Physical setting and mathematical models.

A large number of processes of contact arising in various engineering applications can be cast in the following general physical setting: a deformable body is subjected to the action of body forces and surface tractions, is clamped on part of its surface and is in contact with a foundation on another part of its surface. We are interested in describing the evolution of the mechanical state of the body and, to this end, the first step is to construct a mathematical model which describes the physical setting above. Here and everywhere in this work, by a mathematical model we understand a system of partial differential equations with associated boundary conditions and with possibly initial conditions, for a specific contact process. Such models are constructed based on the general principles of solid mechanics, which can be found in Ciarlet (Reference Ciarlet1988), Khludnev and Sokolowski (Reference Khludnev and Sokolowski1997) and Temam and Miranville (Reference Temam and Miranville2001), for instance.

To present a mathematical model in contact mechanics we need to combine several relations: the constitutive law, the balance equation, the boundary conditions, the interface laws, and for evolutionary problems, the initial conditions. Recall that a constitutive law represents a relation between the stress $\unicode[STIX]{x1D748}$ and the strain $\unicode[STIX]{x1D73A}$ , and the relation may involve derivatives and/or integrals of the the stress and/or strain. The constitutive law describes the mechanical reaction of the material with respect to the action of body forces and boundary tractions. Although the constitutive laws must satisfy some basic axioms and invariance principles, they originate mostly from experiments. We refer the reader to Han and Sofonea (Reference Han and Sofonea2002) for a general description of several diagnostic experiments which provide information needed in constructing constitutive laws for specific materials. The balance equation for the stress field leads either to the equation of motion (used in the modelling of dynamic processes, i.e. processes in which the inertial terms are not neglected) or to the equation of equilibrium (used in the modelling of static and quasistatic processes, i.e. processes in which the inertial terms are neglected). The boundary conditions usually involve the displacement and the surface tractions. They express the fact that the body is held fixed on a part of the boundary and is acted upon by external forces on the other part. The interface laws are to be prescribed on the potential contact surface. These are divided naturally into conditions in the normal direction (called contact conditions) and those in the tangential directions (called friction laws). A comprehensive description of the interface laws used in the mathematical literature dedicated to modelling of contact problems can be found in Sofonea and Matei (Reference Sofonea and Matei2012) and Migórski, Ochal and Sofonea (Reference Migórski, Ochal and Sofonea2013).

Basic notation.

In order to introduce the contact models, we let $\unicode[STIX]{x1D6FA}$ be the reference configuration of the body, assumed to be an open, bounded, connected set in $\mathbb{R}^{d}$ with a Lipschitz boundary $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . The dimension $d=2$ or $3$ for applications. The closure of $\unicode[STIX]{x1D6FA}$ in $\mathbb{R}^{d}$ is denoted by $\overline{\unicode[STIX]{x1D6FA}}$ . To describe the boundary conditions, we split the boundary $\unicode[STIX]{x1D6E4}$ into three disjoint, measurable parts $\unicode[STIX]{x1D6E4}_{1}$ , $\unicode[STIX]{x1D6E4}_{2}$ and $\unicode[STIX]{x1D6E4}_{3}$ . Here, $\operatorname{meas}\,(\unicode[STIX]{x1D6E4}_{1})>0$ and $\operatorname{meas}\,(\unicode[STIX]{x1D6E4}_{3})>0$ , whereas $\unicode[STIX]{x1D6E4}_{2}$ is allowed to be empty. We use boldface letters for vectors and tensors, such as the outward unit normal $\unicode[STIX]{x1D742}$ on $\unicode[STIX]{x1D6E4}$ . A typical point in $\mathbb{R}^{d}$ is denoted by $\boldsymbol{x}=(x_{i})$ . The indices $i$ , $j$ , $k$ , $l$ run between $1$ and $d$ , and, unless stated otherwise, the summation convention over repeated indices is implied. An index that follows a comma indicates a partial derivative with respect to the corresponding component of the spatial variable $\boldsymbol{x}$ . For integrals, we use $\text{d}x$ for the infinitesimal volume element in $\unicode[STIX]{x1D6FA}$ and $\text{d}a$ for the infinitesimal surface element on $\unicode[STIX]{x1D6E4}$ . For a time-dependent contact problem, the time interval of interest will be denoted by $I$ , which can be bounded or unbounded. A dot above a variable will represent the time derivative of the variable.

We are interested in mathematical models which describe the evolution or the equilibrium of the mechanical state of the body within the framework of the linearized strain theory. We use the symbols $\boldsymbol{u}$ , $\unicode[STIX]{x1D748}$ and $\unicode[STIX]{x1D73A}=\unicode[STIX]{x1D73A}(\boldsymbol{u})$ for the displacement vector, the stress tensor and the linearized strain tensor, respectively. The components of the linearized strain tensor $\unicode[STIX]{x1D73A}(\boldsymbol{u})$ are given by

$$\begin{eqnarray}\unicode[STIX]{x1D700}_{ij}(\boldsymbol{u})=(\unicode[STIX]{x1D73A}(\boldsymbol{u}))_{ij}=\displaystyle \frac{1}{2}\,(u_{i,j}+u_{j,i}),\end{eqnarray}$$

where $u_{i,j}=\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j}$ . These are functions of the spatial variable $\boldsymbol{x}$ , and of the time variable $t$ as well for time-dependent problems. Nevertheless, in what follows we do not indicate explicitly the dependence of these quantities on $\boldsymbol{x}$ and $t$ : for example, we write $\unicode[STIX]{x1D748}$ instead of $\unicode[STIX]{x1D748}(\boldsymbol{x})$ or $\unicode[STIX]{x1D748}(\boldsymbol{x},t)$ . The displacement $\boldsymbol{u}$ and the stress $\unicode[STIX]{x1D748}$ play the roles of unknowns in the contact problems.

We let $\mathbb{S}^{d}$ denote the space of second-order symmetric tensors on $\mathbb{R}^{d}$ . Equivalently, $\mathbb{S}^{d}$ can be viewed as the space of symmetric matrices of order $d$ . The canonical inner products and the corresponding norms on $\mathbb{R}^{d}$ and $\mathbb{S}^{d}$ are given by

(2.1) $$\begin{eqnarray}\displaystyle \boldsymbol{u}\cdot \boldsymbol{v} & = & \displaystyle u_{i}v_{i},\quad \Vert \boldsymbol{v}\Vert =(\boldsymbol{v}\cdot \boldsymbol{v})^{1/2}\quad \text{for all}~\boldsymbol{u}=(u_{i}),\boldsymbol{v}=(v_{i})\in \mathbb{R}^{d},\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D748}\cdot \unicode[STIX]{x1D749} & = & \displaystyle \unicode[STIX]{x1D70E}_{ij}\unicode[STIX]{x1D70F}_{ij},\quad \Vert \unicode[STIX]{x1D749}\Vert =(\unicode[STIX]{x1D749}\cdot \unicode[STIX]{x1D749})^{1/2}\quad \text{for all}~\unicode[STIX]{x1D748}=(\unicode[STIX]{x1D70E}_{ij}),\unicode[STIX]{x1D749}=(\unicode[STIX]{x1D70F}_{ij})\in \mathbb{S}^{d},\end{eqnarray}$$

respectively. We use $\mathbf{0}$ for the zero element of the spaces $\mathbb{R}^{d}$ and $\mathbb{S}^{d}$ .

Function spaces.

We will use the standard notation for Sobolev and Lebesgue spaces over $\unicode[STIX]{x1D6FA}$ or on $\unicode[STIX]{x1D6E4}$ . In particular, we will use the spaces $L^{2}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d})$ , $L^{2}(\unicode[STIX]{x1D6E4}_{2};\mathbb{R}^{d})$ , $L^{2}(\unicode[STIX]{x1D6E4}_{3};\mathbb{R}^{d})$ and $H^{1}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d})$ , endowed with their canonical inner products and associated norms. For an element $\boldsymbol{v}\in H^{1}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d})$ we write $\boldsymbol{v}$ for its trace $\unicode[STIX]{x1D6FE}\boldsymbol{v}\in L^{2}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d})$ on $\unicode[STIX]{x1D6E4}$ . Define the function spaces

(2.3) $$\begin{eqnarray}\displaystyle V & = & \displaystyle \{\boldsymbol{v}\in H^{1}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d})\mid \boldsymbol{v}=\mathbf{0}~\text{on }\unicode[STIX]{x1D6E4}_{1}\},\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle Q & = & \displaystyle \{\unicode[STIX]{x1D748}=(\unicode[STIX]{x1D70E}_{ij})\mid \unicode[STIX]{x1D70E}_{ij}=\unicode[STIX]{x1D70E}_{ji}\in L^{2}(\unicode[STIX]{x1D6FA})\},\end{eqnarray}$$

for the displacement and the stress field, respectively. These are real Hilbert spaces endowed with the inner products

$$\begin{eqnarray}(\boldsymbol{u},\boldsymbol{v})_{V}=\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D73A}(\boldsymbol{u})\cdot \unicode[STIX]{x1D73A}(\boldsymbol{v})\,\text{d}x,\quad (\unicode[STIX]{x1D748},\unicode[STIX]{x1D749})_{Q}=\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D748}\cdot \unicode[STIX]{x1D749}\,\text{d}x.\end{eqnarray}$$

The corresponding norms on the spaces are denoted by $\Vert \cdot \Vert _{V}$ and $\Vert \cdot \Vert _{Q}$ , respectively. Since $\operatorname{meas}\,(\unicode[STIX]{x1D6E4}_{1})>0$ , Korn’s inequality holds (see Nečas and Hlaváček Reference Nečas and Hlaváček1981, p. 79):

$$\begin{eqnarray}\Vert \boldsymbol{v}\Vert _{H^{1}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d})}\leq c\,\Vert \unicode[STIX]{x1D73A}(\boldsymbol{v})\Vert _{Q}\quad \text{for all}~\boldsymbol{v}\in V,\end{eqnarray}$$

for some constant $c>0$ . Thus, $\Vert \cdot \Vert _{V}$ defines a norm on $V$ that is equivalent to the standard $H^{1}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d})$ -norm. We will also use the function space

(2.5) $$\begin{eqnarray}H=L^{2}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d})\end{eqnarray}$$

with the canonical inner product and norm.

We let $V^{\ast }$ denote the dual of the space $V$ and $\langle \cdot ,\cdot \rangle$ the corresponding duality pairing. For any element $\boldsymbol{v}\in V$ , we let $v_{\unicode[STIX]{x1D708}}$ and $\boldsymbol{v}_{\unicode[STIX]{x1D70F}}$ denote its normal and tangential components on $\unicode[STIX]{x1D6E4}$ given by $v_{\unicode[STIX]{x1D708}}=\boldsymbol{v}\cdot \unicode[STIX]{x1D742}$ and $\boldsymbol{v}_{\unicode[STIX]{x1D70F}}=\boldsymbol{v}-v_{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D742}$ , respectively. For a regular function $\unicode[STIX]{x1D748}:\overline{\unicode[STIX]{x1D6FA}}\rightarrow \mathbb{S}^{d}$ , we let $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}$ and $\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}$ denote its normal and tangential components on $\unicode[STIX]{x1D6E4}$ , i.e.  $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}=(\unicode[STIX]{x1D748}\unicode[STIX]{x1D742})\cdot \unicode[STIX]{x1D742}$ and $\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D748}\unicode[STIX]{x1D742}-\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D742}$ , and we recall that the following Green’s formula holds:

(2.6) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}\,\unicode[STIX]{x1D748}\cdot \unicode[STIX]{x1D73A}(\boldsymbol{v})\,\text{d}x+\int _{\unicode[STIX]{x1D6FA}}\,\text{Div}\,\unicode[STIX]{x1D748}\cdot \boldsymbol{v}\,\text{d}x=\int _{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D748}\unicode[STIX]{x1D742}\cdot \boldsymbol{v}\,\text{d}a\quad \text{for all}~\boldsymbol{v}\in H^{1}(\unicode[STIX]{x1D6FA},\mathbb{R}^{d}).\end{eqnarray}$$

We also recall that

(2.7) $$\begin{eqnarray}\Vert \boldsymbol{v}\Vert _{L^{2}(\unicode[STIX]{x1D6E4};\mathbb{R}^{d})}\leq \Vert \unicode[STIX]{x1D6FE}\Vert \Vert \boldsymbol{v}\Vert _{V}\quad \text{for all}~\boldsymbol{v}\in V,\end{eqnarray}$$

where $\Vert \unicode[STIX]{x1D6FE}\Vert$ represents the norm of the trace operator $\unicode[STIX]{x1D6FE}:V\rightarrow L^{2}(\unicode[STIX]{x1D6E4};\mathbb{R}^{d})$ . Inequality (2.7) represents a consequence of the Sobolev trace theorem.

We introduce a space of fourth-order tensors:

(2.8) $$\begin{eqnarray}\mathbf{Q_{∞}}=\{{\mathcal{E}}=(e_{ijkl})\mid e_{ijkl}=e_{jikl}=e_{klij}\in L^{\infty }(\unicode[STIX]{x1D6FA}),~1\leq i,j,k,l\leq d\}.\end{eqnarray}$$

It is a Banach space endowed with the norm

$$\begin{eqnarray}\Vert {\mathcal{E}}\Vert _{\mathbf{Q_{∞}}}=\max _{1\leq i,j,k,l\leq d}\Vert e_{ijkl}\Vert _{L^{\infty }(\unicode[STIX]{x1D6FA})}.\end{eqnarray}$$

It is easy to see that

(2.9) $$\begin{eqnarray}\Vert {\mathcal{E}}\unicode[STIX]{x1D749}\Vert _{Q}\leq d\,\Vert {\mathcal{E}}\Vert _{\mathbf{Q_{∞}}}\Vert \unicode[STIX]{x1D749}\Vert _{Q}\quad \text{for all}~{\mathcal{E}}\in \mathbf{Q_{∞}},\,\unicode[STIX]{x1D749}\in Q.\end{eqnarray}$$

Below, $\mathbb{N}$ represents the set of positive integers, $I$ denotes either a bounded interval of the form $[0,T]$ with $T>0$ , or the unbounded interval $\mathbb{R}_{+}=[0,+\infty )$ and $X$ will be a Banach space. We let $C(I;X)$ denote the space of continuous functions on $I$ with values in $X$ . In addition, we let $C^{1}(I;X)$ denote the space of continuously differentiable functions on $I$ with values in $X$ . Therefore, $v\in C^{1}(I;X)$ if and only if $v\in C(I;X)$ and $\dot{v}\in C(I;X)$ where $\dot{v}$ represents the time derivative of the function $v$ . In addition, for a subset $K\subset X$ , we use the notation $C(I;K)$ for the set of functions defined on $I$ with values in $K$ .

In the case $I=[0,T]$ the space $C(I;X)$ is equipped with the norm

$$\begin{eqnarray}\Vert v\Vert _{C([0,T];X)}=\max _{t\in [0,T]}\,\Vert v(t)\Vert _{X}.\end{eqnarray}$$

It is well known that $C(I;X)$ is a Banach space. In the case $I=\mathbb{R}_{+}$ , $C(I;X)$ can be organized in a canonical way as a Fréchet space, i.e., it is a complete metric space in which the corresponding topology is induced by a countable family of seminorms. The convergence of a sequence $\{{v_{k}\}}_{k}$ to an element $v$ , in the space $C(\mathbb{R}_{+};X)$ , can be described as follows:

$$\begin{eqnarray}\displaystyle & & \displaystyle v_{k}\rightarrow v\text{ in }C(\mathbb{R}_{+};X)\text{ as }k\rightarrow \infty \text{if and only if}\nonumber\\ \displaystyle & & \displaystyle \displaystyle \max _{r\in [0,n]}\Vert v_{k}(r)-v(r)\Vert _{V}\rightarrow 0~\text{as }k\rightarrow \infty ,\text{for all }n\in \mathbb{N}.\nonumber\end{eqnarray}$$

In other words, the sequence $\{{v_{k}\}}_{k}$ converges to the element $v$ in the space $C(\mathbb{R}_{+};X)$ if and only if it converges to $v$ in the space $C([0,n];X)$ for all $n\in \mathbb{N}$ .

Let $I=[0,T]$ and $X$ be a Banach space. For $1\leq p\leq \infty$ , we define $L^{p}(I;X)$ to be the space of all measurable functions $v:I\rightarrow X$ such that $\Vert v\Vert _{L^{p}(I;X)}<\infty$ , where the norm is defined by

$$\begin{eqnarray}\Vert v\Vert _{L^{p}(I;X)}=\left\{\begin{array}{@{}ll@{}}\displaystyle \biggl(\int _{I}\Vert v(t)\Vert _{X}^{p}\,\text{d}t\biggr)^{1/p}\quad & \text{if}~1\leq p<\infty ,\\ \displaystyle \underset{t\in I}{\text{ess sup}}\,\Vert v(t)\Vert _{X}\quad & \text{if}~p=\infty .\end{array}\right.\end{eqnarray}$$

The space $L^{p}(I;X)$ is a Banach space. When $X$ is a Hilbert space with the inner product $(\cdot ,\cdot )_{X}$ and $p=2$ , $L^{2}(I;X)$ is a Hilbert space with the inner product

$$\begin{eqnarray}(u,v)=\int _{I}(u(t),v(t))_{X}\,\text{d}t.\end{eqnarray}$$

For $m\in \mathbb{N}$ and $1\leq p\leq \infty$ , we define the space

$$\begin{eqnarray}W^{m,p}(I;X)=\{v\in L^{p}(I;X)\mid \Vert v^{(i)}\Vert _{L^{p}(I;X)}<\infty ,~0\leq i\leq m\},\end{eqnarray}$$

where $v^{(i)}(t)$ is the $i$ th weak derivative of $v$ with respect to $t$ ; the first two weak derivatives are usually also denoted as $\dot{v}$ , $\ddot{v}$ . For $1\leq p<\infty$ , the norm in the space $W^{k,p}(I;X)$ is defined by

$$\begin{eqnarray}\Vert v\Vert _{W^{k,p}(I;X)}=\biggl(\int _{I}\mathop{\sum }_{0\leq i\leq k}\Vert v^{(i)}(t)\Vert _{X}^{p}\,\text{d}t\biggr)^{1/p}.\end{eqnarray}$$

For $p=\infty$ , the norm is defined by

$$\begin{eqnarray}\Vert v\Vert _{W^{k,\infty }(I;X)}=\max _{0\leq i\leq k}\underset{t\in I}{\text{ess sup}}\,\Vert v^{(i)}\Vert _{X}.\end{eqnarray}$$

The space $W^{m,p}(I;X)$ is a Banach space. In the particular case where $X$ is a Hilbert space with the inner product $(\cdot ,\cdot )_{X}$ and $p=2$ , $W^{m,2}(I;X)$ is a Hilbert space, usually written as $H^{m}(I;X)$ , with the inner product

$$\begin{eqnarray}(u,v)_{H^{m}(I;X)}=\mathop{\sum }_{0\leq i\leq m}\int _{I}(u^{(i)}(t),v^{(i)}(t))_{X}\,\text{d}t.\end{eqnarray}$$

With the time interval $I=[0,T]$ and the space $V$ defined in (2.3), we will use the spaces

(2.10) $$\begin{eqnarray}\displaystyle {\mathcal{V}} & = & \displaystyle L^{2}(I;V),\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle \quad {\mathcal{V}}^{\ast } & = & \displaystyle L^{2}(I;V^{\ast }),\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle {\mathcal{W}} & = & \displaystyle \{w\in {\mathcal{V}}\mid {\dot{w}}\in {\mathcal{V}}^{\ast }\}.\end{eqnarray}$$

Subdifferential boundary conditions.

For the contact models, we will use contact laws expressed in terms of the subdifferential. Such conditions are of the form $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}\in \unicode[STIX]{x2202}j(u_{\unicode[STIX]{x1D708}})$ in which $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}$ represents an interface force, $u_{\unicode[STIX]{x1D708}}=\boldsymbol{u}\cdot \unicode[STIX]{x1D742}$ denotes the normal displacement and $\unicode[STIX]{x2202}j$ represents the subdifferential in the sense of Clarke. The definition and some basic properties of this subdifferential for locally Lipschitz functions on Banach spaces will be provided in Section 3. Nevertheless, for the convenience of the reader, we introduce here the Clarke subdifferential for real-valued functions of a real variable, as we need it to describe the interface boundary conditions in this section.

Let $j:\mathbb{R}\rightarrow \mathbb{R}$ be a locally Lipschitz function. The generalized (Clarke) directional derivative of $j$ at $x\in \mathbb{R}$ in the direction $v\in \mathbb{R}$ is defined by

$$\begin{eqnarray}j^{0}(x;v):=\limsup _{y\rightarrow x,\,\unicode[STIX]{x1D706}\downarrow 0}\displaystyle \frac{j(y+\unicode[STIX]{x1D706}v)-j(y)}{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

The generalized subdifferential of $j$ at $x$ is a subset of $\mathbb{R}$ given by

$$\begin{eqnarray}\unicode[STIX]{x2202}j(x):=\{\unicode[STIX]{x1D701}\in \mathbb{R}\mid j^{0}(x;v)\geq \unicode[STIX]{x1D701}v~\text{for all}~v\in \mathbb{R}\}.\end{eqnarray}$$

We are interested in functions $j$ satisfying the following conditions.

(2.13) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\text{(a) } & j:\mathbb{R}\rightarrow \mathbb{R}\text{ is locally Lipschitz.}\\[3.0pt] \text{(b) } & \text{There exists a constant }c_{j}>0\text{ such that}\\ & |\unicode[STIX]{x2202}j(r)|\leq c_{j}(1+|r|)\text{ for all }r\in \mathbb{R},\text{all }\unicode[STIX]{x1D709}\in \unicode[STIX]{x2202}j(r).\\[3.0pt] \text{(c) } & \text{There exists a constant }\unicode[STIX]{x1D6FC}_{j}\geq 0\text{ such that}\\ & (\unicode[STIX]{x1D709}_{1}-\unicode[STIX]{x1D709}_{2})\cdot (r_{1}-r_{2})\geq -\unicode[STIX]{x1D6FC}_{j}|r_{1}-r_{2}|^{2}\\ & \text{for all }r_{1},r_{2}\in \mathbb{R},~\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2}\in \mathbb{R},\text{with }\unicode[STIX]{x1D709}_{i}\in \unicode[STIX]{x2202}j(r_{i}),\,i=1,2.\end{array}\right\}\end{eqnarray}$$

Note that the inequality in (2.13(b)) means

$$\begin{eqnarray}|\unicode[STIX]{x1D709}|\leq c_{j}(1+|r|)\quad \text{for all}~r\in \mathbb{R},~\unicode[STIX]{x1D709}\in \unicode[STIX]{x2202}j(r).\end{eqnarray}$$

We shall use this kind of shorthand notation in various places in the paper. Moreover, note that condition (2.13(c)) is known as the relaxed monotonicity condition in the literature. It is equivalent to the condition

(2.14) $$\begin{eqnarray}j^{0}(r_{1};r_{2}-r_{1})+j^{0}(r_{2};r_{1}-r_{2})\leq \unicode[STIX]{x1D6FC}_{j}|r_{1}-r_{2}|^{2}\quad \text{for all}~r_{1},r_{2}\in \mathbb{R}.\end{eqnarray}$$

If $j:\mathbb{R}\rightarrow \mathbb{R}$ is convex, this condition is satisfied with $\unicode[STIX]{x1D6FC}_{j}=0$ .

Below we present several examples of a function $j$ satisfying the condition (2.13). These examples show that such a function is in general non-convex and its subdifferential can be multivalued with a non-monotone graph in  $\mathbb{R}^{2}$ .

Example 2.1. Let $\unicode[STIX]{x1D6FC}>0$ and $r_{0}>0$ . Define $j:\mathbb{R}\rightarrow \mathbb{R}$ by

(2.15) $$\begin{eqnarray}j(r)=\left\{\begin{array}{@{}ll@{}}{\textstyle \frac{1}{2}}\,\unicode[STIX]{x1D6FC}r_{0}^{2}-\unicode[STIX]{x1D6FC}r_{0}(r+r_{0})\quad & \text{if}~r<-r_{0},\\[3.0pt] {\textstyle \frac{1}{2}}\,\unicode[STIX]{x1D6FC}r^{2}\quad & \text{if}~|r|\leq r_{0},\\[3.0pt] {\textstyle \frac{1}{2}}\,\unicode[STIX]{x1D6FC}r_{0}^{2}+\unicode[STIX]{x1D6FC}r_{0}(r-r_{0})\quad & \text{if}~r>r_{0}.\end{array}\right.\end{eqnarray}$$

It is easy to see that $j$ is a $C^{1}$ function and therefore condition (2.13(a)) is satisfied. From the formula

$$\begin{eqnarray}\unicode[STIX]{x2202}j(r)=\left\{\begin{array}{@{}ll@{}}-\unicode[STIX]{x1D6FC}r_{0}\quad & \text{if}~r<-r_{0},\\ \unicode[STIX]{x1D6FC}r\quad & \text{if}~|r|\leq r_{0},\\ \unicode[STIX]{x1D6FC}r_{0}\quad & \text{if}~r>r_{0},\end{array}\right.\end{eqnarray}$$

we see that $|\unicode[STIX]{x2202}j(r)|\leq \unicode[STIX]{x1D6FC}r_{0}$ for all $r\in \mathbb{R}$ . Hence, condition (2.13(b)) holds with $c_{j}=\unicode[STIX]{x1D6FC}r_{0}$ . Since $\unicode[STIX]{x2202}j$ is a monotone function, we deduce that condition (2.13(c)) holds with $\unicode[STIX]{x1D6FC}_{j}=0$ .

Example 2.2. Let $\unicode[STIX]{x1D6FC}\in [0,1)$ and consider the function

(2.16) $$\begin{eqnarray}j(r)=(\unicode[STIX]{x1D6FC}-1)\,\text{e}^{-|r|}+\unicode[STIX]{x1D6FC}|r|\quad \text{for all}~r\in \mathbb{R}.\end{eqnarray}$$

Then $j$ satisfies condition (2.13(a)). It follows from the definition of the Clarke subdifferential that

$$\begin{eqnarray}\unicode[STIX]{x2202}j(r)=\left\{\begin{array}{@{}ll@{}}(\unicode[STIX]{x1D6FC}-1)\,\text{e}^{r}-\unicode[STIX]{x1D6FC}\quad & \text{if}~r<0,\\ [-1,1]\quad & \text{if}~r=0,\\ (1-\unicode[STIX]{x1D6FC})\,\text{e}^{-r}+\unicode[STIX]{x1D6FC}\quad & \text{if}~r>0.\end{array}\right.\end{eqnarray}$$

Thus, $|\unicode[STIX]{x1D709}|\leq 1$ for all $\unicode[STIX]{x1D709}\in \unicode[STIX]{x2202}j(r)$ and $r\in \mathbb{R}$ , and condition (2.13(b)) holds with $c_{j}=1$ . A simple calculation shows that condition (2.13(c)) holds with $\unicode[STIX]{x1D6FC}_{j}=1$ .

Example 2.3. Let $\unicode[STIX]{x1D6FC}\geq 0$ and let $j:\mathbb{R}\rightarrow \mathbb{R}$ be defined by

(2.17) $$\begin{eqnarray}j(r)=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if}~r<0,\\ -\text{e}^{-r}+\unicode[STIX]{x1D6FC}r+1\quad & \text{if}~r\geq 0.\end{array}\right.\end{eqnarray}$$

It is easy to see that $j$ satisfies condition (2.13(a)). An elementary computation shows that

$$\begin{eqnarray}\unicode[STIX]{x2202}j(r)=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if}~r<0,\\ [0,1+\unicode[STIX]{x1D6FC}]\quad & \text{if}~r=0,\\ \text{e}^{-r}+\unicode[STIX]{x1D6FC}\quad & \text{if}~r>0.\end{array}\right.\end{eqnarray}$$

Thus, $|\unicode[STIX]{x1D709}|\leq 1+\unicode[STIX]{x1D6FC}$ for all $\unicode[STIX]{x1D709}\in \unicode[STIX]{x2202}j(r)$ and $r\in \mathbb{R}$ . In other words, condition (2.13(b)) holds with $c_{j}=1+\unicode[STIX]{x1D6FC}$ . Condition (2.13(c)) holds with $\unicode[STIX]{x1D6FC}_{j}=1$ .

Example 2.4. Let $p:\mathbb{R}\rightarrow \mathbb{R}$ be the function defined by

(2.18) $$\begin{eqnarray}p(r)=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if}~r<0,\\ r\quad & \text{if}~0\leq r<1,\\ 2-r\quad & \text{if}~1\leq r<2,\\ \sqrt{r-2}+r-2\quad & \text{if}~2\leq r<6,\\ r\quad & \text{if}~r\geq 6.\end{array}\right.\end{eqnarray}$$

This function is continuous, yet it is neither monotone nor Lipschitz continuous. Define the function $j:\mathbb{R}\rightarrow \mathbb{R}$ by

(2.19) $$\begin{eqnarray}j(r)=\int _{0}^{r}p(s)\,\text{d}s\quad \text{for all}~r\in \mathbb{R}.\end{eqnarray}$$

Note that $j$ is not convex. Since $j^{\prime }(r)=p(r)$ for $r\in \mathbb{R}$ , $j$ is a $C^{1}$ function and thus is a locally Lipschitz function, that is, condition (2.13(a)) is satisfied. Since $|p(r)|\leq |r|$ for $r\in \mathbb{R}$ , (2.13(b)) is satisfied. Since the function $r\mapsto r+p(r)\in \mathbb{R}$ is non-decreasing,

$$\begin{eqnarray}(p(r_{1})-p(r_{2}))(r_{2}-r_{1})\leq (r_{1}-r_{2})^{2}\quad \text{for all}~r_{1},r_{2}\in \mathbb{R}.\end{eqnarray}$$

We combine this inequality with the equality $j^{0}(r_{1};r_{2})=p(r_{1})\,r_{2}$ , valid for $r_{1}$ , $r_{2}\in \mathbb{R}$ , to see that condition (2.14) is satisfied with $\unicode[STIX]{x1D6FC}_{j}=1$ . Hence, $j$ satisfies condition (2.13(c)).

2.1 A static frictional contact problem

For the first contact problem to be studied in this section we assume that the material is elastic, the process is static, the contact is with normal compliance and unilateral constraint (see Han, Sofonea and Barboteu Reference Han, Sofonea and Barboteu2017), associated with one version of Coulomb’s law of dry friction (see Han, Migórski and Sofonea Reference Han, Migórski and Sofonea2014). These mechanical assumptions lead to the following mathematical model.

Problem 2.5. Find a displacement field $\boldsymbol{u}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}^{d}$ , a stress field $\unicode[STIX]{x1D748}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{S}^{d}$ and an interface function $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}:\unicode[STIX]{x1D6E4}_{3}\rightarrow \mathbb{R}$ such that

(2.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D748}={\mathcal{F}}\unicode[STIX]{x1D73A}(\boldsymbol{u})\quad & \text{in}~\unicode[STIX]{x1D6FA}, & \displaystyle\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle \text{Div}\,\unicode[STIX]{x1D748}+\boldsymbol{f}_{0}=\mathbf{0}\quad & \text{in}~\unicode[STIX]{x1D6FA}, & \displaystyle\end{eqnarray}$$

(2.22) $$\begin{eqnarray}\displaystyle \boldsymbol{u}=\mathbf{0}\quad & \text{on}~\unicode[STIX]{x1D6E4}_{1}, & \displaystyle\end{eqnarray}$$

(2.23) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D748}\unicode[STIX]{x1D742}=\boldsymbol{f}_{2}\quad & \text{on}~\unicode[STIX]{x1D6E4}_{2}, & \displaystyle\end{eqnarray}$$

(2.24) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}ll@{}}u_{\unicode[STIX]{x1D708}}\leq g,\quad \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}+\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}\leq 0, & \quad (u_{\unicode[STIX]{x1D708}}-g)(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}+\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}})=0,\\ \unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}\in \unicode[STIX]{x2202}j_{\unicode[STIX]{x1D708}}(u_{\unicode[STIX]{x1D708}}) & \quad \end{array}\right\}\quad & \text{on}~\unicode[STIX]{x1D6E4}_{3}, & \displaystyle\end{eqnarray}$$

(2.25) $$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}\Vert \leq F_{b}(u_{\unicode[STIX]{x1D708}}),~\quad -\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}=F_{b}(u_{\unicode[STIX]{x1D708}})\displaystyle \frac{\boldsymbol{u}_{\unicode[STIX]{x1D70F}}}{\Vert \boldsymbol{u}_{\unicode[STIX]{x1D70F}}\Vert }~~\text{if}~\boldsymbol{u}_{\unicode[STIX]{x1D70F}}\neq \mathbf{0}\quad & \text{on}~\unicode[STIX]{x1D6E4}_{3}. & \displaystyle\end{eqnarray}$$

We now explain the equations and boundary conditions in Problem 2.5. Equation (2.20) represents the constitutive law of the material where ${\mathcal{F}}:\unicode[STIX]{x1D6FA}\times \mathbb{S}^{d}\rightarrow \mathbb{S}^{d}$ is the elasticity operator, allowed to be nonlinear. For an isotropic linearly elastic material, the constitutive law (2.20) becomes

$$\begin{eqnarray}\unicode[STIX]{x1D748}=\unicode[STIX]{x1D706}\,\text{tr}\,\unicode[STIX]{x1D73A}(\boldsymbol{u})\,\boldsymbol{I}_{d}+2\,\unicode[STIX]{x1D707}\,\unicode[STIX]{x1D73A}(\boldsymbol{u}),\end{eqnarray}$$

where $\unicode[STIX]{x1D706}>0$ and $\unicode[STIX]{x1D707}>0$ are the Lamé coefficients, and $\boldsymbol{I}_{d}\in \mathbb{S}^{d}$ is the identity tensor. Another example of elastic constitutive law of the form (2.20) is provided by

(2.26) $$\begin{eqnarray}\unicode[STIX]{x1D748}={\mathcal{E}}\unicode[STIX]{x1D73A}(\boldsymbol{u})+\unicode[STIX]{x1D6FD}\,(\unicode[STIX]{x1D73A}(\boldsymbol{u})-{\mathcal{P}}_{K}\unicode[STIX]{x1D73A}(\boldsymbol{u})).\end{eqnarray}$$

Here ${\mathcal{E}}$ is a linear or nonlinear operator, $\unicode[STIX]{x1D6FD}>0$ , $K$ is a closed convex subset of $\mathbb{S}^{d}$ such that $\mathbf{0}\in K$ and ${\mathcal{P}}_{K}:\mathbb{S}^{d}\rightarrow K$ denotes the projection operator. The corresponding elasticity operator is nonlinear and is given by

(2.27) $$\begin{eqnarray}{\mathcal{F}}\unicode[STIX]{x1D73A}={\mathcal{E}}\unicode[STIX]{x1D73A}+\unicode[STIX]{x1D6FD}\,(\unicode[STIX]{x1D73A}-{\mathcal{P}}_{K}\unicode[STIX]{x1D73A}).\end{eqnarray}$$

A common choice of the set $K$ is

(2.28) $$\begin{eqnarray}K=\{\unicode[STIX]{x1D73A}\in \mathbb{S}^{d}\mid f(\unicode[STIX]{x1D73A})\leq 0\},\end{eqnarray}$$

where $f:\mathbb{S}^{d}\rightarrow \mathbb{R}$ is a convex continuous function with $f(\mathbf{0})<0$ .

Equation (2.21) is the equation of equilibrium, in which $\boldsymbol{f}_{0}$ represents the density of body forces. This equation is obtained from the equation of motion by neglecting the inertial term. Since the body is fixed on $\unicode[STIX]{x1D6E4}_{1}$ , we impose the homogeneous displacement condition (2.22). If $\unicode[STIX]{x1D6E4}_{2}$ is non-empty, then the body is subject to a surface traction of density $\boldsymbol{f}_{2}$ and (2.23) represents the traction boundary condition.

Conditions (2.24) and (2.25) represent the frictional unilateral contact condition on the contact surface $\unicode[STIX]{x1D6E4}_{3}$ . Condition (2.24) models the contact with a foundation made of a rigid body covered by a layer made of deformable material, say asperities, and is derived based on the following considerations.

First, the thickness of the deformable layer is described by the non-negative valued function $g$ . Penetration of the elastic material to the foundation is allowed but, due to the presence of the rigid body, is limited by the unilateral constraint

(2.29) $$\begin{eqnarray}u_{\unicode[STIX]{x1D708}}\leq g\quad \text{on}~\unicode[STIX]{x1D6E4}_{3}.\end{eqnarray}$$

Next, we assume that the normal stress has an additive decomposition of the form

(2.30) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{D}+\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{R}\quad \text{on}~\unicode[STIX]{x1D6E4}_{3},\end{eqnarray}$$

in which $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{D}$ describes the reaction of the deformable material and $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{R}$ describes the reaction of the rigid body. We assume that $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{D}$ satisfies a normal compliance contact condition in a subdifferential form, that is,

(2.31) $$\begin{eqnarray}-\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{D}\in \unicode[STIX]{x2202}j_{\unicode[STIX]{x1D708}}(u_{\unicode[STIX]{x1D708}})\quad \text{on}~\unicode[STIX]{x1D6E4}_{3},\end{eqnarray}$$

where $\unicode[STIX]{x2202}j_{\unicode[STIX]{x1D708}}$ is the Clarke subdifferential of the potential functional $j_{\unicode[STIX]{x1D708}}$ , which is assumed to be locally Lipschitz. Examples, details and various mechanical interpretations of this condition can be found in Migórski, Ochal and Sofonea (Reference Migórski, Ochal and Sofonea2013). Here we merely mention that such a condition represents a generalization of the well-known normal compliance condition $-\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}=p(u_{\unicode[STIX]{x1D708}})$ introduced in Oden and Martins (Reference Oden and Martins1985). Note that in this condition $p:\mathbb{R}\rightarrow \mathbb{R}$ is usually assumed to be an increasing function, and therefore the contact condition is monotone and single-valued. In contrast, condition (2.31) describes a relation between the stress $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{D}$ and the normal displacement $u_{\unicode[STIX]{x1D708}}$ which can be non-monotone or multivalued.

The part $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{R}$ of the normal stress satisfies the Signorini condition in the form with the gap $g$ , that is,

(2.32) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{R}\leq 0,\quad \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{R}(u_{\unicode[STIX]{x1D708}}-g)=0\quad \text{on}~\unicode[STIX]{x1D6E4}_{3}.\end{eqnarray}$$

The Signorini condition was introduced by Signorini (Reference Signorini1933) to describe the contact with a rigid obstacle. It was used in a large number of papers as explained in Shillor, Sofonea and Telega (Reference Shillor, Sofonea and Telega2004).

We denote $-\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{D}=\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}$ and use (2.30) to see that

(2.33) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}^{R}=\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}+\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}\quad \text{on}~\unicode[STIX]{x1D6E4}_{3}.\end{eqnarray}$$

Then we substitute equality (2.33) in (2.32) and use (2.29), (2.31) to obtain the contact condition (2.24).

Finally, (2.25) represents a version of the static Coulomb law of dry friction. According to this law, the tangential traction $\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}$ is limited in size by $F_{b}$ , the so-called friction bound, which is the maximal frictional resistance that the surface can generate, and once the friction bound is reached, a relative slip motion commences. When slip starts, the frictional resistance has magnitude $F_{b}$ and acts in the direction opposite to the motion. In (2.25) we allow the friction bound to depend on the normal penetration, which appears to be reasonable from the physical point of view, as explained in Han, Migórski and Sofonea (Reference Han, Migórski and Sofonea2014). The original formulation by Coulomb was proposed for the description of contact between rigid bodies, and its use in the description of contact between deformable bodies in the pointwise sense is more recent. More comments on this matter can be found in Shillor, Sofonea and Telega (Reference Shillor, Sofonea and Telega2004).

We now list the assumption on the problem data. For the elasticity operator ${\mathcal{F}}:\unicode[STIX]{x1D6FA}\times \mathbb{S}^{d}\rightarrow \mathbb{S}^{d}$ , assume:

(2.34) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\text{(a) } & \text{there exists a constant }L_{{\mathcal{F}}}>0\text{ such that}\\ & \Vert {\mathcal{F}}(\boldsymbol{x},\unicode[STIX]{x1D73A}_{1})-{\mathcal{F}}(\boldsymbol{x},\unicode[STIX]{x1D73A}_{2})\Vert \leq L_{{\mathcal{F}}}\Vert \unicode[STIX]{x1D73A}_{1}-\unicode[STIX]{x1D73A}_{2}\Vert \\ & \text{for all }\unicode[STIX]{x1D73A}_{1},\unicode[STIX]{x1D73A}_{2}\in \mathbb{S}^{d},~\text{a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6FA};\\[3.0pt] \text{(b) } & \text{there exists a constant }m_{{\mathcal{F}}}>0\text{ such that}\\ & ({\mathcal{F}}(\boldsymbol{x},\unicode[STIX]{x1D73A}_{1})-{\mathcal{F}}(\boldsymbol{x},\unicode[STIX]{x1D73A}_{2}))\cdot (\unicode[STIX]{x1D73A}_{1}-\unicode[STIX]{x1D73A}_{2})\geq m_{{\mathcal{F}}}\,\Vert \unicode[STIX]{x1D73A}_{1}-\unicode[STIX]{x1D73A}_{2}\Vert ^{2}~\\ & \text{for all }\unicode[STIX]{x1D73A}_{1},\unicode[STIX]{x1D73A}_{2}\in \mathbb{S}^{d},~\text{a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6FA};\\[3.0pt] \text{(c) } & {\mathcal{F}}(\cdot ,\unicode[STIX]{x1D73A})\text{ is measurable on }\unicode[STIX]{x1D6FA}\text{ for all }\unicode[STIX]{x1D73A}\in \mathbb{S}^{d};\\[3.0pt] \text{(d) } & {\mathcal{F}}(\boldsymbol{x},\mathbf{0})=\mathbf{0}\text{ for a.e. }\boldsymbol{x}\in \unicode[STIX]{x1D6FA}.\end{array}\right\}\end{eqnarray}$$

For the potential function $j_{\unicode[STIX]{x1D708}}:\unicode[STIX]{x1D6E4}_{3}\times \mathbb{R}\rightarrow \mathbb{R}$ , assume:

(2.35) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\text{(a) } & j_{\unicode[STIX]{x1D708}}(\cdot ,r)\text{ is measurable on }\unicode[STIX]{x1D6E4}_{3}\text{ for all }r\in \mathbb{R}\text{ and there}\\ & \text{exists }\bar{e}\in L^{2}(\unicode[STIX]{x1D6E4}_{3})\text{ such that }j_{\unicode[STIX]{x1D708}}(\cdot ,\bar{e}(\cdot ))\in L^{1}(\unicode[STIX]{x1D6E4}_{3});\\[3.0pt] \text{(b) } & j_{\unicode[STIX]{x1D708}}(\boldsymbol{x},\cdot )\text{ is locally Lipschitz on }\mathbb{R}~\text{for a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3};\\[3.0pt] \text{(c) } & |\unicode[STIX]{x2202}j_{\unicode[STIX]{x1D708}}(\boldsymbol{x},r)|\leq \bar{c}_{0}+\bar{c}_{1}\,|r|~\text{for a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3},\\ & \text{for all }r\in \mathbb{R}\text{ with constants }\bar{c}_{0},\bar{c}_{1}\geq 0;\\[3.0pt] \text{(d) } & j_{\unicode[STIX]{x1D708}}^{0}(\boldsymbol{x},r_{1};r_{2}-r_{1})+j_{\unicode[STIX]{x1D708}}^{0}(\boldsymbol{x},r_{2};r_{1}-r_{2})\leq \unicode[STIX]{x1D6FC}_{j_{\unicode[STIX]{x1D708}}}\,|r_{1}-r_{2}|^{2}\\ & \text{for a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3},~\text{all}~r_{1},r_{2}\in \mathbb{R}\text{ with a constant }\unicode[STIX]{x1D6FC}_{j_{\unicode[STIX]{x1D708}}}\geq 0.\end{array}\right\}\end{eqnarray}$$

For the friction bound $F_{b}:\unicode[STIX]{x1D6E4}_{3}\times \mathbb{R}\rightarrow \mathbb{R}$ , assume:

(2.36) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\text{(a) } & \text{there exists a constant }L_{F_{b}}>0\text{ such that}\\ & |F_{b}(\boldsymbol{x},r_{1})-F_{b}(\boldsymbol{x},r_{2})|\leq L_{F_{b}}|r_{1}-r_{2}|\\ & \text{for all }r_{1},r_{2}\in \mathbb{R},\,\text{a.e. }\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3};\\[3.0pt] \text{(b) } & F_{b}(\cdot ,r)\text{ is measurable on }\unicode[STIX]{x1D6E4}_{3}\text{ for all }r\in \mathbb{R};\\[3.0pt] \text{(c) } & F_{b}(\boldsymbol{x},r)=0\text{ for }r\leq 0,~F_{b}(\boldsymbol{x},r)\geq 0\text{ for }r\geq 0,\text{a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3}.\end{array}\right\}\end{eqnarray}$$

For the densities of body forces and surface tractions, assume

(2.37) $$\begin{eqnarray}\boldsymbol{f}_{0}\in L^{2}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d}),\quad \boldsymbol{f}_{2}\in L^{2}(\unicode[STIX]{x1D6E4}_{2};\mathbb{R}^{d}).\end{eqnarray}$$

Finally, the bound $g$ satisfies

(2.38) $$\begin{eqnarray}g\in L^{2}(\unicode[STIX]{x1D6E4}_{3}),\quad g(\boldsymbol{x})\geq 0\quad \text{a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3}.\end{eqnarray}$$

Problem 2.5 is the classical formulation of the problem, that is, the unknowns and the data are assumed to be smooth functions such that all the derivatives and all the relations are satisfied in the usual sense at each point. However, the frictional contact conditions introduce a mathematical difficulty since they are expressed in terms of non-differentiable functions and belong to the conditions dealt with in the part of mechanics called non-smooth mechanics. In general, the problem does not have a classical solution, that is, a solution which has all the necessary classical derivatives, and some of the conditions will be satisfied in a weak sense that has to be made precise. Moreover, the frictional contact conditions impose a ceiling on the regularity or smoothness of the solutions, even if all the problem data are smooth. This is in contrast with the usual boundary value problems of elliptic partial differential equations where higher regularity on the data leads to higher regularity for the solutions and represents a challenging feature in the analysis of contact problems.

Problem 2.5 is most naturally studied in a weak sense. The weak formulation of the problem is not only a mathematical necessity, but also very useful practically since it leads directly to efficient numerical methods to solve the problem.

To derive the weak formulation, we assume the classical formulation has a solution and all the functions involved are as smooth as is needed for the various mathematical operations to be justified, and so the derivation is formal. We shall return to this point once we obtain the weak formulation. We introduce the set of admissible displacements defined by

(2.39) $$\begin{eqnarray}U=\{\boldsymbol{v}\in V\mid v_{\unicode[STIX]{x1D708}}\leq g~~a.e.~on~\unicode[STIX]{x1D6E4}_{3}\},\end{eqnarray}$$

where the space $V$ is defined by (2.3).

Let $\boldsymbol{v}\in U$ . We multiply the equation (2.21) by $(\boldsymbol{v}-\boldsymbol{u})$ and integrate over  $\unicode[STIX]{x1D6FA}$ :

$$\begin{eqnarray}-\int _{\unicode[STIX]{x1D6FA}}\text{Div}\,\unicode[STIX]{x1D748}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}x=\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{f}_{0}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}x.\end{eqnarray}$$

Apply Green’s formula (2.6) to the integral on the left-hand side:

$$\begin{eqnarray}-\int _{\unicode[STIX]{x1D6FA}}\text{Div}\,\unicode[STIX]{x1D748}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}x=-\int _{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D748}\unicode[STIX]{x1D742}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}a+\int _{\unicode[STIX]{x1D6FA}}\,\unicode[STIX]{x1D748}\cdot (\unicode[STIX]{x1D73A}(\boldsymbol{v})-\unicode[STIX]{x1D73A}(\boldsymbol{u}))\,\text{d}x.\end{eqnarray}$$

Then we split the surface integral into three sub-integrals on $\unicode[STIX]{x1D6E4}_{1}$ , $\unicode[STIX]{x1D6E4}_{2}$ and $\unicode[STIX]{x1D6E4}_{3}$ and use the boundary condition (2.23) to deduce that

$$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6FA}}\,\unicode[STIX]{x1D748}\cdot (\unicode[STIX]{x1D73A}(\boldsymbol{v})-\unicode[STIX]{x1D73A}(\boldsymbol{u}))\,\text{d}x & = & \displaystyle \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{f}_{0}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}x+\int _{\unicode[STIX]{x1D6E4}_{1}}\unicode[STIX]{x1D748}\unicode[STIX]{x1D742}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}a\nonumber\\ \displaystyle & & \displaystyle +\,\int _{\unicode[STIX]{x1D6E4}_{2}}\boldsymbol{f}_{2}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}a+\int _{\unicode[STIX]{x1D6E4}_{3}}\unicode[STIX]{x1D748}\unicode[STIX]{x1D742}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}a.\nonumber\end{eqnarray}$$

Next, using the identities

$$\begin{eqnarray}\displaystyle \boldsymbol{v}-\boldsymbol{u} & = & \displaystyle \mathbf{0}\quad \text{a.e. on }\unicode[STIX]{x1D6E4}_{1},\nonumber\\ \displaystyle \unicode[STIX]{x1D748}\unicode[STIX]{x1D742}\cdot (\boldsymbol{v}-\boldsymbol{u}) & = & \displaystyle \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})+\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}\cdot (\boldsymbol{v}_{\unicode[STIX]{x1D70F}}-\boldsymbol{u}_{\unicode[STIX]{x1D70F}})\quad \text{a.e. on }\unicode[STIX]{x1D6E4}_{3},\nonumber\end{eqnarray}$$

we find that

(2.40) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\unicode[STIX]{x1D6FA}}\,\unicode[STIX]{x1D748}\cdot (\unicode[STIX]{x1D73A}(\boldsymbol{v})-\unicode[STIX]{x1D73A}(\boldsymbol{u}))\,\text{d}x=\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{f}_{0}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}x+\int _{\unicode[STIX]{x1D6E4}_{2}}\boldsymbol{f}_{2}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}a\nonumber\\ \displaystyle & & \displaystyle \quad +\int _{\unicode[STIX]{x1D6E4}_{3}}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})\,\text{d}a+\int _{\unicode[STIX]{x1D6E4}_{3}}\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}\cdot (\boldsymbol{v}_{\unicode[STIX]{x1D70F}}-\boldsymbol{u}_{\unicode[STIX]{x1D70F}})\,\text{d}a.\end{eqnarray}$$

On the other hand, we use the identity

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})=(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}+\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}})(v_{\unicode[STIX]{x1D708}}-g)+(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}+\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}})(g-u_{\unicode[STIX]{x1D708}})-\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})\end{eqnarray}$$

combined with the contact boundary condition (2.24) and the definition of the subdifferential to see that

$$\begin{eqnarray}-\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})\leq j_{\unicode[STIX]{x1D708}}^{0}(u_{\unicode[STIX]{x1D708}};v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})\quad \text{a.e. on }\unicode[STIX]{x1D6E4}_{3}.\end{eqnarray}$$

Therefore,

(2.41) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6E4}_{3}}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})\,\text{d}a\geq -\int _{\unicode[STIX]{x1D6E4}_{3}}j_{\unicode[STIX]{x1D708}}^{0}(u_{\unicode[STIX]{x1D708}};v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})\,\text{d}a.\end{eqnarray}$$

From the friction law (2.25),

$$\begin{eqnarray}\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}\cdot (\boldsymbol{v}_{\unicode[STIX]{x1D70F}}-\boldsymbol{u}_{\unicode[STIX]{x1D70F}})\geq F_{b}(u_{\unicode[STIX]{x1D708}})(\Vert \boldsymbol{u}_{\unicode[STIX]{x1D70F}}\Vert -\Vert \boldsymbol{v}_{\unicode[STIX]{x1D70F}}\Vert )\quad \text{a.e. on }\unicode[STIX]{x1D6E4}_{3}.\end{eqnarray}$$

Therefore

(2.42) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6E4}_{3}}\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}\cdot (\boldsymbol{v}_{\unicode[STIX]{x1D70F}}-\boldsymbol{u}_{\unicode[STIX]{x1D70F}})\,\text{d}a\geq \int _{\unicode[STIX]{x1D6E4}_{3}}F_{b}(u_{\unicode[STIX]{x1D708}})(\Vert \boldsymbol{u}_{\unicode[STIX]{x1D70F}}\Vert -\Vert \boldsymbol{v}_{\unicode[STIX]{x1D70F}}\Vert )\,\text{d}a.\end{eqnarray}$$

We now combine equality (2.40) with inequalities (2.41) and (2.42) to deduce that

(2.43) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D748}\cdot (\unicode[STIX]{x1D73A}(\boldsymbol{v})-\unicode[STIX]{x1D73A}(\boldsymbol{u}))\,\text{d}x+\int _{\unicode[STIX]{x1D6E4}_{3}}F_{b}(u_{\unicode[STIX]{x1D708}})(\Vert \boldsymbol{v}_{\unicode[STIX]{x1D70F}}\Vert -\Vert \boldsymbol{u}_{\unicode[STIX]{x1D70F}}\Vert )\,\text{d}a\nonumber\\ \displaystyle & & \displaystyle \quad +\,\int _{\unicode[STIX]{x1D6E4}_{3}}~j_{\unicode[STIX]{x1D708}}^{0}(u_{\unicode[STIX]{x1D708}};v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})\,\text{d}a\geq \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{f}_{0}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}x+\int _{\unicode[STIX]{x1D6E4}_{2}}\boldsymbol{f}_{2}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}a.\end{eqnarray}$$

Finally, we substitute the constitutive law (2.20) in (2.43) to obtain the following weak formulation of Problem 2.5, in terms of the displacement.

Problem 2.6. Find a displacement field $\boldsymbol{u}\in U$ such that

(2.44) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\unicode[STIX]{x1D6FA}}{\mathcal{F}}\unicode[STIX]{x1D73A}(\boldsymbol{u})\cdot (\unicode[STIX]{x1D73A}(\boldsymbol{v})-\unicode[STIX]{x1D73A}(\boldsymbol{u}))\,\text{d}x\nonumber\\ \displaystyle & & \displaystyle \quad +\int _{\unicode[STIX]{x1D6E4}_{3}}F_{b}(u_{\unicode[STIX]{x1D708}})(\Vert \boldsymbol{v}_{\unicode[STIX]{x1D70F}}\Vert -\Vert \boldsymbol{u}_{\unicode[STIX]{x1D70F}}\Vert )\,\text{d}a+\int _{\unicode[STIX]{x1D6E4}_{3}}~j_{\unicode[STIX]{x1D708}}^{0}(u_{\unicode[STIX]{x1D708}};v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})\,\text{d}a\nonumber\\ \displaystyle & & \displaystyle \quad \quad \geq \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{f}_{0}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}x+\int _{\unicode[STIX]{x1D6E4}_{2}}\boldsymbol{f}_{2}\cdot (\boldsymbol{v}-\boldsymbol{u})\,\text{d}a\quad \text{for all}~\boldsymbol{v}\in U.\end{eqnarray}$$

Thus, if Problem 2.5 has a sufficiently smooth solution for all of the operations above to be justified, it is also a solution of Problem 2.6. However, now we have a weak formulation, Problem 2.6, that may have solutions which do not have the necessary regularity or smoothness, and we call them weak solutions of the original problem. This shows why it is necessary to derive and study weak formulations. This also indicates that once the existence of a weak solution is established, there is considerable interest in establishing its regularity, since if the weak solution is sufficiently smooth, then it is also a classical solution.

We note in passing that even if a problem possesses smooth or classical solutions, the weak formulation is usually the first step in its analysis, since many of the modern mathematical tools are better suited for such a formulation. Moreover, the weak formulation can often be employed directly in the finite element method for numerical approximations of the problem.

The inequality (2.44) involves both convex and locally Lipschitz functions. Moreover, it is time-independent and the corresponding differential operator is elliptic. For this reason, we refer to this inequality as an elliptic variational–hemivariational inequality. The well-posedness and numerical analysis of Problem 2.6 will be provided in Section 7.

2.2 A history-dependent frictionless contact problem

The model presented in this subsection is time-dependent. We assume that the inertial term in the equation of motion can be neglected, that is, the process is quasistatic. We use a viscoelastic constitutive law with long memory and we assume that the contact is frictionless. Moreover, we use the time-dependent version of the contact condition with normal compliance and unilateral constraints in Problem 2.5. Let $I$ be the time interval of interest, which can be either bounded (i.e.  $I=[0,T]$ with $T>0$ ) or unbounded (i.e.  $I=[0,+\infty )$ ). The classical formulation of the problem is as follows.

Problem 2.7. Find a displacement field $\boldsymbol{u}:\unicode[STIX]{x1D6FA}\times I\rightarrow \mathbb{R}^{d}$ , a stress field $\unicode[STIX]{x1D748}:\unicode[STIX]{x1D6FA}\times I\rightarrow \mathbb{S}^{d}$ and an interface function $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}:\unicode[STIX]{x1D6E4}_{3}\times I\rightarrow \mathbb{R}$ such that

(2.45) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D748}(t)={\mathcal{F}}\unicode[STIX]{x1D73A}(\boldsymbol{u}(t))+\int _{0}^{t}{\mathcal{B}}(t-s)\unicode[STIX]{x1D73A}(\boldsymbol{u}(s))\,\text{d}s\quad & \text{in}~\unicode[STIX]{x1D6FA}, & \displaystyle\end{eqnarray}$$

(2.46) $$\begin{eqnarray}\displaystyle \text{Div}\,\unicode[STIX]{x1D748}(t)+\boldsymbol{f}_{0}(t)=\mathbf{0}\quad & \text{in}~\unicode[STIX]{x1D6FA}, & \displaystyle\end{eqnarray}$$

(2.47) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(t)=\mathbf{0}\quad & \text{on}~\unicode[STIX]{x1D6E4}_{1}, & \displaystyle\end{eqnarray}$$

(2.48) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D748}(t)\unicode[STIX]{x1D742}=\boldsymbol{f}_{2}(t)\quad & \text{on}~\unicode[STIX]{x1D6E4}_{2}, & \displaystyle\end{eqnarray}$$

(2.49) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}cc@{}}u_{\unicode[STIX]{x1D708}}(t)\leq g,\quad \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}(t)+\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}(t)\leq 0, & \\ \displaystyle (u_{\unicode[STIX]{x1D708}}(t)-g)(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}(t)+\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}(t))=0, & \quad \unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}(t)\in \unicode[STIX]{x2202}j_{\unicode[STIX]{x1D708}}(u_{\unicode[STIX]{x1D708}}(t))\end{array}\right\}\quad & \text{on}~\unicode[STIX]{x1D6E4}_{3}, & \displaystyle\end{eqnarray}$$

(2.50) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}(t)=\mathbf{0}\quad & \text{on}~\unicode[STIX]{x1D6E4}_{3}, & \displaystyle\end{eqnarray}$$

for all $t\in I$ .

Equation (2.45) represents the viscoelastic constitutive law in which ${\mathcal{F}}$ is the elasticity operator, assumed to satisfy condition (2.34), and ${\mathcal{B}}$ is the relaxation tensor, assumed to have the regularity

(2.51) $$\begin{eqnarray}{\mathcal{B}}\in C(I;\mathbf{Q_{&#x221E;}}).\end{eqnarray}$$

Various results, examples and mechanical interpretations in the study of viscoelastic materials of the form (2.45) can be found in Drozdov (Reference Drozdov1996). Note that for such a constitutive law the current value of the stress may depend on all the strain values up to the current time; therefore, incorporating this constitutive law in the contact model makes the problem history-dependent.

Equation (2.46) is the equilibrium equation, since the process is assumed quasistatic. Conditions (2.47) and (2.48) represent the displacement and the traction boundary condition, respectively. Condition (2.49) is a time-dependent version of contact law used in Problem 2.5. Finally, condition (2.50) represents the frictionless condition. It shows that the friction force, $\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}$ , vanishes during the contact process. This is an idealization of the process, since even completely lubricated surfaces generate shear resistance to tangential motion. However, the frictionless condition (2.50) is a sufficiently good approximation of the reality in some situations, especially when the contact surfaces are lubricated. For this reason it has been used in several publications: see Shillor, Sofonea and Telega (Reference Shillor, Sofonea and Telega2004).

In the study of Problem 2.7 we assume that the potential function $j_{\unicode[STIX]{x1D708}}$ satisfies condition (2.35), the bound $g$ satisfies condition (2.38), and the densities of body forces and surface tractions have the regularity

(2.52) $$\begin{eqnarray}\boldsymbol{f}_{0}\in C(I;L^{2}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d})),\quad \boldsymbol{f}_{2}\in C(I;L^{2}(\unicode[STIX]{x1D6E4}_{2};\mathbb{R}^{d})).\end{eqnarray}$$

We use the space $V$ , (2.3), and the set of admissible displacement fields $U$ , (2.39). Then, the weak formulation of Problem 2.7, obtained by using arguments similar to those used in the previous subsection, is as follows.

Problem 2.8. Find a displacement field $\boldsymbol{u}:I\rightarrow U$ such that

(2.53) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\unicode[STIX]{x1D6FA}}{\mathcal{F}}\unicode[STIX]{x1D73A}(\boldsymbol{u}(t))\cdot (\unicode[STIX]{x1D73A}(\boldsymbol{v})-\unicode[STIX]{x1D73A}(\boldsymbol{u}(t)))\,\text{d}x\nonumber\\ \displaystyle & & \displaystyle \quad +\int _{\unicode[STIX]{x1D6FA}}\biggl(\int _{0}^{t}{\mathcal{B}}(t-s)\,\unicode[STIX]{x1D73A}(\boldsymbol{u}(s))\,\text{d}s\biggr)\cdot (\unicode[STIX]{x1D73A}(\boldsymbol{v})-\unicode[STIX]{x1D73A}(\boldsymbol{u}(t)))\,\text{d}x\nonumber\\ \displaystyle & & \displaystyle \quad +\int _{\unicode[STIX]{x1D6E4}_{3}}~j_{\unicode[STIX]{x1D708}}^{0}(u_{\unicode[STIX]{x1D708}}(t);v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}}(t))\,\text{d}a\nonumber\\ \displaystyle & & \displaystyle \quad \quad \geq \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{f}_{0}(t)\cdot (\boldsymbol{v}-\boldsymbol{u}(t))\,\text{d}x+\int _{\unicode[STIX]{x1D6E4}_{2}}\boldsymbol{f}_{2}(t)\cdot (\boldsymbol{v}-\boldsymbol{u}(t))\,\text{d}a\end{eqnarray}$$

for all $\boldsymbol{v}\in U$ , $t\in I$ .

Note that inequality (2.53) involves both convex and locally Lipschitz functions. Further, it is time-dependent and it includes a history-dependent term. For this reason, we refer to this inequality as a history-dependent variational–hemivariational inequality. The well-posedness and numerical analysis of Problem 2.8 will be provided in Section 8.

2.3 A dynamic frictional contact problem

For the last model, the process is assumed to be dynamic and is considered in a finite time interval $I=[0,T]$ , $T>0$ . The constitutive law is viscoelastic with short memory and the contact is described by a non-monotone version of the normal damped response condition, associated with a subdifferential friction law. The classical formulation of the problem is as follows.

Problem 2.9. Find a displacement field $\boldsymbol{u}:\unicode[STIX]{x1D6FA}\times [0,T]\rightarrow \mathbb{R}^{d}$ and a stress field $\unicode[STIX]{x1D748}:\unicode[STIX]{x1D6FA}\times [0,T]\rightarrow \mathbb{S}^{d}$ such that

(2.54) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D748}(t)={\mathcal{A}}\unicode[STIX]{x1D73A}(\dot{\boldsymbol{u}}(t))+{\mathcal{B}}\unicode[STIX]{x1D73A}(\boldsymbol{u}(t))\quad & \text{in}~\unicode[STIX]{x1D6FA}, & \displaystyle\end{eqnarray}$$

(2.55) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}\,\ddot{\boldsymbol{u}}(t)=\text{Div}\,\unicode[STIX]{x1D748}(t)+\boldsymbol{f}_{0}(t)\quad & \text{in}~\unicode[STIX]{x1D6FA}, & \displaystyle\end{eqnarray}$$

(2.56) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(t)=\mathbf{0}\quad & \text{on}~\unicode[STIX]{x1D6E4}_{1}, & \displaystyle\end{eqnarray}$$

(2.57) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D748}(t)\unicode[STIX]{x1D742}=\boldsymbol{f}_{2}(t)\quad & \text{on}~\unicode[STIX]{x1D6E4}_{2}, & \displaystyle\end{eqnarray}$$

(2.58) $$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D708}}(t)\in \unicode[STIX]{x2202}j_{\unicode[STIX]{x1D708}}(\dot{u}_{\unicode[STIX]{x1D708}}(t))\quad & \text{on}~\unicode[STIX]{x1D6E4}_{3}, & \displaystyle\end{eqnarray}$$

(2.59) $$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}(t)\in \unicode[STIX]{x2202}j_{\unicode[STIX]{x1D70F}}(\dot{\boldsymbol{u}}_{\unicode[STIX]{x1D70F}}(t))\quad & \text{on}~\unicode[STIX]{x1D6E4}_{3}, & \displaystyle\end{eqnarray}$$

for all $t\in [0,T]$ , and

(2.60) $$\begin{eqnarray}\boldsymbol{u}(0)=\boldsymbol{u}_{0},\quad \dot{\boldsymbol{u}}(0)=\boldsymbol{w}_{0}\quad \text{in}~\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

A brief description of equations and boundary conditions in Problem 2.9 follows.

First, equation (2.54) is the constitutive law for viscoelastic materials with short memory in which ${\mathcal{A}}$ represents the viscosity operator and ${\mathcal{B}}$ represents the elasticity operator. In the linear case, the constitutive law (2.54) becomes the well-known Kelvin–Voigt law

(2.61) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{ij}=a_{ijkl}\unicode[STIX]{x1D700}_{kl}(\dot{\boldsymbol{u}})+b_{ijkl}\unicode[STIX]{x1D700}_{kl}(\boldsymbol{u}),\end{eqnarray}$$

where $a_{ijkl}$ represent the components of the viscosity tensor ${\mathcal{A}}$ and $b_{ijkl}$ are the components of the elasticity tensor ${\mathcal{B}}$ . Quasistatic contact problems for viscoelastic materials of the form (2.54) have been considered in Han and Sofonea (Reference Han and Sofonea2002), Shillor, Sofonea and Telega (Reference Shillor, Sofonea and Telega2004) and Sofonea and Matei (Reference Sofonea and Matei2012), in the context of variational inequalities. Numerical analysis of variational inequalities for such contact models can be found in Han and Sofonea (Reference Han and Sofonea2002).

Equation (2.55) is the equation for motion in which $\unicode[STIX]{x1D70C}$ denotes the density of mass. Note that, for simplicity, we assume that $\unicode[STIX]{x1D70C}$ does not depend on $\boldsymbol{x}\in \unicode[STIX]{x1D6FA}$ . On $\unicode[STIX]{x1D6E4}_{1}$ , we have the clamped boundary condition (2.56) and, on $\unicode[STIX]{x1D6E4}_{2}$ , the surface traction boundary condition (2.57). Relation (2.58) represents the contact condition with normal damped response, in which $\unicode[STIX]{x2202}j_{\unicode[STIX]{x1D708}}$ denotes the Clarke subdifferential of the given function $j_{\unicode[STIX]{x1D708}}$ . Such a contact condition was used in Han and Sofonea (Reference Han and Sofonea2002), Shillor, Sofonea and Telega (Reference Shillor, Sofonea and Telega2004) and Migórski, Ochal and Sofonea (Reference Migórski, Ochal and Sofonea2013) in order to model the setting when the foundation is covered with a thin lubricant layer such as oil. Condition (2.59) represents a friction law, written in a general subdifferential form. A particular example of the friction law is given by (2.59) with

(2.62) $$\begin{eqnarray}j_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D743})=\int _{0}^{\Vert \unicode[STIX]{x1D743}\Vert }\unicode[STIX]{x1D707}(s)\,\text{d}s\quad \text{for all}~\unicode[STIX]{x1D743}\in \mathbb{R}^{d},\end{eqnarray}$$

for a suitable friction bound $\unicode[STIX]{x1D707}:[0,+\infty )\rightarrow \mathbb{R}_{+}$ . For this choice of $j_{\unicode[STIX]{x1D70F}}(\cdot )$ , (2.59) represents the subdifferential form of the friction law

(2.63) $$\begin{eqnarray}\Vert \unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}\Vert \leq \unicode[STIX]{x1D707}(\Vert \dot{\boldsymbol{u}}_{\unicode[STIX]{x1D70F}}\Vert ),\quad -\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D70F}}(t)=\unicode[STIX]{x1D707}(\Vert \dot{\boldsymbol{u}}_{\unicode[STIX]{x1D70F}}\Vert )\,\displaystyle \frac{\dot{\boldsymbol{u}}_{\unicode[STIX]{x1D70F}}(t)}{\Vert \dot{\boldsymbol{u}}_{\unicode[STIX]{x1D70F}}(t)\Vert }~\text{if}~\dot{\boldsymbol{u}}_{\unicode[STIX]{x1D70F}}(t)\neq \mathbf{0}\quad \text{on}~\unicode[STIX]{x1D6E4}_{3},\end{eqnarray}$$

which will be used in Section 9.3 of this paper. Additional examples of friction laws of the form (2.59) can be found in Migórski, Ochal and Sofonea (Reference Migórski, Ochal and Sofonea2013), for instance.

Finally, conditions (2.60) are the initial conditions in which $\boldsymbol{u}_{0}$ and $\boldsymbol{w}_{0}$ represent the initial displacement and the initial velocity, respectively.

We now introduce the assumptions of the data of Problem 2.9.

For the viscosity operator ${\mathcal{A}}:\unicode[STIX]{x1D6FA}\times \mathbb{S}^{d}\rightarrow \mathbb{S}^{d}$ , we assume:

(2.64) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\text{(a) } & \text{there exists a constant }L_{{\mathcal{A}}}>0\text{ such that}\\ & \Vert {\mathcal{A}}(\boldsymbol{x},\unicode[STIX]{x1D73A}_{1})-{\mathcal{A}}(\boldsymbol{x},\unicode[STIX]{x1D73A}_{2})\Vert \leq L_{{\mathcal{A}}}\Vert \unicode[STIX]{x1D73A}_{1}-\unicode[STIX]{x1D73A}_{2}\Vert \\ & \text{for all }\unicode[STIX]{x1D73A}_{1},\unicode[STIX]{x1D73A}_{2}\in \mathbb{S}^{d},~\text{a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6FA};\\[3.0pt] \text{(b) } & \text{there exists a constant }m_{{\mathcal{A}}}>0\text{ such that}\\ & ({\mathcal{A}}(\boldsymbol{x},\unicode[STIX]{x1D73A}_{1})-{\mathcal{A}}(\boldsymbol{x},\unicode[STIX]{x1D73A}_{2}))\cdot (\unicode[STIX]{x1D73A}_{1}-\unicode[STIX]{x1D73A}_{2})\geq m_{{\mathcal{A}}}\,\Vert \unicode[STIX]{x1D73A}_{1}-\unicode[STIX]{x1D73A}_{2}\Vert ^{2}~\\ & \text{for all }\unicode[STIX]{x1D73A}_{1},\unicode[STIX]{x1D73A}_{2}\in \mathbb{S}^{d},~\text{a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6FA};\\[3.0pt] \text{(c) } & {\mathcal{A}}(\cdot ,\unicode[STIX]{x1D73A})\text{ is measurable on }\unicode[STIX]{x1D6FA}\text{ for all }\unicode[STIX]{x1D73A}\in \mathbb{S}^{d};\\[3.0pt] \text{(d) } & {\mathcal{A}}(\boldsymbol{x},\mathbf{0})=\mathbf{0}~\text{a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6FA}.\end{array}\right\}\end{eqnarray}$$

For the elasticity operator ${\mathcal{B}}:\unicode[STIX]{x1D6FA}\times \mathbb{S}^{d}\rightarrow \mathbb{S}^{d}$ , we assume:

(2.65) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\text{(a) } & \text{there exists a constant }L_{{\mathcal{B}}}>0\text{ such that}~\\ & \Vert {\mathcal{B}}(\boldsymbol{x},\unicode[STIX]{x1D73A}_{1})-{\mathcal{B}}(\boldsymbol{x},\unicode[STIX]{x1D73A}_{2})\Vert \leq L_{{\mathcal{B}}}\Vert \unicode[STIX]{x1D73A}_{1}-\unicode[STIX]{x1D73A}_{2}\Vert \\ & \text{for all }\unicode[STIX]{x1D73A}_{1},\unicode[STIX]{x1D73A}_{2}\in \mathbb{S}^{d},~\text{a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6FA};\\[3.0pt] \text{(b) } & {\mathcal{B}}(\cdot ,\unicode[STIX]{x1D73A})\text{ is measurable on }\unicode[STIX]{x1D6FA}\text{ for all }\unicode[STIX]{x1D73A}\in \mathbb{S}^{d};\\[3.0pt] \text{(c) } & {\mathcal{B}}(\boldsymbol{x},\mathbf{0})=\mathbf{0}~\text{a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6FA}.\end{array}\right\}\end{eqnarray}$$

For the normal potential function $j_{\unicode[STIX]{x1D708}}:\unicode[STIX]{x1D6E4}_{3}\times \mathbb{R}\rightarrow \mathbb{R}$ , we assume:

(2.66) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\text{(a) } & j_{\unicode[STIX]{x1D708}}(\cdot ,r)\text{ is measurable on }\unicode[STIX]{x1D6E4}_{3}\text{ for all }r\in \mathbb{R}\text{ and there}\\ & \text{exists }\bar{e}_{\unicode[STIX]{x1D708}}\in L^{2}(\unicode[STIX]{x1D6E4}_{3})\text{ such that }j_{\unicode[STIX]{x1D708}}(\cdot ,\bar{e}_{\unicode[STIX]{x1D708}}(\cdot ))\in L^{1}(\unicode[STIX]{x1D6E4}_{3});\\[3.0pt] \text{(b) } & j_{\unicode[STIX]{x1D708}}(\boldsymbol{x},\cdot )\text{ is locally Lipschitz on }\mathbb{R}\text{ for a.e. }\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3};\\[3.0pt] \text{(c) } & |\unicode[STIX]{x2202}j_{\unicode[STIX]{x1D708}}(\boldsymbol{x},r)|\leq \bar{c}_{0\unicode[STIX]{x1D708}}\text{ for a.e. }\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3},\text{ for }r\in \mathbb{R}\text{ with }\bar{c}_{0\unicode[STIX]{x1D708}}\geq 0;~\\[3.0pt] \text{(d) } & j_{\unicode[STIX]{x1D708}}^{0}(\boldsymbol{x},r_{1};r_{2}-r_{1})+j_{\unicode[STIX]{x1D708}}^{0}(\boldsymbol{x},r_{2};r_{1}-r_{2})\leq \unicode[STIX]{x1D6FC}_{j_{\unicode[STIX]{x1D708}}}|r_{1}-r_{2}|^{2}\\ & \text{for a.e. }\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3},~\text{all}~r_{1},\,r_{2}\in \mathbb{R}\text{ with }\unicode[STIX]{x1D6FC}_{j_{\unicode[STIX]{x1D708}}}\geq 0.\end{array}\right\}\end{eqnarray}$$

For the tangential potential function $j_{\unicode[STIX]{x1D70F}}:\unicode[STIX]{x1D6E4}_{3}\times \mathbb{R}^{d}\rightarrow \mathbb{R}$ , we assume:

(2.67) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\text{(a) } & j_{\unicode[STIX]{x1D70F}}(\cdot ,\unicode[STIX]{x1D743})\text{ is measurable on }\unicode[STIX]{x1D6E4}_{3}\text{ for all }\unicode[STIX]{x1D743}\in \mathbb{R}^{d}\text{ and there}\\ & \text{exists }\bar{\boldsymbol{e}}_{\unicode[STIX]{x1D70F}}\in L^{2}(\unicode[STIX]{x1D6E4}_{3})^{d}\text{ such that }j_{\unicode[STIX]{x1D70F}}(\cdot ,\bar{\boldsymbol{e}}_{\unicode[STIX]{x1D70F}}(\cdot ))\in L^{1}(\unicode[STIX]{x1D6E4}_{3});\\[3.0pt] \text{(b) } & j_{\unicode[STIX]{x1D70F}}(\boldsymbol{x},\cdot )\text{ is locally Lipschitz on }\mathbb{R}^{d}\text{ for a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3};\\[3.0pt] \text{(c) } & |\unicode[STIX]{x2202}j_{\unicode[STIX]{x1D70F}}(\boldsymbol{x},\unicode[STIX]{x1D743})|\leq \bar{c}_{0\unicode[STIX]{x1D70F}}\text{ for a.e. }\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3},\text{for }\unicode[STIX]{x1D743}\in \mathbb{R}^{d}\text{ with }\bar{c}_{0\unicode[STIX]{x1D70F}}\geq 0;~\\[3.0pt] \text{(d) } & j_{\unicode[STIX]{x1D70F}}^{0}(\boldsymbol{x},\unicode[STIX]{x1D743}_{1};\unicode[STIX]{x1D743}_{2}-\unicode[STIX]{x1D743}_{1})+j_{\unicode[STIX]{x1D70F}}^{0}(\boldsymbol{x},\unicode[STIX]{x1D743}_{2};\unicode[STIX]{x1D743}_{1}-\unicode[STIX]{x1D743}_{2})\leq \unicode[STIX]{x1D6FC}_{j_{\unicode[STIX]{x1D70F}}}\Vert \unicode[STIX]{x1D743}_{1}-\unicode[STIX]{x1D743}_{2}\Vert ^{2}\\ & \text{for a.e.}~\boldsymbol{x}\in \unicode[STIX]{x1D6E4}_{3},\text{all }\unicode[STIX]{x1D743}_{1},\,\unicode[STIX]{x1D743}_{2}\in \mathbb{R}^{d}\text{ with }\unicode[STIX]{x1D6FC}_{j_{\unicode[STIX]{x1D70F}}}\geq 0.\end{array}\right\}\end{eqnarray}$$

For the mass density assume it is a positive constant,

(2.68) $$\begin{eqnarray}\unicode[STIX]{x1D70C}>0.\end{eqnarray}$$

For the densities of body forces and surface tractions, assume

(2.69) $$\begin{eqnarray}\boldsymbol{f}_{0}\in L^{2}(0,T;L^{2}(\unicode[STIX]{x1D6FA};\mathbb{R}^{d})),\quad \boldsymbol{f}_{2}\in L^{2}(0,T;L^{2}(\unicode[STIX]{x1D6E4}_{2};\mathbb{R}^{d})),\end{eqnarray}$$

and for the initial data,

(2.70) $$\begin{eqnarray}\boldsymbol{u}_{0}\in V,\quad \boldsymbol{w}_{0}\in H.\end{eqnarray}$$

Recall that the spaces $V$ and $H$ are defined by (2.3) and (2.5), respectively. Then, the weak formulation of Problem 2.9, obtained by using arguments similar to those used in Section 2.1, is as follows.

Problem 2.10. Find a displacement field $\boldsymbol{u}:[0,T]\rightarrow V$ such that

(2.71) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D70C}\,\ddot{\boldsymbol{u}}(t)\cdot \boldsymbol{v}\,\text{d}x+\int _{\unicode[STIX]{x1D6FA}}{\mathcal{A}}\unicode[STIX]{x1D73A}(\dot{\boldsymbol{u}}(t))\cdot \unicode[STIX]{x1D73A}(\boldsymbol{v})\,\text{d}x+\int _{\unicode[STIX]{x1D6FA}}{\mathcal{B}}\unicode[STIX]{x1D73A}(\boldsymbol{u}(t))\cdot \unicode[STIX]{x1D73A}(\boldsymbol{v})\,\text{d}x\nonumber\\ \displaystyle & & \displaystyle \quad +\int _{\unicode[STIX]{x1D6E4}_{3}}j_{\unicode[STIX]{x1D708}}^{0}(\dot{u}_{\unicode[STIX]{x1D708}}(t);v_{\unicode[STIX]{x1D708}})\,\text{d}a+\int _{\unicode[STIX]{x1D6E4}_{3}}j_{\unicode[STIX]{x1D70F}}^{0}(\dot{\boldsymbol{u}}_{\unicode[STIX]{x1D70F}}(t);\boldsymbol{v}_{\unicode[STIX]{x1D70F}})\,\text{d}a\nonumber\\ \displaystyle & & \displaystyle \quad \quad \geq \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{f}_{0}(t)\cdot \boldsymbol{v}\,\text{d}x+\int _{\unicode[STIX]{x1D6E4}_{2}}\boldsymbol{f}_{2}(t)\cdot \boldsymbol{v}\,\text{d}a\end{eqnarray}$$

for all $\boldsymbol{v}\in V$ , $t\in [0,T]$ , and

(2.72) $$\begin{eqnarray}\boldsymbol{u}(0)=\boldsymbol{u}_{0},\quad \dot{\boldsymbol{u}}(0)=\boldsymbol{w}_{0}.\end{eqnarray}$$

Note that inequality (2.71) involves the second time derivative of the unknown function $\boldsymbol{u}$ . Consequently, the initial conditions (2.72) are introduced. We refer to such an inequality as an evolutionary hemivariational inequality. The well-posedness analysis and numerical approximations of Problem 2.6 will be provided in Section 9.

3.1 Basics on non-smooth analysis

Nonlinear operators of monotone type.

We start with some definitions concerning single-valued operators.

Definition 3.1. An operator $A:X\rightarrow X^{\ast }$ is called:

1. (a) bounded if $A$ maps bounded sets of $X$ into bounded sets of $X^{\ast }$ ;

2. (b) hemicontinuous if, for all $u,v,w\in X$ , the function $t\mapsto \langle A(u+t\,v),w\rangle$ is continuous for $t\in [0,1]$ ;

3. (c) monotone if $\langle Au-Av,u-v\rangle \geq 0$ for all $u$ , $v\in X$ ;

4. (d) maximal monotone if it is monotone and $\langle w-Av,u-v\rangle \geq 0$ for any $v\in X$ implies that $w=Au$ ;

5. (e) $u_{0}$ -coercive if there exists a function $\unicode[STIX]{x1D6FC}:\mathbb{R}_{+}\rightarrow \mathbb{R}$ with $\lim _{r\rightarrow +\infty }\unicode[STIX]{x1D6FC}(r)=+\infty$ such that $\langle Au,u-u_{0}\rangle \geq \unicode[STIX]{x1D6FC}(\Vert u\Vert _{X})\,\Vert u\Vert _{X}$ for all $u\in X$ , where $u_{0}$ is a given element in $X$ ;

6. (f) pseudomonotone if it is bounded and $u_{n}\rightarrow u$ weakly in $X$ with $\displaystyle \limsup \,\langle Au_{n},u_{n}-u\rangle \leq 0$ imply $\displaystyle \liminf \,\langle Au_{n},u_{n}-v\rangle \geq \langle Au,u-v\rangle$ for all $v\in X$ ;

7. (g) radially continuous if, for any $u,v\in X$ , the function $t\mapsto \langle A(u+t\,v),v\rangle$ is continuous on $[0,1]$ .

Remark 3.2. It can be proved that an operator $A:X\rightarrow X^{\ast }$ is pseudomonotone if and only if it is bounded and $u_{n}\rightarrow u$ weakly in $X$ together with $\displaystyle \limsup \,\langle Au_{n},u_{n}-u\rangle \leq 0$ imply $\displaystyle \lim \,\langle Au_{n},u_{n}-u\rangle =0$ and $Au_{n}\rightarrow Au$ weakly in $X^{\ast }$ . $\Box$

We note that if $A:X\rightarrow X^{\ast }$ is continuous, then it is hemicontinuous. The following result can be deduced from Zeidler (Reference Zeidler1990, Proposition 27.6).

Proposition 3.3. If $A:X\rightarrow X^{\ast }$ is bounded, hemicontinuous and monotone, then it is pseudomonotone.

Next, we move to multivalued operators defined on the space $X$ . Given a multivalued operator $T:X\rightarrow 2^{X^{\ast }}$ , its domain ${\mathcal{D}}(T)$ , range ${\mathcal{R}}(T)$ and graph ${\mathcal{G}}(T)$ are the sets defined by

$$\begin{eqnarray}\displaystyle {\mathcal{D}}(T) & = & \displaystyle \{x\in X\mid Tx\not =\emptyset \},\nonumber\\ \displaystyle {\mathcal{R}}(T) & = & \displaystyle \bigcup \,\{Tx\mid x\in X\},\nonumber\\ \displaystyle {\mathcal{G}}(T) & = & \displaystyle \{(x,x^{\ast })\in X\times X^{\ast }\mid x^{\ast }\in Tx\},\nonumber\end{eqnarray}$$

respectively. We proceed with the following definitions.

Definition 3.4. An operator $T:X\rightarrow 2^{X^{\ast }}$ is called:

1. (a) bounded if the range of any bounded set in $X$ is a bounded set in $X^{\ast }$ ;

2. (b) monotone if $\langle u^{\ast }-v^{\ast },u-v\rangle \geq 0$ for all $(u,u^{\ast })$ , $(v,v^{\ast })\in {\mathcal{G}}(T)$ ;

3. (c) maximal monotone if it is monotone and maximal in the sense of inclusion of graphs in the family of monotone operators from $X$ to $2^{X^{\ast }}$ ;

4. (d) $u_{0}$ -coercive if there exists a function $\unicode[STIX]{x1D6FC}:\mathbb{R}_{+}\rightarrow \mathbb{R}$ with $\lim _{r\rightarrow +\infty }\unicode[STIX]{x1D6FC}(r)=+\infty$ such that for all $(u,u^{\ast })\in {\mathcal{G}}(T)$ we have

   $$\begin{eqnarray}\langle u^{\ast },u-u_{0}\rangle \geq \unicode[STIX]{x1D6FC}(\Vert u\Vert _{X})\,\Vert u\Vert _{X},\end{eqnarray}$$
   where $u_{0}$ is a given element in $X$ ;

5. (e) generalized pseudomonotone if, for any sequences $\{u_{n}\}\!\subset \!X$ and $\{u_{n}^{\ast }\}\!\subset \!X^{\ast }$ such that $u_{n}\rightarrow u$ weakly in $X$ , $u_{n}^{\ast }\in Tu_{n}$ for all $n\in \mathbb{N}$ , $u_{n}^{\ast }\rightarrow u^{\ast }$ weakly in $X^{\ast }$ and $\limsup \,\langle u_{n}^{\ast },u_{n}-u\rangle \leq 0$ , we have $u^{\ast }\in Tu$ and

   $$\begin{eqnarray}\lim \,\langle u_{n}^{\ast },u_{n}\rangle =\langle u^{\ast },u\rangle .\end{eqnarray}$$

Given $u_{0}\in X$ and $T:X\rightarrow 2^{X^{\ast }}$ , we define a multivalued operator $T_{u_{0}}$ by $T_{u_{0}}(v)=T(v+u_{0})$ for $v\in X$ . The following surjectivity result concerns the sum of a generalized pseudomonotone operator and a maximal monotone one. Its proof can be found in Naniewicz and Panagiotopoulos (Reference Naniewicz and Panagiotopoulos1995).

Theorem 3.5. Let $X$ be a reflexive Banach space, $T_{1}:X\rightarrow 2^{X^{\ast }}$ , $T_{2}:X\rightarrow 2^{X^{\ast }}$ and $u_{0}\in X$ . Assume

1. (a) $T_{1}$ is bounded, generalized pseudomonotone and $u_{0}$ -coercive;

2. (b) $T_{1}u$ is a non-empty, closed and convex subset of $X^{\ast }$ , for all $u\in X$ ;

3. (c) $T_{2}$ is a maximal monotone operator, and $u_{0}\in {\mathcal{D}}(T_{2})$ .

Then $T_{1}+T_{2}:X\rightarrow 2^{X^{\ast }}$ is surjective, i.e.  ${\mathcal{R}}(T_{1}+T_{2})=X^{\ast }$ .

Theorem 3.5 will be employed in the proof of the solvability of elliptic variational–hemivariational inequalities on the space $X$ .

Convex functions.

Let $\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{+\infty \}$ . The effective domain of $\unicode[STIX]{x1D711}$ is the set $\text{dom}\,\unicode[STIX]{x1D711}=\{x\in X\mid \unicode[STIX]{x1D711}(x)<+\infty \}$ . We say that $\unicode[STIX]{x1D711}$ is convex if, for all $x$ , $y\in \text{dom}\,\unicode[STIX]{x1D711}$ and all $\unicode[STIX]{x1D706}\in (0,1)$ , we have

$$\begin{eqnarray}\unicode[STIX]{x1D711}(\unicode[STIX]{x1D706}x+(1-\unicode[STIX]{x1D706})y)\leq \unicode[STIX]{x1D706}\unicode[STIX]{x1D711}(x)+(1-\unicode[STIX]{x1D706})\unicode[STIX]{x1D711}(y).\end{eqnarray}$$

The function $\unicode[STIX]{x1D711}$ is said to be proper if $\text{dom}\,\unicode[STIX]{x1D711}\not =\emptyset$ . The function $\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{+\infty \}$ is lower semicontinuous (l.s.c.) if, for any $u\in X$ and for any sequence $\{u_{n}\}\subset X$ such that $u_{n}\rightarrow u$ in $X$ , we have

$$\begin{eqnarray}\liminf \unicode[STIX]{x1D711}(u_{n})\geq \unicode[STIX]{x1D711}(u).\end{eqnarray}$$

Next, we recall the notion of the subdifferential of a convex function.

Definition 3.6. Let $\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{+\infty \}$ be a proper and convex function. The mapping $\unicode[STIX]{x2202}_{c}\unicode[STIX]{x1D711}:X\rightarrow 2^{X^{\ast }}$ defined by

$$\begin{eqnarray}\unicode[STIX]{x2202}_{c}\unicode[STIX]{x1D711}(x)=\{x^{\ast }\in X^{\ast }\mid \langle x^{\ast },v-x\rangle \leq \unicode[STIX]{x1D711}(v)-\unicode[STIX]{x1D711}(x)~\text{for all}~v\in X\}\end{eqnarray}$$

is called the convex subdifferential of $\unicode[STIX]{x1D711}$ . If $\unicode[STIX]{x2202}_{c}\unicode[STIX]{x1D711}(x)$ is non-empty, any element $x^{\ast }\in \unicode[STIX]{x2202}_{c}\unicode[STIX]{x1D711}(x)$ is called a subgradient of $\unicode[STIX]{x1D711}$ at  $x$ .

It is easy to see that $\unicode[STIX]{x2202}_{c}\unicode[STIX]{x1D711}:X\rightarrow 2^{X^{\ast }}$ is a monotone operator. Moreover, we have the following well-known result.

Theorem 3.7. Let $X$ be a Banach space and $\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{+\infty \}$ a proper, convex, l.s.c. function. Then $\unicode[STIX]{x2202}_{c}\unicode[STIX]{x1D711}:X\rightarrow 2^{X^{\ast }}$ is maximal monotone.

A proof of Theorem 3.7 can be found in Denkowski, Migórski and Papageorgiou (Reference Denkowski, Migórski and Papageorgiou2003b ).

In many situations we deal with functions $\unicode[STIX]{x1D711}:K\rightarrow \mathbb{R}$ where $K$ is a non-empty subset of $X$ . In such cases it is convenient to extend the function $\unicode[STIX]{x1D711}$ to the space $X$ by considering the function $\widetilde{\unicode[STIX]{x1D711}}:X\rightarrow \mathbb{R}\cup \{+\infty \}$ defined by

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D711}}(v)=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D711}(v)\quad & \text{if }v\in K,\\ +\infty \quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

We say that the function $\unicode[STIX]{x1D711}:K\rightarrow \mathbb{R}$ is convex if its extension $\widetilde{\unicode[STIX]{x1D711}}:X\rightarrow \mathbb{R}\cup \{+\infty \}$ is convex. It is lower semicontinuous (on $K$ ) if $\widetilde{\unicode[STIX]{x1D711}}$ is lower semicontinuous. The special case $\unicode[STIX]{x1D711}=0$ deserves a more detailed treatment.

Example 3.8. Given a non-empty subset $K$ of $X$ , the function $I_{K}$ on $X$ , defined by

$$\begin{eqnarray}I_{K}(x)=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if}~x\in K,\\ +\infty \quad & \text{if}~x\notin K,\end{array}\right.\end{eqnarray}$$

is called the indicator function of $K$ . It can be proved that the subset $K$ of $X$ is convex if and only if $I_{K}$ is convex. Moreover, $K$ is closed if and only if $I_{K}$ is l.s.c. The subdifferential of $I_{K}$ is the multivalued operator $\unicode[STIX]{x2202}_{c}I_{K}:X\rightarrow 2^{X^{\ast }}$ given by

(3.1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{c}I_{K}(x)=\left\{\begin{array}{@{}ll@{}}\{x^{\ast }\in X^{\ast }\mid \langle x^{\ast },x-v\rangle \geq 0~\,\text{for all}~v\in K\}\quad & \text{if}~x\in K,\\ \emptyset \quad & \text{if}~x\notin K.\end{array}\right.\end{eqnarray}$$

It follows from (3.1) that

(3.2) $$\begin{eqnarray}x^{\ast }\in \unicode[STIX]{x2202}_{c}I_{K}(x)\quad \Longleftrightarrow \quad x\in K,\quad \langle x^{\ast },x-v\rangle \geq 0\quad \text{for all}~v\in K.\end{eqnarray}$$

For detailed discussion of the properties of the convex functions, including the convex subdifferential, we refer the reader to Ekeland and Temam (Reference Ekeland and Temam1976) and Kurdila and Zabarankin (Reference Kurdila and Zabarankin2005), for instance.

Clarke subdifferential.

We now review the notion of the Clarke subdifferential for a locally Lipschitz function. First, we recall that a function $j:X\rightarrow \mathbb{R}$ is said to be locally Lipschitz if, for every $x\in X$ , there exists a neighbourhood of $x$ , denoted by $U_{x}$ , and a $U_{x}$ -dependent constant $L_{x}>0$ such that

$$\begin{eqnarray}|j(y)-j(z)|\leq L_{x}\Vert y-z\Vert _{X}\quad \text{for all}~y,z\in U_{x}.\end{eqnarray}$$

We also recall that a convex continuous function $j:X\rightarrow \mathbb{R}$ is locally Lipschitz.

Definition 3.9. Let $j:X\rightarrow \mathbb{R}$ be a locally Lipschitz function. The generalized (Clarke) directional derivative of $j$ at a point $x\in X$ in a direction $v\in X$ is defined by

(3.3) $$\begin{eqnarray}j^{0}(x;v)=\limsup _{y\rightarrow x,\,\unicode[STIX]{x1D706}\downarrow 0}\displaystyle \frac{j(y+\unicode[STIX]{x1D706}v)-j(y)}{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

The generalized gradient (subdifferential) of $j$ at $x$ is a subset of the dual space $X^{\ast }$ given by

(3.4) $$\begin{eqnarray}\unicode[STIX]{x2202}j(x)=\{\unicode[STIX]{x1D709}\in X^{\ast }\mid j^{0}(x;v)\geq \langle \unicode[STIX]{x1D709},v\rangle ~\text{for all}~v\in X\}.\end{eqnarray}$$

A locally Lipschitz function $j$ is said to be regular (in the sense of Clarke) at the point $x\in X$ if, for all $v\in X$ , the one-sided directional derivative $j^{\prime }(x;v)$ exists and $j^{0}(x;v)=j^{\prime }(x;v)$ .

We now follow Clarke (Reference Clarke1983) and recall the following properties of the generalized directional derivative and the generalized gradient.

Proposition 3.10. Assume that $j:X\rightarrow \mathbb{R}$ is a locally Lipschitz function. Then the following statements are valid.

1. (i) For every $x\in X$ , the function $X\ni v\mapsto j^{0}(x;v)\in \mathbb{R}$ is positively homogeneous and subadditive, that is, $j^{0}(x;\unicode[STIX]{x1D706}v)=\unicode[STIX]{x1D706}j^{0}(x;v)$ for all $\unicode[STIX]{x1D706}\geq 0$ , $v\in X$ and $j^{0}(x;v_{1}+v_{2})\leq j^{0}(x;v_{1})+j^{0}(x;v_{2})$ for all $v_{1}$ , $v_{2}\in X$ , respectively.

2. (ii) For every $v\in X$ , we have $j^{0}(x;v)=\max \{\langle \unicode[STIX]{x1D709},v\rangle \mid \unicode[STIX]{x1D709}\in \unicode[STIX]{x2202}j(x)\}$ .

3. (iii) The function $X\times X\ni (x,v)\mapsto j^{0}(x;v)\in \mathbb{R}$ is upper semicontinuous, that is, for all $x$ , $v\in X$ , $\{x_{n}\}$ , $\{v_{n}\}\subset X$ such that $x_{n}\rightarrow x$ and $v_{n}\rightarrow v$ in $X$ , we have $\limsup j^{0}(x_{n};v_{n})\leq j^{0}(x;v)$ .

4. (iv) For every $x\in X$ , the generalized gradient $\unicode[STIX]{x2202}j(x)$ is a non-empty, convex, and weakly $^{\,\ast }$ compact subset of $X^{\ast }$ .

5. (v) The graph of the generalized gradient $\unicode[STIX]{x2202}j$ is closed in $X\times X_{w^{\ast }}^{\ast }$ topology, that is, if $\{x_{n}\}\subset X$ and $\{\unicode[STIX]{x1D709}_{n}\}\subset X^{\ast }$ are sequences such that $\unicode[STIX]{x1D709}_{n}\in \unicode[STIX]{x2202}j(x_{n})$ and $x_{n}\rightarrow x$ in $X$ , $\unicode[STIX]{x1D709}_{n}\rightarrow \unicode[STIX]{x1D709}$ weakly $^{\ast }$ in $X^{\ast }$ , then $\unicode[STIX]{x1D709}\in \unicode[STIX]{x2202}j(x)$ .

6. (vi) If $j:X\rightarrow \mathbb{R}$ is convex, then the subdifferential in the sense of Clarke $\unicode[STIX]{x2202}j(x)$ at any $x\in X$ coincides with the convex subdifferential $\unicode[STIX]{x2202}_{c}j(x)$ .

Proposition 3.11. Let $j,j_{1},j_{2}:X\rightarrow \mathbb{R}$ be locally Lipschitz functions. Then we have the following.

1. (i) Scalar multiples. The equality $\unicode[STIX]{x2202}(\unicode[STIX]{x1D706}j)(x)=\unicode[STIX]{x1D706}\unicode[STIX]{x2202}j(x)$ holds for all $\unicode[STIX]{x1D706}\in \mathbb{R}$ and all $x\in X$ .

2. (ii) Sum rules. The inclusion

   (3.5) $$\begin{eqnarray}\unicode[STIX]{x2202}(j_{1}+j_{2})(x)\subseteq \unicode[STIX]{x2202}j_{1}(x)+\unicode[STIX]{x2202}j_{2}(x)\end{eqnarray}$$
   holds for all $x\in X$ or, equivalently,
   (3.6) $$\begin{eqnarray}(j_{1}+j_{2})^{0}(x;v)\leq j_{1}^{0}(x;v)+j_{2}^{0}(x;v)~~\end{eqnarray}$$
   for all $x\in X$ , $v\in X$ .

3. (iii) If $j_{1}$ , $j_{2}$ are regular at $x\in X$ , then $j_{1}+j_{2}$ is regular at $x\in X$ and we have equalities in (3.5) and (3.6).

A proof of the following result can be found in Lemma 4.2 of Migórski, Ochal and Sofonea (Reference Migórski, Ochal and Sofonea2010) and follows from the chain rule for the generalized gradient.

Proposition 3.12. Let $X$ and $Y$ be Banach spaces, let $\unicode[STIX]{x1D711}:Y\rightarrow \mathbb{R}$ be locally Lipschitz and let $T:X\rightarrow Y$ be given by $Tx=Ax+y$ for $x\in X$ , where $A\in {\mathcal{L}}(X,Y)$ and $y\in Y$ is fixed. Then the function $j:X\rightarrow \mathbb{R}$ defined by $j(x)=\unicode[STIX]{x1D711}(Tx)$ is locally Lipschitz and

1. (i) $j^{0}(x;v)\leq \unicode[STIX]{x1D711}^{0}(Tx;Av)$ for all $x$ , $v\in X$ ,

2. (ii) $\unicode[STIX]{x2202}j(x)\subseteq A^{\ast }\unicode[STIX]{x2202}\unicode[STIX]{x1D711}(Tx)$ for all $x\in X$ ,

where $A^{\ast }\in {\mathcal{L}}(Y^{\ast },X^{\ast })$ is the adjoint operator of $A$ . Moreover, if $\unicode[STIX]{x1D711}$ (or $-\unicode[STIX]{x1D711}$ ) is regular, then $j$ (or $-j$ ) is regular and in (i) and (ii) we have equalities. These equalities are also true if instead of the regularity condition, we assume that $A$ is surjective.

For detailed discussion of the properties of the Clarke subdifferential, we refer the reader to Clarke (Reference Clarke1975, Reference Clarke1983) and Denkowski, Migórski and Papageorgiou (Reference Denkowski, Migórski and Papageorgiou2003a , Reference Denkowski, Migórski and Papageorgiou2003b ).

3.2 History-dependent operators

In the study of Problems 2.8 and 2.10, we will use the notion of a history-dependent operator. In contact mechanics, a history-dependent operator appears either in the constitutive law or in the contact boundary conditions. Such operators describe various memory effects. Abstract classes of quasivariational inequalities with history-dependent operators were considered in Sofonea and Matei (Reference Sofonea and Matei2011) and Sofonea and Xiao (Reference Sofonea and Xiao2016), where existence, uniqueness and regularity results were proved. A survey of some recent results for history-dependent variational–hemivariational inequalities can be found in Sofonea and Migórski (Reference Sofonea and Migórski2018). There, the abstract well-posedness results are applied to various examples arising in contact mechanics. In this subsection we assume that $X$ and $Y$ are normed spaces endowed with the norms $\Vert \cdot \Vert _{X}$ and $\Vert \cdot \Vert _{Y}$ , respectively. Let $I=[0,T]$ for some $T>0$ or $I=\mathbb{R}_{+}$ be the time interval of interest.

Definition 3.13. An operator ${\mathcal{S}}:C(I;X)\rightarrow C(I;Y)$ is called a history-dependent operator if, for any compact subset $I_{0}\subset I$ , there exists a constant $L_{I_{0}}>0$ such that

(3.7) $$\begin{eqnarray}\displaystyle \Vert {\mathcal{S}}u_{1}(t)-{\mathcal{S}}u_{2}(t)\Vert _{Y} & \leq & \displaystyle L_{I_{0}}\int _{0}^{t}\Vert u_{1}(s)-u_{2}(s)\Vert _{X}\,\text{d}s\nonumber\\ \displaystyle & & \displaystyle \qquad \text{for all}~u_{1},u_{2}\in C(I;X),\,t\in I_{0}.\end{eqnarray}$$

Similarly, an operator ${\mathcal{S}}:L^{2}(I;X)\rightarrow L^{2}(I;Y)$ is called a history-dependent operator if, for any compact subset $I_{0}\subset I$ , there exists a constant $L_{I_{0}}>0$ such that

(3.8) $$\begin{eqnarray}\displaystyle \Vert {\mathcal{S}}u_{1}(t)-{\mathcal{S}}u_{2}(t)\Vert _{Y} & \leq & \displaystyle L_{I_{0}}\int _{0}^{t}\Vert u_{1}(s)-u_{2}(s)\Vert _{X}\,\text{d}s\nonumber\\ \displaystyle & & \displaystyle \qquad \text{for all}~u_{1},u_{2}\in L^{2}(I;X),\text{ a.e. }t\in I_{0}.\end{eqnarray}$$

Note that here and below, when no confusion arises, we use the shorthand notation ${\mathcal{S}}u(t)$ to represent the value of the function ${\mathcal{S}}u$ at a point $t\in I$ , i.e.  ${\mathcal{S}}u(t)=({\mathcal{S}}u)(t)$ . We make some comments regarding Definition 3.13 for ${\mathcal{S}}:C(I;X)\rightarrow C(I;Y)$ ; similar comments can be stated for ${\mathcal{S}}:L^{2}(I;X)\rightarrow L^{2}(I;Y)$ .

Examples of operators satisfying condition (3.7) are given next.

Example 3.15. Let $u_{0}\in X$ and $G:X\rightarrow Y$ be a Lipschitz continuous operator, that is, an operator which satisfies the inequality

$$\begin{eqnarray}\Vert Gu_{1}-Gu_{2}\Vert _{Y}\leq L_{G}\Vert u_{1}-u_{2}\Vert _{X}\quad \text{for all}~u_{1},u_{2}\in X\end{eqnarray}$$

with some constant $L_{G}>0$ . Define an operator ${\mathcal{S}}:C(I;X)\rightarrow C(I;Y)$ by

(3.11) $$\begin{eqnarray}{\mathcal{S}}u(t)=G\biggl(\int _{0}^{t}\,u(s)\,\text{d}s+u_{0}\biggr)\quad \text{for all}~u\in C(I;X),\,t\in I.\end{eqnarray}$$

Then, for $u_{1},u_{2}\in C(I;X)$ , we have

$$\begin{eqnarray}\Vert {\mathcal{S}}u_{1}(t)-{\mathcal{S}}u_{2}(t)\Vert _{Y}\leq L_{G}\int _{0}^{t}\Vert u_{1}(s)-u_{2}(s)\Vert _{X}\,\text{d}s.\end{eqnarray}$$

Thus, condition (3.7) holds with $L_{I_{0}}=L_{G}$ for any compact subset $I_{0}\subset I$ and ${\mathcal{S}}$ is a history-dependent operator. In particular when $X=Y$ and $G$ is the identity operator on $X$ , (3.11) reduces to

(3.12) $$\begin{eqnarray}{\mathcal{S}}u(t)=\int _{0}^{t}u(s)\,\text{d}s+u_{0}\quad \text{for all}~u\in C(I;X),\,t\in I.\end{eqnarray}$$

The operator ${\mathcal{S}}:\!C(I;X)\rightarrow C(I;X)$ defined by (3.12) is a history-dependent operator.

Example 3.16. Let $G\in C(I;{\mathcal{L}}(X,Y))$ and ${\mathcal{S}}:C(I;X)\rightarrow C(I;Y)$ be a Volterra operator given by

(3.13) $$\begin{eqnarray}{\mathcal{S}}u(t)=\int _{0}^{t}G(t-s)\,u(s)\,\text{d}s,\quad u\in C(I;X),\,t\in I.\end{eqnarray}$$

Then for any compact set $I_{0}\subset I$ , inequality (3.7) holds with

$$\begin{eqnarray}L_{I_{0}}=\Vert G\Vert _{C(I_{0};{\mathcal{L}}(X,Y))}=\max _{s\in I_{0}}\Vert G(s)\Vert _{{\mathcal{L}}(X,Y)}.\end{eqnarray}$$

This shows that the operator ${\mathcal{S}}$ defined by (3.13) is a history-dependent operator.

3.3 A class of evolutionary inclusions

We now introduce an abstract result which will be used in the study of the dynamic Problem 2.10. The functional framework is as follows. Let $V\subset H\subset V^{\ast }$ be an evolution triple of spaces, that is, $V$ is a separable, reflexive Banach space, $H$ is a separable Hilbert space, the embedding $V\subset H$ is continuous and $V$ is dense in $H$ . We use $\langle \cdot ,\cdot \rangle$ for the duality pairing between $V^{\ast }$ and $V$ . Given $0<T<+\infty$ , we introduce the function spaces

$$\begin{eqnarray}{\mathcal{V}}=L^{2}(0,T;V)\quad \text{and}\quad {\mathcal{W}}=\{w\in {\mathcal{V}}\mid {\dot{w}}\in {\mathcal{V}}^{\ast }\}.\end{eqnarray}$$

We identify ${\mathcal{H}}=L^{2}(0,T;H)$ with its dual to obtain the following continuous embeddings ${\mathcal{W}}\subset {\mathcal{V}}\subset {\mathcal{H}}\subset {\mathcal{V}}^{\ast }$ . The duality pairing between ${\mathcal{V}}^{\ast }$ and ${\mathcal{V}}$ is denoted by

$$\begin{eqnarray}\langle w,v\rangle _{{\mathcal{V}}^{\ast }\times {\mathcal{V}}}=\int _{0}^{T}\langle w(t),v(t)\rangle \,\text{d}t\quad \text{for}~w\in {\mathcal{V}}^{\ast },\,v\in {\mathcal{V}}.\end{eqnarray}$$

Now consider an operator $A:(0,T)\times V\rightarrow V^{\ast }$ and a function $\unicode[STIX]{x1D713}:(0,T)\times V\rightarrow \mathbb{R}$ , assumed to be locally Lipschitz with respect to its second argument. We let $\unicode[STIX]{x2202}\unicode[STIX]{x1D713}$ denote the Clarke generalized gradient of $\unicode[STIX]{x1D713}$ with respect to its second argument. Given $f:(0,T)\rightarrow V^{\ast }$ and $w_{0}\in V$ , we consider the following evolutionary inclusion.

Problem 3.17. Find $w\in {\mathcal{W}}$ such that

$$\begin{eqnarray}\displaystyle & & \displaystyle {\dot{w}}(t)+A(t,w(t))+\unicode[STIX]{x2202}\unicode[STIX]{x1D713}(t,w(t))\ni f(t)~\text{for a.e.}~t\in (0,T),\nonumber\\ \displaystyle & & \displaystyle w(0)=w_{0}.\nonumber\end{eqnarray}$$

In the study of this inclusion problem, we impose the following hypotheses on the data.

(3.14) $$\begin{eqnarray}\displaystyle & & \displaystyle \left.\begin{array}{@{}ll@{}}A: & (0,T)\times V\rightarrow V^{\ast }\text{is such that:}\\[3.0pt] \text{(a) } & A(\cdot ,v)\text{ is measurable on }(0,T)\text{ for all }v\in V;\\[3.0pt] \text{(b) } & A(t,\cdot )\text{ is demicontinuous on }V\text{ for a.e.}~t\in (0,T),\\ & \text{i.e.}~u_{n}\rightarrow u\text{ in }V\Longrightarrow Au_{n}\rightharpoonup Au\text{ in }V^{\ast };\\[3.0pt] \text{(c) } & \Vert A(t,v)\Vert _{V^{\ast }}\leq a_{0}(t)+a_{1}\Vert v\Vert _{V}\text{ for all }v\in V,\\ & \text{a.e.}~t\in (0,T)\text{ with }a_{0}\in L^{2}(0,T),a_{0}\geq 0\text{ and }a_{1}\geq 0;~\\[3.0pt] \text{(d) } & \text{there is a constant }m_{A}>0\text{ such that}\\ & \langle A(t,v_{1})-A(t,v_{2}),v_{1}-v_{2}\rangle \geq m_{A}\Vert v_{1}-v_{2}\Vert _{V}^{2}\\ & \text{for all }v_{1},v_{2}\in V,\text{a.e.}~t\in (0,T).\end{array}\right\}\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \left.\begin{array}{@{}ll@{}}\unicode[STIX]{x1D713}: & (0,T)\times V\rightarrow \mathbb{R}\text{ is such that:}\\ \text{(a) } & \unicode[STIX]{x1D713}(\cdot ,v)\text{ is measurable on }(0,T)\text{ for all }v\in V;\\[3.0pt] \text{(b) } & \unicode[STIX]{x1D713}(t,\cdot )\text{ is locally Lipschitz on }V\text{ for a.e.}~t\in (0,T);\\[3.0pt] \text{(c) } & \Vert \unicode[STIX]{x2202}\unicode[STIX]{x1D713}(t,v)\Vert _{V^{\ast }}\leq c_{0}(t)+c_{1}\Vert v\Vert _{V}\text{ for all }v\in V,\\ & \text{a.e.}~t\in (0,T)\text{ with }c_{0}\in L^{2}(0,T),c_{0}\geq 0,c_{1}\geq 0;\\[3.0pt] \text{(d) } & \text{there exists a constant }m_{\unicode[STIX]{x1D713}}\geq 0\text{ such that}\\ & \langle z_{1}-z_{2},v_{1}-v_{2}\rangle \geq -m_{\unicode[STIX]{x1D713}}\Vert v_{1}-v_{2}\Vert _{V}^{2}\\ & \text{for all }z_{i}\in \unicode[STIX]{x2202}\unicode[STIX]{x1D713}(t,v_{i}),\,z_{i}\in V^{\ast },\,v_{i}\in V,\,i=1,2,\text{a.e.}~t\in (0,T).\end{array}\right\}\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle & & \displaystyle \,f\in {\mathcal{V}}^{\ast },~w_{0}\in V.\end{eqnarray}$$

(3.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \max \,\{m_{\unicode[STIX]{x1D713}},2\sqrt{2}\,c_{1}\}<m_{A}.\end{eqnarray}$$

We say a function $w\in {\mathcal{W}}$ is a solution of Problem 3.17 if there exists $w^{\ast }\in {\mathcal{V}}^{\ast }$ such that

$$\begin{eqnarray}\displaystyle & & \displaystyle {\dot{w}}(t)+A(t,w(t))+w^{\ast }(t)=f(t)~\text{for a.e.}~t\in (0,T),\nonumber\\ \displaystyle & & \displaystyle w^{\ast }(t)\in \unicode[STIX]{x2202}\unicode[STIX]{x1D713}(t,w(t))~\text{for a.e.}~t\in (0,T),\nonumber\\ \displaystyle & & \displaystyle w(0)=w_{0}.\nonumber\end{eqnarray}$$

We have the following existence and uniqueness result.

A proof of Theorem  3.18 can be found in Sofonea and Migórski (Reference Sofonea and Migórski2018, p. 183). The existence part is based on a surjectivity result with maximal monotone operators in reflexive Banach spaces. The uniqueness part follows from standard arguments and is based on the smallness assumption (3.17).

3.4 Fixed-point theorems

We will apply Banach’s fixed-point theorem in solution existence proofs of various problems: see Zeidler (Reference Zeidler1985, Section 1.1) or Atkinson and Han (Reference Atkinson and Han2009, Section 5.1).

Theorem 3.19. Let $K$ be a non-empty closed set in a Banach space $X$ , and let $\unicode[STIX]{x1D6EC}:K\rightarrow K$ be a contractive mapping, that is, for some constant $\unicode[STIX]{x1D6FC}\in [0,1)$ ,

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6EC}u-\unicode[STIX]{x1D6EC}v\Vert _{X}\leq \unicode[STIX]{x1D6FC}\,\Vert u-v\Vert _{X}\quad \text{for all}~u,v\in K.\end{eqnarray}$$

Then $\unicode[STIX]{x1D6EC}:K\rightarrow K$ has a unique fixed point, that is, there exists a unique $u^{\ast }\in K$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}u^{\ast }=u^{\ast }.\end{eqnarray}$$

History-dependent operators have important fixed-point properties which are useful in proving the solvability of various classes of nonlinear equations and inequalities. The following result is proved in Sofonea, Avramescu and Matei (Reference Sofonea, Avramescu and Matei2008).

In case of a history-dependent operator on $L^{2}(0,T;X)$ , we have a similar result from Sofonea and Migórski (Reference Sofonea and Migórski2018, Theorem 67).

3.5 Some inequalities

We will use the modified Cauchy–Schwarz inequality

(3.18) $$\begin{eqnarray}a\,b\leq \unicode[STIX]{x1D716}\,a^{2}+c\,b^{2}\quad \text{for all}~a,b\in \mathbb{R},\end{eqnarray}$$

where $\unicode[STIX]{x1D716}>0$ is an arbitrary positive number and the constant $c>0$ depends on $\unicode[STIX]{x1D716}$ ; indeed, we may simply take $c=1/(4\,\unicode[STIX]{x1D716})$ .

In the well-posedness analysis of contact problems, we will need the Gronwall inequality: see Han and Sofonea (Reference Han and Sofonea2002, Section 7.4).

Lemma 3.22. Let $a,\,b\in \mathbb{R}$ with $a<b$ and assume $f,g:[a,b]\rightarrow \mathbb{R}$ are continuous functions satisfying

(3.19) $$\begin{eqnarray}f(t)\leq g(t)+c\int _{a}^{t}f(s)\,\text{d}s\quad \text{for all}~t\in [a,b],\end{eqnarray}$$

where $c>0$ is a constant. Then

(3.20) $$\begin{eqnarray}f(t)\leq g(t)+c\int _{a}^{t}g(s)\,\text{e}^{c\,(t-s)}\,\text{d}s\quad \text{for all}~t\in [a,b].\end{eqnarray}$$

Moreover, if $g$ is non-decreasing, then

(3.21) $$\begin{eqnarray}f(t)\leq g(t)\,\text{e}^{c\,(t-a)}\quad \text{for all}~t\in [a,b].\end{eqnarray}$$

In error analysis for numerical solutions of the contact problems, we will need discrete versions of Gronwall’s inequality. For a fixed $T$ and a positive integer $N$ , let $k=T/N$ .

Lemma 3.23. Assume $\{g_{n}\}_{n=1}^{N}$ and $\{e_{n}\}_{n=1}^{N}$ are two sequences of non-negative numbers satisfying

$$\begin{eqnarray}e_{n}\leq c\,g_{n}+c\,k\mathop{\sum }_{i=1}^{n-1}e_{i},\quad n=1,\ldots ,N,\end{eqnarray}$$

for a constant $c>0$ . Then, for a possibly different constant $c>0$ ,

(3.22) $$\begin{eqnarray}\max _{1\leq n\leq N}e_{n}\leq c\,\max _{1\leq n\leq N}g_{n}.\end{eqnarray}$$

Lemma 3.24. Assume $\{g_{n}\}_{n=1}^{N}$ and $\{e_{n}\}_{n=1}^{N}$ are two sequences of non-negative numbers satisfying

$$\begin{eqnarray}e_{n}\leq c\,g_{n}+c\,k\mathop{\sum }_{i=1}^{n}e_{i},\quad n=1,\ldots ,N,\end{eqnarray}$$

for a constant $c>0$ . Then, if $k$ is sufficiently small,

(3.23) $$\begin{eqnarray}\max _{1\leq n\leq N}e_{n}\leq c\,\max _{1\leq n\leq N}g_{n}.\end{eqnarray}$$

More general versions of these results can be found in Han and Sofonea (Reference Han and Sofonea2002, Section 7.4).

4.1 Solution existence and uniqueness

The unique solvability of Problem 4.1 is provided by the following result.

Theorem 4.2. Assume (4.1)–(4.5) and the smallness condition

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}+\unicode[STIX]{x1D6FC}_{j}<m_{A}.\end{eqnarray}$$

Then, Problem 4.1 has a unique solution $u\in K$ .

A version of Theorem 4.2 was proved in Migórski, Ochal and Sofonea (Reference Migórski, Ochal and Sofonea2017), under additional assumptions. The proof of the current version of the result is based on the same arguments used in the above-mentioned paper. Nevertheless, we present it below for the convenience of the reader.

Proof. The proof consists of five steps.

(i) We show the coercivity of the operator $A$ . We first prove that for all $u_{0}\in K$ there exist $\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE}\in \mathbb{R}$ (which depend on $u_{0}$ ) such that

(4.8) $$\begin{eqnarray}\langle Av,v-u_{0}\rangle \geq m_{A}\Vert v\Vert _{X}^{2}-\unicode[STIX]{x1D6FD}\,\Vert v\Vert _{X}-\unicode[STIX]{x1D6FE}\quad \text{for all}~v\in X.\end{eqnarray}$$

Indeed, let $u_{0}\in K$ and $v\in V$ . We write

$$\begin{eqnarray}\displaystyle \langle Av,v-u_{0}\rangle & = & \displaystyle \langle Av-Au_{0},v-u_{0}\rangle +\langle Au_{0},v-u_{0}\rangle \nonumber\\ \displaystyle & \geq & \displaystyle m_{A}\Vert v-u_{0}\Vert _{X}^{2}-\Vert Au_{0}\Vert _{X^{\ast }}\Vert v-u_{0}\Vert _{X}.\nonumber\end{eqnarray}$$

Then we use the inequalities

$$\begin{eqnarray}\displaystyle \big|\Vert v\Vert _{X}-\Vert u_{0}\Vert _{X}\big| & \leq & \displaystyle \Vert v-u_{0}\Vert _{X},\nonumber\\ \displaystyle \Vert Au_{0}\Vert _{X^{\ast }}\Vert v-u_{0}\Vert _{X} & \leq & \displaystyle \Vert Au_{0}\Vert _{X^{\ast }}\Vert v\Vert _{X}+\Vert Au_{0}\Vert _{X^{\ast }}\Vert u_{0}\Vert _{X}\nonumber\end{eqnarray}$$

to obtain

$$\begin{eqnarray}\displaystyle \langle Av,v-u_{0}\rangle & \geq & \displaystyle m_{A}(\Vert v\Vert _{X}-\Vert u_{0}\Vert _{X})^{2}-\Vert Au_{0}\Vert _{X^{\ast }}\Vert v\Vert _{X}-\Vert Au_{0}\Vert _{X^{\ast }}\Vert u_{0}\Vert _{X}\nonumber\\ \displaystyle & = & \displaystyle vm_{A}\Vert v\Vert _{X}^{2}-(2m_{A}\Vert u_{0}\Vert _{X}+\Vert Au_{0}\Vert _{X^{\ast }})\Vert v\Vert _{X}\nonumber\\ \displaystyle & & \displaystyle +\,m_{A}\Vert u_{0}\Vert _{X}^{2}-\Vert Au_{0}\Vert _{X^{\ast }}\Vert u_{0}\Vert _{X},\nonumber\end{eqnarray}$$

which proves inequality (4.8). This inequality shows the $u_{0}$ -coercivity of the operator $A$ in the sense of Definition 3.1(d).

(ii) We introduce an auxiliary inclusion problem and show its unique solvability. For an arbitrarily fixed element $\unicode[STIX]{x1D702}\in K$ , define a function $\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}:X\rightarrow \mathbb{R}\cup \{+\infty \}$ by

(4.9) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}(v)=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702},v)\quad & \text{if}~v\in K,\\ +\infty \quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

Then consider the following problem: find $u_{\unicode[STIX]{x1D702}}\in X$ such that

(4.10) $$\begin{eqnarray}Au_{\unicode[STIX]{x1D702}}+\unicode[STIX]{x2202}j(u_{\unicode[STIX]{x1D702}})+\unicode[STIX]{x2202}_{c}\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}(u_{\unicode[STIX]{x1D702}})\ni f.\end{eqnarray}$$

Let us apply Theorem 3.5 to prove that this inclusion has a solution. We fix an element $u_{0}\in K$ and introduce multivalued operators $T_{1}$ , $T_{2}:X\rightarrow 2^{X^{\ast }}$ defined by

$$\begin{eqnarray}T_{1}v=Av+\unicode[STIX]{x2202}j(v),\quad T_{2}v=\unicode[STIX]{x2202}_{c}\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}(v)\quad \text{for}~v\in X.\end{eqnarray}$$

We check that the operators $T_{1}$ and $T_{2}$ satisfy conditions (a)–(c) in Theorem 3.5.

First, we note that by Proposition 3.3, under the assumptions (4.2), the operator $A$ is pseudomonotone. Therefore, the boundedness of the operator $T_{1}$ follows easily from the boundedness of $A$ and the growth condition (4.4(b)) on $\unicode[STIX]{x2202}j$ .

Next, we show that $T_{1}$ is generalized pseudomonotone. We use hypotheses (4.2(b)), (4.4(c)), (4.7) and inequality (4.6) to see that the operator $T_{1}$ is strongly monotone, that is,

$$\begin{eqnarray}\langle T_{1}v_{1}-T_{1}v_{2},v_{1}-v_{2}\rangle \geq (m_{A}-\unicode[STIX]{x1D6FC}_{j})\,\Vert v_{1}-v_{2}\Vert _{X}^{2}\quad \text{for all}~v_{1},v_{2}\in X.\end{eqnarray}$$

Assume now that $u_{n}\in X$ , $u_{n}\rightarrow u$ weakly in $X$ , $u_{n}^{\ast }\in T_{1}u_{n}$ , $u_{n}^{\ast }\rightarrow u^{\ast }$ weakly in $X^{\ast }$ and $\limsup \,\langle u_{n}^{\ast },u_{n}-u\rangle \leq 0$ . We shall prove that $u^{\ast }\in T_{1}u$ and $\langle u_{n}^{\ast },u_{n}\rangle \rightarrow \langle u^{\ast },u\rangle$ . Using the strong monotonicity of $T_{1}$ , from the relation

$$\begin{eqnarray}(m_{A}-\unicode[STIX]{x1D6FC}_{j})\Vert u_{n}-u\Vert _{X}^{2}\leq \langle u_{n}^{\ast },u_{n}-u\rangle -\langle T_{1}u,u_{n}-u\rangle ,\end{eqnarray}$$

we deduce that $u_{n}\rightarrow u$ in $X$ . From $u_{n}^{\ast }\in T_{1}u_{n}$ , we have $u_{n}^{\ast }=w_{n}+z_{n}$ with $w_{n}=Au_{n}$ and $z_{n}\in \unicode[STIX]{x2202}j(u_{n})$ . Since $A$ and $\unicode[STIX]{x2202}j$ are bounded operators, by passing to a subsequence, if necessary, we may assume that $w_{n}\rightarrow w$ and $z_{n}\rightarrow z$ both weakly in $X^{\ast }$ with some $w$ , $z\in X^{\ast }$ . Therefore, from $u_{n}^{\ast }=w_{n}+z_{n}$ , we find that $u^{\ast }=w+z$ . Exploiting the equivalent condition for the pseudomonotonicity of $A$ in Remark 3.2, we have $Au_{n}\rightarrow Au$ weakly in $X^{\ast }$ , which gives $w=Au$ . On the other hand, by Proposition 3.10(v) it follows that $X\ni v\mapsto \unicode[STIX]{x2202}j(v)\in 2^{X^{\ast }}$ has a closed graph with respect to the strong topology in $X$ and weak topology in $X^{\ast }$ . Exploiting this property we infer that $z\in \unicode[STIX]{x2202}j(u)$ . Hence, $u^{\ast }=w+z\in Au+\unicode[STIX]{x2202}j(u)=T_{1}u$ . Since $u_{n}^{\ast }\rightarrow u^{\ast }$ weakly in $X^{\ast }$ and $u_{n}\rightarrow u$ in $X$ , it is clear that $\langle u_{n}^{\ast },u_{n}\rangle \rightarrow \langle u^{\ast },u\rangle$ . This shows that $T_{1}$ is generalized pseudomonotone.

In order to establish the $u_{0}$ -coercivity of $T_{1}$ , we use inequality (4.8), hypothesis (4.4(c)), inequality (4.6) and the following inequality, which is a consequence of (4.4(b)):

$$\begin{eqnarray}|\langle \unicode[STIX]{x2202}j(u_{0}),v-u_{0}\rangle |\leq (c_{0}+c_{1}\Vert u_{0}\Vert _{X})\Vert v-u_{0}\Vert _{X}.\end{eqnarray}$$

We have

$$\begin{eqnarray}\displaystyle \langle T_{1}v,v-u_{0}\rangle & = & \displaystyle \langle Av,v-u_{0}\rangle +\langle \unicode[STIX]{x2202}j(v)-\unicode[STIX]{x2202}j(u_{0}),v-u_{0}\rangle +\langle \unicode[STIX]{x2202}j(u_{0}),v-u_{0}\rangle \nonumber\\ \displaystyle & \geq & \displaystyle m_{A}\Vert v\Vert _{X}^{2}-\unicode[STIX]{x1D6FD}\,\Vert v\Vert _{X}-\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FC}_{j}\Vert v-u_{0}\Vert _{X}^{2}\nonumber\\ \displaystyle & & \displaystyle -\,(c_{0}+c_{1}\Vert u_{0}\Vert _{X})\Vert v-u_{0}\Vert _{X}\nonumber\\ \displaystyle & \geq & \displaystyle (m_{A}-\unicode[STIX]{x1D6FC}_{j})\Vert v\Vert _{X}^{2}-\Vert v\Vert _{X}(\unicode[STIX]{x1D6FD}+2\unicode[STIX]{x1D6FC}_{j}\Vert u_{0}\Vert _{X}+c_{0}+c_{1}\Vert u_{0}\Vert _{X})\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FC}_{j}\Vert u_{0}\Vert _{X}^{2}-(c_{0}+c_{1}\Vert u_{0}\Vert _{X})\Vert u_{0}\Vert _{X}.\nonumber\end{eqnarray}$$

The $u_{0}$ -coercivity of $T_{1}$ follows now from hypothesis (4.7).

We conclude from the above that $T_{1}$ is bounded, generalized pseudomonotone and $u_{0}$ -coercive and therefore it satisfies condition (a) of Theorem 3.5.

Next, using Proposition 3.10(iv) we see that for all $v\in X$ the set $Av+\unicode[STIX]{x2202}j(v)$ is non-empty, closed and convex in $X^{\ast }$ . Therefore, condition (b) of Theorem 3.5 holds.

From hypothesis (4.3(a)) and the definition of $\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}$ , we know that the function $\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}$ is proper, convex and lower semicontinuous with $\text{dom}\,\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}=K$ . By Theorem 3.7, the operator $T_{2}=\unicode[STIX]{x2202}_{c}\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}:X\rightarrow 2^{X^{\ast }}$ is maximal monotone with ${\mathcal{D}}(\unicode[STIX]{x2202}_{c}\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}})=K$ . This shows that condition (c) of Theorem 3.5 is satisfied.

Summarizing, we can apply Theorem 3.5 to deduce that there exists a solution $u_{\unicode[STIX]{x1D702}}\in X$ to the inclusion problem (4.10).

(iii) We introduce an auxiliary variational–hemivariational inequality and prove that it has a unique solution. Fix an element $\unicode[STIX]{x1D702}\in K$ and consider the auxiliary problem of finding an element $u_{\unicode[STIX]{x1D702}}\in K$ for which

(4.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle Au_{\unicode[STIX]{x1D702}},v-u_{\unicode[STIX]{x1D702}}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702},v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702},u_{\unicode[STIX]{x1D702}})+j^{0}(u_{\unicode[STIX]{x1D702}};v-u_{\unicode[STIX]{x1D702}})\nonumber\\ \displaystyle & & \displaystyle \quad \geq \langle f,v-u_{\unicode[STIX]{x1D702}}\rangle \quad \text{for all}~v\in K.\end{eqnarray}$$

By making use of definition (4.9), we see that (4.11) is equivalent to the problem of finding $u_{\unicode[STIX]{x1D702}}\in X$ such that

(4.12) $$\begin{eqnarray}\langle Au_{\unicode[STIX]{x1D702}},v-u_{\unicode[STIX]{x1D702}}\rangle +\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}(v)-\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}(u_{\unicode[STIX]{x1D702}})+j^{0}(u_{\unicode[STIX]{x1D702}};v-u_{\unicode[STIX]{x1D702}})\geq \langle f,v-u_{\unicode[STIX]{x1D702}}\rangle \quad \text{for all}~v\in X.\end{eqnarray}$$

We claim that every solution of inclusion (4.10) is a solution to problem (4.12). Indeed, let $u_{\unicode[STIX]{x1D702}}\in X$ be such that

(4.13) $$\begin{eqnarray}Au_{\unicode[STIX]{x1D702}}+\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D702}}+\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D702}}=f\end{eqnarray}$$

with $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D702}}\in \unicode[STIX]{x2202}_{c}\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}(u_{\unicode[STIX]{x1D702}})$ and $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D702}}\in \unicode[STIX]{x2202}j(u_{\unicode[STIX]{x1D702}})$ . We have

$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D709}_{\unicode[STIX]{x1D702}},v-u_{\unicode[STIX]{x1D702}}\rangle & \leq & \displaystyle \widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}(v)-\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}(u_{\unicode[STIX]{x1D702}})\quad \text{for all}~v\in X,\nonumber\\ \displaystyle \langle \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D702}},v\rangle & \leq & \displaystyle j^{0}(u_{\unicode[STIX]{x1D702}};v)\quad \text{for all}~v\in X.\nonumber\end{eqnarray}$$

Combining (4.13) with the last two inequalities, we obtain

$$\begin{eqnarray}\langle Au_{\unicode[STIX]{x1D702}},v-u_{\unicode[STIX]{x1D702}}\rangle +\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}(v)-\widetilde{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D702}}(u_{\unicode[STIX]{x1D702}})+j^{0}(u_{\unicode[STIX]{x1D702}};v-u_{\unicode[STIX]{x1D702}})\geq \langle f,v-u_{\unicode[STIX]{x1D702}}\rangle \quad \text{for all}~v\in X.\end{eqnarray}$$

This implies that $u_{\unicode[STIX]{x1D702}}\in X$ solves problem (4.12) which concludes the proof of the claim.

By step (ii), inequality (4.12) has a solution. From the equivalence of inequalities (4.12) and (4.11), inequality (4.11) also has a solution $u_{\unicode[STIX]{x1D702}}\in K$ . To prove its uniqueness assume that $u_{1},u_{2}\in K$ are two solutions of (4.11), that is,

$$\begin{eqnarray}\displaystyle \langle Au_{1},v-u_{1}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702},v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702},u_{1})+j^{0}(u_{1};v-u_{1}) & \geq & \displaystyle \langle f,v-u_{1}\rangle ,\nonumber\\ \displaystyle \langle Au_{2},v-u_{2}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702},v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702},u_{2})+j^{0}(u_{2};v-u_{2}) & \geq & \displaystyle \langle f,v-u_{2}\rangle ,\nonumber\end{eqnarray}$$

for all $v\in K$ . Taking $v=u_{2}$ in the first inequality, $v=u_{1}$ in the second one, and adding the resulting inequalities, we obtain

$$\begin{eqnarray}\langle Au_{1}-Au_{2},u_{2}-u_{1}\rangle +j^{0}(u_{1};u_{2}-u_{1})+j^{0}(u_{2};u_{1}-u_{2})\geq 0.\end{eqnarray}$$

From the strong monotonicity of $A$ and hypothesis (4.4(c)), we have

$$\begin{eqnarray}(m_{A}-\unicode[STIX]{x1D6FC}_{j})\,\Vert u_{1}-u_{2}\Vert _{X}^{2}\leq 0,\end{eqnarray}$$

which, due to the smallness condition (4.7), implies $u_{1}=u_{2}$ . Thus, (4.11) has a unique solution.

(iv) We introduce an operator and apply the Banach fixed-point argument. Define an operator $\unicode[STIX]{x1D6EC}:K\rightarrow K$ by

(4.14) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D702}=u_{\unicode[STIX]{x1D702}}\quad \text{for}~\unicode[STIX]{x1D702}\in K,\end{eqnarray}$$

where $u_{\unicode[STIX]{x1D702}}\in K$ denotes the unique solution of inequality (4.11). We prove that the operator $\unicode[STIX]{x1D6EC}$ has a unique fixed point. For this purpose, let $\unicode[STIX]{x1D702}_{1},\unicode[STIX]{x1D702}_{2}\in K$ and $u_{1}=u_{\unicode[STIX]{x1D702}_{1}},\,u_{2}=u_{\unicode[STIX]{x1D702}_{2}}\in K$ be the unique solutions of (4.11) corresponding to $\unicode[STIX]{x1D702}_{1}$ , $\unicode[STIX]{x1D702}_{2}$ , respectively. From the inequalities

$$\begin{eqnarray}\displaystyle \langle Au_{1},v-u_{1}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{1},v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{1},u_{1})+j^{0}(u_{1};v-u_{1}) & \geq & \displaystyle \langle f,v-u_{1}\rangle ,\nonumber\\ \displaystyle \langle Au_{2},v-u_{2}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{2},v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{2},u_{2})+j^{0}(u_{2};v-u_{2}) & \geq & \displaystyle \langle f,v-u_{2}\rangle ,\nonumber\end{eqnarray}$$

valid for all $v\in K$ , we have

$$\begin{eqnarray}\displaystyle \langle Au_{1}-Au_{2},u_{1}-u_{2}\rangle & \leq & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{1},u_{2})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{1},u_{1})+\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{2},u_{1})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{2},u_{2})\nonumber\\ \displaystyle & & \displaystyle +\,j^{0}(u_{1};u_{2}-u_{1})+j^{0}(u_{2};u_{1}-u_{2}).\nonumber\end{eqnarray}$$

Use the strong monotonicity of $A$ and hypotheses (4.3(b)), (4.4(c)) to obtain

$$\begin{eqnarray}m_{A}\Vert u_{1}-u_{2}\Vert _{X}^{2}\leq \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert \unicode[STIX]{x1D702}_{1}-\unicode[STIX]{x1D702}_{2}\Vert _{X}\Vert u_{1}-u_{2}\Vert _{X}+\unicode[STIX]{x1D6FC}_{j}\Vert u_{1}-u_{2}\Vert _{X}^{2}.\end{eqnarray}$$

Consequently,

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6EC}\unicode[STIX]{x1D702}_{1}-\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D702}_{2}\Vert _{X}=\Vert u_{1}-u_{2}\Vert _{X}\leq \displaystyle \frac{\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}}{m_{A}-\unicode[STIX]{x1D6FC}_{j}}\Vert \unicode[STIX]{x1D702}_{1}-\unicode[STIX]{x1D702}_{2}\Vert _{X}.\end{eqnarray}$$

From condition (4.7), by applying the Banach contraction principle, we deduce that there exists a unique $\unicode[STIX]{x1D702}^{\ast }\in K$ such that $\unicode[STIX]{x1D702}^{\ast }=\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D702}^{\ast }$ .

(v) To prove the existence part of Theorem 4.2, we write inequality (4.11) for $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}^{\ast }$ and observe that $u_{\unicode[STIX]{x1D702}^{\ast }}=\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D702}^{\ast }=\unicode[STIX]{x1D702}^{\ast }$ . So the function $\unicode[STIX]{x1D702}^{\ast }\in K$ is a solution to Problem 4.1.

The uniqueness of a solution to Problem 4.1 is proved directly. Let $u_{1},u_{2}\in K$ be solutions, that is,

$$\begin{eqnarray}\displaystyle \langle Au_{1},v-u_{1}\rangle +\unicode[STIX]{x1D711}(u_{1},v)-\unicode[STIX]{x1D711}(u_{1},u_{1})+j^{0}(u_{1};v-u_{1}) & \geq & \displaystyle \langle f,v-u_{1}\rangle ,\nonumber\\ \displaystyle \langle Au_{2},v-u_{2}\rangle +\unicode[STIX]{x1D711}(u_{2},v)-\unicode[STIX]{x1D711}(u_{2},u_{2})+j^{0}(u_{2};v-u_{2}) & \geq & \displaystyle \langle f,v-u_{2}\rangle ,\nonumber\end{eqnarray}$$

for all $v\in K$ . From these inequalities, we obtain

$$\begin{eqnarray}\displaystyle \langle Au_{1}-Au_{2},u_{1}-u_{2}\rangle & \leq & \displaystyle \unicode[STIX]{x1D711}(u_{1},u_{2})-\unicode[STIX]{x1D711}(u_{1},u_{1})+\unicode[STIX]{x1D711}(u_{2},u_{1})-\unicode[STIX]{x1D711}(u_{2},u_{2})\nonumber\\ \displaystyle & & \displaystyle +\,j^{0}(u_{1};u_{2}-u_{1})+j^{0}(u_{2};u_{1}-u_{2}).\nonumber\end{eqnarray}$$

Conditions (4.2(b)), (4.3(b)) and (4.4(c)) imply

$$\begin{eqnarray}m_{A}\Vert u_{1}-u_{2}\Vert _{X}^{2}\leq \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert u_{1}-u_{2}\Vert _{X}^{2}+\unicode[STIX]{x1D6FC}_{j}\Vert u_{1}-u_{2}\Vert _{X}^{2}\end{eqnarray}$$

from which, due to the smallness assumption (4.7), it follows that $u_{1}=u_{2}$ . This completes the proof of the theorem.◻

We now follow Han (Reference Han2018) and introduce a variant of Problem 4.1, which is more convenient for the numerical analysis as well as for applications in contact mechanics. Besides the reflexive Banach space $X$ , we need two real Banach spaces $X_{\unicode[STIX]{x1D711}}$ and $X_{j}$ , and two operators $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}:X\rightarrow X_{\unicode[STIX]{x1D711}}$ , $\unicode[STIX]{x1D6FE}_{j}:X\rightarrow X_{j}$ . Moreover, we assume that $\unicode[STIX]{x1D711}:X_{\unicode[STIX]{x1D711}}\times X_{\unicode[STIX]{x1D711}}\rightarrow \mathbb{R}$ , $j:X_{j}\rightarrow \mathbb{R}$ . The variational–hemivariational inequality we consider is stated as follows.

Problem 4.3. Find an element $u\in K$ such that

(4.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle Au,v-u\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u)\nonumber\\ \displaystyle & & \displaystyle \quad +j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}u)\geq \langle f,v-u\rangle \quad \text{for all}~v\in K.\end{eqnarray}$$

For applications in contact mechanics, the functionals $\unicode[STIX]{x1D711}(\cdot ,\cdot )$ and $j(\cdot )$ are integrals over the contact boundary $\unicode[STIX]{x1D6E4}_{3}$ . In such a situation, $X_{\unicode[STIX]{x1D711}}$ and $X_{j}$ can be chosen to be $L^{2}(\unicode[STIX]{x1D6E4}_{3})^{d}$ and/or $L^{2}(\unicode[STIX]{x1D6E4}_{3})$ .

For the analysis of Problem 4.3, we consider the following hypotheses on the data, with constants $c_{\unicode[STIX]{x1D711}}$ , $c_{j}$ , $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}$ , $\unicode[STIX]{x1D6FC}_{j}$ , $c_{0}$ and $c_{1}$ .

(4.16) $$\begin{eqnarray}\displaystyle & & \displaystyle \,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}\in {\mathcal{L}}(X,X_{\unicode[STIX]{x1D711}}),\quad \Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v\Vert _{X_{\unicode[STIX]{x1D711}}}\leq c_{\unicode[STIX]{x1D711}}\Vert v\Vert _{X}\quad \text{for all}~v\in X.\end{eqnarray}$$

(4.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \,\unicode[STIX]{x1D6FE}_{j}\in {\mathcal{L}}(X,X_{j}),\quad \Vert \unicode[STIX]{x1D6FE}_{j}v\Vert _{X_{j}}\leq c_{j}\Vert v\Vert _{X}\quad \text{for all}~v\in X.\end{eqnarray}$$

(4.18) $$\begin{eqnarray}\displaystyle & & \displaystyle \left.\begin{array}{@{}ll@{}}\unicode[STIX]{x1D711}:\, & X_{\unicode[STIX]{x1D711}}\times X_{\unicode[STIX]{x1D711}}\rightarrow \mathbb{R}\text{ is such that:}\\[3.0pt] \text{(a) } & \unicode[STIX]{x1D711}(\unicode[STIX]{x1D702},\cdot ):X_{\unicode[STIX]{x1D711}}\rightarrow \mathbb{R}\text{ is convex and l.s.c. for all }\unicode[STIX]{x1D702}\in X_{\unicode[STIX]{x1D711}};\\[3.0pt] \text{(b) } & \text{there exists }\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}>0\text{ such that}\\ & \unicode[STIX]{x1D711}(z_{1},z_{4})-\unicode[STIX]{x1D711}(z_{1},z_{3})+\unicode[STIX]{x1D711}(z_{2},z_{3})-\unicode[STIX]{x1D711}(z_{2},z_{4})\\ & \leq \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert z_{1}-z_{2}\Vert _{X_{\unicode[STIX]{x1D711}}}\Vert z_{3}-z_{4}\Vert _{X_{\unicode[STIX]{x1D711}}}~\text{for all}~z_{1},z_{2},z_{3},z_{4}\in X_{\unicode[STIX]{x1D711}}.\end{array}\right\}\end{eqnarray}$$

(4.19) $$\begin{eqnarray}\displaystyle & & \displaystyle \left.\begin{array}{@{}ll@{}}j:\, & X_{j}\rightarrow \mathbb{R}\text{ is such that:}\\[3.0pt] \text{(a) } & j\text{ is locally Lipschitz;}\\[3.0pt] \text{(b) } & \Vert \unicode[STIX]{x2202}j(z)\Vert _{X_{j}^{\ast }}\leq c_{0}+c_{1}\Vert z\Vert _{X_{j}}\text{ for all }z\in X_{j}\text{ with }c_{0},c_{1}\geq 0;~\\[3.0pt] \text{(c) } & \text{there exists }\unicode[STIX]{x1D6FC}_{j}>0\text{ such that}\\ & j^{0}(z_{1};z_{2}-z_{1})+j^{0}(z_{2};z_{1}-z_{2})\leq \unicode[STIX]{x1D6FC}_{j}\Vert z_{1}-z_{2}\Vert _{X_{j}}^{2}\\ & \text{for all}~z_{1},z_{2}\in X_{j}.\end{array}\right\}\end{eqnarray}$$

The unique solvability of Problem 4.3 is given by the following result.

Theorem 4.4. Assume (4.1), (4.2), (4.5), (4.16)–(4.19) and

(4.20) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}c_{\unicode[STIX]{x1D711}}^{2}+\unicode[STIX]{x1D6FC}_{j}c_{j}^{2}<m_{A}.\end{eqnarray}$$

Then, Problem 4.3 has a unique solution $u\in K$ .

Proof. We prove this result by applying Theorem 4.2. Define functions $\widetilde{\unicode[STIX]{x1D711}}:X\times X\rightarrow \mathbb{R}$ and $\widetilde{\jmath }:X\rightarrow \mathbb{R}$ by

(4.21) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D711}}(u,v) & = & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v)\quad \text{for all}~u,v\in X,\end{eqnarray}$$

(4.22) $$\begin{eqnarray}\displaystyle \widetilde{\jmath }(v) & = & \displaystyle j(\unicode[STIX]{x1D6FE}_{j}v)\quad \text{for all}~v\in X.\end{eqnarray}$$

Then, using assumption (4.18) on the function $\unicode[STIX]{x1D711}$ and inequality (4.16) it is easy to see that the function $\widetilde{\unicode[STIX]{x1D711}}$ satisfies assumption (4.3) on the space $X$ with constant $\unicode[STIX]{x1D6FC}_{\widetilde{\unicode[STIX]{x1D711}}}=\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}c_{\unicode[STIX]{x1D711}}^{2}$ . On the other hand, from arguments similar to those used in the proof of Lemma 6 in Sofonea and Migórski (Reference Sofonea and Migórski2018) it follows that the function $\widetilde{\jmath }$ satisfies condition (4.4) with constant $\unicode[STIX]{x1D6FC}_{\widetilde{\jmath }}=\unicode[STIX]{x1D6FC}_{j}c_{j}^{2}$ . This statement is based on the chain rule for the Clarke subgradient and assumption (4.17) which guarantee that

(4.23) $$\begin{eqnarray}\widetilde{\jmath }^{\;0}(u;v)\leq j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v)\quad \text{for all}~u,v\in X.\end{eqnarray}$$

Assume now that (4.20) holds. Then $\unicode[STIX]{x1D6FC}_{\widetilde{\unicode[STIX]{x1D711}}}+\unicode[STIX]{x1D6FC}_{\widetilde{\jmath }}\leq m_{A}$ . Therefore, we can deduce the existence of a unique element $u\in K$ such that

(4.24) $$\begin{eqnarray}\langle Au,v-u\rangle +\widetilde{\unicode[STIX]{x1D711}}(u,v)-\widetilde{\unicode[STIX]{x1D711}}(u,u)+\widetilde{\jmath }^{\;0}(u;v-u)\geq \langle f,v-u\rangle \quad \text{for all}~v\in K.\end{eqnarray}$$

We now use equality (4.21) and inequalities (4.23), (4.24) to deduce that $u$ is a solution of inequality (4.15). This proves the existence of the solution to Problem 4.3. The uniqueness of the solution follows from the same argument used at the end of the proof of Theorem 4.2.◻

Denote

$$\begin{eqnarray}K_{\unicode[STIX]{x1D711}}=\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(K).\end{eqnarray}$$

We comment that $X_{\unicode[STIX]{x1D711}}$ can be replaced by $K_{\unicode[STIX]{x1D711}}$ in the assumption (4.18) and the statement of Theorem 4.4 is still valid. Moreover, Problem 4.1 may be viewed as a special case of Problem 4.3.

We have an equivalent formulation of Problem 4.3, similar to Minty’s lemma for variational inequalities (see Atkinson and Han Reference Atkinson and Han2009, p. 435).

Theorem 4.5. Assume $K\subset X$ is convex, $A:X\rightarrow X^{\ast }$ is monotone and radially continuous, and for all $z\in K_{\unicode[STIX]{x1D711}}$ , $\unicode[STIX]{x1D711}(z,\cdot )$ is convex on $K_{\unicode[STIX]{x1D711}}$ . Then $u\in K$ is a solution of Problem 4.3 if and only if it satisfies

(4.25) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle Av,v-u\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u)\nonumber\\ \displaystyle & & \displaystyle \quad +j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}u)\geq \langle f,v-u\rangle \quad \text{for all}~v\in K.\end{eqnarray}$$

Proof. By the monotonicity of $A$ , we have

(4.26) $$\begin{eqnarray}\langle Av,v-u\rangle \geq \langle Au,v-u\rangle \quad \text{for all}~u,v\in X.\end{eqnarray}$$

Then it is obvious that a solution of Problem 4.3 satisfies the inequality (4.25).

Conversely, assume $u\in K$ satisfies (4.25). Since $K$ is convex, for any $v\in K$ and any $t\in [0,1]$ , $u+t(v-u)$ belongs to $K$ . We replace $v$ with $u+t(v-u)$ in (4.25):

(4.27) $$\begin{eqnarray}\displaystyle & & \displaystyle t\,\langle A(u+t(v-u)),v-u\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u+t\,(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v-\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u))\nonumber\\ \displaystyle & & \displaystyle \quad -\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u)+t\,j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}u)\geq t\langle f,v-u\rangle .\end{eqnarray}$$

Note that

$$\begin{eqnarray}\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u+t\,(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v-\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u))\leq t\,\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v)+(1-t)\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u).\end{eqnarray}$$

We deduce from (4.27) that for $t\in (0,1)$ ,

$$\begin{eqnarray}\displaystyle & & \displaystyle \langle A(u+t(v-u)),v-u\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u)+j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}u)\nonumber\\ \displaystyle & & \displaystyle \quad \geq \langle f,v-u\rangle .\nonumber\end{eqnarray}$$

We take the limit $t\rightarrow 0+$ in the above inequality to recover the inequality (4.15).◻

4.2 Numerical approximations

In the rest of this section, we assume (4.1), (4.2), (4.5), (4.16)–(4.20) so that Problem 4.3 has a unique solution.

Let $X^{h}\subset X$ be a finite-dimensional subspace with $h>0$ being a spatial discretization parameter. Let $K^{h}$ be a non-empty, closed and convex subset of $X^{h}$ . Then, a Galerkin approximation of Problem 4.3 is as follows.

Problem 4.6. Find an element $u^{h}\in K^{h}$ such that

(4.28) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle Au^{h},v^{h}-u^{h}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\nonumber\\ \displaystyle & & \displaystyle \quad +j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})\geq \langle f,v^{h}-u^{h}\rangle \quad \text{for all}~v^{h}\in K^{h}.\end{eqnarray}$$

The approximation is external if $K^{h}\not \subset K$ , and is internal if $K^{h}\subset K$ . The internal approximation with the choice $K^{h}=X^{h}\cap K$ is considered in Han, Sofonea and Danan (Reference Han, Sofonea and Danan2018).

Remark 4.7. We comment that for the numerical analysis of Problem 2.6 in Section 7, a discrete problem in a form of the type given by Problem 4.6 serves as an intermediate step. For Problem 2.6, the functional $j$ in the inequality (4.15) is defined by the formula

$$\begin{eqnarray}j(\unicode[STIX]{x1D6FE}_{j}\boldsymbol{u})=\int _{\unicode[STIX]{x1D6E4}_{3}}j_{\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D6FE}_{j}\boldsymbol{u})\,\text{d}a,\quad \unicode[STIX]{x1D6FE}_{j}\boldsymbol{u}=u_{\unicode[STIX]{x1D708}}.\end{eqnarray}$$

In the numerical scheme Problem 7.2 for solving Problem 2.6, the term $\int _{\unicode[STIX]{x1D6E4}_{3}}j_{\unicode[STIX]{x1D708}}^{0}(u_{\unicode[STIX]{x1D708}};v_{\unicode[STIX]{x1D708}}-u_{\unicode[STIX]{x1D708}})\,\text{d}a$ in (2.44) is approximated by $\int _{\unicode[STIX]{x1D6E4}_{3}}j_{\unicode[STIX]{x1D708}}^{0}(u_{\unicode[STIX]{x1D708}}^{h};v_{\unicode[STIX]{x1D708}}^{h}-u_{\unicode[STIX]{x1D708}}^{h})\,\text{d}a$ (see (7.15)). We have

(4.29) $$\begin{eqnarray}j^{0}(\unicode[STIX]{x1D6FE}_{j}\boldsymbol{u};\unicode[STIX]{x1D6FE}_{j}\boldsymbol{v})\leq \int _{\unicode[STIX]{x1D6E4}_{3}}j_{\unicode[STIX]{x1D708}}^{0}(\unicode[STIX]{x1D6FE}_{j}\boldsymbol{u};\unicode[STIX]{x1D6FE}_{j}\boldsymbol{v})\,\text{d}a\end{eqnarray}$$

(e.g. Migórski, Ochal and Sofonea Reference Migórski, Ochal and Sofonea2013, Theorem 3.47), and under the additional assumption that $j_{\unicode[STIX]{x1D708}}$ is regular,

$$\begin{eqnarray}j^{0}(\unicode[STIX]{x1D6FE}_{j}\boldsymbol{u};\unicode[STIX]{x1D6FE}_{j}\boldsymbol{v})=\int _{\unicode[STIX]{x1D6E4}_{3}}j_{\unicode[STIX]{x1D708}}^{0}(\unicode[STIX]{x1D6FE}_{j}\boldsymbol{u};\unicode[STIX]{x1D6FE}_{j}\boldsymbol{v})\,\text{d}a.\end{eqnarray}$$

In the latter case, the two numerical schemes are equivalent. In this paper, we do not assume the regularity of $j_{\unicode[STIX]{x1D708}}$ and we get around this assumption by means of the following consideration.

For definiteness in the discussion here, let the functional $j$ in Problem 4.3 be of the form

(4.30) $$\begin{eqnarray}j(\unicode[STIX]{x1D6FE}_{j}u)=\int _{D}j_{0}(\unicode[STIX]{x1D6FE}_{j}u)\end{eqnarray}$$

where the integrand $j_{0}$ is locally Lipschitz, and the integration region $D$ can be a subset of the domain $\unicode[STIX]{x1D6FA}$ or a part of the boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . For applications in the contact problems considered in this paper, $D=\unicode[STIX]{x1D6E4}_{3}$ . Then, the numerical method for implementation in solving Problem 4.3 is to find an element $u^{h}\in K^{h}$ such that

(4.31) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle Au^{h},v^{h}-u^{h}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\nonumber\\ \displaystyle & & \displaystyle \quad +\int _{D}j_{0}^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})\geq \langle f,v^{h}-u^{h}\rangle \quad \text{for all}~v^{h}\in K^{h}.\end{eqnarray}$$

Moreover, we introduce a further intermediate discrete problem of finding an element $u^{h}\in K^{h}$ such that

(4.32) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle Au^{h},v^{h}-u^{h}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\nonumber\\ \displaystyle & & \displaystyle \quad +j^{h,0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})\geq \langle f,v^{h}-u^{h}\rangle \quad \text{for all}~v^{h}\in K^{h}.\end{eqnarray}$$

Here, $j^{h,0}$ is a finite-dimensional version of the Clarke generalized directional derivative (see (3.3))

(4.33) $$\begin{eqnarray}\displaystyle j^{h,0}(z_{1}^{h};z_{2}^{h}) & = & \displaystyle \limsup _{z_{3}^{h}\rightarrow z_{1}^{h},\,\unicode[STIX]{x1D706}\downarrow 0}\displaystyle \frac{j(z_{3}^{h}+\unicode[STIX]{x1D706}z_{2}^{h})-j(z_{3}^{h})}{\unicode[STIX]{x1D706}},\nonumber\\ \displaystyle & & \displaystyle \qquad \text{for all}~z_{1}^{h},z_{2}^{h}\in X_{j}^{h}:=\unicode[STIX]{x1D6FE}_{j}(X^{h})\end{eqnarray}$$

where the limit is taken for $z_{3}^{h}\in X_{j}^{h}$ and $\unicode[STIX]{x1D706}\in \mathbb{R}$ . Easily, we see the inequality

(4.34) $$\begin{eqnarray}j^{h,0}(z_{1}^{h};z_{2}^{h})\leq j^{0}(z_{1}^{h};z_{2}^{h})\quad \text{for all}~z_{1}^{h},z_{2}^{h}\in X_{j}^{h},\end{eqnarray}$$

and from this,

$$\begin{eqnarray}\displaystyle j^{h,0}(z_{1}^{h};z_{2}^{h}-z_{1}^{h})+j^{h,0}(z_{2}^{h};z_{1}^{h}-z_{2}^{h}) & \leq & \displaystyle j^{0}(z_{1}^{h};z_{2}^{h}-z_{1}^{h})+j^{0}(z_{2}^{h};z_{1}^{h}-z_{2}^{h})\nonumber\\ \displaystyle & & \displaystyle \qquad \text{for all}~z_{1}^{h},z_{2}^{h}\in X_{j}^{h}.\nonumber\end{eqnarray}$$

Then, we can apply the arguments of the proof of Theorem 4.4 in the setting of the finite-dimensional space $X^{h}$ to conclude that the discretized hemivariational inequality (4.32) has a unique solution $u^{h}\in K^{h}$ . Using the relation (4.34), we know that the solution $u^{h}\in K^{h}$ of (4.32) is a solution of Problem 4.6 which is also unique. Finally, in the case of (4.30), by (4.29),

$$\begin{eqnarray}j^{0}(z_{1}^{h};z_{2}^{h})\leq \int _{D}j_{0}^{0}(z_{1}^{h};z_{2}^{h}),\end{eqnarray}$$

and we can verify that the solution $u^{h}\in K^{h}$ of (4.32) or Problem 4.6 is a solution of (4.31) which is also unique.

We choose to do numerical analysis of the abstract Problem 4.3 with Problem 4.6 in this section since the main ideas and techniques for convergence analysis and error estimation can be explained more concisely and in general forms, and the analysis of the numerical method such as (4.31) is conducted very similarly. $\Box$

4.3 Convergence under basic solution regularity

In this subsection, we provide a general discussion of convergence for the numerical solution of Problem 4.6. The key point is that the convergence is shown under the minimal solution regularity $u\in K$ that is available from Theorem 4.4. For convergence analysis, we will need ${\{K^{h}\}}_{h}$ to approximate $K$ in the following sense (see Glowinski, Lions and Trémolières Reference Glowinski, Lions and Trémolières1981):

(4.35) $$\begin{eqnarray}\displaystyle & & \displaystyle v^{h}\in K^{h}~\text{and}~v^{h}\rightharpoonup v~\text{in}~X~\text{imply}~v\in K,\end{eqnarray}$$

(4.36) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{for all}~v\in K,~\exists \,v^{h}\in K^{h}~\text{such that }v^{h}\rightarrow v~\text{in}~X~\text{as}~h\rightarrow 0.\end{eqnarray}$$

Note that by (4.18), for any $z\in X_{\unicode[STIX]{x1D711}}$ , $\unicode[STIX]{x1D711}(z,\cdot ):X_{\unicode[STIX]{x1D711}}\rightarrow \mathbb{R}$ is convex and lower semicontinuous. Thus, $\unicode[STIX]{x1D711}(z,\cdot ):X_{\unicode[STIX]{x1D711}}\rightarrow \mathbb{R}$ is continuous.

The following uniform boundedness property will be useful for convergence analysis of the numerical solutions.

Proof. Since $K$ is non-empty, there is an element $u_{0}\in K$ . We fix one such element. Then by (4.36), there exists $u_{0}^{h}\in K^{h}$ such that

$$\begin{eqnarray}u_{0}^{h}\rightarrow u_{0}~\text{in }X~\text{as }h\rightarrow 0.\end{eqnarray}$$

We let $v^{h}=u_{0}^{h}$ in (4.28) to get

$$\begin{eqnarray}\displaystyle & & \displaystyle \langle Au^{h},u_{0}^{h}-u^{h}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\nonumber\\ \displaystyle & & \displaystyle \quad +j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})\geq \langle f,u_{0}^{h}-u^{h}\rangle .\nonumber\end{eqnarray}$$

Then from

$$\begin{eqnarray}m_{A}\Vert u^{h}-u_{0}^{h}\Vert _{X}^{2}\leq \langle Au^{h},u^{h}-u_{0}^{h}\rangle -\langle Au_{0}^{h},u^{h}-u_{0}^{h}\rangle ,\end{eqnarray}$$

we have

(4.37) $$\begin{eqnarray}\displaystyle m_{A}\Vert u^{h}-u_{0}^{h}\Vert _{X}^{2} & \leq & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})+j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}u_{0}^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})\nonumber\\ \displaystyle & & \displaystyle +\,\langle f,u^{h}-u_{0}^{h}\rangle -\langle Au_{0}^{h},u^{h}-u_{0}^{h}\rangle .\end{eqnarray}$$

In (4.18), take $z_{1}=z_{3}=\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}$ and $z_{2}=z_{4}=\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h}$ ,

(4.38) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}) & \leq & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(u^{h}-u_{0}^{h})\Vert _{X_{\unicode[STIX]{x1D711}}}^{2}.\end{eqnarray}$$

Use (4.18) again, this time taking $z_{1}=z_{4}=\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h}$ , $z_{2}=\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}$ and $z_{3}=\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}$ ,

(4.39) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\leq \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h})\nonumber\\ \displaystyle & & \displaystyle \quad -\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})+\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(u_{0}^{h}-u_{0})\Vert _{X_{\unicode[STIX]{x1D711}}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(u^{h}-u_{0})\Vert _{X_{\unicode[STIX]{x1D711}}}.\end{eqnarray}$$

Use the lower bound (see Atkinson and Han Reference Atkinson and Han2009, p. 433)

$$\begin{eqnarray}\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0},z)\geq c_{3}+c_{4}\Vert z\Vert _{X_{\unicode[STIX]{x1D711}}}\quad \text{for all}~z\in X_{\unicode[STIX]{x1D711}}\end{eqnarray}$$

for some constants $c_{3}$ and $c_{4}$ , not necessarily positive. Then

(4.40) $$\begin{eqnarray}-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\leq -c_{3}-c_{4}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}\Vert _{X_{\unicode[STIX]{x1D711}}}.\end{eqnarray}$$

Take $z_{1}=\unicode[STIX]{x1D6FE}_{j}u^{h}$ and $z_{2}=\unicode[STIX]{x1D6FE}_{j}u_{0}^{h}$ in (4.19(c)) to obtain

$$\begin{eqnarray}j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}u_{0}^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})\leq \unicode[STIX]{x1D6FC}_{j}\Vert \unicode[STIX]{x1D6FE}_{j}(u_{0}^{h}-u^{h})\Vert _{X_{j}}^{2}-j^{0}(\unicode[STIX]{x1D6FE}_{j}u_{0}^{h};\unicode[STIX]{x1D6FE}_{j}u^{h}-\unicode[STIX]{x1D6FE}_{j}u_{0}^{h}).\end{eqnarray}$$

By (4.19(b)),

$$\begin{eqnarray}-j^{0}(\unicode[STIX]{x1D6FE}_{j}u_{0}^{h};\unicode[STIX]{x1D6FE}_{j}u^{h}-\unicode[STIX]{x1D6FE}_{j}u_{0}^{h})\leq (c_{0}+c_{1}\Vert \unicode[STIX]{x1D6FE}_{j}u_{0}^{h}\Vert _{X_{j}})\Vert \unicode[STIX]{x1D6FE}_{j}(u^{h}-u_{0}^{h})\Vert _{X_{j}}.\end{eqnarray}$$

Thus,

(4.41) $$\begin{eqnarray}\displaystyle j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}u_{0}^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h}) & \leq & \displaystyle \unicode[STIX]{x1D6FC}_{j}c_{j}^{2}\Vert u_{0}^{h}-u^{h}\Vert _{X}^{2}\nonumber\\ \displaystyle & & \displaystyle +\,c_{j}(c_{0}+c_{1}\Vert \unicode[STIX]{x1D6FE}_{j}u_{0}^{h}\Vert _{X_{j}})\Vert u_{0}^{h}-u^{h}\Vert _{X}.\end{eqnarray}$$

Use (4.38), (4.39), (4.40), and (4.41) in (4.37) to obtain

$$\begin{eqnarray}\displaystyle m_{A}\Vert u^{h}-u_{0}^{h}\Vert _{X}^{2} & \leq & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u_{0}^{h})-c_{3}-c_{4}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}\Vert _{X_{\unicode[STIX]{x1D711}}}+\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(u^{h}-u_{0}^{h})\Vert _{X_{\unicode[STIX]{x1D711}}}^{2}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(u_{0}^{h}-u_{0})\Vert _{X_{\unicode[STIX]{x1D711}}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(u^{h}-u_{0})\Vert _{X_{\unicode[STIX]{x1D711}}}\nonumber\\ \displaystyle & & \displaystyle +\,(c_{0}+c_{1}\Vert \unicode[STIX]{x1D6FE}_{j}u_{0}^{h}\Vert _{X_{j}})\Vert \unicode[STIX]{x1D6FE}_{j}(u^{h}-u_{0}^{h})\Vert _{X_{j}}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FC}_{j}\Vert \unicode[STIX]{x1D6FE}_{j}(u_{0}^{h}-u^{h})\Vert _{X_{j}}^{2}+\langle f-Au_{0}^{h},u^{h}-u_{0}^{h}\rangle .\nonumber\end{eqnarray}$$

Since $u_{0}^{h}\rightarrow u_{0}$ in $X$ , we know that $\Vert u_{0}^{h}\Vert _{X}$ , and then also $\Vert \unicode[STIX]{x1D6FE}_{j}u_{0}^{h}\Vert _{X_{j}}$ and $\Vert Au_{0}^{h}\Vert _{X^{\ast }}$ are uniformly bounded with respect to $h$ . Finally, by the smallness condition (4.20), we conclude that $\Vert u^{h}-u_{0}^{h}\Vert _{X}$ , and then also $\Vert u^{h}\Vert _{X}$ is uniformly bounded in $h$ .◻

We now prove the convergence of the numerical solutions under the minimal solution regularity $u\in K$ . In applications to contact mechanics, $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}$ and $\unicode[STIX]{x1D6FE}_{j}$ are trace operators from an $H^{1}(\unicode[STIX]{x1D6FA})$ -based space to $L^{2}(\unicode[STIX]{x1D6E4}_{3})$ -based spaces, and are thus compact operators.

Theorem 4.9. Assume (4.1), (4.2), (4.5), (4.16)–(4.20), (4.35) and (4.36). Assume further that $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}:X\rightarrow X_{\unicode[STIX]{x1D711}}$ and $\unicode[STIX]{x1D6FE}_{j}:X\rightarrow X_{j}$ are compact operators, and $A:X\rightarrow X^{\ast }$ is continuous. Then,

(4.42) $$\begin{eqnarray}u^{h}\rightarrow u\quad \text{in }X~\text{as }h\rightarrow 0.\end{eqnarray}$$

Proof. The proof consists of two steps. First we show the weak convergence of the numerical solutions. According to Theorem 4.5, the solution $u^{h}\in K^{h}$ of Problem 4.6 is characterized by the inequality

(4.43) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle Av^{h},v^{h}-u^{h}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\nonumber\\ \displaystyle & & \displaystyle \quad +j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})\geq \langle f,v^{h}-u^{h}\rangle \quad \text{for all}~v^{h}\in K^{h}.\end{eqnarray}$$

Note that $\{u^{h}\}$ is bounded in $X$ , by Proposition 4.8. Since $X$ is reflexive and the operators $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}:X\rightarrow X_{\unicode[STIX]{x1D711}}$ and $\unicode[STIX]{x1D6FE}_{j}:X\rightarrow X_{j}$ are compact, there exists a subsequence $\{u^{h^{\prime }}\}\subset \{u^{h}\}$ and an element $w\in X$ such that

$$\begin{eqnarray}u^{h^{\prime }}\rightharpoonup w~\text{in}~X,\quad \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h^{\prime }}\rightarrow \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}w~\text{in}~X_{\unicode[STIX]{x1D711}},\quad \unicode[STIX]{x1D6FE}_{j}u^{h^{\prime }}\rightarrow \unicode[STIX]{x1D6FE}_{j}w~\text{in}~X_{j}.\end{eqnarray}$$

By the assumption (4.35), we know that $w\in K$ .

Fix an arbitrary element $v\in K$ . By the assumption (4.36), we can find a sequence $v^{h^{\prime }}\in K^{h^{\prime }}$ such that $v^{h^{\prime }}\rightarrow v$ in $X$ as $h^{\prime }\rightarrow 0$ . Then, as $h^{\prime }\rightarrow 0$ , the following hold:

$$\begin{eqnarray}\displaystyle & & \displaystyle Av^{h^{\prime }}\rightarrow Av,\quad \langle Av^{h^{\prime }},v^{h^{\prime }}-u^{h^{\prime }}\rangle \rightarrow \langle Av,v-w\rangle ,\nonumber\\ \displaystyle & & \displaystyle j^{0}(\unicode[STIX]{x1D6FE}_{j}w;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}w)\geq \limsup j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h^{\prime }};\unicode[STIX]{x1D6FE}_{j}v^{h^{\prime }}-\unicode[STIX]{x1D6FE}_{j}u^{h^{\prime }}),\nonumber\\ \displaystyle & & \displaystyle \langle f,v^{h^{\prime }}-u^{h^{\prime }}\rangle \rightarrow \langle f,v-w\rangle .\nonumber\end{eqnarray}$$

From (4.18) with $z_{1}=\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}w$ , $z_{2}=z_{4}=\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h^{\prime }}$ , and $z_{3}=\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h^{\prime }}$ , we have

(4.44) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h^{\prime }},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h^{\prime }})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h^{\prime }},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h^{\prime }})\nonumber\\ \displaystyle & & \displaystyle \quad \leq \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}w,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h^{\prime }})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}w,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h^{\prime }})\nonumber\\ \displaystyle & & \displaystyle \quad \quad +\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(w-u^{h^{\prime }})\Vert _{X_{\unicode[STIX]{x1D711}}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(v^{h^{\prime }}-u^{h^{\prime }})\Vert _{X_{\unicode[STIX]{x1D711}}}.\end{eqnarray}$$

Use this inequality in (4.43) with $h=h^{\prime }$ ,

(4.45) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle Av^{h^{\prime }},v^{h^{\prime }}-u^{h^{\prime }}\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}w,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h^{\prime }})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}w,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h^{\prime }})\nonumber\\ \displaystyle & & \displaystyle \quad +\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(w-u^{h^{\prime }})\Vert _{X_{\unicode[STIX]{x1D711}}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(v^{h^{\prime }}-u^{h^{\prime }})\Vert _{X_{\unicode[STIX]{x1D711}}}\nonumber\\ \displaystyle & & \displaystyle \quad +j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h^{\prime }};\unicode[STIX]{x1D6FE}_{j}v^{h^{\prime }}-\unicode[STIX]{x1D6FE}_{j}u^{h^{\prime }})\geq \langle f,v^{h^{\prime }}-u^{h^{\prime }}\rangle .\end{eqnarray}$$

Note that $\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(v^{h^{\prime }}-u^{h^{\prime }})\Vert _{X_{\unicode[STIX]{x1D711}}}$ is bounded whereas $\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(w-u^{h^{\prime }})\Vert _{X_{\unicode[STIX]{x1D711}}}\rightarrow 0$ . Thus, taking the upper limit in (4.45) as $h^{\prime }\rightarrow 0$ , we find that

$$\begin{eqnarray}\langle Av,v-w\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}w,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}w,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}w)+j^{0}(\unicode[STIX]{x1D6FE}_{j}w;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}w)\geq \langle f,v-w\rangle .\end{eqnarray}$$

This inequality holds for any $v\in K$ . By Theorem 4.5, $w$ is the solution $u$ of Problem 4.3. So $u^{h^{\prime }}\rightharpoonup u$ in $X$ . Since the limit $u$ does not depend on the subsequence $\{u^{h^{\prime }}\}$ , the entire family of numerical solutions converges weakly to $u$ .

Next, we show the strong convergence $u^{h}\rightarrow u$ in $X$ as $h\rightarrow 0$ . By the assumption (4.36), there exists a sequence $\{\bar{u}^{h}\}$ , $\bar{u}^{h}\in K^{h}$ , such that $\bar{u}^{h}\rightarrow u$ in $X$ as $h\rightarrow 0$ . Applying (4.2(b)),

$$\begin{eqnarray}m_{A}\Vert u-u^{h}\Vert _{X}^{2}\leq \langle Au-Au^{h},u-u^{h}\rangle .\end{eqnarray}$$

So

$$\begin{eqnarray}m_{A}\Vert u-u^{h}\Vert _{X}^{2}\leq \langle Au,u-u^{h}\rangle -\langle Au^{h},\bar{u}^{h}-u^{h}\rangle -\langle Au^{h},u-\bar{u}^{h}\rangle .\end{eqnarray}$$

From (4.28), we have

$$\begin{eqnarray}\displaystyle -\langle Au^{h},\bar{u}^{h}-u^{h}\rangle & \leq & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}\bar{u}^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\nonumber\\ \displaystyle & & \displaystyle +\,j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}\bar{u}^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})-\langle f,\bar{u}^{h}-u^{h}\rangle .\nonumber\end{eqnarray}$$

As in (4.44), we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}\bar{u}^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}) & \leq & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}\bar{u}^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\nonumber\\ \displaystyle & & \displaystyle ~+\,\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u-\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}\Vert _{X_{\unicode[STIX]{x1D711}}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}\bar{u}^{h}-\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}\Vert _{X_{\unicode[STIX]{x1D711}}}.\nonumber\end{eqnarray}$$

Combining the above three inequalities, we obtain that

(4.46) $$\begin{eqnarray}\displaystyle m_{A}\Vert u-u^{h}\Vert _{X}^{2} & \leq & \displaystyle \langle Au,u-u^{h}\rangle -\langle Au^{h},u-\bar{u}^{h}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}\bar{u}^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u-\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}\Vert _{X_{\unicode[STIX]{x1D711}}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}\bar{u}^{h}-\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}\Vert _{X_{\unicode[STIX]{x1D711}}}\nonumber\\ \displaystyle & & \displaystyle +\,j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}\bar{u}^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})-\langle f,\bar{u}^{h}-u^{h}\rangle .\end{eqnarray}$$

Since $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\cdot ):X_{\unicode[STIX]{x1D711}}\rightarrow \mathbb{R}$ is continuous, as $h\rightarrow 0$ ,

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}\bar{u}^{h})\rightarrow \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u),\nonumber\\ \displaystyle & & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\rightarrow \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u).\nonumber\end{eqnarray}$$

Also note that $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}\rightarrow \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u$ in $X_{\unicode[STIX]{x1D711}}$ , $\unicode[STIX]{x1D6FE}_{j}u^{h}\rightarrow \unicode[STIX]{x1D6FE}_{j}u$ in $X_{j}$ , $\bar{u}^{h}\rightarrow u$ in $X$ and therefore $\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(\bar{u}^{h}-u^{h})\Vert _{X_{\unicode[STIX]{x1D711}}}\rightarrow 0$ , $\Vert \unicode[STIX]{x1D6FE}_{j}(\bar{u}^{h}-u^{h})\Vert _{X_{j}}\rightarrow 0$ , and $\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u-\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}\Vert _{X_{\unicode[STIX]{x1D711}}}\rightarrow 0$ . Consequently, from (4.46),

$$\begin{eqnarray}\limsup _{h\rightarrow 0}\Vert u-u^{h}\Vert _{X}^{2}\leq 0.\end{eqnarray}$$

This implies the strong convergence $u^{h}\rightarrow u$ in $X$ .◻

4.4 Error estimation

We now turn to the derivation of error estimates. In this subsection, we assume (4.1), (4.2), (4.5), (4.16)–(4.20). We will further assume that the operator $A:X\rightarrow X^{\ast }$ is Lipschitz continuous, that is, for a constant $L_{A}>0$ ,

(4.47) $$\begin{eqnarray}\Vert Au-Av\Vert _{X^{\ast }}\leq L_{A}\Vert u-v\Vert _{X}\quad \text{for all}~u,v\in X.\end{eqnarray}$$

We comment that this assumption implies (4.2(a)).

Let $v\in K$ and $v^{h}\in K^{h}$ be arbitrary. By (4.2(b)) with $v_{1}=u$ and $v_{2}=u^{h}$ ,

$$\begin{eqnarray}m_{A}\Vert u-u^{h}\Vert _{X}^{2}\leq \langle Au-Au^{h},u-u^{h}\rangle ,\end{eqnarray}$$

which is rewritten as

(4.48) $$\begin{eqnarray}\displaystyle m_{A}\Vert u-u^{h}\Vert _{X}^{2} & \leq & \displaystyle \langle Au-Au^{h},u-v^{h}\rangle +\langle Au,v^{h}-u\rangle +\langle Au,v-u^{h}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\langle Au,u-v\rangle +\langle Au^{h},u^{h}-v^{h}\rangle .\end{eqnarray}$$

Applying (4.15),

$$\begin{eqnarray}\displaystyle \langle Au,u-v\rangle & \leq & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u)\nonumber\\ \displaystyle & & \displaystyle +\,j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}u)-\langle f,v-u\rangle .\nonumber\end{eqnarray}$$

Applying (4.28),

$$\begin{eqnarray}\displaystyle \langle Au^{h},u^{h}-v^{h}\rangle & \leq & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})\nonumber\\ \displaystyle & & \displaystyle +\,j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})-\langle f,v^{h}-u^{h}\rangle .\nonumber\end{eqnarray}$$

Using these inequalities in (4.48), after some rearrangement of the terms, we have

(4.49) $$\begin{eqnarray}\displaystyle m_{A}\Vert u-u^{h}\Vert _{X}^{2} & \leq & \displaystyle \langle Au-Au^{h},u-v^{h}\rangle +R_{u}(v^{h},u)+R_{u}(v,u^{h})\nonumber\\ \displaystyle & & \displaystyle +\,I_{\unicode[STIX]{x1D711}}(u^{h},v^{h})+I_{j}(v,v^{h}),\end{eqnarray}$$

where

(4.50) $$\begin{eqnarray}\displaystyle R_{u}(v,w) & := & \displaystyle \langle Au,v-w\rangle +\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v)-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}w)\nonumber\\ \displaystyle & & \displaystyle +\,j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}w)-\langle f,v-w\rangle ,\end{eqnarray}$$

(4.51) $$\begin{eqnarray}\displaystyle I_{\unicode[STIX]{x1D711}}(u^{h},v^{h}) & := & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h})+\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h})\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h},\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}),\end{eqnarray}$$

(4.52) $$\begin{eqnarray}\displaystyle I_{j}(v,v^{h}) & := & \displaystyle j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}u)+j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h})\nonumber\\ \displaystyle & & \displaystyle -\,j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u)-j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}u^{h}).\end{eqnarray}$$

Let us bound the first and the last two terms on the right-hand side of (4.49). First,

$$\begin{eqnarray}\langle Au-Au^{h},u-v^{h}\rangle \leq L_{A}\Vert u-u^{h}\Vert _{X}\Vert u-v^{h}\Vert _{X}.\end{eqnarray}$$

So for any $\unicode[STIX]{x1D700}>0$ arbitrarily small,

(4.53) $$\begin{eqnarray}\langle Au-Au^{h},u-v^{h}\rangle \leq \unicode[STIX]{x1D700}\,\Vert u-u^{h}\Vert _{X}^{2}+c\,\Vert u-v^{h}\Vert _{X}^{2}\end{eqnarray}$$

for some constant $c$ depending on $\unicode[STIX]{x1D700}$ . By (4.19(c)), we have

$$\begin{eqnarray}\displaystyle I_{\unicode[STIX]{x1D711}}(u^{h},v^{h}) & \leq & \displaystyle \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u-\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}\Vert _{X_{\unicode[STIX]{x1D711}}}\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}u^{h}-\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}v^{h}\Vert _{X_{\unicode[STIX]{x1D711}}}\nonumber\\ \displaystyle & \leq & \displaystyle \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}c_{\unicode[STIX]{x1D711}}^{2}(\Vert u-u^{h}\Vert _{X}^{2}+\Vert u-u^{h}\Vert _{X}\Vert u-v^{h}\Vert _{X}).\nonumber\end{eqnarray}$$

Thus,

(4.54) $$\begin{eqnarray}I_{\unicode[STIX]{x1D711}}(u^{h},v^{h})\leq (\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}c_{\unicode[STIX]{x1D711}}^{2}+\unicode[STIX]{x1D700})\Vert u-u^{h}\Vert _{X}^{2}+c\,\Vert u-v^{h}\Vert _{X}^{2}\end{eqnarray}$$

for another constant $c$ depending on $\unicode[STIX]{x1D700}>0$ . Applying the subadditivity of the generalized directional derivative (see Proposition 3.10(i)),

$$\begin{eqnarray}j^{0}(z;z_{1}+z_{2})\leq j^{0}(z;z_{1})+j^{0}(z;z_{2})\quad \text{for all}~z,z_{1},z_{2}\in X_{j},\end{eqnarray}$$

we have

$$\begin{eqnarray}\displaystyle j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}u) & \leq & \displaystyle j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v-\unicode[STIX]{x1D6FE}_{j}u^{h})+j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}u^{h}-\unicode[STIX]{x1D6FE}_{j}u),\nonumber\\ \displaystyle j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u^{h}) & \leq & \displaystyle j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u)+j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}u-\unicode[STIX]{x1D6FE}_{j}u^{h}).\nonumber\end{eqnarray}$$

Thus,

$$\begin{eqnarray}\displaystyle I_{j}(v,v^{h}) & \leq & \displaystyle j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u)-j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u)\nonumber\\ \displaystyle & & \displaystyle +\,j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}u^{h}-\unicode[STIX]{x1D6FE}_{j}u)+j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}u-\unicode[STIX]{x1D6FE}_{j}u^{h}).\nonumber\end{eqnarray}$$

By (4.19(c)),

$$\begin{eqnarray}j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}u^{h}-\unicode[STIX]{x1D6FE}_{j}u)+j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}u-\unicode[STIX]{x1D6FE}_{j}u^{h})\leq \unicode[STIX]{x1D6FC}_{j}\Vert \unicode[STIX]{x1D6FE}_{j}u-\unicode[STIX]{x1D6FE}_{j}u^{h}\Vert _{X_{j}}^{2}.\end{eqnarray}$$

Moreover,

$$\begin{eqnarray}\displaystyle |j^{0}(\unicode[STIX]{x1D6FE}_{j}u^{h};\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u)| & \leq & \displaystyle (c_{0}+c_{1}\Vert \unicode[STIX]{x1D6FE}_{j}u^{h}\Vert _{X_{j}})\Vert \unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u\Vert _{X_{j}},\nonumber\\ \displaystyle |j^{0}(\unicode[STIX]{x1D6FE}_{j}u;\unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u)| & \leq & \displaystyle (c_{0}+c_{1}\Vert \unicode[STIX]{x1D6FE}_{j}u\Vert _{X_{j}})\Vert \unicode[STIX]{x1D6FE}_{j}v^{h}-\unicode[STIX]{x1D6FE}_{j}u\Vert _{X_{j}}.\nonumber\end{eqnarray}$$

Combining the above four inequalities and using the fact that $\Vert \unicode[STIX]{x1D6FE}_{j}u^{h}\Vert _{X_{j}}$ is uniformly bounded (see Proposition 4.8), we find that

(4.55) $$\begin{eqnarray}I_{j}(v,v^{h})\leq \unicode[STIX]{x1D6FC}_{j}\Vert \unicode[STIX]{x1D6FE}_{j}u-\unicode[STIX]{x1D6FE}_{j}u^{h}\Vert _{X_{j}}^{2}+c\,\Vert \unicode[STIX]{x1D6FE}_{j}u-\unicode[STIX]{x1D6FE}_{j}v^{h}\Vert _{X_{j}}\end{eqnarray}$$

for some constant $c>0$ independent of $h$ . Using (4.53), (4.54) and (4.55) in (4.49), we have

$$\begin{eqnarray}\displaystyle (m_{A}-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}c_{\unicode[STIX]{x1D711}}^{2}-\unicode[STIX]{x1D6FC}_{j}c_{j}^{2}-2\,\unicode[STIX]{x1D700})\Vert u-u^{h}\Vert _{X}^{2} & \leq & \displaystyle c\,\Vert u-v^{h}\Vert _{X}^{2}+c\,\Vert \unicode[STIX]{x1D6FE}_{j}u-\unicode[STIX]{x1D6FE}_{j}v^{h}\Vert _{X_{j}}\nonumber\\ \displaystyle & & \displaystyle +\,R_{u}(v^{h},u)+R_{u}(v,u^{h}).\nonumber\end{eqnarray}$$

Recall the smallness assumption, $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}c_{\unicode[STIX]{x1D711}}^{2}+\unicode[STIX]{x1D6FC}_{j}c_{j}^{2}<m_{A}$ . We then choose $\unicode[STIX]{x1D700}=(m_{A}-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D711}}c_{\unicode[STIX]{x1D711}}^{2}-\unicode[STIX]{x1D6FC}_{j}c_{j}^{2})/4>0$ and get the inequality

$$\begin{eqnarray}\displaystyle \Vert u-u^{h}\Vert _{X}^{2} & \leq & \displaystyle c\inf _{v^{h}\in K^{h}}[\Vert u-v^{h}\Vert _{X}^{2}+\Vert \unicode[STIX]{x1D6FE}_{j}(u-v^{h})\Vert _{X_{j}}+R_{u}(v^{h},u)]\nonumber\\ \displaystyle & & \displaystyle +\,c\inf _{v\in K}R_{u}(v,u^{h}).\nonumber\end{eqnarray}$$

We summarize the result in the form of a theorem.

Theorem 4.10. Assume (4.1), (4.2), (4.5), (4.16)–(4.20) and (4.47). Then for the solution $u$ of Problem 4.3 and the solution $u^{h}$ of Problem 4.6, we have the Céa-type inequality

(4.56) $$\begin{eqnarray}\displaystyle \Vert u-u^{h}\Vert _{X} & \leq & \displaystyle c\inf _{v^{h}\in K^{h}}[\Vert u-v^{h}\Vert _{X}+\Vert \unicode[STIX]{x1D6FE}_{j}(u-v^{h})\Vert _{X_{j}}^{1/2}+|R_{u}(v^{h},u)|^{1/2}]\nonumber\\ \displaystyle & & \displaystyle +\,c\inf _{v\in K}|R_{u}(v,u^{h})|^{1/2}.\end{eqnarray}$$

We remark that in the literature on error analysis of numerical solutions of variational inequalities, it is standard that Céa-type inequalities involve the square root of the approximation error of the solution in certain norms, due to the inequality form of the problems; see Falk (Reference Falk1974), Kikuchi and Oden (Reference Kikuchi and Oden1988) and Han and Sofonea (Reference Han and Sofonea2002).

To proceed further, we need to bound the residual term (4.50) and this depends on the problem to be solved. We illustrate this point in Section 7 in the context of a contact problem.

In the special case where $K=X$ , we have $K^{h}=X^{h}$ . Then the approximation is always internal, and we have the following reduced Céa-type inequality:

(4.57) $$\begin{eqnarray}\Vert u-u^{h}\Vert _{X}\leq c\inf _{v^{h}\in X^{h}}[\Vert u-v^{h}\Vert _{X}+\Vert \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D711}}(u-v^{h})\Vert _{X_{\unicode[STIX]{x1D711}}}^{1/2}+\Vert \unicode[STIX]{x1D6FE}_{j}(u-v^{h})\Vert _{X_{j}}^{1/2}].\end{eqnarray}$$

Note that in deriving error estimates, the application of (4.57) is straightforward, whereas the application of (4.56) is more involved as one has to bound the residual-type term $R_{u}$ .

5.1 Solution existence and uniqueness

We have the following existence and uniqueness result on Problem 5.1.

Proof. We use a fixed-point argument. Given an $\unicode[STIX]{x1D702}\in C(I;X)$ , define

(5.6) $$\begin{eqnarray}y_{\unicode[STIX]{x1D702}}(t)={\mathcal{S}}\unicode[STIX]{x1D702}(t)\quad \text{for all}~t\in I.\end{eqnarray}$$

Then $y_{\unicode[STIX]{x1D702}}\in C(I;Y)$ . Consider the auxiliary problem of finding a function $u_{\unicode[STIX]{x1D702}}:I\rightarrow K$ such that for all $t\in I$ , the following inequality holds:

(5.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle Au_{\unicode[STIX]{x1D702}}(t),v-u_{\unicode[STIX]{x1D702}}(t)\rangle +\unicode[STIX]{x1D711}(y_{\unicode[STIX]{x1D702}}(t),u_{\unicode[STIX]{x1D702}}(t),v)-\unicode[STIX]{x1D711}(y_{\unicode[STIX]{x1D702}}(t),u_{\unicode[STIX]{x1D702}}(t),u_{\unicode[STIX]{x1D702}}(t))\nonumber\\ \displaystyle & & \displaystyle \quad +j^{0}(u(t);v-u(t))\geq \langle f(t),v-u(t)\rangle \quad \text{for all}~v\in K.\end{eqnarray}$$

Applying Theorem 4.2, we know that there exists a unique element $u_{\unicode[STIX]{x1D702}}(t)\in K$ which solves this inequality, for each $t\in I$ . Moreover, it can be shown that the function $u_{\unicode[STIX]{x1D702}}:I\rightarrow X$ is continuous. Therefore, there exists a unique function $u_{\unicode[STIX]{x1D702}}\in C(I;K)$ such that (5.7) holds, for all $t\in I$ . This allows us to define an operator $\unicode[STIX]{x1D6EC}:C(I;X)\rightarrow C(I;K)$ $\subset C(I;X)$ via the relation

(5.8) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D702}=u_{\unicode[STIX]{x1D702}},\quad \unicode[STIX]{x1D702}\in C(I;X).\end{eqnarray}$$

Let us prove that the operator $\unicode[STIX]{x1D6EC}$ has a unique fixed point $\unicode[STIX]{x1D702}^{\ast }\in C(I;K)$ . For two arbitrary functions $\unicode[STIX]{x1D702}_{1},\unicode[STIX]{x1D702}_{2}\in C(I;X)$ , let $u_{i}$ denote the solution of the variational–hemivariational inequality (5.7) for $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}_{i}$ , i.e.  $u_{i}=u_{\unicode[STIX]{x1D702}_{i}}$ , $i=1,2$ . Let $I_{0}\subset I$ be a compact set and let $t\in I_{0}$ . We use definition (5.8), inequality (5.7) and the assumptions (5.2), (5.3), to derive the inequality

(5.9) $$\begin{eqnarray}\Vert \unicode[STIX]{x1D6EC}\unicode[STIX]{x1D702}_{1}(t)-\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D702}_{2}(t)\Vert _{X}\leq c\,L_{I_{0}}\int _{0}^{t}\Vert \unicode[STIX]{x1D702}_{1}(s)-\unicode[STIX]{x1D702}_{2}(s)\Vert _{X}\,\text{d}s.\end{eqnarray}$$

Here and below, $c$ is a positive constant that depends on $A$ , $\unicode[STIX]{x1D711}$ and $j$ . This shows that the operator $\unicode[STIX]{x1D6EC}:C(I;X)\rightarrow C(I;K)\subset C(I;X)$ is a history-dependent operator. Applying Theorem 3.20, we know that the operator $\unicode[STIX]{x1D6EC}$ has a unique fixed point $\unicode[STIX]{x1D702}^{\ast }\in C(I;X)$ . Since $\unicode[STIX]{x1D6EC}$ takes values in $C(I;K)$ , we have $\unicode[STIX]{x1D702}^{\ast }\in C(I;K)$ . Moreover, by the definition of $\unicode[STIX]{x1D6EC}$ , $\unicode[STIX]{x1D702}^{\ast }$ satisfies the inequality

$$\begin{eqnarray}\displaystyle & & \displaystyle \langle A\unicode[STIX]{x1D702}^{\ast }(t),v-\unicode[STIX]{x1D702}^{\ast }(t)\rangle +\unicode[STIX]{x1D711}({\mathcal{S}}\unicode[STIX]{x1D702}^{\ast }(t),\unicode[STIX]{x1D702}^{\ast }(t),v)-\unicode[STIX]{x1D711}({\mathcal{S}}\unicode[STIX]{x1D702}^{\ast }(t),\unicode[STIX]{x1D702}^{\ast }(t),\unicode[STIX]{x1D702}^{\ast }(t))\nonumber\\ \displaystyle & & \displaystyle \quad +j^{0}(\unicode[STIX]{x1D702}^{\ast }(t);v-\unicode[STIX]{x1D702}^{\ast }(t))\geq \langle f(t),v-\unicode[STIX]{x1D702}^{\ast }(t)\rangle \quad \text{for all}~v\in K\nonumber\end{eqnarray}$$

for all $t\in I$ , that is, $\unicode[STIX]{x1D702}^{\ast }\in C(I;K)$ is a solution of the variational–hemivariational inequality (5.1).

The uniqueness part is a consequence of the uniqueness of the fixed point of the operator $\unicode[STIX]{x1D6EC}$ . A direct proof is also possible and goes as follows. Let $u_{1}$ and $u_{2}$ be two solutions of Problem 5.1. Let $I_{0}\subset I$ be a compact subset and let $t\in I_{0}$ . From (5.1),

$$\begin{eqnarray}\displaystyle & & \displaystyle \langle Au_{1}(t)-Au_{2}(t),u_{1}(t)-u_{2}(t)\rangle \nonumber\\ \displaystyle & & \displaystyle \quad \leq \unicode[STIX]{x1D711}({\mathcal{S}}u_{1}(t),u_{1}(t),u_{2}(t))-\unicode[STIX]{x1D711}({\mathcal{S}}u_{1}(t),u_{1}(t),u_{1}(t))\nonumber\\ \displaystyle & & \displaystyle \quad \quad +\unicode[STIX]{x1D711}({\mathcal{S}}u_{2}(t),u_{2}(t),u_{1}(t))-\unicode[STIX]{x1D711}({\mathcal{S}}u_{2}(t),u_{2}(t),u_{2}(t))\nonumber\\ \displaystyle & & \displaystyle \quad \quad +j^{0}(u_{1}(t);u_{2}(t)-u_{1}(t))+j^{0}(u_{2}(t);u_{1}(t)-u_{2}(t)).\nonumber\end{eqnarray}$$

Using assumptions (4.2), (4.4) and (5.3), after some elementary manipulations, we deduce that

$$\begin{eqnarray}\Vert u_{1}(t)-u_{2}(t)\Vert _{X}\leq c\,\Vert {\mathcal{S}}u_{1}(t)-{\mathcal{S}}u_{2}(t)\Vert _{Y}.\end{eqnarray}$$

We use this inequality and assumption (5.2) to find that

$$\begin{eqnarray}\Vert u_{1}(t)-u_{2}(t)\Vert _{X}\leq cL_{I_{0}}\int _{0}^{t}\Vert u_{1}(s)-u_{2}(s)\Vert _{X}\,\text{d}s.\end{eqnarray}$$

Next, it follows from the Gronwall inequality (see Lemma 3.22) that $u_{1}(t)-u_{2}(t)=0$ for all $t\in I_{0}$ . This implies that $u_{1}(t)=u_{2}(t)$ for all $t\in I$ , that is, the solution is unique.◻