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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 131, 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  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/131\hfil A quasistatic frictional contact problem]
{A quasistatic frictional contact problem with adhesion for
nonlinear elastic materials}

\author[A. Touzaline\hfil EJDE-2008/131\hfilneg]
{Arezki Touzaline} 

\address{Arezki Touzaline \newline
Laboratoire de Syst\`emes Dynamiques \\
Facult\'e de Math\'ematiques, USTHB\\
BP 32 El Alia\\
Bab-Ezzouar, 16111, Alg\'erie}
\email{atouzaline@yahoo.fr}

\thanks{Submitted April 15, 2008. Published September 23, 2008.}
\subjclass[2000]{35J85, 49J40, 47J20, 74M15}
\keywords{Nonlinear elastic materials; adhesion; normal
compliance; \hfill\break\indent
time-discretization; fixed point; quasistatic; weak solution}

\begin{abstract}
 The aim of this paper is to study a quasistatic contact problem between a
 nonlinear elastic body and a foundation. The contact is adhesive and
 frictional and is modelled with a version of normal compliance condition and
 the associated Coulomb's law of dry friction. The evolution of the bonding
 field is described by a first order differential equation. We establish the
 variational formulation of the mechanical problem and prove an existence
 result of the weak solution if the coefficient of friction is sufficiently
 small by passing to the limit with respect to time. The proofs are based on
 arguments of time-discretization, compactness, lower semicontinuity and
 Banach fixed point.
\end{abstract}

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


\section{Introduction}

Contact problems involving deformable bodies are quite frequent in industry
as well as in daily life and play an important role in structural and
mechanical systems. Because of the importance of this process a considerable
effort has been made in its modelling and numerical simulations. A first
study of frictional contact problems within the framework of variational
inequalities was made in \cite{d1}. The mathematical, mechanical and numerical
state of the art can be found in \cite{r2}. Models for dynamic or quasistatic
process of frictionless adhesive contact between a deformable body and a
foundation have been studied in \cite{c2,c3,f3,s3}. In this paper we study a
mathematical model which describes a frictional quasistatic contact problem
with adhesion between a nonlinear elastic body and a foundation. The
adhesive frictional contact is modelled with a version of normal compliance
condition and the associated Coulomb's law of dry friction.
As in \cite{f2,f3}, we
use the bonding field as an additional state variable $\beta $, defined on
the contact surface of the boundary. The variable is restricted to values
$0\leq \beta \leq 1$, when $\beta =0$ all the bonds are severed and there are
no active bonds; when $\beta =1$ all the bonds are active; when $0<\beta <1$
it measures the fraction of active bonds and partial adhesion takes place.
We refer the reader to the extensive bibliography on the subject in
\cite{f4,r1,r3,s2,s3,s4}.
 In \cite{c1} a model of a contact
problem with adhesion and friction was studied in which $\beta $ represents
a continuous transition between total adhesion and pure frictional states.
In \cite{c4} the authors considered the interface model proposed in
\cite{c1} in order to study a quasistatic unilateral contact
problem with local friction and adhesion. They obtained an existence result
under a smallness assumption of the coefficient of friction. In this work,
as in \cite{c4} by applying an implicit time-discretization scheme,
if the coefficient of friction is sufficiently small, we prove that the
time-discretized problem has a unique solution for which appropriate
estimations are established. We finally obtain the existence of a weak
solution by passing to the limit with respect to time. The paper is
structured as follows. In Section 2 we present some notations and give the
variational formulation. In Section 3 we study a time-discretized problem
which admits a unique solution if the coefficient of friction is
sufficiently small, Proposition \ref{prop3.1}. In Section 4 we prove
Theorem \ref{thm2.1}.


\section{Problem Statement and Variational Formulation}

Let $\Omega \subset \mathbb{R}^d;(d=2,3)$, be the domain
initially occupied by an elastic body. $\Omega $ is supposed to be open,
bounded, with a sufficiently regular boundary $\Gamma $. $\Gamma $ is
partitioned into three parts $\Gamma =\bar{\Gamma}_1\cup \bar{\Gamma}
_2\cup \bar{\Gamma}_3$ where $\Gamma _1,\Gamma _2,\Gamma _3$ are
disjoint open sets and $meas$ $\Gamma _1>0$. The body is acted upon by a
volume force of density $\varphi _1$ on $\Omega $ and a surface traction
of density $\varphi _2$ on $\Gamma _2$. On $\Gamma _3$ the body is in
adhesive frictional contact with a foundation.

 Thus, the classical formulation of the mechanical problem is
written as follows.

\subsection*{Problem $P_1$.} Find a displacement field
$u:\Omega \times [0,T]\to \mathbb{R}^d$ and
a bonding field $\beta :\Gamma _3\times [0,T]\to [0,1]$ such that
\begin{gather}
\mathop{\rm div}\sigma +\varphi _1=0\quad\text{in }\Omega \times (0,T),\label{e2.1} \\
\sigma =F\varepsilon (u)\ \quad\text{in }\Omega \times (0,T),  \label{e2.2} \\
u=0\quad\text{on } \Gamma _1\times (0,T),  \label{e2.3} \\
\sigma \nu =\varphi _2 \quad\text{on }\Gamma _2\times (0,T),  \label{e2.4}\\
-\sigma _{\nu }=p(u_{\nu })-c_{\nu }\beta ^2R_{\nu }(
u_{\nu })\quad\text{on }\Gamma _3\times (0,T),  \label{e2.5}
\\
\left\{
\begin{aligned}
&| \sigma _{\tau }+c_{\tau }\beta ^2R_{\tau }(u_{\tau
})|\leq \mu p(u_{\nu }), \\
&| \sigma _{\tau }+c_{\tau }\beta ^2R_{\tau }(u_{\tau
})|<\mu p(u_{\nu })\Longrightarrow \dot{u}
_{\tau }=0, \\
&| \sigma _{\tau }+c_{\tau }\beta ^2R_{\tau }(u_{\tau
})|=\mu p(u_{\nu })\Longrightarrow \\
&\exists \lambda \geq 0\text{ such that }\dot{u}_{\tau }=-\lambda (\sigma
_{\tau }+c_{\tau }\beta ^2R_{\tau }(u_{\tau })),
\end{aligned}
\right. \quad \text{on }\Gamma _3\times (0,T),  \label{e2.6}
\\
\dot{\beta}=-\big[\beta \big(c_{\nu }
\big(R_{\nu }(u_{\nu})\big)^2
+c_{\tau }\big(| R_{\tau }(u_{\tau})|\big)^2\big)
-\varepsilon _{a}\big]_{+}\quad \text{on }\Gamma _3\times (0,T),  \label{e2.7}
\\
u(0)=u_0\quad\text{in }\Omega ,  \label{e2.8}
\\
\beta (0)=\beta _0\quad\text{on }\Gamma _3.  \label{e2.9}
\end{gather}
Equation \eqref{e2.1} represents the equilibrium equation. Equation
\eqref{e2.2} represents the elastic constitutive law of the material
in which $\sigma $ denotes the stress tensor $F$ is a nonlinear elasticity
operator and $\varepsilon (u)$ denotes the strain tensor;
\eqref{e2.3} and \eqref{e2.4} are the displacement and
traction boundary conditions, respectively, in which $\nu $ denotes the unit
outward normal vector on $\Gamma $ and $\sigma \nu $ represents the stress
vector. Condition \eqref{e2.5} represents the normal compliance and
adhesion. Condition \eqref{e2.6} is the associated Coulomb's law of
dry friction. $\dot{u}_{\tau }$ is the tangential velocity on the boundary
$\Gamma _3$. Here $p$ is a given function, $\mu $ is the coefficient of
friction and the parameters $c_{\nu }$, $c_{\tau }$ and $\varepsilon _{a}$
are given adhesion coefficients which may depend on $x\in \Gamma _3$. As
in \cite{s4}, $R_{\nu }$, $R_{\tau }$ are truncation operators
defined by
\[
R_{\nu }(s)=\begin{cases}
L&\text{if }s<-L \\
-s&\text{if }-L\leq s\leq 0 \\
0&\text{if }s>0,
\end{cases}
\quad  R_{\tau }(v)=\begin{cases}
v&\text{if }| v|\leq L \\
L\frac{v}{\quad | v|} &\text{if }|v|>L,
\end{cases}
\]
where $L>0$ is a characteristic length of the bonds. Equation
\eqref{e2.7} represents the ordinary differential equation which describes
the evolution of the bonding field and it was already used in \cite{s4}
where $[s]_{+}=\max (s,0)$ for all $s\in \mathbb{R}$.
Since $\dot{\beta}\leq 0$ on $\Gamma _3\times (0,T)$, once
debonding occurs, bonding cannot be reestablished. Also we wish to make it
clear that from \cite{n1} it follows that the model does not allow
for complete debonding field in finite time. Finally, \eqref{e2.8}
and \eqref{e2.9} are the initial conditions, in which $u_0$ and
$\beta _0$ denotes respectively the initial displacement field and the
initial bonding field. In \eqref{e2.7} a dot above a variable
represents its derivative with respect to time. We recall that the inner
products and the corresponding norms on $\mathbb{R}^d$ and $S_{d}$ are
given by
\begin{gather*}
u.v=u_iv_i,\quad | v|=(v.v)^{1/2}\quad \forall u,v\in \mathbb{R}^d, \\
\sigma .\tau =\sigma _{ij}\tau _{ij}, \quad
| \tau |=(\tau .\tau )^{1/2}\quad \forall \sigma ,\tau \in S_{d},
\end{gather*}
where $S_{d}$ is the space of second order symmetric tensors on
$\mathbb{R}^d$ $(d=2,3)$. Here and below, the indices $i$ and $j$ run
between $1$ and $d$ and the summation convention over repeated indices is
adopted. Now, to proceed with the variational formulation, we need the
following function spaces:
\begin{gather*}
H=(L^2(\Omega ))^d, \quad
Q=\{ \tau =(\tau_{ij}):\tau _{ij}=\tau _{ji}\in L^2(\Omega )\} , \\
H_1=(H^{1}(\Omega ))^d,\quad
Q_1=\{ \sigma \in Q:\mathop{\rm div}\sigma \in H\} .
\end{gather*}
Note that $H$ and $Q$ are real Hilbert spaces endowed with the respective
canonical inner products
\[
\langle u,v\rangle _{H}=\int_{\Omega }u_iv_idx,\quad
(\sigma ,\tau )_{Q}=\int_{\Omega }\sigma _{ij}\tau _{ij}dx.
\]
The small strain tensor is
\[
\varepsilon (u)=(\varepsilon _{ij}(u))
=\frac{1}{2}(u_{i,j}+u_{j,i}), \quad i,j=\{1,\dots ,d\},
\]
where $\mathop{\rm div}\sigma =(\sigma _{ij,j})$ is the divergence of $\sigma $.
For every element $v\in H_1$ we denote by $v_{\nu }$ and $v_{\tau }$ the
normal and the tangential components of $v$ on the boundary $\Gamma $ given
by
\[
v_{\nu }=v.\nu , \quad v_{\tau }=v-v_{\nu }\nu .
\]
Similary, for a regular tensor field $\sigma \in Q_1$, we define its
normal and tangential components by
\[
\sigma _{\nu }=(\sigma \nu ).\nu ,\quad
\sigma _{\tau}=\sigma \nu -\sigma _{\nu }\nu
\]
and we recall that the following Green's formula holds:
\[
(\sigma ,\varepsilon (v))_{Q}+\langle
\mathop{\rm div}\sigma ,v\rangle _{H}=\int_{\Gamma }\sigma \nu .vda\quad
\forall v\in H_1,
\]
where $da$ is the surface measure element. Let $V$ be the closed subspace of
$H_1$ defined by
\[
V=\left\{ v\in H_1:v=0\text{ on }\Gamma _1\right\} .
\]
Since meas$\Gamma _1>0$, the following Korn's inequality holds,
\begin{equation}
\|\varepsilon (v)\|_{Q}\geq c_{\Omega
}\|v\|_{H_1}\quad \forall v\in V,  \label{e2.10}
\end{equation}
where the constant $c_{\Omega }>0$ depends only on $\Omega $ and
$\Gamma_1$; see \cite{d1}. We equip $V$ with the inner product
\[
(u,v)_V =(\varepsilon (u),\varepsilon (v))_{Q}
\]
and $\|.\|_V $ is the associated norm. It follows from
Korn's inequality \eqref{e2.10} that the norms $\|
.\|_{H_1}$ and $\|.\|_V $ are equivalent on
$V$. Then $(V,\|.\|_V )$ is a real Hilbert
space. Moreover by Sobolev's trace theorem, there exists $d_{\Omega }>0$
which depends only on the domain $\Omega $, $\Gamma _1$ and $\Gamma _3$
such that
\begin{equation}
\|v\|_{(L^2(\Gamma _3))^d}\leq d_{\Omega }\|v\|_V \quad \forall v\in V.
\label{e2.11}
\end{equation}
 For $p\in [1,\infty ]$ , we use the standard norm of
$L^{p}(0,T;V)$. We also use the Sobolev space $W^{1,\infty
}(0,T;V)$ equipped with the norm
\[
\|v\|_{W^{1,\infty }(0,T;V)}=\|v\|_{L^{\infty }(0,T;V)}+\|\dot{v}
\|_{L^{\infty }(0,T;V)}.
\]
For every real Banach space $(X,\|.\|_{X})$
and $T>0$ we use the notation $C([0,T];X)$ for the
space of continuous functions from $[0,T]$ to $X$; recall that
$C([0,T];X)$ is a real Banach space with the norm
\[
\|x\|_{C([0,T];X)}=\max_{t\in [0,T]} \|x(t)\|_{X}.
\]
We suppose that the body forces and surface tractions have the regularity
\begin{equation}
\varphi _1\in W^{1,\infty }(0,T;H),\quad
\varphi_2\in W^{1,\infty }(0,T;(L^2(\Gamma _2))^d) \label{e2.12}
\end{equation}
and we denote by $f(t)$ the element of $V$ defined by
\begin{equation}
(f(t),v)_V =\int_{\Omega }\varphi _1(
t).vdx+\int_{\Gamma _2}\varphi _2(t).vda\quad
\forall v\in V,\quad \text{for }t\in [0,T].  \label{e2.13}
\end{equation}
Using \eqref{e2.12} and \eqref{e2.13}, we obtain
$f\in W^{1,\infty }(0,T;V)$.

 In the study of the mechanical problem $P_1$ we assume that
$F:\Omega \times S_{d}$ $\to S_{d}$, satisfies the following four
conditions:
 \begin{itemize}
\item[(a)] there exists $M>0$ such that
$$
| F(x,\varepsilon _1)-F(x,\varepsilon _2)|\leq M| \varepsilon
_1-\varepsilon _2|
$$
for all  $\varepsilon_1,\varepsilon _2$ in $S_{d}$,
a.e. $x$ in $\Omega$;

\item[(b)] there exists $m>0$ such that
\begin{equation}
(F(x,\varepsilon _1)-F(x,\varepsilon _2)).(\varepsilon _1-\varepsilon
_2)\geq m| \varepsilon _1-\varepsilon _2|^2, \label{e2.14}
\end{equation}
for all $\varepsilon _1,\varepsilon _2\text{ in }S_{d}$,
a.e. $x$ in $\Omega$;

\item[(c)] the mapping $x\to F(x,\varepsilon )$
is Lebesgue measurable on $\Omega$ \\
 for any $\varepsilon$ in $S_{d}$;

\item[(d)] $F(x,0)=0$  for a.e. $x$ in $\Omega$.
\end{itemize}

Also we assume that the normal compliance function $p$ satisfies
the following five conditions:
\begin{itemize}
\item[(a)] $p:\Gamma _3\times \mathbb{R}\to \mathbb{R}_{+}$;

\item[(b)] there exists $L_{p}>0$ such that
$| p(x,r_1)-p(x,r_2)|\leq L_{p}| r_1-r_2|$ \\ for all
$r_1,r_2\in \mathbb{R}$, a.e. $x$ in $\Gamma _3$;

\item[(c)]
\begin{equation}
(p(x,r_1)-p(x,r_2))(r_1-r_2)\geq 0\quad\text{for all }
r_1,r_2\in \mathbb{R}, \text{ a.e. $x$ in }\Gamma _3; \label{e2.15}
\end{equation}

\item[(d)]  the mapping $x\to p(x,r)$
is measurable on $\Gamma _3$ for any $r\in \mathbb{R}$;

\item[(e)] $p(x,r)=0$  for all $r\leq 0$, a.e. $x\in \Gamma _3$.

\end{itemize}
We define the functional $j:V\times V\to \mathbb{R}$ by
\[
j(u,v)=\int_{\Gamma _3}(p(u_{\nu })v_{\nu
}+\mu p(u_{\nu })| v_{\tau }|)da \quad \forall (u,v)\in V\times V.
\]
 As in \cite{s3} we suppose that the adhesion coefficients $c_{\nu }$,
$c_{\tau }$ and $\varepsilon _{a}$ satisfy the conditions
\begin{equation}
c_{\nu }, c_{\tau }\in L^{\infty }(\Gamma _3), \quad
\varepsilon_{a}\in L^{\infty }(\Gamma _3), \quad
c_{\nu }, c_{\tau}, \varepsilon _{a}\geq 0\quad \text{a.e. on } \Gamma _3.
\label{e2.16}
\end{equation}
We suppose that $\mu $ satisfies
\begin{equation}
\mu \in L^{\infty }(\Gamma _3),\quad\text{and}\quad
 \mu \geq 0\quad\text{a.e. on }\Gamma _3.  \label{e2.17}
\end{equation}
We need the following set for the bonding fields,
\[
B=\big\{ \beta \in L^{\infty }(0,T;L^2(\Gamma _3)
);\; 0\leq \beta (t)\leq 1\; \forall t\in
[0,T],\quad\text{ a.e. on }\Gamma _3\big\} ,
\]
and finally we assume that the initial displacement field $u_0$ belongs to
$V$ and satisfies
\begin{equation}
(F\varepsilon (u_0),\varepsilon (v-u_0)
)_{Q}+j(u_0,v-u_0)+r(\beta
_0,u_0,v-u_0)\geq (f(0),v-u_0)_V 
 \label{e2.18}
\end{equation}
for all $v\in V$,
where the initial bonding field $\beta _0$ satisfies
\begin{equation}
\beta _0\in L^2(\Gamma _3),\quad 0\leq \beta _0\leq 1
\quad\text{ a.e. on }\Gamma _3.  \label{e2.19}
\end{equation}
 As in \cite{c4}, using Green's formula, we obtain the
following variational formulation to Problem $P_1$.

\subsection*{Problem $P_2$}
 Find a displacement field
$u\in W^{1,\infty }(0,T;V)$ and a bonding field
$\beta \in W^{1,\infty }(0,T;L^2(\Gamma _3))\cap B$ such
that $u(0)=u_0$, $\beta (0)=\beta _0$ and for
almost all $t\in [0,T]$:
\begin{equation}
\begin{aligned}
&\Big(F\varepsilon (u(t)),\varepsilon (v-\dot{u} (t))\Big)_{Q}
+j(u(t),v)-j(u(t),\dot{u}(t))+r(\beta (
t),u(t),v-\dot{u}(t))\\
&\geq (f(t),v-\dot{u}(t))_V \quad \forall v\in V,
\end{aligned}
\label{e2.20}
\end{equation}
\begin{equation}
\dot{\beta}(t)=-[\beta (t)(c_{\nu
}(R_{\nu }(u_{\nu }(t)))^2+c_{\tau
}(| R_{\tau }(u_{\tau }(t))
|)^2)-\varepsilon _{a}]_{+}\quad \text{a.e. on }\Gamma _3,  \label{e2.21}
\end{equation}
where
\begin{gather}
r=r_{\nu }+r_{\tau }, \quad
r_{\nu }(\beta ,u,v)=-\int_{\Gamma _3}c_{\nu }\beta
^2R_{\nu }(u_{\nu })v_{\nu }da,\\
 r_{\tau }(\beta ,u,v)=\int_{\Gamma _3}c_{\tau }\beta ^2R_{\tau }(u_{\tau
}).v_{\tau }da.
\end{gather}
 Our main result of this section, which will be established in the
next is the following theorem.

\begin{theorem} \label{thm2.1}
Let $T>0$  and assume \
\eqref{e2.12}, \eqref{e2.14}, \eqref{e2.15}, \eqref{e2.16}, \eqref{e2.17},
\eqref{e2.18}, and \eqref{e2.19}.
Then there exists a constant $\mu _{\ast}>0$ such that for
$\|\mu \|_{L^{\infty }(\Gamma _3)}<\mu_{\ast }$,
Problem $P_2$ has at least one solution.
\end{theorem}

\section{A time-discretization}

As in \cite{c4}, we adopt the following time-discretization. For all
$n\in \mathbf{N}^{\ast }$, we set $t_i=i\Delta t$, $0\leq i\leq n$, and
$\Delta t=T/n$. We denote respectively by $u^{i}$ and $\beta^i$ 
the approximation  of $u$ adn $\beta$ at time $t_i$ and  
$\Delta u^{i}=u^{i+1}-u^i$,
$\Delta \beta^{i}=\beta^{i+1}-\beta^i$.
For a continuous function $w(t)$, we use the
notation $w^{i}=w(t_i)$. Then we obtain a sequence of
time-discretized problems $P_{n}^{i}$ defined for $u^{0}=u_0$
and $\beta^)=\beta_0$ by:

\subsection*{Problem $P_{n}^{i}$}
 For $u^{i}\in V$, $\beta ^{i}\in L^{\infty }(\Gamma _3)$, find
$u^{i+1}\in V$, $\beta ^{i+1}\in L^{\infty }(\Gamma _3)$ such that
\begin{equation}
\begin{aligned}
&(F\varepsilon (u^{i+1}),\varepsilon (w-u^{i+1})
)_{Q}+j(u^{i+1},w-u^{i})\\
&-j(u^{i+1},\Delta u^{i})
+r(\beta ^{i+1},u^{i+1},w-u^{i+1})
\\
&\geq (f^{i+1},w-u^{i+1})_V \quad \forall w\in V,
\\
&\frac{\beta ^{i+1}-\beta ^{i}}{\Delta t}=-[\beta ^{i+1}(c_{\nu
}(R_{\nu }(u_{\nu }^{i+1}))^2+c_{\tau }(
| R_{\tau }(u_{\tau }^{i+1})|)
^2)-\varepsilon _{a}]_{+}\quad \text{a.e. on }\Gamma _3.
\end{aligned}
\label{e3.1}
\end{equation}
We have the following result.

\begin{proposition} \label{prop3.1}
There exists $\mu _{c}>0$ such that for
$\|\mu \|_{L^{\infty }(\Gamma _3)}<\mu _{c}$,  Problem
$P_{n}^{i}$ has a unique solution.
\end{proposition}

 To show this proposition we introduce an intermediate problem. For
$\eta \in V$, we introduce the following problem

 \subsection*{Problem $P_{\eta n}^{i}$}
 For $u^{i}\in V$, $\beta ^{i}\in L^{\infty }(\Gamma _3)$, find
$u_{\eta }\in V$, $\beta _{\eta }\in L^{\infty }(\Gamma _3)$ such
that
\begin{equation}
\begin{aligned}
&(F\varepsilon (u_{\eta }),\varepsilon (w-u_{\eta })
)_{Q}+j(\eta ,w-u^{i})-j(\eta ,u_{\eta }-u^{i})
+r(\beta _{\eta },u_{\eta },v-u_{\eta })\\
&\geq  (f^{i+1},w-u_{\eta })_V \quad \forall w\in V,
\\
&\frac{\beta _{\eta }-\beta ^{i}}{\Delta t}
=-[\beta _{\eta }(c_{\nu }(R_{\nu }(u_{\eta \nu }))
^2+c_{\tau }(| R_{\tau }(u_{\eta \tau })
|)^2)-\varepsilon _{a}]_{+}\quad \text{a.e. on }\Gamma _3.
\end{aligned}
 \label{e3.2}
\end{equation}
As in \cite{c4} we have the following lemma.

\begin{lemma} \label{lem3.2}
 For any $\eta \in V$,
Problem $P_{\eta n}^{i}$ has a unique solution
$(u_{\eta},\beta _{\eta })$, if $\Delta t$ is small enough.
\end{lemma}

 To prove this lemma we introduce the following auxiliary problem.

 \subsection*{Problem $P_{1\beta }$}
 For $u^{i}\in V$, $\beta \in L^{\infty }(\Gamma _3)$, find
$u_{\beta }\in V$ such that
\begin{equation}
\begin{aligned}
&(F\varepsilon (u_{\beta }),\varepsilon (v-u_{\beta
}))_{Q}+j(\eta ,v-u^{i})-j(\eta ,u_{\beta
}-u^{i})+r(\beta ,u_{\beta },v-u_{\beta })\\
&\geq (f^{i+1},v-u_{\beta })_V \quad \forall \; v\in V.
\end{aligned} \label{e3.3}
\end{equation}
We have the following lemma.

\begin{lemma} \label{lem3.3}
Problem $P_{1\beta }$ has a
unique solution.
\end{lemma}

\begin{proof}
 Let $A:V\to V$ be the operator given by
\[
(Au,v)_V 
=(F\varepsilon (u),\varepsilon (v))_{Q}
+\int_{\Gamma _3}(-c_{\nu }\beta ^2R_{\nu }(u_{\nu
})v_{\nu }+c_{\tau }\beta ^2R_{\tau }(u_{\tau }).v_{\tau })da.
\]
Using \eqref{e2.14}(a), \eqref{e2.11}, \eqref{e2.16},
the properties  of the operators $R_{\nu }$ and
$R_{\tau }$ (see \cite{s2}) such that
\begin{equation}
| R_{\nu }(a)-R_{\nu }(b)|\leq
| a-b|, \forall a,b\in \mathbb{R}\text{; }
| R_{\tau }(a)-R_{\tau }(b)|
\leq | a-b|, \forall a,b\in \mathbb{R}^d,
\label{e3.4}
\end{equation}
it follows that $A$ satisfies
\[
| (Au-Av,w)_V |\leq [M+(
\|c_{\nu }\|_{L^{\infty }(\Gamma _3)
}+\|c_{\tau }\|_{L^{\infty }(\Gamma _3)
})d_{\Omega }^2]\|u-v\|_V \| w\|_V .
\]
Also, we use \eqref{e2.14}(b) to see that
\begin{align*}
(Au-Av,u-v)_V 
&\geq m\|u-v\| _V ^2-\int_{\Gamma _3}\beta ^2c_{\nu }(R_{\nu }(u_{\nu
})-R_{\nu }(v_{\nu }))(u_{\nu }-v_{\nu
})da \\
&\quad +\int_{\Gamma _3}\beta ^2c_{\tau }(R_{\tau }(u_{\tau
})-R_{\tau }(v_{\tau })).(u_{\tau }-v_{\tau })da.
\end{align*}
As
\begin{equation}
\begin{gathered}
(R_{\nu }(u_{\nu })-R_{\nu }(v_{\nu })
)(u_{\nu }-v_{\nu })\leq 0\quad \text{a.e.  on }\Gamma_3, \\
(R_{\tau }(u_{\tau })-R_{\tau }(v_{\tau })
).(u_{\tau }-v_{\tau })\geq 0\quad \text{a.e.  on }\Gamma _3,
\end{gathered} \label{e3.5}
\end{equation}
we get
\[
(Au-Av,u-v)_V \geq m\|u-v\|_V ^2,
\]
which implies that $A$ is strongly monotone. Therefore, $A$ is an operator
strongly monotone and Lipschitz continuous. On the other hand the functional
$j_{\eta }:V\to \mathbb{R}$ defined by
\[
j_{\eta }(v)= j(\eta, v-u^i)\quad \forall v\in V,
\]
is convex, proper and lowly semicontinuous, then by a classical argument of
elliptic variational inequalities \cite{b1}, we deduce that the
problem $P_{1\beta }$ has a unique solution $u_{\beta }$.
\end{proof}

 We also consider the following problem.

 \subsection*{Problem $P_{2\beta }$}
 For $\beta ^{i}\in L^{\infty}(\Gamma _3)$, $u\in V$, find
$\beta \in L^{\infty }(\Gamma _3)$ such that
\[
\frac{\beta -\beta ^{i}}{\Delta t}
=-\big[\beta (c_{\nu }(R_{\nu }(u_{\beta \nu }))^2+c_{\tau }(
| R_{\tau }(u_{\beta \tau })|)
^2)-\varepsilon _{a}\big]_{+}\text{a.e. on }\Gamma _3.
\]
Obviously, Problem $P_{2\beta }$ has a unique solution which is given by
\[
\beta =\begin{cases}
\beta ^{i},\quad \text{if }(c_{\nu }(R_{\nu }(u_{\beta \nu
}))^2+c_{\tau }(| R_{\tau }(
u_{\beta \tau })|)^2)\beta^i -\varepsilon_{a}<0, \\[4pt]
\dfrac{\beta ^{i}+\varepsilon _{a}\Delta t}{1+\Delta t(c_{\nu }(
R_{\nu }(u_{\beta \nu }))^2+c_{\tau }(
| R_{\tau }(u_{\beta \tau })|)
^2)}, \\
\text{if }(c_{\nu }(R_{\nu }(u_{\beta \nu })
)^2+c_{\tau }(| R_{\tau }(u_{\beta \tau
})|)^2)\beta^i-\varepsilon _{a}>0,
\end{cases}
\]
and it satisfies $\beta \in [0,1]$. 
To complete the proof of Lemma \ref{lem3.2}, let $v\in V$
and  $\beta(v)$ be the corresponding solution of Problem
$P_{2\beta}$.
Let $u_{\beta(v)}$ be the corresponding solution of Problem
$P_{1\beta}$, and define the mapping  $\Psi:V\to V$ as
\[
v \to \Psi (v)=u_{\beta(v)}\,.
\]
Take $v=u_i$, $i=1,2$. As in \cite[Lemma 2.3]{c4}, there exists a
positive constant $C$ such that
\[
\|\Psi(u_2)-\Psi(u_1)\|_V \leq C \Delta t \|u_2-u_1\|,\quad
\forall u_1, u_2 \in V\,.
\]
Then we conclude by a contraction argument that for $\Delta t$
sufficiently small, Problem $P_{\eta n}^i$ has a unique solution
$(u_\eta,\beta_\eta)$.
Next, we shall establish
the proof of Proposition \ref{prop3.1}. Indeed, write the inequality
\eqref{e3.2} for $\eta=\eta_i$ and take $v=u_{\eta _{j}}$, $i,j=1,2$. Adding the
two inequalities we have
\begin{align*}
&(F\varepsilon (u_{\eta _1})-F\varepsilon (u_{\eta
_2}),\varepsilon (u_{\eta _1}-u_{\eta _2}))_{Q}\\
&\leq r(\beta _{\eta _1},u_{\eta _1},u_{\eta _2}-u_{\eta
_1})+r(\beta _{\eta _2},u_{\eta _2},u_{\eta _1}-u_{\eta
_2})+j(\eta _1,u_{\eta _2}-u^{i})\\
&\quad -j(\eta _1,u_{\eta
_1}-u^{i})+j(\eta _2,u_{\eta _1}-u^{i})-j(
\eta _2,u_{\eta _2}-u^{i}).
\end{align*}
Then
\begin{align*}
&r(\beta _{g_1},u_{g_1},u_{g_2}-u_{g_1})+r(\beta
_{g_2},u_{g_2},u_{g_1}-u_{g_2})\\
&= \int_{\Gamma _3}[c_{\tau }(\beta _{\eta _1}-\beta _{_{\eta
_2}})(\beta _{\eta _1}+\beta _{_{\eta _2}})R(
u_{\eta _{1\tau }})(u_{\eta _{2\tau }}-u_{\eta _{1\tau }})]da\\
&\quad  -\int_{\Gamma _3}[c_{\nu }(\beta _{\eta
_1}-\beta _{_{\eta _2}})(\beta _{\eta _1}+\beta _{_{\eta
_2}})R(u_{\eta _{1\nu }})(u_{\eta _{1\nu
}}-u_{\eta _{2\nu }})]da \\
&\quad +\int_{\Gamma _3}[c_{\nu }\beta _{_{\eta _2}}^2(R(
u_{\eta _{1\nu }})-R(u_{\eta _{2\nu }}))(
u_{\eta _{1\nu }}-u_{\eta _{2\nu }})]da\\
&\quad+ \int_{\Gamma _3}[c_{\tau }\beta _{_{\eta _2}}^2(R(
u_{\eta _{_{1\tau }}})-R(u_{\eta _{2\tau }}))
(u_{\eta _{2\tau }}-u_{\eta _{1\tau }})]da.
\end{align*}
Using the properties \eqref{e3.5}, we deduce
\begin{align*}
&(F\varepsilon (u_{\eta _1})-F\varepsilon (u_{\eta
_2}),\varepsilon (u_{\eta _1}-u_{\eta _2}))
_{Q}\\
&\leq \int_{\Gamma _3}[c_{\tau }(\beta _{\eta _1}-\beta _{\eta
_2})(\beta _{\eta _1}+\beta _{\eta _2})R(
u_{\eta _{1\tau }})(u_{\eta _{2\tau }}-u_{\eta _{1\tau
}})]da
\\
&\quad -\int_{\Gamma _3}[c_{\nu }(\beta _{\eta _1}-\beta _{\eta
_2})(\beta _{u_{\eta _21}}+\beta _{\eta _2})
R(u_{\eta _{1\nu }})(u_{\eta _{1\nu }}-u_{\eta _{2\nu
}})]da \\
&\quad +j(\eta _1,u_{\eta _2}-u^{i})-j(\eta _1,u_{\eta
_1}-u^{i})+j(\eta _2,u_{\eta _1}-u^{i})-j(
\eta _2,u_{\eta _2}-u^{i}).
\end{align*}
Now, using \eqref{e2.11}, \eqref{e2.14}(b),
\eqref{e2.15}(b), \eqref{e2.15}(c), the properties \eqref{e3.4},
$| R_{\nu }(u_{\nu })|\leq L$, and
$| R_{\tau }(u_{\tau })|\leq L$, it follows
that
\begin{equation}
\begin{aligned}
m\|u_{\eta _1}-u_{\eta _2}\|_V 
&\leq Ld_{\Omega }(\|c_{\nu }\|_{L^{\infty }(\Gamma
_3)}+\|c_{\tau }\|_{L^{\infty }(\Gamma
_3)})\|\beta _{\eta _1}-\beta _{\eta
_2}\|_{L^2(\Gamma _3)} \\
&\quad +L_{p}d_{\Omega}^2\|\mu \|_{L^{\infty }(\Gamma _3)
}\|\eta _1-\eta _2\|_V .
\end{aligned}
\label{e3.6}
\end{equation}
On the other hand using that for $a,b\in \mathbb{R}$,
$|a_{+}-b_{+}|\leq | a-b|$, we deduce from the
equality to relation \eqref{e3.2} that
\begin{align*}
\|\frac{\beta _{\eta _1}-\beta _{\eta _2}}{\Delta t}
\|_{L^2(\Gamma _3)}
&\leq \|(\beta _{\eta _1}-\beta _{\eta _2})\Big(
c_{\nu }(R_{\nu }(u_{\beta _{\eta _1}\nu })
)^2+c_{\tau }(| R_{\tau }(u_{\beta _{\eta
_1}\tau })|)^2\Big)\|
_{L^2(\Gamma _3)} \\
&\quad + \|\beta _{\eta _2}\Big[\Big(c_{\nu }(R_{\nu }(
u_{\beta _{\eta _1}\nu }))^2+c_{\tau }(
| R_{\tau }(u_{\beta _{\eta _1}\tau })
|)^2\Big)\\
&\quad -\Big(c_{\nu }(R_{\nu }(
u_{\beta _{\eta _2}\nu }))^2+c_{\tau }(
| R_{\tau }(u_{\beta _{\eta _2}\tau })
|)^2\Big)\Big]\|_{L^2(\Gamma_3)}.
\end{align*}
The above inequality implies
\begin{align*}
\|\frac{\beta _{\eta _1}-\beta _{\eta _2}}{\Delta t}
\|_{L^2(\Gamma _3)}
&\leq L^2(\|c_{\nu }\|_{L^{\infty }(\Gamma
_3)}+\|c_{\tau }\|_{L^{\infty }(\Gamma
_3)})\|\beta _{\eta _1}-\beta _{\eta
_2}\|_{L^2(\Gamma _3)}\\
&\quad +2L(\|
c_{\nu }\|_{L^{\infty }(\Gamma _3)}+\|
c_{\tau }\|_{L^{\infty }(\Gamma _3)})
d_{\Omega }\|u_{\eta _1}-u_{\eta _2}\|_V .
\end{align*}
Therefore,
\begin{align*}
&[1-\Delta tL^2(\|c_{\nu }\|_{L^{\infty
}(\Gamma _3)}+\|c_{\tau }\|_{L^{\infty
}(\Gamma _3)})]\|\beta _{\eta
_1}-\beta _{\eta _2}\|_{L^2(\Gamma _3)}\\
&\leq 2L(\|c_{\nu }\|_{L^{\infty }(\Gamma
_3)}+\|c_{\tau }\|_{L^{\infty }(\Gamma
_3)})d_{\Omega }\Delta t\|u_{\eta _1}-u_{\eta
_2}\|_V \,.
\end{align*}
If
\[
\Delta t<\frac{1}{L^2(\|c_{\nu }\|_{L^{\infty
}(\Gamma _3)}+\|c_{\tau }\|_{L^{\infty
}(\Gamma _3)})}\,,
\]
there exists a constant $C_1>0$ such that
\[
\|\beta _{\eta _1}-\beta _{\eta _2}\|_{L^2(
\Gamma _3)}\leq C_1\Delta t\|u_{\eta _1}-u_{\eta
_2}\|_V \,.
\]
Then from \eqref{e3.6} we get
\begin{align*}
&[m-C_1Ld_{\Omega }(\|c_{\nu }\|_{L^{\infty
}(\Gamma _3)}+\|c_{\tau }\|_{L^{\infty
}(\Gamma _3)})\Delta t]\|u_{\eta
_1}-u_{\eta _2}\|_V  \\
&\leq L_{p}d_{\Omega}^2\|\mu \|_{L^{\infty }(\Gamma _3)
}\|\eta _1-\eta _2\|_V ,
\end{align*}
and thus for
\[ 
\Delta t<\min \Big(\frac{m}{C_1Ld_{\Omega }(\|c_{\nu
}\|_{L^{\infty }(\Gamma _3)}+\|c_{\tau
}\|_{L^{\infty }(\Gamma _3)})}\,,\;
\frac{1}{L^2(\|c_{\nu }\|_{L^{\infty }(\Gamma
_3)}+\|c_{\tau }\|_{L^{\infty }(\Gamma _3)})}\Big),
\] 
there exists a constant $C_2>0$ such that
\begin{equation}
\|u_{\eta _1}-u_{\eta _2}\|_V \leq
C_2L_{p}d_{\Omega }^2\|\mu \|_{L^{\infty }(
\Gamma _3)}\|\eta _1-\eta _2\|_V \,.
\label{e3.7}
\end{equation}
To complete the proof let us define the mapping $\Phi :V\to V$ as
$ \Phi (\eta )=u_{\eta }$.
Then from \eqref{e3.7} it follows
\[
\|\Phi (\eta _1)-\Phi (\eta _2)
\|_V \leq C_2L_{p}d_{\Omega }^2\|\mu \|
_{L^{\infty }(\Gamma _3)}\|\eta _1-\eta_2\|_V \,,\quad
\forall \eta_1, \eta_2 \in V\,.
\]
Then when $\mu _{c}=\frac{1}{L_{p}d_{\Omega }^2C_2}$,
 the mapping $\Phi $ is a contraction for
$\|\mu \| _{L^{\infty }(\Gamma _3)}<\mu _{c}$, thus it admits a unique
fixed point $\eta _{c}$ and $(u_{\eta _{c}},\beta _{\eta _{c}})$
is a unique solution to Problem $P_{n}^{i}$.

 Now, to prove Theorem \ref{thm2.1} it is necessary to establish the
following estimates.

\begin{lemma} \label{lem3.7}
There exist two constants $C_3>0$,
$C_{4}>0$  such that
\begin{equation}
\|u^{i+1}\|_V \leq C_3\|f^{i+1}\|_V ,\quad
\|\Delta u^{i}\|_V \leq C_{4}(\|\Delta f^{i}\|_V +\Delta t).
\label{e3.8}
\end{equation}
\end{lemma}

\begin{proof}
We take $v=0$ in \eqref{e3.1} to deduce
\begin{align*}
&\Big(F\varepsilon (u^{i+1}),\varepsilon (u^{i+1})
\Big)_{Q}\\
&\leq j(u^{i+1},-u^{i})-j(u^{i+1},\Delta u^{i})
+r(\beta ^{i+1},u^{i+1},-u^{i+1})+(f^{i+1},u^{i+1})_V .
\end{align*}
Using the properties of $j$ we have
\[
j(u^{i+1},-u^{i})-j(u^{i+1},\Delta u^{i})\leq
d_{\Omega }^2L_{p}\|\mu \|_{L^{\infty }(\Gamma
_3)}\|u^{i+1}\|_V ^2.
\]
On the other hand using $| R(u_{\nu })|\leq L$,
$| R(u_{\tau })|\leq L$, and the
relation \eqref{e2.11}, we have
\[
| r(\beta ^{i+1},u^{i+1},-u^{i+1})|\leq
d_{\Omega }L\sqrt{\mathop{\rm meas} \Gamma _3}\big(\|c_{\nu
}\|_{L^{\infty }(\Gamma _3)}+\|c_{\tau
}\|_{L^{\infty }(\Gamma _3)}\big)\|
u^{i+1}\|_V 
\]
Using \eqref{e2.14}(b), we get
\begin{align*}
m\|u^{i+1}\|_V ^2
&\leq d_{\Omega }^2L_{p}\|\mu \|_{L^{\infty }(\Gamma _3)}\|
u^{i+1}\|_V ^2.+\|f^{i+1}\|_V \|u^{i+1}\|_V \\
&\quad + d_{\Omega }L\sqrt{\mathop{\rm meas}
\Gamma _3}(\|c_{\nu }\|_{L^{\infty }(\Gamma _3)}+\|c_{\tau
}\|_{L^{\infty }(\Gamma _3)})\|u^{i+1}\|_V .
\end{align*}
Therefore, if we take
\[
\mu _{\ast }=\min (\mu _{c},\frac{m}{L_{p}d_{\Omega }^2}),
\]
we deduce that for
\begin{equation}
\|\mu \|_{L^{\infty }(\Gamma _3)}<\mu_{\ast },  \label{e3.9}
\end{equation}
there exists a constant $C_3>0$ such that the first inequality holds. To
show the second inequality \eqref{e3.8} we consider the translated
inequality to relation \eqref{e3.1} at time $t_i$; that is,
\begin{equation}
\begin{aligned}
&(F\varepsilon (u^{i}),\varepsilon (w-u^{i})
)_{Q}+j(u^{i},w-u^{i-1})
-j(u^{i},u^{i}-u^{i-1}) +r(\beta ^{i},u^{i},w-u^{i})\\
&\geq (f^{i},w-u^{i})_V \quad \forall  w\in V.
\end{aligned}
\label{e3.10}
\end{equation}
Taking $w=u^{i}$ in the inequality to relation \eqref{e3.1} and
$w=u^{i+1}$ in the inequality \eqref{e3.10} and adding the two
inequalities, we obtain the inequality
\begin{align*}
&-(F\varepsilon (u^{i+1})-F\varepsilon (u^{i})
,\varepsilon (\Delta u^{i}))_{Q}-j(u^{i+1},\Delta
u^{i})+j(u^{i},u^{i+1}-u^{i-1})\\
&-j(u^{i},u^{i}-u^{i-1})+ r(\beta ^{i+1},u^{i+1},u^{i}-u^{i+1})+r(\beta
^{i},u^{i},u^{i+1}-u^{i})\\
&\geq (-\Delta f^{i},\Delta u^{i})_V .
\end{align*}
Then using the inequality
\[
\big| | u_{\tau }^{t_{i+1}}-u_{\tau }^{t_{i-1}}|
-| u_{\tau }^{t_i}-u_{\tau }^{t_{i-1}}|\big|
\leq | u_{\tau }^{t_{i+1}}-u_{\tau }^{t_i}|,
\]
we have
\[
j(u^{i},u^{i+1}-u^{i-1})-j(u^{i},u^{i}-u^{i-1})
\leq \ j(u^{i},\Delta u^{i}).
\]
Therefore,
\begin{equation}
\begin{aligned}
(F\varepsilon (u^{i+1})-F\varepsilon (u^{i}),
 \varepsilon (\Delta u^{i}))_{Q}
&\leq j(u^{i},\Delta
u^{i})-j(u^{i+1},\Delta u^{i})+r(\beta ^{i+1},u^{i+1},-\Delta u^{i})\\
&\quad +r(\beta ^{i},u^{i},\Delta u^{i})+(\Delta f^{i},\Delta
u^{i})_V .
\end{aligned}
\label{e3.11}
\end{equation}
Using \eqref{e2.11}, \eqref{e2.15}(b) and \eqref{e2.15}(c), it
follows that
\[
j(u^{i+1},\Delta u^{i})-j(u^{i},\Delta u^{i})\leq
\|\mu \|_{L^{\infty }(\Gamma _3)
}L_{p}d_{\Omega }^2\|\Delta u^{i}\|_V ^2.
\]
Moreover, using \eqref{e2.11},
$| R_{\nu }(u^{j})|\leq L$,
$| R_{\tau }(u^{j})|\leq L$, $j=i,i+1$, and \eqref{e3.5}, we have
\begin{align*}
&r(\beta ^{i+1},u^{i+1},-\Delta u^{i})+r(\beta
^{i},u^{i},\Delta u^{i})\\
&\leq Ld_{\Omega }(\|c_{\nu }\|_{L^{\infty }(\Gamma
_3)}+\|c_{\tau }\|_{L^{\infty }(\Gamma
_3)})\|\Delta u^{i}\|_V \|
\Delta \beta ^{i}\|_{L^2(\Gamma _3)}.
\end{align*}
On the other hand,
\[
\|\Delta \beta ^{i}\|_{L^2(\Gamma _3)}\leq \Delta td_1,
\]
where $d_1>0$. Combining the previous relations, we obtain from inequality
\eqref{e3.11} that for the same condition \eqref{e3.9},
there exists a constant $C_{4}>0$ such that
\[
\|\Delta u^{i}\|_V \leq C_{4}(\|\Delta f^{i}\|_V +\Delta t).
\]
\end{proof}


\section{Existence}

In this section we prove our main result, Theorem \ref{thm2.1}.
 We consider the sequences of functions $(u^n)$, $(\beta ^n)$
defined on $[0,T]$ by
\[
u^n(t)=u^{i}+\frac{(t-t_i)}{\Delta t}\Delta u^{i}, \quad
\beta ^n(t)=\beta ^{i}+\frac{(t-t_i)}{\Delta t}\Delta \beta ^{i}
\]
 for $t\in [t_i,t_{i+1}]$, $i=0,\dots ,n-1$.
As in \cite[Proposition 4.2]{t1} we have the following lemma.

\begin{lemma} \label{lem4.1}
There exists $u\in W^{1,\infty}(0,T;V)$ and a subsequence
$(u^n)$, still denoted $(u^n)$, such that
\[
u^n\to u\quad \text{weak $\ast$ in }W^{1,\infty }(0,T;V).
\]
\end{lemma}

\begin{proof}
 From \eqref{e3.5} it follows that there exists a constant
$C_{5}>0$ such that
\[
\|u^n\|_{W^{1,\infty }(0,T;V)}\leq
C_{5}(\|f\|_{W^{1,\infty }(0,T;V)}+1)
\]
Consequently the sequence $(u^n)$ is bounded in
$W^{1,\infty }(0,T;V)$. Therefore, there exists a function $u$ $\in
W^{1,\infty }(0,T;V)$ and a subsequence, still denoted by $
(u^n)$, such that
\[
u^n\to u\quad \text{weak $\ast$ in }W^{1,\infty }(0,T;V)
\quad \text{as }n\to \infty .
\]
\end{proof}

 \begin{remark} \label{rmk4.2}\rm
 As $W^{1,\infty }(0,T;V) \hookrightarrow C([0,T];V)$ we have
$u^n(t)\to u(t)$ weakly in $V$ for all $t\in [0,T]$.
\end{remark}

 Now let us introduce the sequences of functions
$(\widetilde{u}^n)$, $(\tilde{f}^n)$,
 $(\tilde{\beta}^n)$ defined on $[0,T]$ by
\[
\widetilde{u}^n(t)=u^{i+1}, \quad
\widetilde{f}^n(t)=f(t_{i+1}),\quad
\tilde{\beta}^n(t)=\beta ^{i+1}
\]
for $t\in (t_i,t_{i+1}]$, $i=0,\dots ,n-1$
and $\widetilde{u}^n(0)=u_0$,
$\widetilde{f}^n(0)=f(0)$,
$\tilde{\beta}^n(0)=\beta _0$.
As in \cite{t1} we have the following result.

\begin{lemma} \label{lem4.3}
Passing to a subsequence again
denoted $(\tilde{u}^n)$ we have
\begin{itemize}
\item[(i)] $\widetilde{u}^n\to u$ weak $\ast$ in
$L^{\infty }(0,T;V)$,
\item[(ii)] $\widetilde{u}^n(t)\to u(t)$
 weakly in $V$  a.e. $t\in \lbrack 0,T]$,
\item[(iii)] $\widetilde{u}^n\to u$ strongly
in $L^2(0,T;V)$.
\end{itemize}
\end{lemma}

\begin{proof}
For (i) and (ii) we refer the reader to
\cite[lemma 4.3]{t1}. For (iii) it suffices to give only some partial
proof. Indeed, from the inequality of the relation \eqref{e3.1} we
deduce the inequality
\begin{align*}
&(F\varepsilon (u^{i+1}),\varepsilon (w-u^{i+1})
)_{Q}+j(u^{i+1},w-u^{i+1})+r(\beta
^{i+1},u^{i+1},w-u^{i+1})\\
&\geq (f^{i+1},w-u^{i+1})_V \quad \forall w\in V,
\end{align*}
which implies the inequality
\begin{equation}
\begin{aligned}
&(F\varepsilon (\tilde{u}^n(t)),\varepsilon
(w-\tilde{u}^n(t)))_{Q}+j(\tilde{u}
^n(t),w-\tilde{u}^n(t))\\
&\quad +r\Big(\tilde{
\beta}^n(t),\tilde{u}^n(t),w-\tilde{u}
^n(t)\Big) \\
&\geq (\tilde{f}^n(t),w-\tilde{u}^n(t)
)_V \quad \forall w\in V,\quad \text{a.e. }t\in [0,T].
\end{aligned}
\label{e4.1}
\end{equation}
To show the strong convergence, we take $w=\tilde{u}^{n+m}(t)$
in \eqref{e4.1} and $v=\tilde{u}^n(t)$ in the same
inequality satisfied by $\tilde{u}^{n+m}(t)$, and adding the
two inequalities, we obtain by using \eqref{e2.15}(c)
\begin{align*}
&(F\varepsilon (\tilde{u}^{n+m}(t))-F\varepsilon
(\tilde{u}^n(t)),\varepsilon (\tilde{u}
^{n+m}(t)-\tilde{u}^n(t)))_{Q}\\
&\leq \int_{\Gamma _3}\mu (p(\tilde{u}_{\nu }^{n+m}(t)
)+p(\tilde{u}_{\nu }^n(t)))
| \tilde{u}^{n+m}(t)-\tilde{u}^n(t)|da\\
&\quad +r\Big(\tilde{\beta}^n(t),\tilde{u}^n(
t),\tilde{u}^{n+m}(t)-\tilde{u}^n(t) \Big)
\\
&\quad +r\Big(\tilde{\beta}^{n+m}(t),\tilde{u}^{n+m}(t),
\tilde{u}^n(t)-\tilde{u}^{n+m}(t)\Big),\quad \text{a.e. } t\in [0,T].
\end{align*}
Using \eqref{e2.15}(b) and \eqref{e2.11} we
deduce that there exists a constant $C_{6}>0$ such that
\begin{align*}
&\int_{\Gamma _3}\mu (p(\tilde{u}_{\nu }^{n+m}(t))
+p(\tilde{u}_{\nu }^n(t)))| \tilde{u}_{\tau }^{n+m}(t)-\tilde{u}_{\tau
}^n(t)|da\\
&\leq C_{6}\|\tilde{u}_{\tau }^{n+m}(t)-\tilde{u}_{\tau }^n(t)\|
_{(L^2(\Gamma _3))^d}.
\end{align*}
In the same away there exists a constant $C_{7}>0$ such that
\begin{align*}
&r\Big(\tilde{\beta}^n(t),\tilde{u}^n(t),
\tilde{u}^{n+m}(t)-\tilde{u}^n(t)\Big)\\
&+r\Big(\tilde{\beta}^{n+m}(t),\tilde{u}^{n+m}(t),
\tilde{u}^n(t)-\tilde{u}^{n+m}(t)\Big)\\
&\leq C_{7}\Big(\|\tilde{u}_{\tau }^{n+m}(t)-\tilde{u}
_{\tau }^n(t)\|_{(L^2(\Gamma
_3))^d}+\|\tilde{u}_{\nu }^{n+m}(t)-
\tilde{u}_{\nu }^n(t)\|_{L^2(\Gamma
_3)}\Big).
\end{align*}
Using \eqref{e2.14}(b) it follows that there exists a
constant $C_{8}>0$ such that
\begin{align*}
&m\|\tilde{u}^{n+m}(t)-\tilde{u}^n(t)\|_V ^2\\
&\leq C_{8}\Big(\|\tilde{u}_{\tau
}^{n+m}(t)-\tilde{u}_{\tau }^n(t)\|
_{(L^2(\Gamma _3))^d}+\|\tilde{u}
_{\nu }^{n+m}(t)-\tilde{u}_{\nu }^n(t)
\|_{L^2(\Gamma _3)}\Big), \quad
\text{a.e. } t\in [0,T].
\end{align*}
Now, to complete the proof we refer the reader to
\cite[Proposition 4.5]{t1}.
\end{proof}

 Next, we consider the  problem.

 \subsection*{Problem $P_{a}$}
 Find a bonding field $\beta :[0,T
]\to L^{\infty }(\Gamma _3)$ such that
\begin{gather*}
\dot{\beta}(t)=-[\beta (t)(c_{\nu
}(R_{\nu }(u_{\nu }(t)))^2+c_{\tau
}(| R_{\tau }(u_{\tau }(t))
|)^2)-\varepsilon _{a}]_{+}\quad
\text{a.e. } t\in (0,T),\\
\beta (0)=\beta _0\quad \text{on }\Gamma _3,
\end{gather*}
where $u$ is a weak limit founded in Lemma \ref{lem4.1}.
We have the following result.

\begin{proposition} \label{prop4.4}
There exists a unique solution
to Problem $P_{a}$  and it satisfies
\[
\beta \in W^{1,\infty }(0,T;L^2(\Gamma _3)) \cap B.
\]
\end{proposition}

 \begin{proof}
 As in \cite{c4} let $k>0$ and
\[
X=\big\{ \beta \in C([0,T];L^2(\Gamma
_3));\quad \sup_{t\in [0,T]}[\exp (-kt)\|\beta (t)\|
_{L^2(\Gamma _3)}]<+\infty \big\} .
\]
$X$ is a Banach space with the norm
\[
\|\beta \|_{X}=\sup_{t\in [0,T]}
[\exp (-kt)\|\beta (t)\|_{L^2(\Gamma _3)}],
\]
and consider the mapping $T:X\to X$ given by
\[
T\beta (t)=\beta _0-\int_0^{t}[\beta (s)
(c_{\nu }(R_{\nu }(u_{\nu }(s)))
^2+c_{\tau }(| R_{\tau }(u_{\tau }(s)
)|)^2)-\varepsilon _{a}]_{+}ds.
\]
Then there exists a constant $c_1'>0$ such that
\begin{align*}
&| T\beta _1(t)-T\beta _2(t)|^2 \\
&\leq c_1'\int_0^{t}(c_{\nu }(R_{\nu }(u_{\nu
}(s)))^2+c_{\tau }| R_{\tau }(
u_{\tau }(s))|^2)(\beta _1(
s)-\beta _2(s))^2ds.
\end{align*}
Using $| R_{\nu }(u_{\nu }(s))|\leq L$,
$| R_{\tau }(u_{\tau }(s))|\leq L$, it follows that
\begin{align*}
\|T\beta _1(t)-T\beta _2(t)\|_{L^2(\Gamma _3)}^2
&\leq c_2'\int_0^{t}\|\beta _1(s)-\beta _2(s)\|_{L^2(\Gamma _3)}^2ds \\
&\leq c_2'\|\beta _1-\beta _2\|_{X}^2 \frac{\exp (2kt)}{2k},
\end{align*}
for some constant $c_2'>0$. So we obtain
\[
\|T\beta _1-T\beta _2\|_{X}\leq \sqrt{\frac{
c_2'}{2k}}\|\beta _1-\beta _2\|_{X},
\]
and then for $k$ sufficiently large $T$ has a unique fixed point $\beta$.
To show that $\beta \in [0,1]$ for all $t\in [0,T]$, a.e. on $\Gamma_3$,
we refer the reader to \cite[Remark 3.1]{s3}.
\end{proof}

 Next, we prove a convergence result.

\begin{lemma} \label{lem4.5}
Let $\beta $ be the unique
solution to Problem $P_{a}$. Then we have:
\begin{itemize}
\item[(i)] $\beta ^n\to \beta$ strongly in
$L^{\infty }(0,T;L^2(\Gamma _3))$,

\item[(ii)] $\tilde{\beta}^n\to \beta$ strongly in
$L^{\infty }(0,T;L^2(\Gamma _3))$.
\end{itemize}
\end{lemma}

\begin{proof}
 (i), Since $\dot{\beta}^n(t)=\frac{\Delta \beta
^{i}}{\Delta t}$, for all $t\in ]t_i,t_{i+1}[$, we have
\begin{gather*}
\beta ^n(t)=\beta ^i-\int_{t_i}^{t}[\tilde{\beta}
^n(s)(c_{\nu }(R_{\nu }(\tilde{u}_{\nu
}^n(s)))^2+c_{\tau }(| R_{\tau
}(\tilde{u}_{\tau }^n(s))|)
^2)-\varepsilon _{a}]_{+}ds,
\\
\beta (t)=\beta (t_i)-\int_{t_i}^{t}[\beta (s)
(c_{\nu }(R_{\nu }(u_{\nu }(s)))
^2+c_{\tau }(| R_{\tau }(u_{\tau }(s)
)|)^2)-\varepsilon _{a}]_{+}ds.
\end{gather*}
Then
\begin{align*}
\beta ^n(t)-\beta (t)
&=\beta^i-\beta(t_i) -\int_{t_i}^{t}[\tilde{
\beta}^n(s)(c_{\nu }(R_{\nu }(\tilde{u}
_{\nu }^n(s)))^2+c_{\tau }(|
R_{\tau }(\tilde{u}_{\tau }^n(s))|
)^2)-\varepsilon _{a}]_{+}ds \\
&\quad +\int_{t_i}^{t}[\beta (s)(c_{\nu }(R_{\nu
}(u_{\nu }(s)))^2+c_{\tau }(
| R_{\tau }(u_{\tau }(s))|
)^2)-\varepsilon _{a}]_{+}ds.
\end{align*}
Thus,
\begin{align*}
\|\beta ^n(t)-\beta (t)\|_{L^2(\Gamma _3)}
&\leq \|\beta^i -\beta(t_i)\|_{L^2(\Gamma_3)}\\
&\quad + \int_0^{t}\|\tilde{\beta}^n(s)\Big(c_{\nu
}(R_{\nu }(\tilde{u}_{\nu }^n(s)))
^2+c_{\tau }(| R_{\tau }(\tilde{u}_{\tau }^n(s))|)^2\Big)\\
&\quad -\beta (s)
\Big(c_{\nu }(R_{\nu }(u_{\nu }(s)))
^2+c_{\tau }(| R_{\tau }(u_{\tau }(s)
)|)^2\Big)\|_{_{L^2(\Gamma_3)}}ds.
\end{align*}
Using the properties of $R_{l}$, $l=\nu ,\tau $ (see \cite{s2})
such that $| R_{l}(u_{l})|\leq L$ and
\eqref{e3.4}, we have
\begin{align*}
&\|c_{\nu }\tilde{\beta}^n(s)(R_{\nu }(
\tilde{u}_{\nu }^n(s)))^2-c_{\nu }\beta
(s)(R_{\nu }(u_{\nu }(s)) )^2\|_{L^2(\Gamma _3)}\\
&\leq  \big\|\tilde{\beta}^n(s)c_{\nu }\Big((R_{\nu
 }(\tilde{u}_{\nu }^n(s)))^2-(
 R_{\nu }(u_{\nu }(s)))^2\Big)\\
&\quad +\Big(\tilde{\beta}^n(s)-\beta ^n(s)\Big)
 c_{\nu}(R_{\nu }(u_{\nu }(s)))^2\big\|_{L^2(\Gamma _3)} \\
&\quad + \|(\beta ^n(s)-\beta (s))
 c_{\nu }(R_{\nu }(u_{\nu }(s)))^2\|_{L^2(\Gamma _3)}\\
&\leq 2L\|c_{\nu}\|_{L^{\infty }(\Gamma _3)}\|\tilde{u}
 _{\nu }^n(s)-u_{\nu }(s)\|_{L^2(\Gamma _3)}\\
&\quad + L^2\|c_{\nu }\|_{L^{\infty }(\Gamma _3)
}\Delta tc_1'+L^2\|c_{\nu }\|_{L^{\infty
}(\Gamma _3)}\|\beta ^n(s)-\beta
(s)\|_{L^2(\Gamma _3)}.
\end{align*}
Also we have
\begin{align*}
&\|\tilde{\beta}^n(s)c_{\tau }(|R_{\tau }(\tilde{u}_{\nu }^n(s))|
)^2-c_{\tau }\beta (t)(| R_{\tau}(u_{\tau }(s))|)^2\|
_{L^2(\Gamma _3)}\\
&\leq  \|\tilde{\beta}^n(s)c_{\tau }\Big((
 | R_{\tau }(\tilde{u}_{\tau }^n(s))
|)^2-(| R_{\tau }(u_{\tau }(s))|)^2\Big)\\
&\quad +\Big(\tilde{\beta} ^n(s)-\beta ^n(s)\Big)
c_{\tau }(| R_{\tau }(u_{\tau }(s))|)^2\|_{L^2(\Gamma _3)} \\
&\quad +\|(\beta ^n(s)-\beta (s))
c_{\tau }(| R_{\tau }(u_{\tau }(s))|)^2\|_{L^2(\Gamma _3)}\\
&\leq 2L\|c_{\tau }\|_{L^{\infty }(\Gamma _3)
}\|\tilde{u}_{\tau }^n(s)-u_{\tau }(s)
\|_{(L^2(\Gamma _3))^d}\\
&\quad + L^2\|c_{\tau }\|_{L^{\infty }(\Gamma _3)
}\Delta tc_1'+L^2\|c_{\tau }\|_{L^{\infty
}(\Gamma _3)}\|\beta ^n(s)-\beta
(s)\|_{L^2(\Gamma _3)}.
\end{align*}
 From the above inequalities, we deduce
\begin{align*}
&\|\beta ^n(t)-\beta (t)\|_{L^2(\Gamma _3)}\\
&\leq \|\beta^i-\beta(t_i)\|_{L^2(\Gamma_3)}
+ 2L\Big(\|c_{\nu }\|_{L^{\infty }(\Gamma
 _3)}\int_0^{t}\|\tilde{u}_{\nu }^n(s)
 -u_{\nu }(s)\|_{L^2(\Gamma _3)}ds \\
&\quad +\|c_{\tau }\|_{L^{\infty }(\Gamma _3)
}\int_0^{t}\|\tilde{u}_{\tau }^n(s)-u_{\tau
}(s)\|_{(L^2(\Gamma _3)) ^d}ds\Big)
\\
&\quad +L^2\Big(\|c_{\nu }\|_{L^{\infty }(\Gamma
_3)}+\|c_{\tau }\|_{L^{\infty }(\Gamma
_3)}\Big)\int_0^{t}\|\beta ^n(s)-\beta
(s)\|_{L^2(\Gamma _3)}ds\\
&\quad +\Big(
\|c_{\nu }\|_{L^{\infty }(\Gamma _3)
}+\|c_{\tau }\|_{L^{\infty }(\Gamma _3)}\Big)TL^2\Delta tc_1'.
\end{align*}
Now using a Gronwall-type argument it follows that there exists a constant
$C_{9}>0$ such that
\begin{align*}
\|\beta ^n(t)-\beta (t)\|_{L^2(\Gamma _3)}
&\leq C_{9}\Big(\|\beta^i-\beta(t_i)\|_{L^2(\Gamma_3)}
+\int_0^{t}\Big(\|\tilde{u}_{\nu }^n(s)-u_{\nu }(s)\|_{L^2(\Gamma
_3)}\\
&\quad +\|\tilde{u}_{\tau }^n(s)-u_{\tau
}(s)\|_{(L^2(\Gamma _3))^d}\Big)ds+\Delta t\Big).
\end{align*}
Using \eqref{e2.11}, the above inequality implies
\begin{align*}
&\max_{t\in [0,T]} \|\beta ^n(t)-\beta (t)\|_{L^2(\Gamma _3)}\\
&\leq C_{9}\Big(\max_{i=0,\dots,n}  \|\beta^i-\beta(t_i)\|_{L^2(\Gamma_3)}
+2d_{\Omega }\int_0^T  \|\tilde{u}^n(s)-u(s)\|_V ds
+\Delta t\Big)
\end{align*}
and
\begin{align*}
&\max_{t\in [0,T]} \|\beta ^n( t)-\beta (t)\|_{L^2(\Gamma _3)}\\
&\leq C_{9}\Big(\max_{i=0,\dots,n}  \|\beta^i-\beta(t_i)\|_{L^2(\Gamma_3)}
+2d_{\Omega }\sqrt{T}\|\tilde{u}^n-u\|
_{L^2(0,T;V)}+\Delta t\Big).
\end{align*}
As in \cite[Lemma 3.5]{c4}, we still have
\[
\lim_{n\to+\infty} \max_{i=0,\dots,n} 
\|\beta^i-\beta(t_i)\|_{L^2(\Gamma_3)}=0\,.
\]
Using (iii) of Lemma \ref{lem4.3}, one obtains
\[
\lim_{n\to+\infty} \max_{t\in[0,T]} 
\|\beta^n(t)-\beta(t)\|_{L^2(\Gamma_3)}=0\,.
\]
So (i) is proved. To prove (ii) it suffices to remark that there exists
a constant $C_{10}>0$ such that
\begin{align*}
\|\tilde{\beta}^n(t)-\beta (t) \|_{L^2(\Gamma _3)}
&\leq \|\tilde{\beta}^n(t)-\beta ^n(t)\|_{L^2(
\Gamma _3)}+\|\beta ^n(t)-\beta (
t)\|_{L^2(\Gamma _3)} \\
&\leq C_{10}\Delta t+\|\beta ^n(t)-\beta (
t)\|_{L^2(\Gamma _3)}.
\end{align*}
\end{proof}

 Now we have all the ingredients to prove the following proposition.

\begin{proposition} \label{prop4.6}
$(u,\beta )$ is a solution to Problem $P_2$.
\end{proposition}

\begin{proof}
 In the inequality \eqref{e3.1}, for $v\in V$ set
$w=u^{i}+v\Delta t$ and divide by $\Delta t;$ we obtain the inequality
\begin{align*}
&(F\varepsilon (u^{i+1}),\varepsilon
 (v-\frac{\Delta u^{i}}{\Delta t}))_{Q}+j(u^{i+1},v)-j(
u^{i+1},\frac{\Delta u^{i}}{\Delta t})
+r(\beta ^{i+1},u^{i+1},v-\frac{\Delta u^{i}}{\Delta t})\\
&\geq (f^{i+1},v-\frac{\Delta u^{i}}{\Delta t})_V 
\end{align*}
Whence for any $v\in L^2(0,T;V)$, we have
\begin{align*}
&(F\varepsilon (\widetilde{u}^n(t)),\varepsilon
(v(t)-\dot{u}^n(t)))_{Q}+j(
\widetilde{u}^n(t),v(t))-j(
\widetilde{u}^n(t),\dot{u}^n(t))\\
&+r(\tilde{\beta}^n(t),\widetilde{u}^n(t)
,v(t)-\dot{u}^n(t))\\
&\geq (\widetilde{f}^n(t),v(t)-\dot{u}^n(t))_V \quad
\text{a.e. }t\in [0,T].
\end{align*}
Integrating both sides of the above inequality on $(0,T)$,
we obtain
\begin{equation}
\begin{aligned}
&\int_0^T (F\varepsilon (\widetilde{u}^n(t))
,\varepsilon (v(t)-\dot{u}^n(t))
)_{Q}dt+\int_0^T j(\widetilde{u}^n(t),v(t))dt\\
&-\int_0^T j(\widetilde{u}^n(t),\dot{u}^n(t))dt
+\int_0^T r\Big(\tilde{\beta}^n(t),\widetilde{u}
^n(t),v(t)-\dot{u}^n(t)\Big) dt\\
&\geq \int_0^T (\widetilde{f}^n(t),v(t)
-\dot{u}^n(t))_V dt.
\end{aligned} \label{e4.2}
\end{equation}
To pass in the limit in this inequality we need to establish the following
properties.
\end{proof}

\begin{lemma} \label{lem4.7}
We have the following properties
for $v\in L^2(0,T;V)$:
\begin{gather}
\lim_{n\to \infty }\int_0^T (F\varepsilon (
\widetilde{u}^n(t)),\varepsilon (v(
t)-\dot{u}^n))_{Q}dt=\int_0^T (F\varepsilon (
u(t)),\varepsilon (v(t)-\dot{u}(t)))_{Q}dt,  \label{e4.3}
\\
\liminf_{n\to \infty } \int_0^T j(\widetilde{u}^n(t),\dot{u}^n(t))dt\geq
\int_0^T j(u(t),\dot{u}(t))dt,  \label{e4.4}
\\
\lim_{n\to \infty } \int_0^T j(\widetilde{u}^n(t),v(t))dt=\int_0^T j(u(
t),v(t))dt,\label{e4.5}
\\
\lim_{n\to \infty } \int_0^T (\tilde{f}^n(t),v(t)-\dot{u}^n(t))
_V dt=\int_0^T (f(t),v(t)-\dot{u}(t))_V dt, \label{e4.6}
\\
\lim_{n\to \infty } \int_0^T r(\tilde{\beta}
^n(t),u^n(t),v(t)-\dot{u}
^n(t))dt=\int_0^T r(\beta (t)
,u(t),v(t)-\dot{u}(t))dt.\label{e4.7}
\end{gather}
\end{lemma}

\begin{proof}
 For the proof of \eqref{e4.3} and \eqref{e4.6} we
refer the reader to \cite[Lemma 4.6]{t1}. To prove
\eqref{e4.4}, it suffices to see \cite[Lemma 3.5]{m1}.
To prove \eqref{e4.5}, it suffices to use (iii) of Lemma \ref{lem4.3}.
Finally for the proof of \eqref{e4.7} we refer the reader to
\cite[Lemma 3.8]{c4} and use the properties \eqref{e3.4}.

 Now using lemma \ref{lem4.5} (ii) and Lemma \ref{lem4.7} we pass to the limit as
$n\to +\infty $ in the inequality \eqref{e4.2} to obtain
\begin{align*}
&\int_0^T (F\varepsilon (u(t)),\varepsilon
(v(t)-\dot{u}(t)))_{Q}dt+\int_0^T j(u(t),v(t))dt\\
&-\int_0^T j(u(t),\dot{u}(t))dt
+\int_0^T r(\beta (t),u(t),v(t)-\dot{u}(t))dt\\
&\geq \int_0^T (f(t),v(t)-\dot{u}(t))_V dt,
\end{align*}
from which we deduce the inequality \eqref{e2.20} and also
that $\beta $
is the unique solution of the differential equation \eqref{e2.21}.
\end{proof}


 \begin{remark} \label{rmk4.8} \rm
 We can consider another quasistatic
frictional contact problem with adhesion. In Problem $P_1$ the contact
conditions on $\Gamma _3$ \eqref{e2.5} and \eqref{e2.6}
are modified as follows.
\begin{gather*}
-\sigma _{\nu }=p(u_{\nu })-c_{\nu }\beta ^2R_{\nu }(
u_{\nu })\quad\text{on }\Gamma _3\times (0,T), % \label{e4.8}
\\
\left\{
\begin{aligned}
&| \sigma _{\tau }+c_{\tau }\beta ^2R_{\tau }(u_{\tau
})|\leq \mu | p(u_{\nu })-c_{\nu
}\beta ^2R_{\nu }(u_{\nu })|,
\\
&| \sigma _{\tau }+c_{\tau }\beta ^2R_{\tau }(u_{\tau
})|<\mu | p(u_{\nu })-c_{\nu }\beta
^2R_{\nu }(u_{\nu })|\Longrightarrow \dot{u}_{\tau}=0,
\\
&| \sigma _{\tau }+c_{\tau }\beta ^2R_{\tau }(u_{\tau
})|=\mu | p(u_{\nu })-c_{\nu }\beta
^2R_{\nu }(u_{\nu })|\Longrightarrow
\\
&\exists \lambda \geq 0\text{ such that }\dot{u}_{\tau }=-\lambda (\sigma
_{\tau }+c_{\tau }\beta ^2R_{\tau }(u_{\tau })),
\end{aligned}
\right.\quad \text{on }\Gamma _3\times (0,T),  %\label{e4.9}
\end{gather*}
Using the new contact conditions,  %\eqref{e4.8} and \eqref{e4.9},
as in Problem $P_2$ the corresponding variational problem is written
with the functional $j:V\times V\to \mathbb{R}$ defined by
\[
j(u,v)=\int_{\Gamma _3}(p(u_{\nu })v_{\nu
}+\mu | p(u_{\nu })-c_{\nu }\beta ^2R_{\nu }(
u_{\nu })|| v_{\tau }|)da\quad \forall u,v\in V.
\]
In the same away we show that if there exists a constant $\mu _{\ast }>0$,
this problem admits at least one solution for
\[
\|\mu \|_{L^{\infty }(\Gamma _3)}<\mu _{\ast }.
\]
\end{remark}

\subsection*{Conclusion}

In this paper we have studied a mathematical model which describes a
quasistatic frictional contact problem with adhesion for nonlinear elastic
materials. The adhesive and frictional contact is modelled with a normal
compliance condition and the associated version of Coulomb's law of dry
friction. An existence result of a weak solution was proved under a
smallness assumption of the friction coefficient. Finally, we note that the
important question of uniqueness of the solution is not resolved here, and
remains still open.


\subsection*{Acknowledgements}
The author wants to express his gratitude to the anonymous referee
of this journal for his/her helpful suggestions and comments.



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\section*{Addendum posted on January 8, 2009.}

The author wants to correct some misprints:

\begin{itemize}

\item Page 7: In the first displayed inequality, the norm of $\beta$ has been
included:
\[
\vert (Au-Av,w) _{V}\vert 
\leq \big[M+\big(\Vert c_{\nu }\Vert _{L^{\infty }(\Gamma _{3})}
+\Vert c_{\tau }\Vert _{L^{\infty }(\Gamma _{3})}\big) 
\Vert \beta \Vert _{L^{\infty }(\Gamma _{3})}^{2}d_{\Omega }^{2}\big] 
\Vert u-v\Vert _{V}\Vert w\Vert _{V}
\]

\item Page 7, Problem $P_{2\beta }$: 
In the second part of the definition of $\beta$,
$>0$ has been replaced by $\geq 0$:
\[
\beta =\begin{cases}
\beta ^{i},\quad \text{if }(c_{\nu }(R_{\nu }(u_{\beta \nu
}))^2+c_{\tau }(| R_{\tau }(
u_{\beta \tau })|)^2)\beta^i -\varepsilon_{a}<0, \\[4pt]
\dfrac{\beta ^{i}+\varepsilon _{a}\Delta t}{1+\Delta t(c_{\nu }(
R_{\nu }(u_{\beta \nu }))^2+c_{\tau }(
| R_{\tau }(u_{\beta \tau })|)
^2)}, \\
\text{if }(c_{\nu }(R_{\nu }(u_{\beta \nu })
)^2+c_{\tau }(| R_{\tau }(u_{\beta \tau
})|)^2)\beta^i-\varepsilon _{a} \geq 0,
\end{cases}
\]
 
\item Page 9, Lemma \ref{lem3.7}: $+1$ has been attached to the right-hand side 
of the first inequality:
\begin{equation}
\|u^{i+1}\|_V \leq C_3\big(\|f^{i+1}\|_V+1\big) ,\quad
\|\Delta u^{i}\|_V \leq C_{4}(\|\Delta f^{i}\|_V +\Delta t).
\tag{3.8}
\end{equation}

\item Page 11, in the last displayed inequality:
The second and third terms on the right-hand side have been 
modified as follows:
\begin{align*}
&(F\varepsilon (\tilde{u}^{n+m}(t))-F\varepsilon
(\tilde{u}^n(t)),\varepsilon (\tilde{u}
^{n+m}(t)-\tilde{u}^n(t)))_{Q}\\
&\leq \int_{\Gamma _3}\mu (p(\tilde{u}_{\nu }^{n+m}(t)
)+p(\tilde{u}_{\nu }^n(t)))
| \tilde{u}^{n+m}(t)-\tilde{u}^n(t)|da
\\
&\quad +r\Big( \tilde{\beta}^{n+m}(t) ,\tilde{u}^{n+m}(t),
 \tilde{u}^{n}(t) -\tilde{u}^{n+m}(t) \Big) \\
&\quad +\Big( \tilde{f}^{n+m}(t) -\tilde{f}^{n}(t),
 \tilde{u}^{n+m}(t) -\tilde{u}^{n}(t) \Big)_V\,, 
\quad\text{ a.e. }t\in [ 0,T].
\end{align*}


\item Page 12, lines 9 and 10, after 
``Using \eqref{e2.14}(b) it follows that there exists a
constant $C_{8}>0$ such that'':
 Replace the displayed inequality by
\begin{align*}
\Vert \tilde{u}^{n+m}(t) -\tilde{u}^{n}(t)\Vert _{V}^{2}
&\leq C_{8}\Big(\Vert \tilde{u}_{\tau }^{n+m}(t) 
 -\tilde{u}_{\tau }^{n}(t)\Vert _{(L^{2}(\Gamma _3))^d}
 +\Vert \tilde{u}_{\nu}^{n+m}(t) -\tilde{u}_{\nu }^{n}(t)
  \Vert_{L^{2}\Gamma _3) } \\ 
&\quad +\Vert \tilde{f}^{n+m}(t) -\tilde{f}^{n}(t)\Vert_V^2\Big),
\quad\text{a.e. }t\in [0,T] .
\end{align*}
 
\end{itemize}
End of addendum.

\end{document}