#  Well-posedness of Constrained Evolutionary Differential
Variational–Hemivariational Inequalities
with Applications ††thanks:  The project has received funding from the
European Union’s Horizon 2020 Research and Innovation Programme under the
Marie Skłodowska-Curie grant agreement No. 823731 CONMECH. It is supported by
Natural Science Foundation of Guangxi (Grant No: 2018GXNSFAA281353), Beibu
Gulf University Project No. 2018KYQD06, and the projects financed by the
Ministry of Science and Higher Education of Republic of Poland under Grants
Nos. 4004/GGPJII/H2020/2018/0 and 440328/PnH2/2019, and the National Science
Centre of Poland under Project No. 2021/41/B/ST1/01636.

Stanisław Migórski 111 College of Applied Mathematics, Chengdu University of
Information Technology, Chengdu 610225, Sichuan Province, P.R. China, and
Jagiellonian University in Krakow, Chair of Optimization and Control, ul.
Lojasiewicza 6, 30348 Krakow, Poland. Tel.: +48-12-6646666. E-mail address:
<EMAIL_ADDRESS>
###### Abstract

A system of a first order history-dependent evolutionary variational-
hemivariational inequality with unilateral constraints coupled with a
nonlinear ordinary differential equation in a Banach space is studied. Based
on a fixed point theorem for history dependent operators, results on the well-
posedness of the system are proved. Existence, uniqueness, continuous
dependence of the solution on the data, and the solution regularity are
established. Two applications of dynamic problems from contact mechanics
illustrate the abstract results. First application is a unilateral
viscoplastic frictionless contact problem which leads to a hemivariational
inequality for the velocity field, and the second one deals with a
viscoelastic frictional contact problem which is described by a variational
inequality.

Key words. Variational-hemivariational inequality, history-dependent operator,
unilateral constraint, evolution triple, inclusion, subgradient, contact
problem.

2010 Mathematics Subject Classification. 35K86, 35M86, 35R70, 47J20, 49J40,
74M10, 74M15.

##  1 Introduction

In this paper we study the following system of an evolutionary variational-
hemivariational inequality coupled with an ordinary differential equation in a
Banach space.

Find ${\mathtt{x}}\colon(0,T)\to E$ and $w\colon(0,T)\to V$ with $w(t)\in K$
for a.e. $t\in(0,T)$ such that

$\displaystyle{\mathtt{x}}^{\prime}(t)=F(t,{\mathtt{x}}(t),w(t),(Sw)(t))\ \
\mbox{\rm for a.e.}\ \ t\in(0,T),$ (1) $\displaystyle\langle
w^{\prime}(t)+A(t,{\mathtt{x}}(t),w(t))+(R_{1}w)(t),v-w(t)\rangle+\,j^{0}(t,{\mathtt{x}}(t),(R_{2}w)(t),Mw(t);Mv-
Mw(t))$ $\displaystyle\quad\
\,+\,\varphi(t,{\mathtt{x}}(t),(R_{3}w)(t),Mv)-\varphi(t,{\mathtt{x}}(t),(R_{3}w)(t),Mw(t))\geq\langle
f(t,{\mathtt{x}}(t)),v-w(t)\rangle$ $\displaystyle\qquad\ \ \ \ \mbox{\rm for
all}\ v\in K,\ \mbox{\rm a.e.}\ t\in(0,T),$ (2)
$\displaystyle{\mathtt{x}}(0)={\mathtt{x}}_{0},\ \ w(0)=w_{0}.$ (3)

Here, $E$ is a Banach space, and $(V,H,V^{*})$ is an evolution triple of
spaces. Problem (1)–(3) represents a system which couples the ordinary
differential equation (1) with the constrained variational-hemivariational
inequality (2), supplemented with the initial conditions (3). Following the
terminology in [2, 16, 17, 31], we refer to (1)–(3) as an evolutionary
differential variational-hemivariational inequality. The inequality (2)
couples with equation (1) through the solution $w$ and its history value $Sw$,
$j^{0}$ denotes the generalized directional derivative in the last variable of
a nonconvex and nondifferentiable function $j\colon(0,T)\times E\times Z\times
X\to\mathbb{R}$, $\varphi\colon(0,T)\times E\times Y\times X\to\mathbb{R}$ is
a function convex in the last variable, $M\colon V\rightarrow X$ is an affine
linear operator, $S$, $R_{1}$, $R_{2}$, $R_{3}$ are the so-called history-
dependent operators, and $K\subset V$ represents a set of constraints. For a
history-dependent operator, the current value for a given function at the time
instant $t$ depends on the values of the function at the time instants from
$0$ to $t$. For this reason, the probem (1)–(3) is also called a history-
dependent differential variational-hemivariational inequality. The main
features of the system are the presence of the constraints set $K$ and the
strong coupling between the differential equation and the inequality which
leads to a complex time dependent dynamics. Problem (1)–(3) is new and, to the
best of our knowledge, has not been studied in the literature. Our main
existence and uniqueness result, in its particular case, solves an open
problem stated in [39, Section 10.4], and extends the recent result in [30,
Theorem 20] obtained for a purely variational-hemivariational inequality.

There are two main goals of the paper. First, we establish a well-posedness
result which includes existence, uniqueness and continuous dependence of
solution to the system (1)–(3) on the data. In particular, if $f$ is assumed
to be independent of ${\mathtt{x}}$, then we show, see Theorem 4, that the map
$({\mathtt{x}}_{0},w_{0},f)\mapsto({\mathtt{x}},w)$ is Lipschitz continuous.
The second goal of the paper is to provide two applications to new classes of
dynamic problems in contact mechanics. In the first application we study a
dynamic unilateral viscoplastic frictionless contact problem involving a
nonmonotone Clarke subdifferential boundary condition. Its weak formulation is
a hemivariational inequality coupled with an ordinary differential equation.
The second application concerns a dynamic frictional viscoelastic contact
problem with adhesion which leads to a variational inequality combined with a
differential equation on the contact surface.

Only special versions of the system (1)-(3) have been explored in the
literature. For instance, the evolutionary variational-hemivariational
inequalities without a constraint set $K$, and with and without history-
dependent operators and their variants are discussed in [39, Chapter 7]. If
$K=V$, $\varphi=0$, and $A$, $j$, $f$ are independent of ${\mathtt{x}}$, then
(2)-(3) reduces to the evolution hemivariational inequality considered in
[29]. If the operator $A$ and a locally Lipschitz function $j$ are assumed to
be independent of ${\mathtt{x}}$ and $R_{1}$, then the problem reduces to the
parabolic hemivariational inequality studied, for example, in [15, 18, 23].
Very recently the system has been treated in [44] with $K=V$, $\varphi=0$, and
$F$, $j$ independent of the history-dependent operators, and $f$ independent
of the variable ${\mathtt{x}}$. A general dynamic variational-hemivariational
inequality with history-dependent operators without constraints can be found
in [12]. All aforementioned papers treat the special case of (2) with $K=V$.
In contact mechanics, the ordinary differential equation (1) supplementing (2)
appears naturally. For example, the system is met in rate-type viscoplastic
constitutive laws, including the internal state variables in viscoplasticity,
see [39, Chapter 3], and in modeling of additional effects like wear and
adhesion phenomena in contact problems, see [35, Chapter 11], [37, Chapter 2]
and the references therein.

We remark that the notion of a history-dependent operator was introduced in
[38] and used in a several recent papers [12, 20, 25, 27, 29, 36, 38, 40, 42,
43]. Differential variational inequalities was initiated and first
systematically discussed in [34] in finite dimensional spaces. Note also that
a related result on a class of differential hemivariational inequalities which
consists of a hemivariational inequality of parabolic type combined with a
nonlinear evolution equation has been delivered by the Rothe method in [31].
The differential parabolic-parabolic variational inequalities are examined in
[2] by using the technique of measure of noncompactness. Results on partial
differential variational inequalities with nonlocal boundary conditions can be
found in [16]. The literature on hemivariational inequalities has been
significantly enlarged in the last forty years, see monographs [4, 9, 26, 33,
39], for analysis of various classes of such inequalities, see e.g. [3, 10,
12, 14, 15, 16, 17, 19, 22, 21, 31, 44]. By our approach, we relax the
assumptions on similar problems treated in the aforementioned papers and
considerably improve some results by allowing the history-dependent operators
to appear in the convex term and in the generalized directional derivative of
a nonconvex potential.

The paper is structured as follows. In Section 2, we recall some prerequisites
needed throughout this paper. In Section 3, we state and prove the main
results on the well-posedness of the system (1)–(3). Next, in Section 4, we
consider a model of contact which leads to a differential hemivariational
inequality for the velocity field of the form (1)–(3). Finally, for the second
contact problem in Section 5, under appropriate hypotheses on the data, we
prove the well-posedness for a differential variational inequality with
constraints which is a weak form of the contact model.

##  2 Preliminaries

Let $(X,\|\cdot\|_{X})$ be a normed space, $X^{*}$ be the dual of $X$ and
$\langle\cdot,\cdot\rangle_{X^{*}\times X}$ denote the duality brackets for
the pair $(X^{*},X)$. For simplicity, when no confusion arises, we often skip
the subscripts. For Banach spaces $X$, $Y$, we will use the notation
${\mathcal{L}}(X,Y)$ for the set of all linear continuous operators from $X$
to $Y$. The notation $\|A\|$ stands for the operator norm of $A\colon X\to Y$
in ${\mathcal{L}(X,Y)}$. Given a set $D\subset X$, we write
$\|D\|_{X}=\sup\\{\|x\|_{X}\mid x\in D\\}$. We denote by $C([0,T];X)$ the
space of continuous functions on $[0,T]$ with values in $X$. It is well known
that if $X$ is a Banach space, then $C([0,T];X)$ is also a Banach space.

Let $h\colon X\to\mathbb{R}$ be a locally Lipschitz function. The generalized
(Clarke) directional derivative of $h$ at the point $x\in X$ in the direction
$v\in X$ is defined by

$h^{0}(x;v)=\limsup_{y\to x,\ \lambda\downarrow 0}\frac{h(y+\lambda
v)-h(y)}{\lambda}.$

The generalized subgradient of $h$ at $x$ is a subset of the dual space
$X^{*}$ given by

$\partial h(x)=\\{\,\zeta\in X^{*}\mid h^{0}(x;v)\geq{\langle\zeta,v\rangle}\
\mbox{for all}\ v\in X\,\\}.$

A locally Lipschitz function $h$ is said to be regular (in the sense of
Clarke) at the point $x\in X$ if for all $v\in X$ the derivative
$h^{\prime}(x;v)$ exists and $h^{0}(x;v)=h^{\prime}(x;v)$. Throughout the
paper, the generalized derivative and the generalized subgradient are always
taken with respect to the last variable of a given function.

We say that a map $A\colon X\to X^{*}$ is demicontinuous if the function
$u\mapsto\langle Au,v\rangle_{X^{*}\times X}$ is continuous for all $v\in X$,
i.e., $A$ is continuous as a mapping from $X$ to $X^{*}$ endowed with
$w^{*}$-topology. It is hemicontinuous, if for all $u$, $v$, $w\in X$, the
function $t\mapsto\langle A(u+tv),w\rangle_{X^{*}\times X}$ is continuous on
$[0,1]$. A map $A$ is monotone if $\langle Au-Av,u-v\rangle_{X^{*}\times
X}\geq 0$ for all $u$, $v\in X$. It is strongly monotone with constant $c>0$
if $\langle Au-Av,u-v\rangle_{X^{*}\times X}\geq c\,\|u-v\|^{2}_{X}$ for all
$u$, $v\in X$. For more details, we refer to [6, 7, 8, 26].

The following definition and a fixed point result (being a consequence of the
Banach contraction principle) can be found in [39, Definition 30 and Theorem
67], respectively.

######  Definition 1.

Let ${\mathbb{X}}$ and ${\mathbb{Y}}$ be normed spaces. An operator
${\mathbb{S}}\colon L^{2}(0,T;{\mathbb{X}})\to L^{2}(0,T;{\mathbb{Y}})$ is
called a history-dependent operator if there exists $M>0$ such that

$\|({\mathbb{S}}u_{1})(t)-({\mathbb{S}}u_{2})(t)\|_{{\mathbb{Y}}}\leq
M\int_{0}^{t}\|u_{1}(s)-u_{2}(s)\|_{{\mathbb{X}}}\,ds$

for all $u_{1}$, $u_{2}\in L^{2}(0,T;{\mathbb{X}})$, a.e. $t\in(0,T)$.

######  Lemma 2.

Let ${\mathbb{X}}$ be a Banach space and $0<T<\infty$. Let $F\colon
L^{2}(0,T;{\mathbb{X}})\to L^{2}(0,T;{\mathbb{X}})$ be an operator such that

$\|(F\eta_{1})(t)-(F\eta_{2})(t)\|^{2}_{\mathbb{X}}\leq
c\int_{0}^{t}\|\eta_{1}(s)-\eta_{2}(s)\|^{2}_{\mathbb{X}}\,ds$

for all $\eta_{1}$, $\eta_{2}\in L^{2}(0,T;{\mathbb{X}})$, a.e. $t\in(0,T)$
with a constant $c>0$. Then $F$ has a unique fixed point in
$L^{2}(0,T;{\mathbb{X}})$, i.e., there exists a unique $\eta^{*}\in
L^{2}(0,T;{\mathbb{X}})$ such that $F\eta^{*}=\eta^{*}$.

##  3 Well-posedness of the abstract problem

In this section we provide results on the well-posedness of the differential
variational-hemivariational inequalities. We shall establish existence and
uniqueness of solution and prove continuous dependence of solution on the
initial conditions and the function $f$.

Let $E$, $U$, $X$, $Y$ and $Z$ be Banach spaces and $(V,H,V^{*})$ be an
evolution triple of spaces. The latter means that $V$ is a separable reflexive
Banach space and $H$ is a separable Hilbert space such that the embedding
$V\subset H$ is continuous and dense. Then $H$ is embedded continuously and
densely in $V^{*}$, and the duality brackets $\langle\cdot,\cdot\rangle$ for
the pair $(V^{*},V)$ and the inner product $\langle\cdot,\cdot\rangle_{H}$ on
$H$ coincide on $H\times V$. In what follows we denote by $\|\cdot\|$ the norm
in $V$. We set

${\mathbb{W}}=\\{\,w\in L^{2}(0,T;V)\mid w^{\prime}\in L^{2}(0,T;V^{*})\,\\},$

where the time derivative $w^{\prime}$ is understood in the distributional
sense. It is known that $L^{2}(0,T;V)^{*}\simeq L^{2}(0,T;V^{*})$ and the
space ${\mathbb{W}}$ endowed with the norm
$\|w\|_{{\mathbb{W}}}=\|w\|_{L^{2}(0,T;V)}+\|w^{\prime}\|_{L^{2}(0,T;V^{*})}$
is a separable reflexive Banach space.

We deal with the following system.

######  Problem 3.

Find ${\mathtt{x}}\in H^{1}(0,T;E)$ and $w\in{\mathbb{W}}$ such that $w(t)\in
K$ for a.e. $t\in(0,T)$ and

$\begin{cases}\displaystyle{\mathtt{x}}^{\prime}(t)=F(t,{\mathtt{x}}(t),w(t),(Sw)(t))\
\ \mbox{\rm for a.e.}\ \,t\in(0,T),\\\\[2.84526pt] \langle
w^{\prime}(t)+A(t,{\mathtt{x}}(t),w(t))+(R_{1}w)(t),v-w(t)\rangle+\,j^{0}(t,{\mathtt{x}}(t),(R_{2}w)(t),Mw(t);Mv-
Mw(t))\\\\[2.84526pt] \ \
\quad\qquad+\,\varphi(t,{\mathtt{x}}(t),(R_{3}w)(t),Mv)-\varphi(t,{\mathtt{x}}(t),(R_{3}w)(t),Mw(t))\geq\langle
f(t,{\mathtt{x}}(t)),v-w(t)\rangle\\\\[2.84526pt] \qquad\qquad\qquad\qquad\ \
\mbox{\rm for all}\ v\in K,\ \mbox{\rm a.e.}\ t\in(0,T),\\\
{\mathtt{x}}(0)={\mathtt{x}}_{0},\ \ w(0)=w_{0}.\end{cases}$

We need the following hypotheses on the data of Problem 3.

$\underline{H({F})}:$ $\displaystyle F\colon(0,T)\times E\times V\times U\to
E$ is such that

  * (a)

$F(\cdot,{\mathtt{x}},v,u)$ belongs to $L^{2}(0,T;E)$ for all ${\mathtt{x}}\in
E$, $v\in V$, $u\in U$.

  * (b)

$\|F(t,{\mathtt{x}}_{1},v_{1},u_{1})-F(t,{\mathtt{x}}_{2},v_{2},u_{2})\|_{E}\leq
L_{F}\,(\|{\mathtt{x}}_{1}-{\mathtt{x}}_{2}\|_{E}+\|v_{1}-v_{2}\|+\|u_{1}-u_{2}\|_{U})$
for all ${\mathtt{x}}_{1}$, ${\mathtt{x}}_{2}\in E$, $v_{1}$, $v_{2}\in V$,
$u_{1}$, $u_{2}\in U$, a.e. $t\in(0,T)$.

$\underline{H({A})}:$ $\displaystyle A\colon(0,T)\times E\times V\to V^{*}$ is
such that

  * (a)

$A(\cdot,{\mathtt{x}},v)$ is measurable for all ${\mathtt{x}}\in E$, $v\in V$.

  * (b)

$A(t,\cdot,v)$ is continuous for all $v\in V$, a.e. $t\in(0,T)$.

  * (c)

$\|A(t,{\mathtt{x}},v)\|_{V^{*}}\leq
a_{0}(t)+a_{1}\|{\mathtt{x}}\|_{E}+a_{2}\|v\|$ for all ${\mathtt{x}}\in E$,
$v\in V$, a.e. $t\in(0,T)$ with $a_{0}\in L^{2}(0,T)$, $a_{0}$, $a_{1}$,
$a_{2}\geq 0$.

  * (d)

$A(t,{\mathtt{x}},\cdot)$ is demicontinuous and there are constants $m_{A}>0$,
${\overline{m}}_{A}\geq 0$ such that

$\langle
A(t,{\mathtt{x}}_{1},v_{1})-A(t,{\mathtt{x}}_{2},v_{2}),v_{1}-v_{2}\rangle\geq
m_{A}\,\|v_{1}-v_{2}\|^{2}-{\overline{m}}_{A}\,\|{\mathtt{x}}_{1}-{\mathtt{x}}_{2}\|_{E}\,\|v_{1}-v_{2}\|$

for all ${\mathtt{x}}_{1}$, ${\mathtt{x}}_{2}\in E$, $v_{1}$, $v_{2}\in V$,
a.e. $t\in(0,T)$.

$\underline{H(j)}:$ $j\colon(0,T)\times E\times Z\times X\to\mathbb{R}$ is
such that

  * (a)

$j(\cdot,{\mathtt{x}},z,v)$ is measurable for all ${\mathtt{x}}\in E$, $z\in
Z$, $v\in X$.

  * (b)

$j(t,\cdot,\cdot,v)$ is continuous for all $v\in X$, a.e. $t\in(0,T)$.

  * (c)

$j(t,{\mathtt{x}},z,\cdot)$ is locally Lipschitz for all ${\mathtt{x}}\in E$,
$z\in Z$, a.e. $t\in(0,T)$.

  * (d)

$\|\partial j(t,{\mathtt{x}},z,v)\|_{X^{*}}\leq
c_{0j}(t)+c_{1j}\|{\mathtt{x}}\|_{E}+c_{2j}\|z\|_{Z}+c_{3j}\|v\|_{X}$ for all
${\mathtt{x}}\in E$, $z\in Z$, $v\in X$, a.e. $t\in(0,T)$ with $c_{0j}\in
L^{2}(0,T)$, $c_{0j}$, $c_{1j}$, $c_{2j}$, $c_{3j}\geq 0$.

  * (e)

$j^{0}(t,{\mathtt{x}}_{1},z_{1},v_{1};v_{2}-v_{1})+j^{0}(t,{\mathtt{x}}_{2},z_{2},v_{2};v_{1}-v_{2})\\\
~{}~{}\qquad\qquad\qquad\qquad\leq
m_{j}\,\|v_{1}-v_{2}\|^{2}_{X}+{\overline{m}}_{j}\,(\|{\mathtt{x}}_{1}-{\mathtt{x}}_{2}\|_{E}+\|z_{1}-z_{2}\|_{Z})\,\|v_{1}-v_{2}\|_{X}$
for all ${\mathtt{x}}_{i}\in E$, $z_{i}\in Z$, $v_{i}\in X$, $i=1$, $2$, a.e.
$t\in(0,T)$ with $m_{j}$, ${\overline{m}}_{j}\geq 0$.

$\underline{H(\varphi)}:$ $\varphi\colon(0,T)\times E\times Y\times
X\to\mathbb{R}$ is such that

  * (a)

$\varphi(\cdot,{\mathtt{x}},y,v)$ is measurable for all ${\mathtt{x}}\in E$,
$y\in Y$, $v\in X$.

  * (b)

$\varphi(t,\cdot,\cdot,v)$ is continuous for all $v\in X$, a.e. $t\in(0,T)$.

  * (c)

$\varphi(t,{\mathtt{x}},y,\cdot)$ is convex and lower semicontinuous for all
${\mathtt{x}}\in E$, $y\in Y$, a.e. $t\in(0,T)$.

  * (d)

$\|\partial\varphi(t,{\mathtt{x}},y,v)\|_{X^{*}}\leq
c_{0\varphi}(t)+c_{1\varphi}\|{\mathtt{x}}\|_{E}+c_{2\varphi}\|y\|_{Y}+c_{3\varphi}\|v\|_{X}$
for all ${\mathtt{x}}\in E$, $y\in Y$, $v\in X$, a.e. $t\in(0,T)$ with
$c_{0\varphi}\in L^{2}(0,T)$, $c_{0\varphi}$, $c_{1\varphi}$, $c_{2\varphi}$,
$c_{3\varphi}\geq 0$.

  * (e)

$\varphi(t,{\mathtt{x}}_{1},y_{1},v_{2})-\varphi(t,{\mathtt{x}}_{1},y_{1},v_{1})+\varphi(t,{\mathtt{x}}_{2},y_{2},v_{1})-\varphi(t,{\mathtt{x}}_{2},y_{2},v_{2})\\\
~{}~{}\qquad\qquad\qquad\qquad\leq
m_{\varphi}\,(\|{\mathtt{x}}_{1}-{\mathtt{x}}_{2}\|_{E}+\|y_{1}-y_{2}\|_{Y})\,\|v_{1}-v_{2}\|_{X}$
for all ${\mathtt{x}}_{i}\in E$, $y_{i}\in Y$, $v_{i}\in X$, $i=1$, $2$, a.e.
$t\in(0,T)$ with $m_{\varphi}\geq 0$.

$\underline{H(K)}:$ $K$ is a closed and convex subset of $V$ with $0\in K$.

$\underline{H(M)}:$ $M\colon V\rightarrow X$ is such that

  * (a)

$M$ is an affine operator.

  * (b)

the Nemitsky operator ${\mathcal{M}}\colon L^{2}(0,T;V)\to L^{2}(0,T;X)$
corresponding to $M$ is compact.

$\underline{H(f)}:$ $f\colon(0,T)\times E\to V^{*}$ is such that

  * (a)

$f(\cdot,{\mathtt{x}})$ belongs to $L^{2}(0,T;V^{*})$ for all ${\mathtt{x}}\in
E$.

  * (b)

$\|f(t,{\mathtt{x}}_{1})-f(t,{\mathtt{x}}_{2})\|_{V^{*}}\leq
L_{f}\,\|{\mathtt{x}}_{1}-{\mathtt{x}}_{2}\|_{E}$ for all ${\mathtt{x}}_{1}$,
${\mathtt{x}}_{2}\in E$, a.e. $t\in(0,T)$ with $L_{f}>0$.

$\underline{(H_{1})}:$ $m_{j}\|M_{1}\|^{2}<m_{A}$, where $M_{1}\colon V\to X$
is defined by $M_{1}v=Mv-M0$ for $v\in V$.

$\underline{(H_{2})}:$ ${\mathtt{x}}_{0}\in E$, $w_{0}\in V$.

$\underline{(H_{3})}:$ $S\colon L^{2}(0,T;V)\to L^{2}(0,T;U)$, ${R}_{1}\colon
L^{2}(0,T;V)\to L^{2}(0,T;V^{*})$,

${R}_{2}\colon L^{2}(0,T;V)\to L^{2}(0,T;Z)$, ${R}_{3}\colon L^{2}(0,T;V)\to
L^{2}(0,T;Y)$ are such that

  * (a)

$\displaystyle\|(Sv_{1})(t)-(Sv_{2})(t)\|_{U}\leq
c_{S}\int_{0}^{t}\|v_{1}(s)-v_{2}(s)\|\,ds$,

  * (b)

$\displaystyle\|({R}_{1}v_{1})(t)-({R}_{1}v_{2})(t)\|_{V^{*}}\leq
c_{R_{1}}\int_{0}^{t}\|v_{1}(s)-v_{2}(s)\|\,ds$,

  * (c)

$\displaystyle\|({R}_{2}v_{1})(t)-({R}_{2}v_{2})(t)\|_{Z}\leq
c_{R_{2}}\int_{0}^{t}\|v_{1}(s)-v_{2}(s)\|\,ds$,

  * (d)

$\displaystyle\|({R}_{3}v_{1})(t)-({R}_{3}v_{2})(t)\|_{Y}\leq
c_{R_{3}}\int_{0}^{t}\|v_{1}(s)-v_{2}(s)\|\,ds$

for all $v_{1}$, $v_{2}\in L^{2}(0,T;V)$, a.e. $t\in(0,T)$ with $c_{S}$,
$c_{R_{1}}$, $c_{R_{2}}$, $c_{R_{3}}>0$.

######  Remark 3.1.

In hypothesis $H(M)$, the operator $M$ is an affine operator if and only if
the operator $M_{1}$ defined in $(H_{1})$ is linear. The operator $M_{1}$ is
called the linear part of $M$. In $(H_{1})$ and in what follows, the constant
$\|M_{1}\|$ denotes the norm in ${\mathcal{L}}(V,X)$ of the linear part
$M_{1}$ of $M$. It is clear that if $M\in{\mathcal{L}}(V,X)$, then $M_{1}=M$.

######  Remark 3.2.

The following conditions are useful while checking the hypothesis $H(A)$(d).
If $A(t,{\mathtt{x}},\cdot)$ is strongly monotone with $m_{A}>0$ and
$A(t,\cdot,v)$ is Lipschitz with $L_{A}>0$, i.e.,

$\displaystyle\langle
A(t,{\mathtt{x}},v_{1})-A(t,{\mathtt{x}},v_{2}),v_{1}-v_{2}\rangle\geq
m_{A}\|v_{1}-v_{2}\|^{2}\ \ \mbox{for all}\ \ {\mathtt{x}}\in E,\
v_{1},v_{2}\in V,$
$\displaystyle\|A(t,{\mathtt{x}}_{1},v)-A(t,{\mathtt{x}}_{2},v)\|_{V^{*}}\leq
L_{A}\,\|{\mathtt{x}}_{1}-{\mathtt{x}}_{2}\|_{E}\ \ \mbox{for all}\ \
{\mathtt{x}}_{1},{\mathtt{x}}_{2}\in E,\ v\in V,$

for a.e. $t\in(0,T)$, then

$\langle
A(t,{\mathtt{x}}_{1},v_{1})-A(t,{\mathtt{x}}_{2},v_{2}),v_{1}-v_{2}\rangle\geq
m_{A}\,\|v_{1}-v_{2}\|^{2}-L_{A}\,\|{\mathtt{x}}_{1}-{\mathtt{x}}_{2}\|_{E}\|v_{1}-v_{2}\|$

for all ${\mathtt{x}}_{1}$, ${\mathtt{x}}_{2}\in E$, $v_{1}$, $v_{2}\in V$,
a.e. $t\in(0,T)$.

The main result on the well-posedness of Problem 3 reads as follows.

######  Theorem 4.

Under hypotheses $H(F)$, $H(A)$, $H(j)$, $H(\varphi)$, $H(K)$, $H(M)$, $H(f)$,
$(H_{1})$–$(H_{3})$, for each $({\mathtt{x}}_{0},w_{0})\in E\times V$, Problem
3 has a unique solution $({\mathtt{x}},w)\in H^{1}(0,T;E)\times{\mathbb{W}}$
with $w(t)\in K$ for a.e. $t\in(0,T)$. Further, for each initial conditions
$({\mathtt{x}}_{0},w_{0})$,
$(\widetilde{\mathtt{x}}_{0},{\widetilde{w}}_{0})\in E\times V$, there is a
constant $c>0$ such that

$\|{\mathtt{x}}-{\widetilde{{\mathtt{x}}}}\|_{H^{1}(0,T;E)}+\|w-{\widetilde{w}}\|_{L^{2}(0,T;V)}\leq
c\,(\|{\mathtt{x}}_{0}-\widetilde{\mathtt{x}}_{0}\|_{E}+\|w_{0}-\widetilde{w}_{0}\|),$
(4)

where $({\mathtt{x}},w)$ and $({\widetilde{{\mathtt{x}}}},{\widetilde{w}})$
denote the unique solutions to Problem 3 corresponding to
$({\mathtt{x}}_{0},w_{0})$ and
$(\widetilde{\mathtt{x}}_{0},{\widetilde{w}}_{0})$, respectively. Further, the
solution has the regularity $({\mathtt{x}},w)\in C(0,T;E\times H)$.

Proof. The proof is carried out in several steps and it is inspired by a
recent result from [30, Theorem 20] combined with a fixed-point argument of
Lemma 2.

Step 1. Let $\lambda\in L^{2}(0,T;E)$, $\xi\in L^{2}(0,T;V^{*})$, $\zeta\in
L^{2}(0,T;Z)$, and $\eta\in L^{2}(0,T;Y)$ be fixed and consider the following
auxiliary problem.

######  Problem 5.

Find $w=w_{\lambda\xi\zeta\eta}\in{\mathbb{W}}$ with $w(t)\in K$ for a.e.
$t\in(0,T)$ such that

$\begin{cases}\displaystyle\langle
w^{\prime}(t)+A(t,\lambda(t),w(t))+\xi(t),v-w(t)\rangle+\,j^{0}(t,\lambda(t),\zeta(t),Mw(t);Mv-
Mw(t))\\\\[2.84526pt] \ \
\quad\qquad+\,\varphi(t,\lambda(t),\eta(t),Mv)-\varphi(t,\lambda(t),\eta(t),Mw(t))\geq\langle
f(t,\lambda(t)),v-w(t)\rangle\\\\[2.84526pt] \qquad\qquad\qquad\qquad\ \
\mbox{\rm for all}\ v\in K,\ \mbox{\rm a.e.}\ t\in(0,T),\\\
w(0)=w_{0}.\end{cases}$ (5)

We define operator ${\widetilde{A}}\colon(0,T)\times V\to V^{*}$, functions
${\widetilde{j}}\colon(0,T)\times X\to\mathbb{R}$,
${\widetilde{\varphi}}\colon(0,T)\times X\to\mathbb{R}$, and
${\widetilde{f}}\colon(0,T)\to V^{*}$ by

$\displaystyle{\widetilde{A}}(t,v)=A(t,\lambda(t),v)\ \ \mbox{for}\ \ v\in
V,\,\mbox{a.e.}\ \,t\in(0,T),$
$\displaystyle\quad{\widetilde{j}}(t,v)=j(t,\lambda(t),\zeta(t),v)\ \
\mbox{for}\ \ v\in X,\,\mbox{a.e.}\ \,t\in(0,T),$
$\displaystyle\qquad{\widetilde{\varphi}}(t,v)=\varphi(t,\lambda(t),\eta(t),v)\
\ \mbox{for}\ \ v\in X,\,\mbox{a.e.}\ \,t\in(0,T),$
$\displaystyle\qquad\quad{\widetilde{f}}(t)=f(t,\lambda(t))-\xi(t)\ \
\mbox{for a.e.}\ \,t\in(0,T).$

With this notation Problem 5 can be equivalently reformulated as follows. Find
element $w=w_{\lambda\xi\zeta\eta}\in{\mathbb{W}}$ with $w(t)\in K$ for a.e.
$t\in(0,T)$ such that

$\begin{cases}\displaystyle\langle
w^{\prime}(t)+{\widetilde{A}}(t,w(t)),v-w(t)\rangle+\,{\widetilde{j}}^{0}(t,Mw(t);Mv-
Mw(t))\\\\[2.84526pt]
\quad\qquad+\,{\widetilde{\varphi}}(t,Mv)-{\widetilde{\varphi}}(t,Mw(t))\geq\langle{\widetilde{f}}(t),v-w(t)\rangle\
\ \mbox{\rm for all}\ v\in K,\ \mbox{\rm a.e.}\ t\in(0,T),\\\\[2.84526pt]
w(0)=w_{0}.\end{cases}$ (6)

We shall establish the properties of the data in Problem 6.

By hypothesis $H(A)$(a), (b), it is clear that $A(\cdot,\cdot,v)$ is a
Carathéodory function for all $v\in V$. Therefore, the measurability of
$t\mapsto\lambda(t)$ entails, see [8, Corollary 2.5.24], that
${\widetilde{A}}(\cdot,v)$ is measurable for all $v\in V$. We use $H(A)$(c),
(d) to get that ${\widetilde{A}}(t,\cdot)$ is demicontinuous for a.e.
$t\in(0,T)$ and

$\|{\widetilde{A}}(t,v)\|_{V^{*}}\leq{\widetilde{a}}_{0}(t)+a_{2}\|v\|\ \
\mbox{for all}\ \ v\in V,\ \mbox{a.e.}\ \,t\in(0,T),$

where ${\widetilde{a}}_{0}\in L^{2}(0,T)$. Further, we employ $H(A)$(d) to
infer that ${\widetilde{A}}(t,\cdot)$ is strongly monotone with constant
$m_{A}>0$ for a.e. $t\in(0,T)$. The function ${\widetilde{j}}(\cdot,v)$ is
measurable because of $H(j)$(c) and of the measurability of
$t\mapsto\lambda(t)$ and $t\mapsto\zeta(t)$. It is clear that
${\widetilde{j}}(t,\cdot)$ is locally Lipschitz for a.e. $t\in(0,T)$ and

$\|\partial{\widetilde{j}}(t,v)\|_{X^{*}}\leq{\widetilde{c}}_{0j}+c_{3j}\|v\|_{X}\
\ \mbox{for all}\ \ v\in X,\ \mbox{a.e.}\ \,t\in(0,T).$

with ${\widetilde{c}}_{0j}\in L^{2}(0,T)$. From $H(j)$(e), we know that

${\widetilde{j}}^{0}(t,v_{1};v_{2}-v_{1})+{\widetilde{j}}^{0}(t,v_{2};v_{1}-v_{2})\leq
m_{j}\,\|v_{1}-v_{2}\|_{X}^{2}\ \ \mbox{for all}\ \ v_{1},v_{2}\in X,\
\mbox{a.e.}\ \,t\in(0,T).$

By $H(\varphi)$, we deduce that ${\widetilde{\varphi}}(\cdot,v)$ is measurable
for all $v\in X$, ${\widetilde{\varphi}}(t,\cdot)$ is convex and lower
semicontinuous for a.e. $t\in(0,T)$, and

$\|\partial{\widetilde{\varphi}}(t,v)\|_{X^{*}}\leq{\widetilde{c}}_{0\varphi}(t)+c_{3\varphi}\|v\|_{X}\
\ \mbox{for all}\ \ v\in X,\ \mbox{a.e.}\ \,t\in(0,T)$

with ${\widetilde{c}}_{0\varphi}\in L^{2}(0,T)$. Since $\lambda\in
L^{2}(0,T;E)$ and $\xi\in L^{2}(0,T;V^{*})$, we use $H(f)$ to obtain
${\widetilde{f}}\in L^{2}(0,T;V^{*})$. Having verified the above properties of
the data, and having in mind hypotheses $H(K)$, $(H_{1})$–$(H_{3})$, we are in
a position to apply [30, Theorem 20] to deduce that the problem (6), and
equivalently Problem 5, is uniquely solvable.

Step 2. Let $(\lambda_{i},\xi_{i},\zeta_{i},\eta_{i})\in L^{2}(0,T;E\times
V^{*}\times Z\times Y)$, $i=1$, $2$ and
$w_{1}=w_{\lambda_{1}\xi_{1}\zeta_{1}\eta_{1}}$,
$w_{2}=w_{\lambda_{2}\xi_{2}\zeta_{2}\eta_{2}}\in{\mathbb{W}}$ with
$w_{1}(t)$, $w_{2}(t)\in K$ for a.e. $t\in(0,T)$, be the unique solutions to
(6) corresponding to $(\lambda_{1},\xi_{1},\zeta_{1},\eta_{1})$ and
$(\lambda_{2},\xi_{2},\zeta_{2},\eta_{2})$, respectively. We shall prove the
following estimate

$\displaystyle\|w_{1}-w_{2}\|_{L^{2}(0,t;V)}\leq
c\,\big{(}\|\lambda_{1}-\lambda_{2}\|_{L^{2}(0,t;E)}+\|\xi_{1}-\xi_{2}\|_{L^{2}(0,t;V^{*})}$
$\displaystyle\qquad\qquad+\|\zeta_{1}-\zeta_{2}\|_{L^{2}(0,t;Z)}+\|\eta_{1}-\eta_{2}\|_{L^{2}(0,t;Y)}\big{)}$
(7)

for all $t\in[0,T]$, where $c>0$ is a constant. From Problem 5 it follows that

$\displaystyle\displaystyle\langle
w_{1}^{\prime}(t)+A(t,\lambda_{1}(t),w_{1}(t))+\xi_{1}(t),w_{2}(t)-w_{1}(t)\rangle$
$\displaystyle\quad+\,j^{0}(t,\lambda_{1}(t),\zeta_{1}(t),Mw_{1}(t);Mw_{2}(t)-Mw_{1}(t))$
$\displaystyle\qquad+\,\varphi(t,\lambda_{1}(t),\eta_{1}(t),Mw_{2}(t))-\varphi(t,\lambda_{1}(t),\eta_{1}(t),Mw_{1}(t))\geq\langle
f(t,\lambda_{1}(t)),w_{2}(t)-w_{1}(t)\rangle$

for a.e. $t\in(0,T)$ and

$\displaystyle\displaystyle\langle
w_{2}^{\prime}(t)+A(t,\lambda_{2}(t),w_{2}(t))+\xi_{2}(t),w_{1}(t)-w_{2}(t)\rangle$
$\displaystyle\quad+\,j^{0}(t,\lambda_{2}(t),\zeta_{2}(t),Mw_{2}(t);Mw_{1}(t)-Mw_{2}(t))$
$\displaystyle\qquad+\,\varphi(t,\lambda_{2}(t),\eta_{2}(t),Mw_{1}(t))-\varphi(t,\lambda_{2}(t),\eta_{2}(t),Mw_{2}(t))\geq\langle
f(t,\lambda_{2}(t)),w_{1}(t)-w_{2}(t)\rangle$

for a.e. $t\in(0,T)$, and $w_{1}(0)-w_{2}(0)=0$. Next, we sum up the last two
inequalities to get

$\displaystyle\langle
w^{\prime}_{1}(t)-w^{\prime}_{2}(t),w_{1}(t)-w_{2}(t)\rangle+\langle
A(t,\lambda_{1}(t),w_{1}(t))-A(t,\lambda_{2}(t),w_{2}(t)),w_{1}(t)-w_{2}(t)\rangle$
$\displaystyle\leq\langle\xi_{1}(t)-\xi_{2}(t),w_{2}(t)-w_{1}(t)\rangle+\langle
f(t,\lambda_{1}(t))-f(t,\lambda_{2}(t)),w_{1}(t)-w_{2}(t)\rangle$
$\displaystyle+j^{0}(t,\lambda_{1}(t),\zeta_{1}(t),Mw_{1}(t);Mw_{2}(t)-Mw_{1}(t))+j^{0}(t,\lambda_{2}(t),\zeta_{2}(t),Mw_{2}(t);Mw_{1}(t)-Mw_{2}(t))$
$\displaystyle+\,\varphi(t,\lambda_{1}(t),\eta_{1}(t),Mw_{2}(t))-\varphi(t,\lambda_{1}(t),\eta_{1}(t),Mw_{1}(t))$
$\displaystyle\hskip
2.84544pt+\,\varphi(t,\lambda_{2}(t),\eta_{2}(t),Mw_{1}(t))-\varphi(t,\lambda_{2}(t),\eta_{2}(t),Mw_{2}(t))$

for a.e. $t\in(0,T)$. We integrate the above inequality on $(0,t)$, apply the
integration by parts formula, see [8, Proposition 3.4.14], to the first term,
use hypotheses $H(A)$(d), $H(j)$(e), $H(\varphi)$(e) to obtain

$\displaystyle\frac{1}{2}\|w_{1}(t)-w_{2}(t)\|^{2}_{H}-\frac{1}{2}\|w_{1}(0)-w_{2}(0)\|^{2}_{H}+m_{A}\,\int_{0}^{t}\|w_{1}(s)-w_{2}(s)\|^{2}\,ds$
$\displaystyle\leq{\overline{m}}_{A}\int_{0}^{t}\|\lambda_{1}(s)-\lambda_{2}(s)\|_{E}\,\|w_{1}(s)-w_{2}(s)\|\,ds+\int_{0}^{t}\|\xi_{1}(s)-\xi_{2}(s)\|_{V^{*}}\,\|w_{1}(s)-w_{2}(s)\|\,ds$
$\displaystyle+\int_{0}^{t}\|f(s,\lambda_{1}(s))-f(s,\lambda_{2}(s))\|_{V^{*}}\,\|w_{1}(s)-w_{2}(s)\|\,ds+m_{j}\int_{0}^{t}\|Mw_{2}(s)-Mw_{1}(s)\|_{X}^{2}$
$\displaystyle\hskip
2.84544pt+\,{\overline{m}}_{j}\int_{0}^{t}\Big{(}\|\lambda_{1}(s)-\lambda_{2}(s)\|_{E}+\|\zeta_{1}(s)-\zeta_{2}(s)\|_{Z}\Big{)}\|w_{1}(s)-w_{2}(s)\|\,ds$
$\displaystyle\hskip
8.5359pt+\,m_{\varphi}\int_{0}^{t}\Big{(}\|\lambda_{1}(s)-\lambda_{2}(s)\|_{E}+\|\eta_{1}(s)-\eta_{2}(s)\|_{Y}\Big{)}\|w_{1}(s)-w_{2}(s)\|\,ds.$

for all $t\in[0,T]$. Next, using hypothesis $H(f)$(b), condition
$w_{1}(0)-w_{2}(0)=0$ and the inequality $\|Mv\|_{X}\leq\|M_{1}\|\,\|v\|$ for
all $v\in V$, and the Hölder inequality, we have

$\displaystyle\big{(}m_{A}-m_{j}\|M_{1}\|^{2}\big{)}\,\|w_{1}-w_{2}\|_{L^{2}(0,t;V)}^{2}\leq\|\xi_{1}-\xi_{2}\|_{L^{2}(0,t;V^{*})}\|w_{1}-w_{2}\|_{L^{2}(0,t;V)}$
$\displaystyle\quad+\,\Big{(}{\overline{m}}_{A}+L_{f}+{\overline{m}}_{j}\|M_{1}\|+m_{\varphi}\|M_{1}\|\Big{)}\,\|\lambda_{1}-\lambda_{2}\|_{L^{2}(0,t;E)}\|w_{1}-w_{2}\|_{L^{2}(0,t;V)}$
$\displaystyle\qquad+\,\Big{(}{\overline{m}}_{j}\|M_{1}\|\,\|\zeta_{1}-\zeta_{2}\|_{L^{2}(0,t;Z)}+m_{\varphi}\|M_{1}\|\|\eta_{1}-\eta_{2}\|_{L^{2}(0,t;Y)}\Big{)}\,\|w_{1}-w_{2}\|_{L^{2}(0,t;V)}$

for all $t\in[0,T]$. Hence, by $(H_{1})$, the inequality (3) follows.

Step 3. Let $w\in L^{2}(0,T;V)$ be fixed. We claim that there exists a unique
function ${\mathtt{x}}\in H^{1}(0,T;E)$ solution to the Cauchy problem

$\displaystyle\begin{cases}\displaystyle{\mathtt{x}}^{\prime}(t)=F(t,{\mathtt{x}}(t),w(t),(Sw)(t))\
\ \mbox{for a.e.}\ \ t\in(0,T)\\\
{\mathtt{x}}(0)={\mathtt{x}}_{0}.\end{cases}$ (8)

Further, if $w_{1}$, $w_{2}\in L^{2}(0,T;E)$ and ${\mathtt{x}}_{1}$,
${\mathtt{x}}_{2}\in H^{1}(0,T;E)$ denote the unique solutions to (8)
corresponding to $w_{1}$ and $w_{2}$, respectively, then

$\|{\mathtt{x}}_{1}(t)-{\mathtt{x}}_{2}(t)\|_{E}\leq
m\int_{0}^{t}\|w_{1}(s)-w_{2}(s)\|\,ds\ \ \mbox{for all}\ \ t\in(0,T)$ (9)

with a constant $m>0$. In fact, let $w\in L^{2}(0,T;V)$ and define the
function ${\widetilde{F}}\colon(0,T)\times E\to E$ by

${\widetilde{F}}(t,{\mathtt{x}})=F(t,{\mathtt{x}},w(t),(Sw)(t))\ \ \mbox{for}\
\ {\mathtt{x}}\in E,\ \mbox{a.e.}\ \ t\in(0,T).$

By hypotheses $H(F)$, we obtain

$\|{\widetilde{F}}(t,{\mathtt{x}}_{1})-{\widetilde{F}}(t,{\mathtt{x}}_{2})\|_{E}=\|F(t,{\mathtt{x}}_{1},w(t),(Sw)(t))-F(t,{\mathtt{x}}_{2},w(t),(Sw)(t))\|_{E}\leq
L_{f}\|{\mathtt{x}}_{1}-{\mathtt{x}}_{2}\|_{E}$

for all ${\mathtt{x}}_{1}$, ${\mathtt{x}}_{2}\in E$, a.e. $t\in(0,T)$, and

$(0,T)\ni t\mapsto{\widetilde{F}}(t,{\mathtt{x}})\in E$

is a $L^{2}$-function for any ${\mathtt{x}}\in E$. Hence, we are able to apply
the classical Cauchy-Lipschitz theorem, see e.g., [35, Theorem 2.30] to deduce
the unique solvability of (8). Next, for $w_{1}$, $w_{2}\in L^{2}(0,T;V)$, we
integrate equation in (8) to get

${\mathtt{x}}_{i}(t)={\mathtt{x}}_{0}+\int_{0}^{t}F(s,{\mathtt{x}}_{i}(s),w_{i}(s),(Sw_{i})(s))\,ds\
\ \mbox{for}\ \ i=1,2.$

Thus

$\displaystyle\|{\mathtt{x}}_{1}(t)-{\mathtt{x}}_{2}(t)\|_{E}\leq\int_{0}^{t}\|F(s,{\mathtt{x}}_{1}(s),w_{1}(s),(Sw_{1})(s))-F(s,{\mathtt{x}}_{2}(s),w_{2}(s),(Sw_{2})(s))\|_{E}\,ds$
$\displaystyle\quad\leq
L_{F}\int_{0}^{t}\|{\mathtt{x}}_{1}(s)-{\mathtt{x}}_{2}(s)\|_{E}+\|w_{1}(s)-w_{2}(s)\|+\|(Sw_{1})(s)-(Sw_{2})(s)\|_{U}\,ds$
$\displaystyle\qquad\leq
L_{F}\int_{0}^{t}\|{\mathtt{x}}_{1}(s)-{\mathtt{x}}_{2}(s)\|_{E}+\|w_{1}(s)-w_{2}(s)\|\,ds+c_{S}L_{F}T\int_{0}^{t}\|w_{1}(s)-w_{2}(s)\|\,ds$

for all $t\in[0,T]$. Finally, we use the standard argument based on the
Growall inequality to deduce inequality (9) with a constant $m>0$.

We denote by $R_{0}\colon L^{2}(0,T;V)\to H^{1}(0,T;E)\subset L^{2}(0,T;E)$
the operator $R_{0}(w)={\mathtt{x}}$ which to any $w\in L^{2}(0,T;V)$ assigns
the unique solution ${\mathtt{x}}\in L^{2}(0,T;E)$ to the problem (8). The
inequality (9) can be restated as follows

$\|(R_{0}w_{1})(t)-(R_{0}w_{2})(t)\|_{E}\leq
m\int_{0}^{t}\|w_{1}(s)-w_{2}(s)\|\,ds\ \ \mbox{for all}\ \ t\in(0,T),$ (10)

i.e., $R_{0}$ is a history-dependent operator.

Step 4. In this part of the proof we apply a fixed point argument. We define
the operator $\Lambda\colon L^{2}(0,T;E\times V^{*}\times Z\times Y)\to
L^{2}(0,T;E\times V^{*}\times Z\times Y)$ by

$\Lambda(\lambda,\xi,\zeta,\eta)=(R_{0}w_{\lambda\xi\zeta\eta},{R}_{1}w_{\lambda\xi\zeta\eta},{R_{2}}w_{\lambda\xi\zeta\eta},{R_{3}}w_{\lambda\xi\zeta\eta})$

for all $(\lambda,\xi,\zeta,\eta)\in L^{2}(0,T;E\times V^{*}\times Z\times
Y)$, where $w_{\lambda\xi\zeta\eta}\in{\mathbb{W}}$ denotes the unique
solution to Problem 5 corresponding to $(\lambda,\xi,\zeta,\eta)$. From
hypothesis $(H_{3})$(b)–(d), inequalities (3) and (10), and by the Hölder
inequality, we find a constant $c>0$ such that

$\displaystyle\|\Lambda(\lambda_{1},\xi_{1},\zeta_{1},\eta_{1})(t)-\Lambda(\lambda_{1},\xi_{2},\zeta_{2},\eta_{2})(t)\|^{2}_{E\times
V^{*}\times Z\times Y}$
$\displaystyle=\|(R_{0}w_{1})(t)-(R_{0}w_{2})(t)\|_{E}^{2}+\|({R}_{1}w_{1})(t)-({R}_{1}w_{2})(t)\|_{V^{*}}^{2}$
$\displaystyle+\|({R_{2}}w_{1})(t)-({R_{2}}w_{2})(t)\|_{Z}^{2}+\|({R_{3}}w_{1})(t)-({R_{3}}w_{2})(t)\|_{Y}^{2}$
$\displaystyle\leq\Big{(}m\int_{0}^{t}\|w_{1}(s)-w_{2}(s)\|\,ds\Big{)}^{2}+\Big{(}c_{R_{1}}\int_{0}^{t}\|w_{1}(s)-w_{2}(s)\|\,ds\Big{)}^{2}$
$\displaystyle\hskip
5.69046pt+\Big{(}c_{R_{2}}\int_{0}^{t}\|w_{1}(s)-w_{2}(s)\|\,ds\Big{)}^{2}+\Big{(}c_{R_{3}}\int_{0}^{t}\|w_{1}(s)-w_{2}(s)\|\,ds\Big{)}^{2}\leq
c\,\|w_{1}-w_{2}\|_{L^{2}(0,t;V)}^{2}$ $\displaystyle\hskip 11.38092pt\leq
c\,\Big{(}\|\lambda_{1}-\lambda_{2}\|^{2}_{L^{2}(0,t;E)}+\|\xi_{1}-\xi_{2}\|^{2}_{L^{2}(0,t;V^{*})}+\|\zeta_{1}-\zeta_{2}\|^{2}_{L^{2}(0,t;Z)}+\|\eta_{1}-\eta_{2}\|^{2}_{L^{2}(0,t;Y)}\Big{)},$

which implies

$\displaystyle\|\Lambda(\lambda_{1},\xi_{1},\zeta_{1},\eta_{1})(t)-\Lambda(\lambda_{2},\xi_{2},\zeta_{2},\eta_{2})(t)\|^{2}_{E\times
V^{*}\times Z\times Y}$ $\displaystyle\qquad\qquad\leq
c\,\int_{0}^{t}\,\|(\lambda_{1},\xi_{1},\zeta_{1},\eta_{1})(s)-(\lambda_{2},\xi_{2},\zeta_{2},\eta_{2})(s)\|^{2}_{E\times
V^{*}\times Z\times Y}\,ds$ (11)

for a.e. $t\in(0,T)$. By Lemma 2, we deduce that there exists a unique fixed
point $(\lambda^{*},\xi^{*},\zeta^{*},\eta^{*})$ of $\Lambda$, that is

$(\lambda^{*},\xi^{*},\zeta^{*},\eta^{*})\in L^{2}(0,T;E\times V^{*}\times
Z\times Y)\ \ {\rm and}\ \
\Lambda(\lambda^{*},\xi^{*},\zeta^{*},\eta^{*})=(\lambda^{*},\xi^{*},\zeta^{*},\eta^{*}).$

Let $(\lambda^{*},\xi^{*},\zeta^{*},\eta^{*})\in L^{2}(0,T;E\times V^{*}\times
Z\times Y)$ be the unique fixed point of the operator $\Lambda$. We define
$w_{\lambda^{*}\xi^{*}\zeta^{*}\eta^{*}}\in{\mathbb{W}}$ to be the unique
solution to Problem 5 corresponding to
$(\lambda^{*},\xi^{*},\zeta^{*},\eta^{*})$. From the definition of the
operator $\Lambda$, we have

$\lambda^{*}=R_{0}(w_{\lambda^{*}\xi^{*}\zeta^{*}\eta^{*}}),\ \
\xi^{*}={R}_{1}(w_{\lambda^{*}\xi^{*}\zeta^{*}\eta^{*}}),\ \
\eta^{*}={R_{2}}(w_{\lambda^{*}\xi^{*}\zeta^{*}\eta^{*}})\ \ \mbox{and}\ \
\zeta^{*}={R_{3}}(w_{\lambda^{*}\xi^{*}\zeta^{*}\eta^{*}}).$

Finally, we use these relations in Problem 5, and conclude that
$w_{\lambda^{*}\xi^{*}\zeta^{*}\eta^{*}}\in{\mathbb{W}}$ is the unique
solution to Problem 5.

Step 5. Finally, we shall prove (4). Let $({\mathtt{x}},w)$ and
$({\widetilde{{\mathtt{x}}}},{\widetilde{w}})$ be the unique solutions to
Problem 3 corresponding to $({\mathtt{x}}_{0},w_{0})$,
$(\widetilde{\mathtt{x}}_{0},{\widetilde{w}}_{0})\in E\times V$, respectively.
We apply the arguments used in the inequalities in Steps 2 and 3. We write two
inequalities for $w$ and ${\widetilde{w}}$ as in (6), then we choose
$v={\widetilde{w}}(t)$ and $v=w(t)$ in the inequality satisfied by $w$ and
${\widetilde{w}}$, respectively, and add the resulting inequalities to get

$\displaystyle\langle
w^{\prime}(t)-{\widetilde{w}}^{\prime}(t),w(t)-{\widetilde{w}}(t)\rangle+\langle
A(t,\lambda_{1}(t),w(t))-A(t,\lambda_{2}(t),{\widetilde{w}}(t)),w(t)-{\widetilde{w}}(t)\rangle$
$\displaystyle\leq\langle(R_{1}w)(t)-(R_{1}{\widetilde{w}})(t),{\widetilde{w}}(t)-w(t)\rangle+\langle
f(t,{\mathtt{x}}(t))-f(t,{\widetilde{{\mathtt{x}}}}(t)),{\widetilde{w}}(t)-w(t)\rangle$
$\displaystyle+\,j^{0}(t,{\mathtt{x}}(t),(R_{2}w)(t);M{\widetilde{w}}(t)-Mw(t))+j^{0}(t,{\widetilde{{\mathtt{x}}}}(t),(R_{2}{\widetilde{w}})(t);Mw(t)-M{\widetilde{w}}(t))$
$\displaystyle+\,\varphi(t,{\mathtt{x}}(t),(R_{3}w)(t),M{\widetilde{w}}(t))-\varphi(t,{\mathtt{x}}(t),(R_{3}w)(t),Mw(t))$
$\displaystyle\hskip
2.84544pt+\,\varphi(t,{\widetilde{{\mathtt{x}}}}(t),(R_{3}{\widetilde{w}})(t),Mw(t))-\varphi(t,{\widetilde{{\mathtt{x}}}}(t),(R_{3}{\widetilde{w}})(t),M{\widetilde{w}}(t))$

for a.e. $t\in(0,T)$. Integrating the latter on $(0,t)$, we use the
integration by parts formula to the first term, hypotheses $H(A)$(d),
$H(j)$(e), $H(\varphi)$(e), $H(f)$(b), and the Young inequality
$ab\leq\frac{\varepsilon^{2}}{2}a^{2}+\frac{1}{2\varepsilon^{2}}b^{2}$ for
$a$, $b\in\mathbb{R}$ to obtain

$\displaystyle(m_{A}-m_{j}\|M_{1}\|^{2})\,\int_{0}^{t}\|w(s)-{\widetilde{w}}(s)\|^{2}\,ds\leq\frac{1}{2}\|w_{0}-\widetilde{w}_{0}\|^{2}_{H}$
$\displaystyle+\,\frac{\varepsilon^{2}}{2}\left({\overline{m}}_{A}+2{\overline{m}}_{j}\|M_{1}\|+2m_{\varphi}\|M_{1}\|+2\right)\int_{0}^{t}\|w(s)-{\widetilde{w}}(s)\|^{2}\,ds$
$\displaystyle+\,\frac{1}{2\varepsilon^{2}}\left({\overline{m}}_{A}+{\overline{m}}_{j}\|M_{1}\|+m_{\varphi}\|M_{1}\|+L_{f}^{2}\right)\int_{0}^{t}\|{\mathtt{x}}(s)-{\widetilde{{\mathtt{x}}}}(s)\|^{2}\,ds$
$\displaystyle+\,\frac{1}{2\varepsilon^{2}}\int_{0}^{t}\|(R_{1}w)(s)-(R_{1}{\widetilde{w}})(s)\|_{V^{*}}^{2}\,ds$
$\displaystyle+\,\frac{1}{2\varepsilon^{2}}{\overline{m}}_{j}\|M_{1}\|\int_{0}^{t}\|(R_{2}w)(s)-(R_{2}{\widetilde{w}})(s)\|_{Z}^{2}\,ds$
$\displaystyle+\,\frac{1}{2\varepsilon^{2}}m_{\varphi}\|M_{1}\|\int_{0}^{t}\|(R_{3}w)(s)-(R_{3}{\widetilde{w}})(s)\|_{V^{*}}^{2}\,ds.$

for all $t\in[0,T]$. By the smallness condition $(H_{1})$, we choose
$\varepsilon>0$ such that

$m_{A}-m_{j}\|M_{1}\|^{2}-\frac{\varepsilon^{2}}{2}\left({\overline{m}}_{A}+2{\overline{m}}_{j}\|M_{1}\|+2m_{\varphi}\|M_{1}\|+2\right)>0.$

Next, we employ $(H_{3})$(b)–(d) and get

$\displaystyle
d_{1}\int_{0}^{t}\|w(s)-{\widetilde{w}}(s)\|^{2}\,ds\leq\frac{1}{2}\|w_{0}-\widetilde{w}_{0}\|^{2}_{H}$
(12)
$\displaystyle\quad+\,d_{2}\int_{0}^{t}\|{\mathtt{x}}(s)-{\widetilde{{\mathtt{x}}}}(s)\|_{E}^{2}\,ds+\,d_{3}\int_{0}^{t}\left(\int_{0}^{s}\|w(\tau)-{\widetilde{w}}(\tau)\|^{2}\,d\tau\right)\,ds$

with some positive constants $d_{i}$, $i=1$, $2$, $3$. On the other hand, we
use the ordinary differential equation in Problem 3 and, analogously as in
(9), by the Gronwall inequality, we have

$\|{\mathtt{x}}(t)-{\widetilde{{\mathtt{x}}}}(t)\|_{E}\leq
d_{4}\left(\|{\mathtt{x}}_{0}-\widetilde{\mathtt{x}}_{0}\|_{E}+\int_{0}^{t}\|w(s)-{\widetilde{w}}(s)\|\,ds\right)\
\ \mbox{for all}\ \ t\in(0,T)$ (13)

with a constant $d_{4}>0$. We combine (12) and (13), and again use the
Gronwall inequality to deduce

$\|w-{\widetilde{w}}\|_{L^{2}(0,t;V)}\leq
d_{5}\left(\|{\mathtt{x}}_{0}-\widetilde{\mathtt{x}}_{0}\|_{E}+\|w_{0}-\widetilde{w}_{0}\|\right)$
(14)

for all $t\in[0,T]$ with $d_{5}>0$. By $H(F)$(b), it follows

$\|{\mathtt{x}}^{\prime}-{\widetilde{{\mathtt{x}}}}^{\prime}\|_{L^{2}(0,t;E)}\leq
d_{6}\left(\|{\mathtt{x}}-{\widetilde{{\mathtt{x}}}}\|_{L^{2}(0,t;E)}+\|w-{\widetilde{w}}\|_{L^{2}(0,t;V)}\right)\
\ \mbox{for all}\ \ t\in[0,T]$

with a constant $d_{6}>0$. The latter together with (13) and (14) imply (4).
The regularity of the solution follows from the following embeddings
$W^{1,2}(0,T;E)\subset C([0,T];E)$ and ${\mathbb{W}}\subset C([0,T];H)$. This
completes the proof of the theorem. $\Box$

We conclude this section with a corollary of Theorem 4 when the term $f$ is
independent of the second variable and $f\in L^{2}(0,T;V^{*})$.

######  Corollary 3.1.

Let hypotheses $H(F)$, $H(A)$, $H(j)$, $H(\varphi)$, $H(K)$, $H(M)$,
$(H_{1})$–$(H_{3})$ hold, and $f\in L^{2}(0,T;V^{*})$. Then Problem 3 is
uniquely solvable with $({\mathtt{x}},w)\in H^{1}(0,T;E)\times{\mathbb{W}}$
and $w(t)\in K$ for a.e. $t\in(0,T)$. Moreover, for every
$({\mathtt{x}}_{0},w_{0},f)$,
$(\widetilde{\mathtt{x}}_{0},{\widetilde{w}}_{0},{\widetilde{f}})\in E\times
V\times L^{2}(0,T;V^{*})$, there exists a constant $c>0$ such that

$\|{\mathtt{x}}-{\widetilde{{\mathtt{x}}}}\|_{H^{1}(0,T;E)}+\|w-{\widetilde{w}}\|_{L^{2}(0,T;V)}\leq
c\,(\|{\mathtt{x}}_{0}-\widetilde{\mathtt{x}}_{0}\|_{E}+\|w_{0}-\widetilde{w}_{0}\|+\|f-{\widetilde{f}}\|_{L^{2}(0,T;V^{*})}),$
(15)

where $({\mathtt{x}},w)$ and $({\widetilde{{\mathtt{x}}}},{\widetilde{w}})$
are the unique solutions to Problem 3 corresponding to
$({\mathtt{x}}_{0},w_{0},f)$ and
$(\widetilde{\mathtt{x}}_{0},{\widetilde{w}}_{0},\widetilde{f})$,
respectively.

##  4 A dynamic frictionless viscoplastic contact problem

In this section we shall illustrate the applicability of results of Section 3.
We examine an example of a dynamic unilateral viscoplastic frictionless
contact problem. We describe the physical setting and give the classical
formulation of the contact problem, provide its variational formulation, and
prove a result on its well-posedness.

A viscoplastic body occupies a bounded domain $\Omega$ of $\mathbb{R}^{d}$,
$d=1$, $2$, $3$. The boundary $\Gamma$ of $\Omega$ is supposed to be Lipschitz
continuous and splitted into three mutually disjoint and measurable parts
$\Gamma=\overline{\Gamma}_{D}\cup\overline{\Gamma}_{N}\cup\overline{\Gamma}_{C}$
with $|\Gamma_{D}|>0$. The outward unit normal at $\Gamma$, denoted by $\nu$,
is defined a.e. on $\Gamma$. Let $Q=\Omega\times(0,T)$, where $0<T<\infty$.

The symbol $\mathbb{S}^{d}$ denotes the space of $d\times d$ symmetric
matrices. We use the standard notation for inner products and norms on
$\mathbb{R}^{d}$ and $\mathbb{S}^{d}$, i.e.,
$\mbox{\boldmath{$u$}}\cdot\mbox{\boldmath{$v$}}=u_{i}v_{i}$,
$\|\mbox{\boldmath{$u$}}\|^{2}=\mbox{\boldmath{$u$}}\cdot\mbox{\boldmath{$u$}}$
for $\mbox{\boldmath{$u$}}=(u_{i})$,
$\mbox{\boldmath{$v$}}=(v_{i})\in\mathbb{R}^{d}$, and
$\mbox{\boldmath{$\sigma$}}\cdot\mbox{\boldmath{$\tau$}}=\sigma_{i}\tau_{i}$,
$\|\mbox{\boldmath{$\sigma$}}\|^{2}=\mbox{\boldmath{$\sigma$}}\cdot\mbox{\boldmath{$\sigma$}}$
for $\mbox{\boldmath{$\sigma$}}=(\sigma_{ij})$,
$\mbox{\boldmath{$\tau$}}=(\tau_{ij})\in\mathbb{S}^{d}$, respectively. The
normal and tangential components of vectors and tensors at the boundary are
expressed by the following notation

$v_{\nu}=\mbox{\boldmath{$v$}}\cdot\mbox{\boldmath{$\nu$}},\ \ \
\mbox{\boldmath{$v$}}_{\tau}=\mbox{\boldmath{$v$}}-v_{\nu}\,\mbox{\boldmath{$\nu$}},\
\ \
\sigma_{\nu}=(\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}})\cdot\mbox{\boldmath{$\nu$}},\
\ \
\mbox{\boldmath{$\sigma$}}_{\tau}=\mbox{\boldmath{$\sigma$}}\,\mbox{\boldmath{$\nu$}}-\sigma_{\nu}\,\mbox{\boldmath{$\nu$}}.$

Since
$\mbox{\boldmath{$v$}}_{\tau}\cdot\mbox{\boldmath{$\nu$}}=\mbox{\boldmath{$\sigma$}}_{\tau}\cdot\mbox{\boldmath{$\nu$}}=0$,
the following decomposition formula holds

$\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}\cdot\mbox{\boldmath{$v$}}=(\sigma_{\nu}\,\mbox{\boldmath{$\nu$}}+\mbox{\boldmath{$\sigma$}}_{\tau})\cdot(v_{\nu}\,\mbox{\boldmath{$\nu$}}+\mbox{\boldmath{$v$}}_{\tau})=\sigma_{\nu}\,v_{\nu}+\mbox{\boldmath{$\sigma$}}_{\tau}\cdot\mbox{\boldmath{$v$}}_{\tau}.$
(16)

The classical model for the contact process has the following formulation.

######  Problem 6.

Find a displacement field $\mbox{\boldmath{$u$}}\colon Q\to\mathbb{R}^{d}$ and
a stress field $\mbox{\boldmath{$\sigma$}}\colon Q\rightarrow\mathbb{S}^{d}$
such that for all $t\in(0,T)$,

$\displaystyle\mbox{\boldmath{$\sigma$}}(t)$
$\displaystyle={\mathscr{A}}\mbox{\boldmath{$\varepsilon$}}({\mbox{\boldmath{$u$}}}^{\prime}(t))+{\mathscr{B}}\mbox{\boldmath{$\varepsilon$}}({\mbox{\boldmath{$u$}}}(t))+\int_{0}^{t}{\mathscr{G}}\big{(}\mbox{\boldmath{$\sigma$}}(s)-{\mathscr{A}}\mbox{\boldmath{$\varepsilon$}}({\mbox{\boldmath{$u$}}}^{\prime}(s)),\mbox{\boldmath{$\varepsilon$}}({\mbox{\boldmath{$u$}}}(s)\big{)}\,ds\quad$
$\displaystyle{\rm in}\ $ $\displaystyle\Omega,$ (17)
$\displaystyle{\mbox{\boldmath{$u$}}}^{\prime\prime}(t)$ $\displaystyle={\rm
Div}\,\mbox{\boldmath{$\sigma$}}(t)+\mbox{\boldmath{$f$}}_{0}(t)\quad$
$\displaystyle{\rm in}\ $ $\displaystyle\Omega,$ (18)
$\displaystyle\mbox{\boldmath{$u$}}(t)$ $\displaystyle=\mbox{\boldmath{$0$}}$
$\displaystyle{\rm on}\ $ $\displaystyle\Gamma_{D},$ (19)
$\displaystyle\mbox{\boldmath{$\sigma$}}(t)\mbox{\boldmath{$\nu$}}$
$\displaystyle=\mbox{\boldmath{$f$}}_{N}(t)\quad$ $\displaystyle{\rm on}\ $
$\displaystyle\Gamma_{N},$ (20) $\displaystyle u^{\prime}_{\nu}(t)$
$\displaystyle\leq g,\ \sigma_{\nu}(t)+\xi(t)\leq 0,\
(u^{\prime}_{\nu}(t)-g)(\sigma_{\nu}(t)+\xi(t))=0$ $\displaystyle\ \ \ {\rm
with}\ \xi(t)\in k(u_{\nu}(t))\,\partial j_{\nu}(u^{\prime}_{\nu}(t))\quad$
$\displaystyle{\rm on}\ $ $\displaystyle\Gamma_{C},$ (21)
$\displaystyle\mbox{\boldmath{$\sigma$}}_{\tau}(t)$
$\displaystyle=\mbox{\boldmath{$0$}}\quad$ $\displaystyle{\rm on}\ $
$\displaystyle\Gamma_{C},$ (22)

and

$\mbox{\boldmath{$u$}}(0)=\mbox{\boldmath{$u$}}_{0},\qquad{\mbox{\boldmath{$u$}}}^{\prime}(0)=\mbox{\boldmath{$w$}}_{0}\quad\
{\rm in}\quad\Omega.$ (23)

In this problem, equation (17) represents a general viscoplastic constitutive
law in which $\mathscr{A}$, $\mathscr{B}$ and $\mathscr{G}$ stand for the
viscosity operator, the elasticity operator and the viscoplastic constitutive
function, respectively. Recall that the components of the linearized strain
tensor $\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}})$ are given by

$\mbox{\boldmath{$\varepsilon$}}_{ij}(\mbox{\boldmath{$u$}})=(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}))_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})\
\ \mbox{in}\ \ \Omega,$

where $u_{i,j}=\partial u_{i}/\partial x_{j}$. Equation (18) is the equation
of motion which governs the evolution of the mechanical state of the body.
Here $u^{\prime\prime}(t)=\partial^{2}\mbox{\boldmath{$u$}}/\partial t^{2}$
represents the acceleration field, ${\rm
Div}\,\mbox{\boldmath{$\sigma$}}=(\sigma_{ij,j})$, and
$\mbox{\boldmath{$f$}}_{0}$ denotes the density of body forces. The
displacement homogeneous boundary condition (19) means that the body is fixed
on $\Gamma_{D}$, while (20) is the traction boundary condition with surface
tractions of density $\mbox{\boldmath{$f$}}_{N}$ acting on $\Gamma_{N}$.
Condition (21) is the Signorini unilateral contact boundary condition for the
normal velocity in which $g>0$ and $\partial j_{\nu}$ denotes the Clarke
subgradient of a prescribed function $j_{\nu}$. Condition $\xi(t)\in
k(u_{\nu}(t))\,\partial j_{\nu}(u^{\prime}_{\nu}(t))$ on $\Gamma_{C}$
represents the normal damped response condition where $k$ is a given damper
coefficient depending on the normal displacement. Condition (22) is called the
frictionless condition in which the tangential part of the stress vanishes.
Examples and details on mechanical interpretation of conditions (21) and (22)
can be found in [12, 26] and references therein. Finally, equations (23) are
the initial conditions in which $\mbox{\boldmath{$u$}}_{0}$ and
$\mbox{\boldmath{$w$}}_{0}$ represent the initial displacement and the initial
velocity, respectively.

To cope with the weak formulation of Problem 6, we introduce the spaces

$V=\\{\,\mbox{\boldmath{$v$}}\in
H^{1}(\Omega;\mathbb{R}^{d})\mid\mbox{\boldmath{$v$}}={\bf{0}}~{}\textrm{on}~{}\Gamma_{D}\,\\}\
\ \mbox{and}\ \ \mathcal{H}=L^{2}(\Omega;\mathbb{S}^{d}).$ (24)

The inner product and the corresponding norm on $V$ are given by

$(\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}})_{V}=(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}))_{\mathcal{H}},~{}~{}\|\mbox{\boldmath{$v$}}\|_{V}=\|\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})\|_{\mathcal{H}}\
\ \mbox{for all}\ \ \mbox{\boldmath{$u$}},\,\mbox{\boldmath{$v$}}\in V.$ (25)

The space $\mathcal{H}$ is a Hilbert space endowed with the inner product

$\langle\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$\tau$}}\rangle_{\mathcal{H}}=\int_{\Omega}\mbox{\boldmath{$\sigma$}}(\mbox{\boldmath{$x$}})\cdot\mbox{\boldmath{$\tau$}}(\mbox{\boldmath{$x$}})\,dx\
\ \mbox{for}\ \
\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$\tau$}}\in{\mathcal{H}},$

and the associated norm $\|\cdot\|_{\mathcal{H}}$. Since $|\Gamma_{D}|>0$, the
Korn inequality implies that the norm $\|\cdot\|_{V}$ defined by (25) is
equivalent to the usual norm $\|\cdot\|_{H^{1}(\Omega;\mathbb{R}^{d})}$.
Recall that the trace operator $\gamma\colon V\to
L^{2}(\Gamma_{C};\mathbb{R}^{d})$ is linear and continuous, i.e.,
$\|\mbox{\boldmath{$v$}}\|_{L^{2}(\Gamma;\mathbb{R}^{d})}\leq\|\gamma\|\|\mbox{\boldmath{$v$}}\|_{V}$
for $\mbox{\boldmath{$v$}}\in V$, where $\|\gamma\|$ denotes the norm of the
trace operator in ${\cal L}(V,L^{2}(\Gamma_{C};\mathbb{R}^{d}))$.

We need the following hypotheses on the data to Problem 6.

$\underline{H({\mathscr{A}})}:$ ${\mathscr{A}}\colon
Q\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}$ is such that

  * (1)

${\mathscr{A}}(\cdot,\cdot,\mbox{\boldmath{$\varepsilon$}})$ is measurable on
$Q$ for all $\mbox{\boldmath{$\varepsilon$}}\in\mathbb{S}^{d}$.

  * (2)

${\mathscr{A}}(\mbox{\boldmath{$x$}},t,\cdot)$ is continuous on
$\mathbb{S}^{d}$ for a.e. $t\in(0,T)$.

  * (3)

$\|{\mathscr{A}}(\mbox{\boldmath{$x$}},t,\mbox{\boldmath{$\varepsilon$}})\|\leq{\widetilde{a}}_{0}(\mbox{\boldmath{$x$}},t)+{\widetilde{a}}_{2}\,\|\mbox{\boldmath{$\varepsilon$}}\|$
for all $\mbox{\boldmath{$\varepsilon$}}\in\mathbb{S}^{d}$, a.e.
$(\mbox{\boldmath{$x$}},t)\in Q$ with ${\widetilde{a}}_{0}$,
${\widetilde{a}}_{2}\geq 0$, ${\widetilde{a}}_{0}\in L^{2}(Q)$.

  * (4)

$\mbox{there exists}\ m_{\mathscr{A}}>0\ \mbox{such that}\ \
(\mathscr{A}(\mbox{\boldmath{$x$}},t,\mbox{\boldmath{$\varepsilon$}}_{1})-\mathscr{A}(\mbox{\boldmath{$x$}},t,\mbox{\boldmath{$\varepsilon$}}_{2}))\cdot(\mbox{\boldmath{$\varepsilon$}}_{1}-\mbox{\boldmath{$\varepsilon$}}_{2})\geq
m_{\mathscr{A}}\,\|\mbox{\boldmath{$\varepsilon$}}_{1}-\mbox{\boldmath{$\varepsilon$}}_{2}\|^{2}$
$\ \ \qquad\mbox{for all}\
\mbox{\boldmath{$\varepsilon$}}_{1},\mbox{\boldmath{$\varepsilon$}}_{2}\in\mathbb{S}^{d},\
\mbox{a.e.}\ (\mbox{\boldmath{$x$}},t)\in Q$.

  * (5)

${\mathscr{A}}(\mbox{\boldmath{$x$}},t,{\bf{0}})={\bf{0}}$ for a.e.
$(\mbox{\boldmath{$x$}},t)\in Q$.

$\underline{H({\mathscr{B}})}:$ $\mathscr{B}\colon
Q\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}$ is such that

  * (1)

$\mathscr{B}(\cdot,\cdot,\mbox{\boldmath{$\varepsilon$}})$ is measurable on
$Q$ for all $\mbox{\boldmath{$\varepsilon$}}\in\mathbb{S}^{d}$.

  * (2)

there exists $L_{\mathscr{B}}>0$ such that
$\|\mathscr{B}(\mbox{\boldmath{$x$}},t,\mbox{\boldmath{$\varepsilon$}}_{1})-\mathscr{B}(\mbox{\boldmath{$x$}},t,\mbox{\boldmath{$\varepsilon$}}_{2})\|\leq
L_{\mathscr{B}}\|\mbox{\boldmath{$\varepsilon$}}_{1}-\mbox{\boldmath{$\varepsilon$}}_{2}\|$
for all
$\mbox{\boldmath{$\varepsilon$}}_{1}$,
$\mbox{\boldmath{$\varepsilon$}}_{2}\in\mathbb{S}^{d}$, a.e.
$(\mbox{\boldmath{$x$}},t)\in Q$.

  * (3)

${\mathscr{B}}(\cdot,\cdot,{\bf{0}})\in L^{2}(Q;{\mathbb{S}}^{d})$.

$\underline{H({\mathscr{G}})}:$ $\mathscr{G}\colon
Q\times\mathbb{S}^{d}\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}$ is such
that

  * (1)

$\mathscr{G}(\cdot,\cdot,\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$\varepsilon$}})$
is measurable on $Q$ for all $\sigma$,
$\mbox{\boldmath{$\varepsilon$}}\in\mathbb{S}^{d}$.

  * (2)

there exists $L_{\mathscr{G}}>0$ such that for all
$\mbox{\boldmath{$\sigma$}}_{1}$, $\mbox{\boldmath{$\sigma$}}_{2}$,
$\mbox{\boldmath{$\varepsilon$}}_{1}$,
$\mbox{\boldmath{$\varepsilon$}}_{2}\in\mathbb{S}^{d}$, a.e.
$(\mbox{\boldmath{$x$}},t)\in Q$, it holds
$\|\mathscr{G}(\mbox{\boldmath{$x$}},t,\mbox{\boldmath{$\sigma$}}_{1},\mbox{\boldmath{$\varepsilon$}}_{1})-\mathscr{G}(\mbox{\boldmath{$x$}},t,\mbox{\boldmath{$\sigma$}}_{2},\mbox{\boldmath{$\varepsilon$}}_{2})\|\leq
L_{\mathscr{G}}\,(\|\mbox{\boldmath{$\sigma$}}_{1}-\mbox{\boldmath{$\sigma$}}_{2}\|+\|\mbox{\boldmath{$\varepsilon$}}_{1}-\mbox{\boldmath{$\varepsilon$}}_{2}\|)$.

  * (3)

${\mathscr{G}}(\cdot,\cdot,{\bf{0}},\mbox{\boldmath{$0$}})\in
L^{2}(Q;{\mathbb{S}}^{d})$.

$\underline{H({k})}:$ $k\colon\Gamma_{C}\times\mathbb{R}\to\mathbb{R}$ is such
that

  * (1)

$k(\cdot,r)$ is measurable on $\Gamma_{C}$ for all $r\in\mathbb{R}$.

  * (2)

there exist $k_{1}$, $k_{2}$ such that $0<k_{1}\leq
k(\mbox{\boldmath{$x$}},t)\leq k_{2}$ for all $r\in\mathbb{R}$, a.e.
$\mbox{\boldmath{$x$}}\in\Gamma_{C}$.

  * (3)

there is $L_{k}>0$ such that
$|k(\mbox{\boldmath{$x$}},r_{1})-k(\mbox{\boldmath{$x$}},r_{2})|\leq
L_{k}|r_{1}-r_{2}|$ for all $r_{1}$, $r_{2}\in\mathbb{R}$, a.e.
$\mbox{\boldmath{$x$}}\in\Gamma_{C}$.

$\underline{H({j_{\nu}})}:$
$j_{\nu}\colon\Gamma_{C}\times\mathbb{R}\to\mathbb{R}$ is such that

  * (1)

$j_{\nu}(\cdot,r)$ is measurable on $\Gamma_{C}$ for all $r\in\mathbb{R}$, and
$j_{\nu}(\cdot,{e}(\cdot))\in L^{1}(\Gamma_{C})$ for some ${e}\in
L^{2}(\Gamma_{C})$.

  * (2)

$j_{\nu}(\mbox{\boldmath{$x$}},\cdot)$ is locally Lipschitz on $\mathbb{R}$
for a.e. $\mbox{\boldmath{$x$}}\in\Gamma_{C}$.

  * (3)

there is $c_{0}\geq 0$ such that $|\partial
j_{\nu}(\mbox{\boldmath{$x$}},r)|\leq c_{0}$ for all $r\in\mathbb{R}$, a.e.
$\mbox{\boldmath{$x$}}\in\Gamma_{C}$.

  * (4)

there exists $\alpha_{j_{\nu}}\geq 0$ such that
$j_{\nu}^{0}(\mbox{\boldmath{$x$}},r_{1};r_{2}-r_{1})+j_{\nu}^{0}(\mbox{\boldmath{$x$}},r_{2};r_{1}-r_{2})\leq\alpha_{j_{\nu}}|r_{1}-r_{2}|^{2}$
for all $r_{1}$, $r_{2}\in\mathbb{R}$, a.e.
$\mbox{\boldmath{$x$}}\in\Gamma_{C}$.

$\underline{(H_{4})}:$ $\mbox{\boldmath{$f$}}_{0}\in
L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))$, $\mbox{\boldmath{$f$}}_{N}\in
L^{2}(0,T;L^{2}(\Gamma_{N};\mathbb{R}^{d}))$, $\mbox{\boldmath{$u$}}_{0}$,
$\mbox{\boldmath{$w$}}_{0}\in V$.

We introduce the set $K$ of admissible velocity fields defined by

$K=\\{\,\mbox{\boldmath{$v$}}\in V\mid v_{\nu}\leq g\ \ \mbox{\rm on}\ \
\Gamma_{C}\,\\},$ (26)

and an element $\mbox{\boldmath{$f$}}\in L^{2}(0,T;V^{*})$ given by

$\langle\mbox{\boldmath{$f$}}(t),\mbox{\boldmath{$v$}}\rangle=\langle\mbox{\boldmath{$f$}}_{0}(t),\mbox{\boldmath{$v$}}\rangle_{L^{2}(\Omega;\mathbb{R}^{d})}+\langle\mbox{\boldmath{$f$}}_{N}(t),\mbox{\boldmath{$v$}}\rangle_{L^{2}(\Gamma_{N};\mathbb{R}^{d})}$
(27)

for all $\mbox{\boldmath{$v$}}\in V$, a.e. $t\in(0,T)$.

We now turn to the weak formulation of Problem 6. Let
$(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})$ be a smooth solution to
this problem which means that the data are smooth functions such that all the
derivatives and all the conditions are satisfied in the usual sense at each
point. Let $\mbox{\boldmath{$v$}}\in K$ and $t\in(0,T)$. We multiply (18) by
$\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t)$, use Green’s
formula, see [26, Theorem 2.25], and apply the boundary conditions (19) and
(20) to obtain

$\displaystyle\int_{\Omega}\mbox{\boldmath{$u$}}^{\prime\prime}(t)\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t))\,dx+\int_{\Omega}\mbox{\boldmath{$\sigma$}}(t)\cdot\big{(}\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}^{\prime}(t))\big{)}\,dx$
$\displaystyle\ \
=\int_{\Omega}\mbox{\boldmath{$f$}}_{0}(t)\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t))dx+\int_{\Gamma_{N}}\mbox{\boldmath{$f$}}_{N}(t)\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t))\,d\Gamma+\int_{\Gamma_{C}}\mbox{\boldmath{$\sigma$}}(t)\mbox{\boldmath{$\nu$}}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t))\,d\Gamma.$

By (21) and the definition of the Clarke subgradient, we have

$\displaystyle\sigma_{\nu}(t)(v_{\nu}-u_{\nu}^{\prime}(t))=(\sigma_{\nu}(t)+\xi(t))(v_{\nu}-g)-(\sigma_{\nu}(t)+\xi(t))(u_{\nu}^{\prime}(t)-g)$
$\displaystyle\qquad-\xi(t)(v_{\nu}-u_{\nu}^{\prime}(t))\geq-k(u_{\nu}(t))\,j_{\nu}^{0}(u_{\nu}^{\prime}(t);v_{\nu}-u_{\nu}^{\prime}(t))\
\ \mbox{on}\ \ \Gamma_{C}.$ (28)

On the other hand, (16) and the frictionless condition (22) imply

$\mbox{\boldmath{$\sigma$}}(t)\mbox{\boldmath{$\nu$}}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t))=\sigma_{\nu}(t)(v_{\nu}-u_{\nu}^{\prime}(t))\
\ \mbox{on}\ \ \Gamma_{C}.$ (29)

Combining (27), (28) and (29), we have

$\displaystyle\int_{\Omega}\mbox{\boldmath{$u$}}^{\prime\prime}(t)\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t))\,dx+\langle\mbox{\boldmath{$\sigma$}}(t),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}^{\prime}(t))\rangle_{\mathcal{H}}$
$\displaystyle\qquad\qquad+\int_{\Gamma_{C}}k(u_{\nu}(t))\,j_{\nu}^{0}(u^{\prime}_{\nu}(t);v_{\nu}-u^{\prime}_{\nu}(t))\,d\Gamma\geq\langle\mbox{\boldmath{$f$}}(t),\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t)\rangle.$
(30)

From the constitutive law (17), we get

$\mbox{\boldmath{$\sigma$}}(t)={\mathscr{A}}\mbox{\boldmath{$\varepsilon$}}({\mbox{\boldmath{$u$}}}^{\prime}(t))+{\mathscr{B}}\mbox{\boldmath{$\varepsilon$}}({\mbox{\boldmath{$u$}}}(t))+\mbox{\boldmath{$\eta$}}(t)$
(31)

with

$\mbox{\boldmath{$\eta$}}(t)=\int_{0}^{t}{\mathscr{G}}\big{(}\mbox{\boldmath{$\sigma$}}(s)-{\mathscr{A}}\mbox{\boldmath{$\varepsilon$}}({\mbox{\boldmath{$u$}}}^{\prime}(s)),\mbox{\boldmath{$\varepsilon$}}({\mbox{\boldmath{$u$}}}(s)\big{)}\,ds.$
(32)

Inserting (31) into (30) and using (32), we obtain the following variational
formulation of Problem 6.

######  Problem 7.

Find $\mbox{\boldmath{$\eta$}}\colon(0,T)\to{\mathcal{H}}$ and
$\mbox{\boldmath{$u$}}\colon(0,T)\to V$ such that
$\mbox{\boldmath{$u$}}^{\prime}(t)\in K$ a.e. $t\in(0,T)$ and

$\displaystyle\mbox{\boldmath{$\eta$}}^{\prime}(t)={\mathscr{G}}(\mbox{\boldmath{$\eta$}}(t)+\mathscr{B}(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}(t))),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}(t)))\
\ \mbox{\rm for all}\ \ t\in(0,T),$
$\displaystyle\int_{\Omega}\mbox{\boldmath{$u$}}^{\prime\prime}(t)\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t))\,dx+\langle\mathscr{A}(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}^{\prime}(t)))+\mathscr{B}(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}(t)))+\mbox{\boldmath{$\eta$}}(t),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}^{\prime}(t))\rangle_{\mathcal{H}}$
$\displaystyle\quad+\int_{\Gamma_{C}}k(u_{\nu}(t))\,j_{\nu}^{0}(u^{\prime}_{\nu}(t);v_{\nu}-u^{\prime}_{\nu}(t))\,d\Gamma\geq\langle\mbox{\boldmath{$f$}}(t),\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t)\rangle\
\ \mbox{\rm for all}\ \ \mbox{\boldmath{$v$}}\in K,\ \mbox{\rm a.e.}\
t\in(0,T),$ $\displaystyle\mbox{\boldmath{$\eta$}}(0)=\mbox{\boldmath{$0$}},\
\mbox{\boldmath{$u$}}(0)=\mbox{\boldmath{$u$}}_{0},\
\mbox{\boldmath{$u$}}^{\prime}(0)=\mbox{\boldmath{$w$}}_{0}.$

Problem 7 couples the ordinary differential equation with a hemivariational
inequality with constraints. The result below concerns the unique solvability,
continuous dependence, and regularity of solution to Problem 7.

######  Theorem 8.

Assume hypotheses $H({\mathscr{A}})$, $H({\mathscr{B}})$, $H({\mathscr{G}})$,
$H(k)$, $H(j_{\nu})$, $(H_{4})$ and the smallness condition

$\alpha_{j_{\nu}}k_{2}\|\gamma\|^{2}<m_{\mathscr{A}}.$ (33)

Then Problem 7 has a solution $\mbox{\boldmath{$\eta$}}\in
H^{1}(0,T;{\mathcal{H}})$ and $\mbox{\boldmath{$u$}}\in C([0,T];V)$ such that
$\mbox{\boldmath{$u$}}^{\prime}\in{\mathbb{W}}$ with
$\mbox{\boldmath{$u$}}^{\prime}(t)\in K$ for a.e. $t\in(0,T)$. If, in
addition,

$\mbox{\rm either}\ \ j_{\nu}(\mbox{\boldmath{$x$}},\cdot)\ \ \mbox{\rm or}\ \
-j_{\nu}(\mbox{\boldmath{$x$}},\cdot)\ \ \mbox{\rm is regular for a.e.}\
\mbox{\boldmath{$x$}}\in\Gamma_{C},$ (34)

then the solution to Problem 7 is unique, and there exists a constant $c>0$
such that

$\displaystyle\|\mbox{\boldmath{$\eta$}}_{1}-\mbox{\boldmath{$\eta$}}_{2}\|_{H^{1}(0,T;E)}+\|\mbox{\boldmath{$u$}}_{1}-\mbox{\boldmath{$u$}}_{2}\|_{C(0,T;V)}+\|\mbox{\boldmath{$u$}}^{\prime}_{1}-\mbox{\boldmath{$u$}}^{\prime}_{2}\|_{L^{2}(0,T;V)}$
(35) $\displaystyle\quad\leq
c\,(\|\mbox{\boldmath{$u$}}_{0}-\widetilde{\mbox{\boldmath{$u$}}}_{0}\|+\|\mbox{\boldmath{$w$}}_{0}-\widetilde{\mbox{\boldmath{$w$}}}_{0}\|+\|\mbox{\boldmath{$f$}}_{0}-{\widetilde{\mbox{\boldmath{$f$}}}_{0}}\|_{L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))}+\|\mbox{\boldmath{$f$}}_{N}-{\widetilde{\mbox{\boldmath{$f$}}}_{N}}\|_{L^{2}(0,T;L^{2}(\Gamma_{N};\mathbb{R}^{d}))}),$

where $(\mbox{\boldmath{$\eta$}}_{1},\mbox{\boldmath{$u$}}_{1})$ and
$(\mbox{\boldmath{$\eta$}}_{2},\mbox{\boldmath{$u$}}_{2})$ are the unique
solutions to Problem 7 corresponding to

$(\mbox{\boldmath{$u$}}_{0},\mbox{\boldmath{$w$}}_{0},\mbox{\boldmath{$f$}}_{0},\mbox{\boldmath{$f$}}_{N}),(\widetilde{\mbox{\boldmath{$u$}}}_{0},{\widetilde{\mbox{\boldmath{$w$}}}}_{0},\widetilde{\mbox{\boldmath{$f$}}},\widetilde{\mbox{\boldmath{$f$}}}_{N})\in
V\times V\times L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d})\times
L^{2}(\Gamma_{N};\mathbb{R}^{d})),$

respectively. Moreover, we have the regularity
$(\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$u$}}^{\prime})\in
C([0,T];{\mathcal{H}}\times L^{2}(\Omega;\mathbb{R}^{d}))$.

Proof. Let $\mbox{\boldmath{$w$}}(t)=\mbox{\boldmath{$u$}}^{\prime}(t)$ for
all $t\in(0,T)$. Then
$\mbox{\boldmath{$u$}}(t)=\mbox{\boldmath{$u$}}_{0}+\int_{0}^{t}\mbox{\boldmath{$w$}}(s)\,ds$
for all $t\in(0,T)$ and Problem 7 can be reformulated as follows.

######  Problem 9.

Find $\mbox{\boldmath{$\eta$}}\colon(0,T)\to{\mathcal{H}}$ and
$\mbox{\boldmath{$w$}}\colon(0,T)\to V$ such that $\mbox{\boldmath{$w$}}(t)\in
K$ a.e. $t\in(0,T)$, $\mbox{\boldmath{$w$}}(0)=\mbox{\boldmath{$w$}}_{0}$,
$\mbox{\boldmath{$\eta$}}(0)=\mbox{\boldmath{$0$}}$ and

$\displaystyle\mbox{\boldmath{$\eta$}}^{\prime}(t)={\mathscr{G}}\left(\mbox{\boldmath{$\eta$}}(t)+\mathscr{B}\left(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}_{0}+\int_{0}^{t}\mbox{\boldmath{$w$}}(s)\,ds)\right),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}_{0}+\int_{0}^{t}\mbox{\boldmath{$w$}}(s)\,ds)\right)\
\ \mbox{\rm for all}\ \ t\in(0,T),$
$\displaystyle\int_{\Omega}\mbox{\boldmath{$w$}}^{\prime}(t)\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$w$}}(t))\,dx+\langle\mathscr{A}(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$w$}}(t)))+\mathscr{B}(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}_{0}+\int_{0}^{t}\mbox{\boldmath{$w$}}(s)\,ds))+\mbox{\boldmath{$\eta$}}(t),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$w$}}(t))\rangle_{\mathcal{H}}$
$\displaystyle\quad+\int_{\Gamma_{C}}k\left(u_{0\nu}+\int_{0}^{t}w_{\nu}(s)\,ds\right)\,j_{\nu}^{0}(w_{\nu}(t);v_{\nu}-w_{\nu}(t))\,d\Gamma\geq\langle\mbox{\boldmath{$f$}}(t),\mbox{\boldmath{$v$}}-\mbox{\boldmath{$w$}}(t)\rangle$
$\displaystyle\ \ \ \ \mbox{\rm for all}\ \ \mbox{\boldmath{$v$}}\in K,\
\mbox{\rm a.e.}\ t\in(0,T).$

Let $E:={\mathcal{H}}$, $U:=V$, $Z=X:=L^{2}(\Gamma_{C})$, and
$A\colon(0,T)\times E\times V\to V^{*}$, $S\colon L^{2}(0,T;V)\to
L^{2}(0,T;U)$, $R_{1}\colon L^{2}(0,T;V)\to L^{2}(0,T;V^{*})$, $R_{2}\colon
L^{2}(0,T;V)\to L^{2}(0,T;Z)$, $j\colon Z\times X\to\mathbb{R}$,
$F\colon(0,T)\times E\times U\to E$, $M\colon V\to X$ be defined by

$\displaystyle\langle
A(t,\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$v$}}),\mbox{\boldmath{$z$}}\rangle:=\langle\mathscr{A}(t,\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}))+\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$z$}})\rangle_{\mathcal{H}}\
\ \mbox{for}\ \ \mbox{\boldmath{$v$}},\mbox{\boldmath{$z$}}\in V,\
\mbox{\boldmath{$\eta$}}\in{\mathcal{H}},\ \mbox{a.e.}\ t\in(0,T),$
$\displaystyle(S\mbox{\boldmath{$w$}})(t):=\mbox{\boldmath{$u$}}_{0}+\int_{0}^{t}\mbox{\boldmath{$w$}}(s)\,ds\
\ \mbox{for}\ \ \mbox{\boldmath{$w$}}\in L^{2}(0,T;V),\ \mbox{a.e.}\
t\in(0,T),$
$\displaystyle\langle(R_{1}\mbox{\boldmath{$w$}})(t),\mbox{\boldmath{$v$}}\rangle:=\langle{\mathscr{B}}\mbox{\boldmath{$\varepsilon$}}((S\mbox{\boldmath{$w$}})(t)),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})\rangle_{\mathcal{H}}\
\ \mbox{for}\ \ \mbox{\boldmath{$w$}}\in L^{2}(0,T;V),\
\mbox{\boldmath{$v$}}\in V,\ \mbox{a.e.}\ t\in(0,T),$
$\displaystyle(R_{2}\mbox{\boldmath{$w$}})(t):=((S\mbox{\boldmath{$w$}})(t))_{\nu}=u_{0\nu}+\int_{0}^{t}w_{\nu}(s)\,ds\
\ \mbox{for}\ \ \mbox{\boldmath{$w$}}\in L^{2}(0,T;V),\ \mbox{a.e.}\
t\in(0,T),$ $\displaystyle
j(z,v)=\int_{\Gamma_{C}}k(\mbox{\boldmath{$x$}},z)\,j_{\nu}(\mbox{\boldmath{$x$}},v)\,d\Gamma\
\ \mbox{for}\ \ z\in Z,\ v\in X,\ \mbox{a.e.}\ t\in(0,T),$ $\displaystyle
F(t,\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$w$}})(\mbox{\boldmath{$x$}}):={\mathscr{G}}(\mbox{\boldmath{$x$}},t,\mbox{\boldmath{$\eta$}}(\mbox{\boldmath{$x$}})+{\mathscr{B}}(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$w$}}(\mbox{\boldmath{$x$}}))),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$w$}}(\mbox{\boldmath{$x$}})))\
\mbox{for}\ \mbox{\boldmath{$\eta$}}\in E,\ \mbox{\boldmath{$w$}}\in U,\
\mbox{a.e.}\ t\in(0,T),$ $\displaystyle M\mbox{\boldmath{$v$}}:=v_{\nu}\ \
\mbox{for}\ \ \mbox{\boldmath{$v$}}\in V,$

and $\varphi\equiv 0$. With the above notation, we consider the following
auxiliary system associated with Problem 9.

######  Problem 10.

Find $\mbox{\boldmath{$\eta$}}\in H^{1}(0,T;E)$ and
$\mbox{\boldmath{$w$}}\in\mathbb{W}$ such that $\mbox{\boldmath{$w$}}(t)\in K$
for a.e. $t\in(0,T)$, $\mbox{\boldmath{$w$}}(0)=\mbox{\boldmath{$w$}}_{0}$,
$\mbox{\boldmath{$\eta$}}(0)=\mbox{\boldmath{$0$}}$ and

$\begin{cases}\displaystyle\mbox{\boldmath{$\eta$}}^{\prime}(t)=F(t,\mbox{\boldmath{$\eta$}}(t),(S\mbox{\boldmath{$w$}})(t))\
\ \mbox{\rm for all}\ \ t\in(0,T),\\\\[2.84526pt]
\langle\mbox{\boldmath{$w$}}^{\prime}(t)+A(t,\mbox{\boldmath{$\eta$}}(t),\mbox{\boldmath{$w$}}(t))+(R_{1}\mbox{\boldmath{$w$}})(t)-\mbox{\boldmath{$f$}}(t),\mbox{\boldmath{$v$}}-\mbox{\boldmath{$w$}}(t)\rangle\\\\[2.84526pt]
\qquad\ \
\,+\,j^{0}((R_{2}\mbox{\boldmath{$w$}})(t),M\mbox{\boldmath{$w$}}(t);M\mbox{\boldmath{$v$}}-M\mbox{\boldmath{$w$}}(t))\geq
0\ \ \mbox{\rm for all}\ \mbox{\boldmath{$v$}}\in K,\ \mbox{\rm a.e.}\
t\in(0,T).\end{cases}$

First, we apply Corollary 3.1 to deduce that Problem 10 is solvable. We will
verify hypotheses $H(F)$, $H(A)$, $H(j)$, $H(\varphi)$, $H(K)$, $H(M)$, and
$(H_{1})$–$(H_{3})$. We use assumptions $H(\mathscr{G})$ and $H(\mathscr{B})$
to obtain the estimate

$\displaystyle\|F(t,\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$w$}})\|_{\mathcal{H}}\leq\|F(t,\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$w$}})-F(t,\mbox{\boldmath{$0$}},\mbox{\boldmath{$0$}})\|_{\mathcal{H}}+\|F(t,\mbox{\boldmath{$0$}},\mbox{\boldmath{$0$}})\|_{\mathcal{H}}$
$\displaystyle\quad\leq\sqrt{3}\,L_{\mathscr{G}}\,(\|\mbox{\boldmath{$\eta$}}\|_{\mathcal{H}}+L_{\mathscr{B}}\|\mbox{\boldmath{$w$}}\|+\|\mbox{\boldmath{$w$}}\|)+\sqrt{2}\,(L_{\mathscr{G}}\,\|{\mathscr{B}}(\cdot,t,\mbox{\boldmath{$0$}})\|_{\mathcal{H}}+\|{\mathscr{G}}(\cdot,t,\mbox{\boldmath{$0$}},\mbox{\boldmath{$0$}})\|_{\mathcal{H}})$

for all $\mbox{\boldmath{$\eta$}}\in{\mathcal{H}}$, $\mbox{\boldmath{$w$}}\in
V$, a.e. $t\in(0,T)$. Hence, the function $t\mapsto
F(t,\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$w$}})$ belongs to
$L^{2}(0,T;{\mathcal{H}})$ for all $\mbox{\boldmath{$\eta$}}\in{\mathcal{H}}$,
$\mbox{\boldmath{$w$}}\in V$. Similarly, we find that

$\|F(t,\mbox{\boldmath{$\eta$}}_{1},\mbox{\boldmath{$w$}}_{1})-F(t,\mbox{\boldmath{$\eta$}}_{2},\mbox{\boldmath{$w$}}_{2})\|_{\mathcal{H}}\leq\sqrt{2}\,L_{\mathscr{G}}\,(\sqrt{2}\,\|\mbox{\boldmath{$\eta$}}_{1}-\mbox{\boldmath{$\eta$}}_{2}\|_{\mathcal{H}}+\sqrt{2}\,L_{\mathscr{B}}\|\mbox{\boldmath{$w$}}_{1}-\mbox{\boldmath{$w$}}_{2}\|+\|\mbox{\boldmath{$w$}}_{1}-\mbox{\boldmath{$w$}}_{2}\|)$

for all $\mbox{\boldmath{$\eta$}}_{1}$,
$\mbox{\boldmath{$\eta$}}_{2}\in{\mathcal{H}}$, $\mbox{\boldmath{$w$}}_{1}$,
$\mbox{\boldmath{$w$}}_{2}\in V$, a.e. $t\in(0,T)$. We deduce that the map $F$
satisfies $H(F)$.

Next, we show that operator ${\mathscr{A}}$ satisfies $H(A)$. It is clear from
$H({\mathscr{A}})$ that
$A(\cdot,\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$v$}})$ is measurable for
all $\mbox{\boldmath{$\eta$}}\in E$, $\mbox{\boldmath{$v$}}\in V$, and
$A(t,\cdot,\mbox{\boldmath{$v$}})$ is continuous for all
$\mbox{\boldmath{$v$}}\in V$, a.e. $t\in(0,T)$. From $H({\mathscr{A}})$(3), we
have
$\|{\mathscr{A}}(t,\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}))\|_{\mathcal{H}}\leq\sqrt{2}\,(\|{\widetilde{a}}_{0}(t)\|_{L^{2}(\Omega)}+{\widetilde{a}}_{2}\,\|\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})\|_{\mathcal{H}})$
for all $\mbox{\boldmath{$v$}}\in V$, a.e. $t\in(0,T)$, and consequently

$\displaystyle\|A(t,\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$v$}})\|_{V^{*}}=\sup_{\|z\|\leq
1}\langle{\mathscr{A}}(t,\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}))+\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$z$}})\rangle_{\mathcal{H}}\leq\sup_{\|z\|\leq
1}\left(\|{\mathscr{A}}(t,\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}))\|_{\mathcal{H}}+\|\mbox{\boldmath{$\eta$}}\|_{\mathcal{H}}\right)\,\|\mbox{\boldmath{$z$}}\|$
$\displaystyle\qquad\leq\sqrt{2}\,\|{\widetilde{a}}_{0}(t)\|_{L^{2}(\Omega)}+\|\mbox{\boldmath{$\eta$}}\|_{\mathcal{H}}+\sqrt{2}\,{\widetilde{a}}_{2}\,\|\mbox{\boldmath{$v$}}\|,$

which implies condition $H(A)$(c) with
$a_{0}(t)=\sqrt{2}\,\|{\widetilde{a}}_{0}(t)\|_{L^{2}(\Omega)}$, $a_{1}=1$ and
$a_{2}=\sqrt{2}\,{\widetilde{a}}_{2}$. To demonstrate $H(A)$(d), we observe
that

$\|A(t,\mbox{\boldmath{$\eta$}}_{1},\mbox{\boldmath{$v$}})-A(t,\mbox{\boldmath{$\eta$}}_{2},\mbox{\boldmath{$v$}})\|_{V^{*}}=\sup_{\|z\|\leq
1}\langle\mbox{\boldmath{$\eta$}}_{1}-\mbox{\boldmath{$\eta$}}_{2},\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$z$}})\rangle_{\mathcal{H}}\leq\sup_{\|z\|\leq
1}\|\mbox{\boldmath{$\eta$}}_{1}-\mbox{\boldmath{$\eta$}}_{2}\|_{\mathcal{H}}\|\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$z$}})\|_{\mathcal{H}}\leq\|\mbox{\boldmath{$\eta$}}_{1}-\mbox{\boldmath{$\eta$}}_{2}\|_{\mathcal{H}}$

for all $\mbox{\boldmath{$\eta$}}_{1}$,
$\mbox{\boldmath{$\eta$}}_{2}\in{\mathcal{H}}$, $\mbox{\boldmath{$v$}}\in V$,
a.e. $t\in(0,T)$, and from $H({\mathscr{A}})$(4), we deduce

$\displaystyle\langle
A(t,\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$v$}}_{1})-A(t,\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$v$}}_{2})\rangle=\langle{\mathscr{A}}(t,\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}_{1}))-{\mathscr{A}}(t,\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}_{2})),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}_{1})-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}_{2})\rangle_{\mathcal{H}}$
$\displaystyle\qquad\geq
m_{\mathscr{A}}\int_{\Omega}\|\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}_{1})-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}_{2})\|^{2}_{\mathbb{S}^{d}}\,dx=m_{\mathscr{A}}\,\|\mbox{\boldmath{$v$}}_{1}-\mbox{\boldmath{$v$}}_{2}\|^{2}$

for all $\mbox{\boldmath{$\eta$}}\in{\mathcal{H}}$,
$\mbox{\boldmath{$v$}}_{1}$, $\mbox{\boldmath{$v$}}_{2}\in V$, a.e.
$t\in(0,T)$. Exploiting the last two inequalities, by Remark 3.2, we obtain
$H(A)$(d) with $m_{A}=m_{\mathscr{A}}$ and ${\overline{m}}_{A}=0$. Hence
$H(A)$ holds.

From hypotheses $H(k)$, $H(j_{\nu})$, and [26, Theorem 3.47], it follows that
function $j$ satisfies $H(j)$(a)–(d), and

$j^{0}(z,v;w)\leq\int_{\Gamma_{C}}k(\mbox{\boldmath{$x$}},z)\,j_{\nu}^{0}(\mbox{\boldmath{$x$}},v;w)\,d\Gamma\
\ \mbox{for all}\ \ z,v,w\in X.$ (36)

We use (36) with $H(k)$ and $H(j_{\nu})$ to get

$\displaystyle j^{0}(z_{1},v_{1};v_{2}-v_{1})+j^{0}(z_{2},v_{2};v_{1}-v_{2})$
$\displaystyle\leq\int_{\Gamma_{C}}\Big{(}k(z_{1})j_{\nu}^{0}(v_{1};v_{2}-v_{1})+k(z_{2})j_{\nu}^{0}(v_{2};v_{1}-v_{2})\Big{)}\,d\Gamma$
$\displaystyle\leq\int_{\Gamma_{C}}\big{(}k(z_{1})-k(z_{2})\big{)}j_{\nu}^{0}(v_{1};v_{2}-v_{1})+\,k(z_{2})\Big{(}j_{\nu}^{0}(v_{1};v_{2}-v_{1})+j_{\nu}^{0}(v_{2};v_{1}-v_{2})\Big{)}\,d\Gamma$
$\displaystyle\leq
c_{0}\,L_{k}\int_{\Gamma_{C}}|z_{1}(\mbox{\boldmath{$x$}})-z_{2}(\mbox{\boldmath{$x$}})|\,\|v_{1}(\mbox{\boldmath{$x$}})-v_{2}(\mbox{\boldmath{$x$}})\|\,d\Gamma+\alpha_{j_{\nu}}\,k_{2}\int_{\Gamma_{C}}\|v_{1}(\mbox{\boldmath{$x$}})-v_{2}(\mbox{\boldmath{$x$}})\|^{2}\,d\Gamma$
$\displaystyle\leq{\overline{m}}_{j}\|z_{1}-z_{2}\|_{Z}\|v_{1}-v_{2}\|_{X}+m_{j}\|v_{1}-v_{2}\|^{2}_{X},$

where $m_{j}=\alpha_{j_{\nu}}k_{2}$ and ${\overline{m}}_{j}=c_{0}L_{k}$. Hence
$H(j)$(e) follows, and therefore, $j$ satisfies $H(j)$.

Condition $H(\varphi)$ holds trivially. It is clear that the set $K$ is a
closed and convex subset of $V$ with $\mbox{\boldmath{$0$}}\in K$, i.e.,
$H(K)$ holds. By the linearity and continuity of the normal trace operator, it
is obvious that $M_{1}=M$ and $H(M)$(a) is satisfied. Condition $H(M)$(b)
holds as a consequence of the proof of [28, Theorem 2.18, p.59]. Using the
regularity of $\mbox{\boldmath{$f$}}_{0}$ and $\mbox{\boldmath{$f$}}_{N}$ in
$(H_{4})$, we readily obtain that the functional defined by (27) satisfies
$\mbox{\boldmath{$f$}}\in L^{2}(0,T;V^{*})$.

Condition $(H_{1})$ is a consequence of the smallness hypothesis (33) while
condition $(H_{2})$ follows from $(H_{4})$. Finally, we argue as in [41,
Theorem 14.2, p.367] to demonstrate that operators $S$, $R_{1}$ and $R_{2}$
satisfy the inequalities $(H_{3})$(a)–(c), respectively.

Having verified all hypotheses of Corollary 3.1, we deduce from it that
Problem 10 has a unique solution $\mbox{\boldmath{$\eta$}}\in H^{1}(0,T;E)$,
$\mbox{\boldmath{$w$}}\in\mathbb{W}$ such that $\mbox{\boldmath{$w$}}(t)\in K$
for a.e. $t\in(0,T)$. By the inequality (36) it follows that any solution to
Problem 10 is a solution to Problem 9. Moreover, from the relation
$\mbox{\boldmath{$u$}}(t)=\int_{0}^{t}\mbox{\boldmath{$w$}}(s)\,ds+\mbox{\boldmath{$u$}}_{0}$
for all $t\in(0,T)$, we conclude that $\mbox{\boldmath{$u$}}\in C([0,T];V)$,
$\mbox{\boldmath{$u$}}^{\prime}\in\mathbb{W}$ with
$\mbox{\boldmath{$u$}}^{\prime}(t)\in K$ for a.e. $t\in(0,T)$. The proof of
existence of solution to Problem 7 is completed.

Next, we suppose the regularity hypothesis (34). Invoking [26, Theorem
3.47(vii)], we deduce that (36) holds with equality. This implies that
Problems 9 and 10 are equivalent. Therefore, in this case, Problem 7 is
uniquely solvable.

Let $(\mbox{\boldmath{$\eta$}}_{1},\mbox{\boldmath{$u$}}_{1})$ and
$(\mbox{\boldmath{$\eta$}}_{2},\mbox{\boldmath{$u$}}_{2})$ be the unique
solutions to Problem 7 corresponding to the data

$(\mbox{\boldmath{$u$}}_{0},\mbox{\boldmath{$w$}}_{0},\mbox{\boldmath{$f$}}_{0},\mbox{\boldmath{$f$}}_{N}),(\widetilde{\mbox{\boldmath{$u$}}}_{0},{\widetilde{\mbox{\boldmath{$w$}}}}_{0},\widetilde{\mbox{\boldmath{$f$}}},\widetilde{\mbox{\boldmath{$f$}}}_{N})\in
V\times V\times L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d})\times
L^{2}(\Gamma_{N};\mathbb{R}^{d})),$

respectively. From (27) and $(H_{4})$, we readily have

$\|\mbox{\boldmath{$f$}}-\widetilde{\mbox{\boldmath{$f$}}}\|_{L^{2}(0,T;V^{*})}\leq
c\,\left(\|\mbox{\boldmath{$f$}}_{0}-{\widetilde{\mbox{\boldmath{$f$}}}_{0}}\|_{L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))}+\|\mbox{\boldmath{$f$}}_{N}-{\widetilde{\mbox{\boldmath{$f$}}}_{N}}\|_{L^{2}(0,T;L^{2}(\Gamma_{N};\mathbb{R}^{d}))}\right)$
(37)

with a constant $c>0$, which by (15) of Corollary 3.1 implies

$\displaystyle\|\mbox{\boldmath{$\eta$}}_{1}-\mbox{\boldmath{$\eta$}}_{2}\|_{H^{1}(0,T;E)}+\|\mbox{\boldmath{$w$}}_{1}-\mbox{\boldmath{$w$}}_{2}\|_{L^{2}(0,T;V)}$
(38) $\displaystyle\quad\leq
c\,(\|\mbox{\boldmath{$w$}}_{0}-\widetilde{\mbox{\boldmath{$w$}}}_{0}\|+\|\mbox{\boldmath{$f$}}_{0}-{\widetilde{\mbox{\boldmath{$f$}}}_{0}}\|_{L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))}+\|\mbox{\boldmath{$f$}}_{N}-{\widetilde{\mbox{\boldmath{$f$}}}_{N}}\|_{L^{2}(0,T;L^{2}(\Gamma_{N};\mathbb{R}^{d}))}).$

Using again the equation
$\mbox{\boldmath{$u$}}(t)=\mbox{\boldmath{$u$}}_{0}+\int_{0}^{t}\mbox{\boldmath{$w$}}(s)\,ds$
for all $t\in(0,T)$, we obtain

$\|\mbox{\boldmath{$u$}}_{1}-\mbox{\boldmath{$u$}}_{2}\|_{C(0,T;V)}\leq\|\mbox{\boldmath{$u$}}_{0}-\widetilde{\mbox{\boldmath{$u$}}}_{0}\|+\sqrt{T}\,\|\mbox{\boldmath{$w$}}_{1}-\mbox{\boldmath{$w$}}_{2}\|_{L^{2}(0,T;V)}.$
(39)

Adding (38) and (39), and combining with
$\mbox{\boldmath{$w$}}_{i}=\mbox{\boldmath{$u$}}^{\prime}_{i}$ for $i=1$, $2$,
we deduce the inequality (35). Finally, we use the continuity of the
embeddings

$H^{1}(0,T;{\mathcal{H}})\subset C([0,T];{\mathcal{H}})\ \ \mbox{and}\ \
{\mathbb{W}}\subset C([0,T];L^{2}(\Omega;\mathbb{R}^{d}))$

to conclude the desired regularity of the solution
$(\mbox{\boldmath{$\eta$}},\mbox{\boldmath{$u$}}^{\prime})\in
C([0,T];{\mathcal{H}}\times L^{2}(\Omega;\mathbb{R}^{d}))$. This completes the
proof. $\Box$

##  5 A dynamic frictional viscoelastic contact problem

In this section we consider the dynamic contact problem for viscoelastic
materials with friction and adhesion. The physical framework is similar to the
one considered in the previous section. The difference arises in the fact that
now the body $\Omega\subset\mathbb{R}^{d}$ is assumed to be viscoelastic with
long memory, the process is frictional and it is governed by a convex
potential, and the adhesion process is modeled by an ordinary differential
equation for the bonding variable on a contact surface $\Gamma_{C}$. The
classical formulation of the contact problem we study in this section is the
following.

######  Problem 11.

Find a displacement field $\mbox{\boldmath{$u$}}\colon Q\to\mathbb{R}^{d}$ and
a stress field $\mbox{\boldmath{$\sigma$}}\colon Q\rightarrow\mathbb{S}^{d}$
and a bonding field $\beta\colon\Gamma_{C}\times(0,T)\to[0,1]$ such that for
all $t\in(0,T)$,

$\displaystyle\mbox{\boldmath{$\sigma$}}(t)$
$\displaystyle={\mathscr{A}}\mbox{\boldmath{$\varepsilon$}}({\mbox{\boldmath{$u$}}}^{\prime}(t))+{\mathscr{B}}\mbox{\boldmath{$\varepsilon$}}({\mbox{\boldmath{$u$}}}(t))+\int_{0}^{t}{\mathscr{C}}(t-s)\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}^{\prime}(s))\,ds\quad$
$\displaystyle{\rm in}\ $ $\displaystyle\Omega,$ (40)
$\displaystyle{\mbox{\boldmath{$u$}}}^{\prime\prime}(t)$ $\displaystyle={\rm
Div}\,\mbox{\boldmath{$\sigma$}}(t)+\mbox{\boldmath{$f$}}_{0}(t)\quad$
$\displaystyle{\rm in}\ $ $\displaystyle\Omega,$ (41)
$\displaystyle\mbox{\boldmath{$u$}}(t)$ $\displaystyle=\mbox{\boldmath{$0$}}$
$\displaystyle{\rm on}\ $ $\displaystyle\Gamma_{D},$ (42)
$\displaystyle\mbox{\boldmath{$\sigma$}}(t)\mbox{\boldmath{$\nu$}}$
$\displaystyle=\mbox{\boldmath{$f$}}_{N}(t)\quad$ $\displaystyle{\rm on}\ $
$\displaystyle\Gamma_{N},$ (43) $\displaystyle u^{\prime}_{\nu}(t)$
$\displaystyle\leq g,\ \sigma_{\nu}(t)\leq 0,\
(u^{\prime}_{\nu}(t)-g)\,\sigma_{\nu}(t)=0\quad$ $\displaystyle{\rm on}\ $
$\displaystyle\Gamma_{C},$ (44)
$\displaystyle-\mbox{\boldmath{$\sigma$}}_{\tau}(t)$
$\displaystyle\in\mu(u_{\nu}(t))\,h_{1}(\beta(t))\,\partial
h_{2}(\mbox{\boldmath{$u$}}^{\prime}_{\tau}(t))\quad$ $\displaystyle{\rm on}\
$ $\displaystyle\Gamma_{C},$ (45) $\displaystyle\beta^{\prime}(t)$
$\displaystyle=G(t,\beta(t),\mbox{\boldmath{$u$}}(t))\quad$ $\displaystyle{\rm
on}\ $ $\displaystyle\Gamma_{C},$ (46) $\displaystyle\beta(0)$
$\displaystyle=\beta_{0}\quad$ $\displaystyle{\rm on}\ $
$\displaystyle\Gamma_{C},$ (47) $\displaystyle\mbox{\boldmath{$u$}}(0)$
$\displaystyle=\mbox{\boldmath{$u$}}_{0},\ \
\mbox{\boldmath{$u$}}^{\prime}(t)=\mbox{\boldmath{$w$}}_{0}\quad$
$\displaystyle{\rm in}\ $ $\displaystyle\Omega.$ (48)

In this problem, the viscoelastic constitutive law is of the form (40), where
$\mathscr{A}$, $\mathscr{B}$ and $\mathscr{C}$ denote the viscosity operator,
the elasticity operator, and the relaxation tensor, respectively.

In the study of Problem 11 we suppose that the viscosity operator
${\mathscr{A}}$, the elasticity operator ${\mathscr{B}}$ satisfy condition
$H({\mathscr{A}})$ and $H({\mathscr{B}})$, respectively. We also assume that
the densities of the body forces ${\mbox{\boldmath{$f$}}_{0}}$ and tractions
${\mbox{\boldmath{$f$}}_{N}}$, and the initial data
$\mbox{\boldmath{$u$}}_{0}$ and $\mbox{\boldmath{$w$}}_{0}$ satisfy hypothesis
$(H_{4})$. The condition (45) represents the friction condition which is a law
between the tangential velocity and the tangential stress corresponding to the
friction force. This law involves the product of a function $h_{1}$ which
depends on the adhesion field $\beta$ and the subdifferential $\partial h_{2}$
of a convex potential $h_{2}(\mbox{\boldmath{$x$}},\cdot)$. The coefficient
$\mu$ is a given function which depends on the normal displacement. Since
condition (44) is the Signorini unilateral contact boundary condition for the
normal velocity with a constant $g>0$, analogously as in the previous section,
we need the set $K$ of admissible unilateral velocity constraints defined by
(26).

The unknown surface variable $\beta$ is called the adhesion field, see [37].
It describes intensity of adhesion on the contact surface and is governed by
an ordinary differential equation (46) with an initial condition (47). The
value $\beta=1$ means that the adhesion is complete, $\beta=0$ means that
there is no adhesion, and $0<\beta<1$ means that the adhesion is partial.
Further, we need the following hypotheses on the relaxation tensor
${\mathscr{C}}$, the potential $h$, the coefficient $\mu$, and the adhesive
evolution rate function $G$.

$\underline{H({\mathscr{C}})}:$ $\mathscr{C}\colon
Q\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}$ is such that

  * (1)

$\mathscr{C}(\mbox{\boldmath{$x$}},t,\mbox{\boldmath{$\varepsilon$}})=(c_{ijkl}(\mbox{\boldmath{$x$}},t)\varepsilon_{kl})$
for all $\mbox{\boldmath{$\varepsilon$}}=(\varepsilon_{ij})\in\mathbb{S}^{d}$,
a.e. $(\mbox{\boldmath{$x$}},t)\in Q$.

  * (2)

$c_{ijkl}(\cdot,t)=c_{jikl}(\cdot,t)=c_{lkij}(\cdot,t)\in L^{\infty}(\Omega)$
for a.e. $t\in(0,T)$.

  * (3)

$t\mapsto c_{ijkl}(\cdot,t)\in L^{\infty}(0,T;L^{\infty}(\Omega))$.

$\underline{H(h_{1})}:$
$h_{1}\colon\Gamma_{C}\times\mathbb{R}\rightarrow\mathbb{R}$ is such that

  * (1)

$h_{1}(\cdot,r)$ is measurable on $\Gamma_{C}$, for all $r\in\mathbb{R}$.

  * (2)

$h_{1}(\mbox{\boldmath{$x$}},\cdot)$ is Lipschitz continuous on $\mathbb{R}$,
a.e. $\mbox{\boldmath{$x$}}\in\Gamma_{C}$.

  * (3)

$0\leq h_{1}(\mbox{\boldmath{$x$}},r)\leq h_{0}$ for all $r\in\mathbb{R}$,
a.e. $\mbox{\boldmath{$x$}}\in\Gamma_{C}$ with $h_{0}>0$.

$\underline{H(h_{2})}:$
$h_{2}\colon\Gamma_{C}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ is such that

  * (1)

$h_{2}(\cdot,\mbox{\boldmath{$\xi$}})$ is measurable on $\Gamma_{C}$ for
$\mbox{\boldmath{$\xi$}}\in\mathbb{R}^{d}$.

  * (2)

$h_{2}(\mbox{\boldmath{$x$}},\cdot)$ is convex, lower semicontinuous, and
Lipschitz, for a.e. $\mbox{\boldmath{$x$}}\in\Gamma_{C}$.

  * (3)

$\|\partial
h_{2}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$\xi$}})\|\leq\rho_{1}(\mbox{\boldmath{$x$}})+\rho_{2}\|\mbox{\boldmath{$\xi$}}\|$
for all $\mbox{\boldmath{$\xi$}}\in\mathbb{R}^{d}$, a.e.
$\mbox{\boldmath{$x$}}\in\Gamma_{C}$ with $\rho_{1}\in L^{2}(\Gamma_{C})$,
$\rho_{1}$, $\rho_{2}\geq 0$.

$\underline{H(\mu)}:$
$\mu\colon\Gamma_{C}\times\mathbb{R}\rightarrow\mathbb{R}$ is such that

  * (1)

$\mu(\cdot,r)$ is measurable on $\Gamma_{C}$ for all $r\in\mathbb{R}$.

  * (2)

$\mu(\mbox{\boldmath{$x$}},\cdot)$ is continuous on $\mathbb{R}$ for a.e.
$\mbox{\boldmath{$x$}}\in\Gamma_{C}$.

  * (3)

$0\leq\mu(\mbox{\boldmath{$x$}},r)\leq\mu_{0}$ for all $r\in\mathbb{R}$, a.e.
$\mbox{\boldmath{$x$}}\in\Gamma_{C}$ with $\mu_{0}>0$.

$\underline{H(G)}:$
$G\colon\Gamma_{C}\times(0,T)\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
is such that

  * (1)

$G(\cdot,\cdot,r,\mbox{\boldmath{$\xi$}})$ is measurable on
$\Gamma_{C}\times(0,T)$ for all $r\in\mathbb{R}$,
$\mbox{\boldmath{$\xi$}}\in\mathbb{R}^{d}$.

  * (2)

$|G(\mbox{\boldmath{$x$}},t,r_{1},\xi_{1})-G(\mbox{\boldmath{$x$}},t,r_{2},\mbox{\boldmath{$\xi$}}_{2})|\leq
L_{G}\,\left(|r_{1}-r_{2}|+\|\mbox{\boldmath{$\xi$}}_{1}-\mbox{\boldmath{$\xi$}}_{2}\|\right)$
for all
$(r_{i},\mbox{\boldmath{$\xi$}}_{i})\in\mathbb{R}\times\mathbb{R}^{d}$, $i=1$,
$2$, a.e. $(\mbox{\boldmath{$x$}},t)\in\Gamma_{C}\times(0,T)$ with $L_{G}>0$.

  * (3)

$G(\mbox{\boldmath{$x$}},t,0,\mbox{\boldmath{$\xi$}})=0$,
$G(\mbox{\boldmath{$x$}},t,r,\mbox{\boldmath{$\xi$}})\geq 0$ for $r\leq 0$,
and $G(\mbox{\boldmath{$x$}},t,r,\mbox{\boldmath{$\xi$}})\leq 0$ for $r\geq
1$, all $\mbox{\boldmath{$\xi$}}\in\mathbb{R}^{d}$, a.e.
$(\mbox{\boldmath{$x$}},t)\in\Gamma_{C}\times(0,T)$.

Let $V$ and ${\mathcal{H}}$ be the spaces defined by (24), and
$\mbox{\boldmath{$f$}}\in L^{2}(0,T;V^{*})$ be given by (27). We use the
standard procedure as in Section 4 to obtain the following weak formulation of
Problem 11.

######  Problem 12.

Find $\beta\colon(0,T)\to[0,1]$ and $\mbox{\boldmath{$u$}}\colon(0,T)\to V$
such that $\mbox{\boldmath{$u$}}^{\prime}(t)\in K$ a.e. $t\in(0,T)$ and

$\displaystyle\beta^{\prime}(t)=G(t,\beta(t),\mbox{\boldmath{$u$}}(t))\ \
\mbox{\rm on}\ \ \Gamma_{C}\ \mbox{\rm for all}\ \,t\in(0,T),$
$\displaystyle\langle\mbox{\boldmath{$u$}}^{\prime\prime}(t),\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t)\rangle+\langle\mathscr{A}(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}^{\prime}(t)))+\mathscr{B}(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}(t)))+\int_{0}^{t}{\mathscr{C}}(t-s)\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}^{\prime}(s))\,ds,\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}^{\prime}(t))\rangle_{\mathcal{H}}$
$\displaystyle\quad+\int_{\Gamma_{C}}\mu(u_{\nu}(t))\,h_{1}(\beta(t))\,\left(h_{2}(\mbox{\boldmath{$v$}}_{\tau})-h_{2}(\mbox{\boldmath{$u$}}^{\prime}_{\tau}(t))\right)\,d\Gamma\geq\langle\mbox{\boldmath{$f$}}(t),\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}^{\prime}(t)\rangle$
$\displaystyle\qquad\ \ \mbox{\rm for all}\ \,\mbox{\boldmath{$v$}}\in K,\
\mbox{\rm a.e.}\ t\in(0,T),$ $\displaystyle\beta(0)=\beta_{0},\
\mbox{\boldmath{$u$}}(0)=\mbox{\boldmath{$u$}}_{0},\
\mbox{\boldmath{$u$}}^{\prime}(0)=\mbox{\boldmath{$w$}}_{0}.$

Problem 12 consists with the ordinary differential equation on the contact
surface and a dynamic variational inequality with constraints. The following
result deals with the unique solvability, continuous dependence, and
regularity of solution to Problem 12.

######  Theorem 13.

Assume hypotheses $H({\mathscr{A}})$, $H({\mathscr{B}})$, $H({\mathscr{C}})$,
$H(h_{1})$, $H(h_{2})$, $H(\mu)$, $H(G)$, $(H_{4})$ and $\beta_{0}\in
L^{2}(\Gamma_{C})$ with $0\leq\beta_{0}(\mbox{\boldmath{$x$}})\leq 1$ for a.e.
$\mbox{\boldmath{$x$}}\in\Gamma_{C}$. Then Problem 12 has a unique solution
$\beta\in H^{1}(0,T;L^{2}(\Gamma_{C}))$ and $\mbox{\boldmath{$u$}}\in
C([0,T];V)$ such that $0\leq\beta(\mbox{\boldmath{$x$}},t)\leq 1$ for all
$t\in(0,T)$, a.e. $\mbox{\boldmath{$x$}}\in\Gamma_{C}$,
$\mbox{\boldmath{$u$}}^{\prime}\in{\mathbb{W}}$ with
$\mbox{\boldmath{$u$}}^{\prime}(t)\in K$ for a.e. $t\in(0,T)$. Moreover, let
$(\beta_{1},\mbox{\boldmath{$u$}}_{1})$ and
$(\beta_{2},\mbox{\boldmath{$u$}}_{2})$ be the unique solutions to Problem 12
corresponding to the data

$(\beta_{0},\mbox{\boldmath{$u$}}_{0},\mbox{\boldmath{$w$}}_{0},\mbox{\boldmath{$f$}}_{0},\mbox{\boldmath{$f$}}_{N}),(\widetilde{\beta}_{0},\widetilde{\mbox{\boldmath{$u$}}}_{0},{\widetilde{\mbox{\boldmath{$w$}}}}_{0},\widetilde{\mbox{\boldmath{$f$}}},\widetilde{\mbox{\boldmath{$f$}}}_{N})\in
L^{2}(\Gamma_{C})\times V\times V\times
L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d})\times
L^{2}(\Gamma_{N};\mathbb{R}^{d})),$

respectively. Then, there is a constant $c>0$ such that

$\displaystyle\|\beta_{1}-\beta_{2}\|_{H^{1}(0,T;L^{2}(\Gamma_{C}))}+\|\mbox{\boldmath{$u$}}_{1}-\mbox{\boldmath{$u$}}_{2}\|_{C(0,T;V)}+\|\mbox{\boldmath{$u$}}^{\prime}_{1}-\mbox{\boldmath{$u$}}^{\prime}_{2}\|_{L^{2}(0,T;V)}\leq
c\,(\|\beta_{0}-\widetilde{\beta}_{0}\|_{L^{2}(\Gamma_{C})}$
$\displaystyle\quad+\,\|\mbox{\boldmath{$u$}}_{0}-\widetilde{\mbox{\boldmath{$u$}}}_{0}\|+\|\mbox{\boldmath{$w$}}_{0}-\widetilde{\mbox{\boldmath{$w$}}}_{0}\|+\|\mbox{\boldmath{$f$}}_{0}-{\widetilde{\mbox{\boldmath{$f$}}}_{0}}\|_{L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))}+\|\mbox{\boldmath{$f$}}_{N}-{\widetilde{\mbox{\boldmath{$f$}}}_{N}}\|_{L^{2}(0,T;L^{2}(\Gamma_{N};\mathbb{R}^{d}))}),$

Moreover, $(\beta,\mbox{\boldmath{$u$}}^{\prime})\in
C([0,T];L^{2}(\Gamma_{C})\times L^{2}(\Omega;\mathbb{R}^{d}))$.

Proof. Let $\mbox{\boldmath{$w$}}(t)=\mbox{\boldmath{$u$}}^{\prime}(t)$ for
all $t\in(0,T)$, hence
$\mbox{\boldmath{$u$}}(t)=\mbox{\boldmath{$u$}}_{0}+\int_{0}^{t}\mbox{\boldmath{$w$}}(s)\,ds$
for all $t\in(0,T)$. We reformulate Problem 12 in terms of the velocity $w$
using the notation below.

Let $E:=Y=L^{2}(\Gamma_{C})$, $U:=V$, $X=L^{2}(\Gamma_{C};\mathbb{R}^{d})$,
$A\colon(0,T)\times V\to V^{*}$, $I\colon L^{2}(0,T;V)\to L^{2}(0,T;V)$,
$S\colon L^{2}(0,T;V)\to L^{2}(0,T;U)$, $R_{1}\colon L^{2}(0,T;V)\to
L^{2}(0,T;V^{*})$, $R_{3}\colon L^{2}(0,T;V)\to L^{2}(0,T;Y)$,
$\varphi\colon(0,T)\times E\times Y\times X\to\mathbb{R}$, $F\colon(0,T)\times
E\times U\to E$, $M\colon V\to X$ be defined by

$\displaystyle\langle
A(t,\mbox{\boldmath{$v$}}),\mbox{\boldmath{$z$}}\rangle:=\langle\mathscr{A}(t,\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$z$}})\rangle_{\mathcal{H}}\
\ \mbox{for}\ \ \mbox{\boldmath{$v$}},\mbox{\boldmath{$z$}}\in V,\
\mbox{a.e.}\ t\in(0,T),$
$\displaystyle(I\mbox{\boldmath{$w$}})(t):=\mbox{\boldmath{$u$}}_{0}+\int_{0}^{t}\mbox{\boldmath{$w$}}(s)\,ds\
\ \mbox{for}\ \ \mbox{\boldmath{$w$}}\in L^{2}(0,T;V),\ \mbox{a.e.}\
t\in(0,T),$
$\displaystyle(S\mbox{\boldmath{$w$}})(t):=((I\mbox{\boldmath{$w$}})(t))_{\tau}=\mbox{\boldmath{$u$}}_{0\tau}+\int_{0}^{t}\mbox{\boldmath{$w$}}_{\tau}(s)\,ds\
\ \mbox{for}\ \ \mbox{\boldmath{$w$}}\in L^{2}(0,T;V),\ \mbox{a.e.}\
t\in(0,T),$
$\displaystyle\langle(R_{1}\mbox{\boldmath{$w$}})(t),\mbox{\boldmath{$v$}}\rangle:=\langle{\mathscr{B}}\mbox{\boldmath{$\varepsilon$}}((I\mbox{\boldmath{$w$}})(t))+\int_{0}^{t}{\mathscr{C}}(t-s)\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$w$}}(s))\,ds,\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})\rangle_{\mathcal{H}}$
$\displaystyle\qquad\qquad\qquad\qquad\ \ \mbox{for}\ \
\mbox{\boldmath{$w$}}\in L^{2}(0,T;V),\ \mbox{\boldmath{$v$}}\in V,\
\mbox{a.e.}\ t\in(0,T),$
$\displaystyle(R_{3}\mbox{\boldmath{$w$}})(t):=((I\mbox{\boldmath{$w$}})(t))_{\nu}=u_{0\nu}+\int_{0}^{t}w_{\nu}(s)\,ds\
\ \mbox{for}\ \ \mbox{\boldmath{$w$}}\in L^{2}(0,T;V),\ \mbox{a.e.}\
t\in(0,T),$
$\displaystyle\varphi(t,\beta,y,\mbox{\boldmath{$v$}}):=\int_{\Gamma_{C}}\mu(y)\,h_{1}(\beta)\,h_{2}(\mbox{\boldmath{$v$}}_{\tau})\,d\Gamma\
\ \mbox{for}\ \ \beta\in E,\ y\in Y,\ \mbox{\boldmath{$v$}}\in X,\
\mbox{a.e.}\ t\in(0,T),$ $\displaystyle
F(t,\beta,\mbox{\boldmath{$w$}})(\mbox{\boldmath{$x$}}):=G(\mbox{\boldmath{$x$}},t,\beta(\mbox{\boldmath{$x$}},t),\mbox{\boldmath{$w$}}_{\tau}(\mbox{\boldmath{$x$}},t))\
\mbox{for}\ \beta\in E,\ \mbox{\boldmath{$w$}}\in U,\ \mbox{a.e.}\ t\in(0,T),$
$\displaystyle M\mbox{\boldmath{$v$}}:=\mbox{\boldmath{$v$}}_{\tau}\ \
\mbox{for}\ \ \mbox{\boldmath{$v$}}\in V,$

and $j\equiv 0$. Under this notation Problem 12 is equivalently written as

######  Problem 14.

Find $\beta\in H^{1}(0,T;E)$ and $\mbox{\boldmath{$w$}}\in\mathbb{W}$ such
that $\mbox{\boldmath{$w$}}(t)\in K$ for a.e. $t\in(0,T)$,
$\mbox{\boldmath{$w$}}(0)=\mbox{\boldmath{$w$}}_{0}$, $\beta(0)=\beta_{0}$ and

$\begin{cases}\displaystyle\beta^{\prime}(t)=F(t,\beta(t),(S\mbox{\boldmath{$w$}})(t))\
\ \mbox{\rm for all}\ \ t\in(0,T),\\\\[2.84526pt]
\langle\mbox{\boldmath{$w$}}^{\prime}(t)+A(t,\mbox{\boldmath{$w$}}(t))+(R_{1}\mbox{\boldmath{$w$}})(t)-\mbox{\boldmath{$f$}}(t),\mbox{\boldmath{$v$}}-\mbox{\boldmath{$w$}}(t)\rangle+\,\varphi(t,\beta(t),(R_{3}\mbox{\boldmath{$w$}})(t),M\mbox{\boldmath{$v$}})\\\\[2.84526pt]
\qquad\ \
\,-\,\varphi(t,\beta(t),(R_{3}\mbox{\boldmath{$w$}})(t),M\mbox{\boldmath{$w$}}(t))\geq
0\ \ \mbox{\rm for all}\ \mbox{\boldmath{$v$}}\in K,\ \mbox{\rm a.e.}\
t\in(0,T).\end{cases}$

Now we apply Corollary 3.1 to deduce the well-posedness of Problem 14. To this
end, we are able to verify hypotheses $H(F)$, $H(A)$, $H(j)$, $H(\varphi)$,
$H(K)$, $H(M)$, $\mbox{\boldmath{$f$}}\in L^{2}(0,T;V^{*})$, and
$(H_{1})$–$(H_{3})$. Note that hypotheses $H(j)$ and $(H_{1})$ hold trivially
and no smallness condition is needed. Condition $H(F)$ is a consequence of
$H(G)$ and can be proved analogously as in the proof of Theorem 8.
Analogously, we can also show that $H(A)$, $H(K)$, $H(M)$, and $(H_{3})$ are
satisfied, while $(H_{2})$ follows from $(H_{4})$. Additionally, a careful
examination of [24, Lemma 5] implies that
$0\leq\beta(\mbox{\boldmath{$x$}},t)\leq 1$ for all $t\in(0,T)$, a.e.
$\mbox{\boldmath{$x$}}\in\Gamma_{C}$. Next, we use $H(h_{1})$, $H(h_{2})$ and
$H(\mu)$ to check that $H(\varphi)$(a)–(d) holds with
$c_{0\varphi}=h_{0}\mu_{0}\|\rho\|_{L^{2}(\Gamma_{C})}$,
$c_{1\varphi}=c_{2\varphi}=0$ and $c_{3\varphi}=h_{0}\mu_{0}\rho_{2}$.
Condition $H(\varphi)$(e) follows from the inequality

$\displaystyle\varphi(t,\beta_{1},y_{1},v_{2})-\varphi(t,\beta_{1},y_{1},v_{1})+\varphi(t,\beta_{2},y_{2},v_{1})-\varphi(t,\beta_{2},y_{2},v_{2})$
$\displaystyle\leq\mu_{0}\int_{\Gamma_{C}}|h_{1}(\beta_{1})-h_{1}(\beta_{2})|\,|h_{2}(\mbox{\boldmath{$v$}}_{1\tau})-h_{2}(\mbox{\boldmath{$v$}}_{2\tau})|\,d\Gamma\leq
c\,\|\beta_{1}-\beta_{2}\|_{E}\,\|\mbox{\boldmath{$v$}}_{1}-\mbox{\boldmath{$v$}}_{2}\|_{X}$

for all $\beta_{1}$, $\beta_{2}\in E$, $\mbox{\boldmath{$v$}}_{1}$,
$\mbox{\boldmath{$v$}}_{2}\in X$ with $c>0$. Hence hypothesis $H(\varphi)$ is
verified. By Corollary 3.1 we deduce the unique solvability of Problem 14.

Moreover, due to (15) in Corollary 3.1, we have

$\displaystyle\|\beta_{1}-\beta_{2}\|_{H^{1}(0,T;L^{2}(\Gamma_{C}))}+\|\mbox{\boldmath{$w$}}_{1}-\mbox{\boldmath{$w$}}_{2}\|_{L^{2}(0,T;V)}\leq
c\,(\|\beta_{0}-\widetilde{\beta}_{0}\|_{L^{2}(\Gamma_{C})}$ (49)
$\displaystyle\quad+\,\|\mbox{\boldmath{$w$}}_{0}-\widetilde{\mbox{\boldmath{$w$}}}_{0}\|+\|\mbox{\boldmath{$f$}}_{0}-{\widetilde{\mbox{\boldmath{$f$}}}_{0}}\|_{L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))}+\|\mbox{\boldmath{$f$}}_{N}-{\widetilde{\mbox{\boldmath{$f$}}}_{N}}\|_{L^{2}(0,T;L^{2}(\Gamma_{N};\mathbb{R}^{d}))}).$

We combine (39) with (49) to deduce the inequality in the thesis of the
theorem. As in Theorem 8, we use the continuous embeddings
$H^{1}(0,T;L^{2}(\Gamma_{C}))\subset C([0,T];L^{2}(\Gamma_{C}))$ and
${\mathbb{W}}\subset C([0,T];L^{2}(\Omega;\mathbb{R}^{d}))$ to conclude the
desired regularity of the solution $(\beta,\mbox{\boldmath{$u$}}^{\prime})\in
C([0,T];L^{2}(\Gamma_{C})\times L^{2}(\Omega;\mathbb{R}^{d}))$. This completes
the proof. $\Box$

We refer to [35, Chapter 11.4], [37, Chapter 5], and [3, 11, 24, 32] for
examples of the adhesive evolution rate function $G$ which depends on both the
bonding field $\beta$ and the displacement and may change sign. This allows
for rebonding to take place after debonding, and it allows for possible cycles
of debonding and rebonding. Note also that the choice
$h_{2}(\mbox{\boldmath{$\xi$}})=\|\mbox{\boldmath{$\xi$}}\|$ for
$\mbox{\boldmath{$\xi$}}\in\mathbb{R}^{d}$ leads to a modified version of
Coulomb’s law which is usually used to model the frictional contact.

##  6 Final comments

We note that similar well-posedness results for more complicated contact
models can be obtained when various conditions on different parts of the
contact boundary are considered.

Furthermore, by the application of Theorem 4 and Corollary 3.1, one can deal
with the well-posedness of dynamic frictional and frictionless contact models
involving the internal state variables in viscoplasticity, see [39, Chapter
3], and the wear phenomena in contact problems, see e.g. [35, Chapter 11],
[37, Chapter 2].

It would be interesting to address the following open issues related to the
results of this paper. First it is of interest to examine the penalty methods
for the system (1)–(3) which will improve earlier results obtained recently in
[5, 39, 40, 42]. Also, an interesting topic is to study optimal control
problems for the system (1)–(3) including the necessary conditions of
optimality for the control problems, see [20]. Finally, it is an extensive and
important project to provide the numerical analysis of the system, see [36]
and the references therein.

## References

  * [1]
  * [2] N.T.V. Anh, T.D. Ke, On the differential variational inequalities of parabolic-parabolic type, Acta Appl. Math. 176 (2021), 5.
  * [3] K. Bartosz, Hemivariational inequalities modeling dynamic contact problems with adhesion, Nonlinear Anal. Theory Methods Appl. 71 (2009), 1747–1762.
  * [4] S. Carl, V.K. Le, D. Motreanu, Nonsmooth Variational Problems and their Inequalities, Springer, New York, 2007.
  * [5] J.X. Cen, L. Li, S. Migórski, Van Thien Nguyen, Convergence of a generalized penalty and regularization method for quasi-variational-hemivariational inequalities, Communications in Nonlinear Science and Numerical Simulation 103 (2021), 105998.
  * [6] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
  * [7] Z. Denkowski, S. Migórski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.
  * [8] Z. Denkowski, S. Migórski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.
  * [9] D. Goeleven, D. Motreanu, Y. Dumont, M. Rochdi, Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003.
  * [10] J. Gwinner, On a new class of differential variational inequalities and a stability result, Math. Program. 139 (2013), 205–221.
  * [11] J. Han, Y. Li, S. Migórski, Analysis of an adhesive contact problem for viscoelastic materials with long memory, Journal of Mathematical Analysis and Applications 427 (2015), 646–668.
  * [12] W. Han, S. Migórski, M. Sofonea, Analysis of a general dynamic history-dependent variational-hemivariational inequality, Nonlinear Analysis: Real World Applications 36 (2017), 69–88.
  * [13] W. Han, M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30, Americal Mathematical Society, Providence, RI–International Press, Somerville, MA, 2002.
  * [14] A. Kulig, S. Migórski, Solvability and continuous dependence results for second order nonlinear inclusion with Volterra-type operator, Nonlinear Analysis 75 (2012), 4729–4746.
  * [15] Z.H. Liu, Existence results for quasilinear parabolic hemivariational inequalities, J. Differential Equations 244 (2008), 1395–1409.
  * [16] Z.H. Liu, S. Migórski, S.D. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations 263 (2017), 3989–4006.
  * [17] Z.H. Liu, S.D. Zeng, D. Motreanu, Partial differential hemivariational inequalities, Adv. Nonlinear Anal. 7 (2017), 571–586.
  * [18] M. Miettinen, A parabolic hemivariational inequality, Nonlinear Analysis: Theory, Methods and Applications 26 (1996), 725–734.
  * [19] S. Migórski, Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems, Computers & Mathematics with Applications 52 (2006), 677–698.
  * [20] S. Migórski, Optimal control of history-dependent evolution inclusions with applications to frictional contact, J. Optimization Theory and Applications 185 (2020), 574–596.
  * [21] S. Migórski, A class of history-dependent systems of evolution inclusions with applications, Nonlinear Analysis: Real World Applications 59 (2021), 103246.
  * [22] S. Migórski, W. Han, S.D. Zeng, A new class of hyperbolic variational-hemivariational inequalities driven by non-linear evolution equations, European J. Appl. Math. 32 (2021), 59–88.
  * [23] S. Migórski, A. Ochal, Boundary hemivariational inequality of parabolic type, Nonlinear Analysis: Theory Methods and Appl. 57 (2004), 579–596.
  * [24] S. Migórski, A. Ochal, Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion, Nonlinear Analysis: Theory, Methods and Applications 69 (2008), 495–509.
  * [25] S. Migórski, A. Ochal, M. Sofonea, History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics, Nonlinear Analysis: Real World Applications 12 (2011), 3384–3396.
  * [26] S. Migórski, A. Ochal, M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013.
  * [27] S. Migórski, A. Ochal, M. Sofonea, History-dependent variational-hemivariational inequalities in contact mechanics, Nonlinear Analysis: Real World Applications 22 (2015), 604–618.
  * [28] S. Migórski, A. Ochal, M. Sofonea, Evolutionary inclusions and hemivariational inequalities, Chapter 2 in _Advances in Variational and Hemivariational Inequalities: Theory, Numerical Analysis, and Applications_ , edited by W. Han, et al., Advances in Mechanics and Mathematics Series, vol. 33 (2015), 39–64, Springer.
  * [29] S. Migórski, J. Ogorzaly, Dynamic history-dependent variational-hemivariational inequalities with applications to contact mechanics, Zeitschrift für angewandte Mathematik und Physik 68 (2017), Article ID.15, 22p.
  * [30] S. Migórski, B. Zeng, A new class of history–dependent evolutionary variational–hemivariational inequalities with unilateral constraints, Applied Mathematics & Optimization 84 (2021), 2671–2697.
  * [31] S. Migórski, S. Zeng, A class of differential hemivariational inequalities in Banach spaces, J. Global Optimization 72 (2018), 761–779.
  * [32] S. Migórski, S.D. Zeng, Hyperbolic hemivariational inequalities controled by evolution equations with application to adhesive contact model, Nonlinear Analysis: Real World Applications 43 (2018), 121–143.
  * [33] Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995.
  * [34] J.S. Pang, D.E. Stewart, Differential variational inequalities, Math. Program. 113 (2008), 345–424.
  * [35] M. Shillor, M. Sofonea, J.J. Telega, Models and Analysis of Quasistatic Contact, Lect. Notes Phys. 655, Springer, Berlin, Heidelberg, 2004.
  * [36] M. Sofonea, W. Han, S. Migórski, Numerical analysis of history-dependent variational-hemivariational inequalities with applications to contact problems, European Journal of Applied Mathematics 26 (2015), 427–452.
  * [37] M. Sofonea, W. Han, M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics 276, Chapman-Hall/CRC Press, New York, 2006.
  * [38] M. Sofonea, A. Matei, History-dependent quasivariational inequalities arising in Contact Mechanics, European Journal of Applied Mathematics 22 (2011), 471–491.
  * [39] M. Sofonea, S. Migórski, Variational-Hemivariational Inequalities with Applications, Chapman & Hall/CRC, Boca Raton, 2018.
  * [40] M. Sofonea, S. Migórski, W. Han, A penalty method for history-dependent variational-hemivariational inequalities, Computers & Mathematics with Applications 75 (2018), 2561–2573.
  * [41] M. Sofonea, S. Migórski, A. Ochal, Two history-dependent contact problems, Chapter 14 in _Advances in Variational and Hemivariational Inequalities: Theory, Numerical Analysis, and Applications_ , edited by W. Han, et al., Advances in Mechanics and Mathematics Series, vol. 33 (2015), 355–380, Springer.
  * [42] M. Sofonea, F. Pǎtrulescu, Penalization of history-dependent variational inequalities, European Journal of Applied Mathematics 25 (2014), 155–176.
  * [43] M. Sofonea, Y. Xiao, Fully history-dependent quasivariational inequalities in contact mechanics, Applicable Analysis 95 (2016), 2464–2484.
  * [44] S.D. Zeng, S. Migórski, Dynamic history-dependent hemivariational inequalities controlled by evolution equations with application to contact mechanics, Journal of Dynamics and Differential Equations, 2021, in press, https://doi.org/10.1007/s10884-021-10088-0.