ with $g(t)=g_1(t)+g_2(t)$ and 
$(g_1(t),g_2(t))\in \partial J(z(t), z'(t))$ for a.e. $t\in I $.
Consequently, for all $w\in V$ and a.e. $t\in I $, one has
\begin{equation}\label{6-inequality2}
\begin{split}
 \big\langle (g_1(t),g_2(t)), (w,w)\big\rangle_{H\times H}
 &\leq J^0(z(t),z'(t);w,w)\\
 &\leq \int_{\Omega}j^\circ(x,z(x,t),z'(x,t); w(x),w(x))dx,
\end{split}
\end{equation}
by the definition of Clarke generalized gradient and \eqref{6-inequality1}.
Furthermore, since
$$
\big\langle g(t), w\big\rangle_{V}=\big\langle g_1(t)+g_2(t),w\big\rangle_{H}
 =\big\langle (g_1(t),g_2(t)), (w,w)\big\rangle_{H\times H}
$$
we have
\begin{equation}\label{6-inequality3}
\big\langle g(t), w\big\rangle_{V}\leq \int_{\Omega}j^\circ(x,z(x,t),z'(x,t); 
w(x),w(x))dx.
 \end{equation}
Finally, multiplying by $w\in V$ on both sides of equation in \eqref{6-equation1}, 
then hemivariational inequality \eqref{S-HVI} follows from \eqref{6-inequality3} 
immediately. We complete the proof of Theorem \ref{Thm-sec6}.
\end{proof}

\subsection*{Acknowledgements}
The authors would like to thank the anonymous reviewers for their careful
 reviewing and valuable suggestions. 
This work is supported by National Science Foundation of China 
(Grant No. 11426071), Guangxi Natural Science Foundation 
(Grant No. 2014GXNSFBA118001) and the Project of Guangxi Education 
Department (Grant No. 2013YB067).
The second author is also supported by the Natural Science Foundation 
of the Department of Education of Hunan Province (Grant No. 13A013).

\begin{thebibliography}{95}

\bibitem{Brezis1968} H. Brezis;
\emph{Equations et inequations non lineaires dans les espaces vectoriels
en dualite}, Ann. Inst. Fourier, 18(1968) 115-175.

\bibitem{Browder1972} F. E. Browder, P. Hess;
\emph{Nonlinear mappings of monotone type in Banach spaces},
J. Funct. Anal., 11(1972) 251-294.

\bibitem{Ciulcu-Motreanu-Radulescu} C. Ciulcu, D. Motreanu, V. Radulescu;
\emph{Multiplicity of solutions for a class of
non-symemtric eigenvalue hemivariational inequalities}, Math. Methods Appl.
Sci., 26 (2003) 801-814.

\bibitem{Carl-heikkila} S. Carl, S. Heikkil\"{a};
\emph{Nonlinear Differential Equations in Ordered Spaces},
Chapman \& Hall/CRC, Boca Raton, 2000.

\bibitem{Carl-Le-Motreanubook} S. Carl, V. K. Le, D. Motreanu;
\emph{Nonsmooth Variational Problems and
Their Inequalities: Comparison Principles and Applications}, Springer
Monographs in Mathematics, Springer, New York, 2007.

\bibitem{CarlMo2003} S. Carl, D. Motreanu;
\emph{Extremal solutions of quasilinear parabolic inclusions with generalized
Clarke's gradient},  J. Differential Equations, 191 (2003) 206-233.

\bibitem{Clarkebook} F. H. Clarke;
\emph{Optimization and Nonsmooth Analysis}, Wiley, New York, 1983.

\bibitem{Eklandbook} I. Ekeland, R. T\'{e}mam;
\emph{Convex Analysis and Variational Problems}, SIAM., Philadelphia, 1999.

\bibitem{Gasinski2002} L. Gasi\'{n}ski, M. Smolka;
\emph{An existence theorem for wave-type hyperbolic
hemivariational inequalities}, Math. Nachr., 242 (2002), 79-90.

\bibitem{GoelevenMotreanuPana1997} D. Goeleven, D. Motreanu, P. D. Panagiotopoulos;
\emph{Semicoercive variational hemivariational inequalities},
Applicable Analysis, 65 (1997) 119-134.

\bibitem{Kalita-JMAA} P. Kalita;
\emph{Regularity and Rothe method error estimates for parabolic
hemivariatioal inequality}, J. Math. Anal. Appl., 389 (2012) 618-631.

\bibitem{Kalita-IJNAM} P. Kalita;
\emph{Convergence of Rothe scheme for hemivariational inequalities of
parabolic type}, Int J. Numerical Analysis and Modeling, 10 (2013), 445-465.

\bibitem{Yunxiang Li JMAA2011} Y. Li, Z. Liu;
\emph{Dynamic contact problem for viscoelastic piezoelectric materials
with normal damped response and friction},
J. Math. Anal. Appl., 373 (2011) 726-738.

\bibitem{Lionsbook} J. L. Lions;
\emph{Quelques Methodes de Resolution des Problemes aux Limites non Lineaires},
Dunod, Gauthier-Villars, Pairs, 1969.

\bibitem{Liu2008} Z. Liu;
\emph{Existence results for quasilinear parabolic hemivariational inequalities},
J. Differential Equations, 244 (2008) 1395-1409.

\bibitem{Liu2005} Z. Liu;
\emph{Browder-Tikhonov regularization of non-coercive evolution
hemivariational inequalities}, Inverse Problems, 21 (2005) 13-20.

\bibitem{Migorski-AA} S. Mig\'{o}rski;
\emph{Dynamic hemivariational inequalities modeling viscoelastic
contact problem with normal damped response and friction},
Applicable Analysis, 84 (2005) 669-699.

\bibitem{Migorski2006} S. Mig\'{o}rski, A. Ochal;
\emph{A unified approach to dynamic contact
problems in viscoelasticity}, J. Elasticity, 83 (2006) 247-275.

\bibitem{Migorski2004} S. Mig\'{o}rski, A. Ochal;
\emph{Boundary hemivariational inequality of parabolic type},
Nonlinear Anal., 57 (2004) 579-596.

\bibitem{Migorski-Ochal-SIAM} S. Mig\'{o}rski, A. Ochal;
\emph{Quasi-static hemivariational inequality via vanishing acceleration approach},
SIAM J. Math. Anal., 41 (2009) 1415-1435.

\bibitem{Migorskibook2013} S. Migorski, A. Ochal, M. Sofonea;
\emph{Nonlinear Inclusions and Hemivariational Inequalities:
Models and Analysis of Contact Problems}, New York: Springer, 2013.

\bibitem{Motreanubook} D. Motreanu, P. D. Panagiotopoulos;
\emph{Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational
Inequalities and Applications}, Boston, Dordrecht, London: Kluwer
Academic Publishers, 1999.

\bibitem{Motreanu-Radulescu2} D. Motreanu, V. Radulescu;
\emph{Variational and Nonvariational Methods in Nonlinear
Analysis and Boundary Value Problems, Nonconvex Optimization and Its
Applications}, Vol. 67, Kluwer Academic Publishers, Dordrecht, 2003.

\bibitem{Motreanu-Radulescu} D. Motreanu, V. Radulescu;
\emph{Existence results for inequality problems with lack
of convexity}, Numer. Funct. Anal. Optimiz., 21 (2000) 869-884.

\bibitem{Naniewicz1995} Z. Naniewicz, P. D. Panagiotopoulos;
\emph{Mathematical Theory of Hemivariational Inequalities and Applications},
Marcel Dekker, New York, 1995.

\bibitem{pana1985} P. D. Panagiotopoulos;
\emph{Inequality Problems in Mechanics and Applications,
Convex and Nonconvex Energy Functions}, Birkh\"{a}user, Basel, 1985.

\bibitem{pana1993} P. D. Panagiotopoulos;
\emph{Hemivariational Inequalities, Applications in Mechanics and Engineering},
Springer, Berlin, 1993.

\bibitem{papag1999} N. S. Papageorgiou, F. Papalini, F. Renzacci;
\emph{Existence of solutions and periodic solutions for nonlinear evolution
inclusions}, Rendiconti del Circolo Matematico di Palermo Serie II, Tomo XlVIII
(1999) 341-364.

\bibitem{papag2005} N. S. Papageorgiou, N. Yannakakis;
\emph{Second order nonlinear evolution inclusions I: Existence and
relaxation results},
Acta Mathematica Sinica, English Series, 21 (2005) 977-996.

\bibitem{PengJGO} Z. Peng, Z. Liu;
\emph{Evolution hemivariational inequality problems with doubly nonlinear 
operators}, J. Global Optim., 51 (2011) 413-427.

\bibitem{Peng Math2013} Z. Peng, Z. Liu, Xiaoyou Liu;
\emph{Boundary hemivariational inequality problems
with doubly nonlinear operators}, Mathematische Annalen, 356(4) (2013) 1339-1358.

\bibitem{Radulescu-Repovs} V. Radulescu, D. Repovs;
\emph{Existence results for variational-hemivariational problems with lack of
convexity}, Nonlinear Analysis, 73 (2010) 99-104.

\bibitem{Simon1987} J. Simon;
\emph{Compact sets in the space $L^p(0,T;B)$},
Ann. Mat. Pura Appl.(4), 146 (1987) 65-96.

\end{thebibliography}
\end{document}