11institutetext:  Y. Bai 22institutetext: School of Science, Guangxi
University of Science and Technology, Liuzhou 545006, Guangxi, China
22email<EMAIL_ADDRESS>33institutetext: N. Costea 44institutetext:
Department of Mathematics and Computer Science, Politehnica University of
Bucharest, 313 Splaiul Independenţei, 060042 Bucharest, Romania
44email<EMAIL_ADDRESS><EMAIL_ADDRESS>55institutetext:
S. Zeng 66institutetext: Guangxi Colleges and Universities Key Laboratory of
Complex System Optimization and Big Data Processing, Yulin Normal University,
Yulin 537000, Guangxi, P.R. China & Jagiellonian University in Krakow, Faculty
of Mathematics and Computer Science, ul. Lojasiewicza 6, 30-348 Krakow, Poland
66email<EMAIL_ADDRESS><EMAIL_ADDRESS>

# Existence results for variational-quasi-hemivariational inequality systems
with nonlinear couplings ††thanks: a Partially supported by NNSF of China
Grant No. 12101143, and Natural Science Foundation of Guangxi Grants No.
GKAD21220144.
b Partially supported by CNCS-UEFISCDI Grant No. PN-III-P1-1.1-TE-2019-0456.
c Partially supported by NNSF of China Grant No. 12001478, the European
Union’s Horizon 2020 Research and Innovation Programme under the Marie
Sklodowska-Curie grant agreement No. 823731 CONMECH, National Science Center
of Poland under Preludium Project No. 2017/25/N/ST1/00611, the Startup Project
of Doctor Scientific Research of Yulin Normal University No. G2020ZK07, and
the Natural Science Foundation of Guangxi Grants Nos. 2021GXNSFFA196004 and
2020GXNSFBA297137.

Yunru Baia Nicuşor Costeab Shengda Zengc

###### Abstract

In this paper we investigate a system of coupled inequalities consisting of a
variational-hemivariational inequality and a quasi-hemivariational inequality
on Banach spaces. The approach is topological, and a wide variety of existence
results is established for both bounded and unbounded constraint sets in real
reflexive Banach spaces. Applications to Contact Mechanics are provided in the
last section of the paper. More precisely, we consider a contact model with
(possibly) multivalued constitutive law whose variational formulation leads to
a coupled system of inequalities. The weak solvability of the problem is
proved via employing the theoretical results obtained in the previous section.
The novelty of our approach comes from the fact that we consider two potential
contact zones and the variational formulation allows us to determine
simultaneously the displacement field and the Cauchy stress tensor. 35J88
39B72 58E50 70G75 74M15

###### keywords:

Hemivariational inequalities Nonlinear coupling functional Bounded and
unbounded constraint sets Contact problems Weak solution via bipotentials

##  1 Introduction and preliminaries

Let $X,{Y}$ be two real reflexive Banach spaces and $K\subseteq X$,
$\Lambda\subseteq{Y}$ be nonempty, closed and convex subsets. Assume $B\colon
X\times{Y}\to{\mathbb{R}}$ is a nonlinear bifunction, $\phi\colon
X\to(-\infty,\infty]$ is a proper, convex and lower semicontinuous such that
$D(\phi)\cap K\neq\emptyset$, $Z_{1},Z_{2}$ are Banach spaces and
$\gamma_{1}\colon X\to Z_{1}$, $\gamma_{2}\colon Y\to Z_{2}$ are linear and
compact operators, $J\colon Z_{1}\to\mathbb{R}$ and $L\colon
Z_{2}\to\mathbb{R}$ are locally Lipschitz functionals, $H\colon
Y\to(0,\infty)$ is a given function, and $F\colon X\to X^{\ast}$, $G\colon
Y\to Y^{\ast}$ are prescribed nonlinear operators. In the present, we are
interested in the following system of inequalities:

Find $(u,\sigma)\in\left(K\cap D(\phi)\right)\times\Lambda$ such that

$\displaystyle\left\\{\begin{array}[]{ll}B(v,\sigma)-B(u,\sigma)+\phi(v)-\phi(u)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))\geq\langle
F(u),v-u\rangle,\\\
B(u,\mu)-B(u,\sigma)+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))\geq\langle
G(\sigma),\mu-\sigma\rangle,\end{array}\right.$ (1.3)

for all $v\in K$ and all $\mu\in\Lambda$.

We point out the fact that the first line of problem (1.3) represents a
nonlinear variational-hemivariational inequality, while the second line is a
quasi-hemivariational inequality. Observe that the unknown pair $(u,\sigma)$
appears in both inequalities of (1.3) via the functional $B$ which will be
called coupling functional, so, the system (1.3) is called coupled.

Very recently, Bai, Migórski & Zeng Bai-Mig-Zeng20  established existence and
uniqueness results for the following system with bilinear coupling function:

Find $(u,\sigma)\in X\times\Lambda$ such that

$\displaystyle\left\\{\begin{array}[]{ll}\langle
A(u),v-u\rangle+b(v-u,\sigma)+J^{0}(\gamma(u);\gamma(v)-\gamma(u))\geq\langle
f,v-u\rangle,&\mbox{ for all }v\in X,\\\ b(u,\mu-\sigma)\leq 0,&\mbox{ for all
}\mu\in\Lambda,\end{array}\right.$ (1.6)

where $X,Y$ are reflexive Banach spaces, $Z$ is a Banach space, $f\in
X^{\ast}$, $A\colon X\to X^{\ast}$ is a nonlinear operator, $J\colon
Z\to\mathbb{R}$ is locally Lipschitz, $\gamma\colon X\to Z$ is a linear and
continuous operator and $b\colon X\times Y\to\mathbb{R}$ is a bilinear
continuous function. In fact, system (1.6) can be a useful mathematical model
to solve various complicated engineering problems, such as thermo-
piezoelectric media problems with semi-permeable thermal boundary conditions,
fluid mechanics problems with nonmonotone constitutive law, population dynamic
problems, and so forth, see for instance, Ekeland-Temam99 ; han2017numerical ;
Has-Hla-Necas96 ; Matei-RWA14 ; Matei15 ; Mig-Bai-Zeng19 ; zeng2021nonlinear .

Note that, if we choose $B(u,\sigma):=b(u,\sigma)$, $F(u):=f-A(u)$, $K:=X$,
$\phi\equiv 0$, $H\equiv 0$ and $G\equiv 0$, then the system (1.3) reduces to

Find $(u,\sigma)\in X\times\Lambda$ such that

$\left\\{\begin{array}[]{l}\langle
A(u),v-u\rangle+b(v-u,\sigma)+J^{0}(\gamma(u);\gamma(v)-\gamma(u))\geq\langle
f,v-u\rangle,\ \forall v\in X,\\\ b(u,\mu-\sigma)\geq 0,\
\forall\mu\in\Lambda,\end{array}\right.$ (1.7)

which is very similar to $\eqref{eqn2}$, but the second inequality is
reversed, therefore, even for bilinear coupling functions our results are new
and supplement the existing knowledge in the literature.

To our best knowledge, systems with nonlinear coupling functions were only
recently considered in Cos23  in the framework of variational inequalities.
More precisely, the following problem was investigated:

Find $(u,\sigma)\in K\times\Lambda$ such that

$\displaystyle\left\\{\begin{array}[]{ll}B(v,\sigma)-B(u,\sigma)+\chi(u,v-u)\geq\langle
f,v-u\rangle,\forall v\in K\\\
B(u,\mu)-B(u,\sigma)+\psi(\sigma,\mu-\sigma)\leq\langle
g,\mu-\sigma\rangle,\forall\mu\in\Lambda\end{array}\right.$ (1.10)

where $\chi:X\times X\to\mathbb{R},\psi:Y\times Y\to\mathbb{R}$ are nonlinear
functionals and $f\in X^{\ast}$, $g\in Y^{\ast}$. Note that the second
inequality of (1.3) and (1.10), respectively, have opposite sign, therefore
even for particular cases the solution sets of the two problems do not
coincide. Moreover, besides the fact that hemivariational inequalities are a
generalization of variational inequalities further difficulties occur due to
the presence of the nonlinear operators $F$ and $G$ in each inequality of
(1.3) and the fact that the second inequality is of quasi-hemivariational type
(since, in general, there does not exist a locally Lipschitz functional
$T\colon Y\to\mathbb{R}$ such that
$\partial_{C}T(\sigma)=H(\sigma)\partial_{C}J^{0}(\gamma_{2}(\sigma))$).

Finally, we point out the fact that choosing opposite sign for the second
inequality of (1.3) comes from applications point of view as it allows to
tackle the weak solvability of general 3D contact models with multivalued
constitutive law via bipotentials, an approach that has captured special
attention, see e.g., Bod-Sax ; Bul-deSax-Vall10 ; BSV ; Cos-Csi-Var15 ; Sax-
Feng98 ; Mat13 ; Mat-Nic11 , since this method was introduced at the beginning
of 1990’s by de Saxcé & Feng Sax-Feng91 . Such an application is provided in
the last section of the paper.

We fix next basic notations and recall some concepts and facts which we need
in this paper. For more details and connections one can consult the monographs
Clarke ; Cos-Kri-Var ; Ekeland-Temam99 ; sofonea2017variational .

For a real Banach space $E$ we denote its norm by $\|\cdot\|_{E}$ and by
$E^{\ast}$ its dual space. The duality pairing between $E^{\ast}$ and $E$ will
be denoted by $\langle\cdot,\cdot\rangle_{E^{\ast}\times E}$. If there is no
danger of confusion the subscripts will be ignored, i.e., we will simply write
$\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ from time to time. Particularly,
if $E$ is a Hilbert space, then $(\cdot,\cdot)_{E}$ stands for the inner
product of $E$.

Let $E,F$ be two real Banach spaces. If for each $u\in E$ there exists a
corresponding subset $A(u)\subseteq F$, then we say $A\colon E\to 2^{F}$ is a
set-valued mapping (or multifunction) from $E$ to $F$. For a set-valued
mapping $A\colon E\to 2^{F}$ we define domain of $A$ to be the set

$D(A):=\\{x\in E:\ A(x)\neq\emptyset\\}.$

The range of $A$ is the set

$R(A):=\bigcup_{x\in E}A(x),$

and the graph of $A$ is the set

$Gr(A):=\\{(x,y)\in E\times F:\ \ y\in A(x)\mbox{ and }x\in D(A)\\}.$

The inverse of the set-valued mapping $A\colon E\to 2^{F}$ is the set-valued
mapping $A^{-1}\colon F\to 2^{E}$ defined by

$A^{-1}(y):=\\{x\in E:\ \ y\in A(x)\\}.$

We point out the fact that every single-valued map $A:E\rightarrow F$ may be
regarded as the set-valued map $E\ni x\mapsto\\{A(x)\\}\in 2^{F}$. In this
case we identify $A(x)$ with its unique element.

######  Definition 1.1.

Let $\beta\colon E\to\mathbb{R}$ be a given functional. A set-valued mapping
$A\colon E\to 2^{E^{\ast}}$ is called $\beta$-relaxed dissipative if

$\langle\zeta_{y}-\zeta_{x},y-x\rangle\leq\beta(y-x)\mbox{ for all
}(x,\zeta_{x}),(y,\zeta_{y})\in Gr(A).$ (1.11)

######  Remark 1.2.

If $\beta(x):=-m\|x\|^{2}$ for some $m>0$, then $A$ is said to be strongly
dissipative, while for $\beta\equiv 0$ the set-valued mapping $A$ is called
dissipative. If $A$ is single-valued, then (1.11) reduces to

$\langle A(y)-A(x),y-x\rangle\leq\beta(y-x)\mbox{ for all }x,y\in E.$

Moreover, we have

  * $\bullet$

if $-A$ is $-\beta$-relaxed dissipative, then $A$ is called $\beta$-relaxed
monotone, i.e.,

$\langle\zeta_{y}-\zeta_{x},y-x\rangle\geq\beta(y-x)\mbox{ for all
}(x,\zeta_{x}),(y,\zeta_{y})\in Gr(A).$

  * $\bullet$

if $-A$ is strongly dissipative (namely, $A$ is $-\beta$-relaxed dissipative
and $\beta(x):=m\|x\|^{2}$ for some $m>0$), then $A$ is called strongly
monotone, i.e.,

$\langle\zeta_{y}-\zeta_{x},y-x\rangle\geq m\|y-x\|^{2}\mbox{ for all
}(x,\zeta_{x}),(y,\zeta_{y})\in Gr(A).$

  * $\bullet$

if $-A$ is dissipative (i.e., $\beta\equiv 0$), then $A$ is called monotone,
i.e.,

$\langle\zeta_{y}-\zeta_{x},y-x\rangle\geq 0\mbox{ for all
}(x,\zeta_{x}),(y,\zeta_{y})\in Gr(A).$

  * $\bullet$

if $-A$ is $-\beta$-relaxed dissipative and $\beta(x):=-m\|x\|^{2}$ for some
$m>0$, then $A$ is called relaxed monotone with constant $m$, i.e.,

$\langle\zeta_{y}-\zeta_{x},y-x\rangle\geq-m\|y-x\|^{2}\mbox{ for all
}(x,\zeta_{x}),(y,\zeta_{y})\in Gr(A).$

######  Definition 1.3.

A single-valued operator $A\colon E\to E^{\ast}$ is called hemicontinuous, if
for any $u,v,w\in E$ one has

$\lim_{t\to 0_{+}}\langle A(u+t(v-u)),w\rangle=\langle A(u),w\rangle.$

Next, we recall two kinds of set-valued mappings that play an important role
in Nonsmooth Analysis, namely, the subdifferential of a convex functional and
the Clarke subdifferential of a locally Lipschitz functional.

Let $\phi\colon E\to(-\infty,+\infty]$ be a prescribed functional. We define
its effective domain to be the set

$D(\phi):=\\{x\in E:\ \ \phi(x)<+\infty\\}.$

######  Definition 1.4.

A functional $\phi\colon E\to(-\infty,\infty]$ is said to be:

$\bullet$ proper, if $D(\phi)\neq\emptyset$;

$\bullet$ lower semicontinuous, if
$\liminf_{n\to\infty}\phi(x_{n})\geq\phi(x)$, whenever $x_{n}\to x$;

$\bullet$ convex, if $\phi(tx+(1-t)y)\leq t\phi(x)+(1-t)\phi(y)$, for all
$x,y\in E$ and all $t\in[0,1]$;

$\bullet$ Gâteaux differentiable at $x\in D(\phi)$, if there exists an element
$\nabla\phi(x)\in E^{\ast}$ (called the gradient of $\phi$ at $x$) such that

$\lim\limits_{t\rightarrow
0_{+}}\frac{\phi(x+ty)-\phi(x)}{t}=\left\langle\nabla\phi(x),y\right\rangle,\mbox{
for all }y\in E.$

Conversely, we say that $\phi\colon[-\infty,\infty)\to\mathbb{R}$ is upper
semicontinuous (resp. concave) if $-\phi$ is lower semicontinuous (resp.
convex).

The following result characterizes the Gâteaux differentiability of convex
functionals.

######  Proposition 1.5.

Let $\phi\colon E\to{\mathbb{R}}$ be a Gâteaux differentiable functional. The
following statements are equivalent:

  1. (i)

$\phi$ is convex;

  2. (ii)

$\phi(y)-\phi(x)\geq\langle\nabla\phi(x),y-x\rangle$, for all $y\in E$;

  3. (iii)

$\langle\nabla\phi(y)-\nabla\phi(x),y-x\rangle\geq 0$, for all $x,y\in E$.

A direct consequence of the above result is that convex and Gâteaux
differentiable functions are in fact lower semicontinuous. Proposition 1.5
also suggests the following generalization of the gradient of a convex
function.

######  Definition 1.6.

Let $\phi:E\rightarrow(-\infty,+\infty]$ be a convex function. The
subdifferential of $\phi$ at a point $x\in D(\phi)$ is the set

$\partial\phi(x):=\left\\{\xi\in E^{\ast}:\ \
\langle\xi,y-x\rangle\leq\phi(y)-\phi(x)\mbox{ for all }y\in E\right\\},$
(1.12)

and $\partial\phi(x):=\emptyset$ if $x\not\in D(\phi)$.

It is well known that if $\phi$ is convex and Gâteaux differentiable at a
point $x\in{\rm int\;}D(\phi)$, then $\partial\phi(x)$ contains exactly one
element, namely, $\partial\phi(x)=\\{\nabla\phi(x)\\}$. Moreover, it is well-
known that the set-valued mapping $\partial\phi\colon E\to 2^{E^{*}}$ is
monotone, i.e., $-\partial\phi$ is dissipative.

The Fenchel conjugate of $\phi\colon E\to(-\infty,+\infty]$ is the functional
$\phi^{\ast}\colon E^{\ast}\to(-\infty,+\infty]$ defined by

$\phi^{\ast}(\xi):=\sup\limits_{x\in
E}\left\\{\langle\xi,x\rangle-\phi(x)\right\\}\mbox{ for all $\xi\in E^{*}$}.$

######  Proposition 1.7.

Let $\phi\colon E\to(-\infty,+\infty]$ be a proper, convex and lower
semicontinuous function. Then, the following statements are true:

  1. (i)

$\phi^{\ast}$ is proper, convex and lower semicontinuous;

  2. (ii)

$\phi(x)+\phi^{\ast}(\xi)\geq\langle\xi,x\rangle$, for all $x\in E,\ \xi\in
E^{\ast}$;

  3. (iii)

$\xi\in\partial\phi(x)\Leftrightarrow
x\in\partial\phi^{\ast}(\xi)\Leftrightarrow\phi(x)+\phi^{\ast}(\xi)=\langle\xi,x\rangle$.

######  Definition 1.8.

A bipotential is a function $b\colon E\times E^{\ast}\to(-\infty,+\infty]$
satisfying the following conditions:

  1. (i)

for any $x\in E$, if $D(b(x,\cdot))\neq\varnothing$, then $b(x,\cdot)$ is
convex and lower semicontinuous, and for any $\xi\in E^{\ast}$, if
$D(b(\cdot,\xi))\neq\varnothing$, then $b(\cdot,\xi)$ is convex and lower
semicontinuous;

  2. (ii)

for all $x\in E$ and all $\xi\in E^{\ast}$ it holds
$b(x,\xi)\geq\langle\xi,x\rangle$;

  3. (iii)

$\xi\in\partial b(\cdot,\xi)(x)\Leftrightarrow x\in\partial
b(x,\cdot)(\xi)\Leftrightarrow b(x,\xi)=\langle\xi,x\rangle$.

We recall that a function $h\colon E\to{\mathbb{R}}$ is called locally
Lipschitz if for every $x\in E$ there exists a neighborhood $U$ of $x$ and a
positive constant $c_{x}=c_{x}(U)$ such that

$|h(z)-h(y)|\leq c_{x}\|z-y\|\mbox{ for all }y,z\in U.$

######  Definition 1.9.

Let $h:E\rightarrow{\mathbb{R}}$ be a locally Lipschitz function. The
generalized derivative (in the sense of Clarke) of $h$ at a point $x$ in the
direction $y$, denoted by $h^{0}(x;y)$, is defined by

$h^{0}(x;y):=\limsup\limits_{\stackrel{{\scriptstyle z\rightarrow
x}}{{t\downarrow 0}}}\frac{h(z+ty)-h(z)}{t}.$

Let $h\colon E_{1}\times\ldots\times E_{n}\to{\mathbb{R}}$ be locally
Lipschitz with respect to the $k^{th}$ variable. So, in the sequel, we denote
by $h^{0}_{,k}(x_{1},\ldots,x_{k},\ldots,x_{n};y_{k})$ the partial generalized
derivative of $h$ with respect to the $k^{th}$ variable, that is,

$h^{0}_{,k}(x_{1},\ldots,x_{k},\ldots,x_{n};y_{k}):=\limsup\limits_{\stackrel{{\scriptstyle
z_{k}\rightarrow x_{k}}}{{t\downarrow
0}}}\frac{h(x_{1},\ldots,z_{k}+ty_{k},\ldots,x_{n})-h(x_{1},\ldots,z_{k},\ldots,x_{n})}{t}.$

The next results recall some important properties of the generalized
derivative and subgradient in the sense of Clarke, (for the details proof see
Clarke Clarke ).

######  Proposition 1.10.

Let $h\colon E\to{\mathbb{R}}$ be a locally Lipschitz function such that
$c_{x}>0$ is the constant near the point $x\in E$. Then, we have

  1. (i)

the function $y\mapsto h^{0}(x;y)$ is finite, positively homogeneous,
subadditive and satisfies

$|h^{0}(x;y)|\leq c_{x}\|y\|\mbox{ for all $x,y\in E$};$

  2. (ii)

the function $(x,y)\mapsto h^{0}(x;y)$ is upper semicontinuous;

  3. (iii)

$(-h)^{0}(x;y)=h^{0}(x;-y)$.

######  Definition 1.11.

The Clarke subdifferential of a locally Lipschitz function $h\colon
E\to{\mathbb{R}}$ at a point $x$, denoted by $\partial_{C}h(x)$, is the
following subset of $E^{\ast}$

$\partial_{C}h(x):=\left\\{\xi\in E^{\ast}:\ \ \langle\xi,y\rangle\leq
h^{0}(x;y)\mbox{ for all }y\in E\right\\}.$

Let $h\colon E_{1}\times\ldots\times E_{n}\to{\mathbb{R}}$ be locally
Lipschitz with respect to the $k^{th}$ variable. In what follows, we use the
similar way to denote the partial Clarke subdifferential of $h$ with respect
to the $k^{th}$ variable, thus,

$\partial_{C}^{k}h(x_{1},\ldots,x_{n}):=\left\\{\xi_{k}\in E_{k}^{\ast}:\ \
\langle\xi_{k},x_{k}\rangle\leq
h^{0}_{,k}(x_{1},\dots,x_{k},\ldots,x_{n};y_{k})\mbox{ for all }y_{k}\in
E_{k}\right\\}.$

######  Proposition 1.12.

Let $h\colon E\to{\mathbb{R}}$ be a locally Lipschitz functional. Then the
following properties hold:

  1. (i)

For each $x\in E$, $\partial_{C}h(x)$ is a nonempty, convex, weak∗-compact
subset of $E^{\ast}$ and

$\|\zeta\|_{E^{*}}\leq c_{x},\mbox{ for all }\zeta\in\partial_{C}h(x),$

where $c_{x}$ is the Lipschitz constant near $x$;

  2. (ii)

For each $y\in E$ one has
$h^{0}(x;y)=\max\limits_{\zeta\in\partial_{C}h(u)}\langle\zeta,y\rangle$;

  3. (iii)

The set-valued mapping $x\mapsto\partial_{C}h(x)$ is weakly∗-closed;

  4. (iv)

The set-valued mapping $x\mapsto\partial_{C}h(x)$ is upper semicontinuous from
$E$ into $E^{\ast}$ endowed with the weak∗-topology.

We end this section with an alternative theorem for equilibrium problems due
to Mosco mosco  which will play a center role in proving the existence of
solutions for the coupled system of inequalities (1.3).

######  Theorem 1.13 (Mosco mosco ).

Let $\mathcal{K}$ be a nonempty, compact and convex subset of a topological
vector space $E$ and $\varphi:E\rightarrow(-\infty,\infty]$ be a proper,
convex and lower semicontinuous functional such that
${D}(\varphi)\cap\mathcal{K}\neq\emptyset$. Assume $g,h\colon E\times
E\to{\mathbb{R}}$ are two functions that satisfy the following conditions:

  1. (a)

$g(x,y)\geq h(x,y)\mbox{ for all }x,y\in E$;

  2. (b)

$y\mapsto h(x,y)$ is convex for each $y\in E$;

  3. (c)

$x\mapsto g(x,y)$ is upper semicontinous for each $x\in E$.

Then, for each $a\in\mathbb{R}$ the following alternative holds:

$\bullet$ either there exists $x_{0}\in{D}(\varphi)\cap\mathcal{K}$ such that

$\displaystyle g(x_{0},y)+\varphi(y)-\varphi(x_{0})\geq a\mbox{ for all }y\in
E,$

$\bullet$ or there exists $y_{0}\in E$ such that $h(x_{0},x_{0})<a$.

##  2 Existence results for bounded and unbounded constraint sets

The primary goal of this section is to establish the existence of solutions
for the nonlinear coupled system (1.3). Our approach is topological. More
precisely we employ the alternative Theorem 1.13 for general equilibrium
problems to obtain the existence theorems. The first set of assumptions under
which we are able to establish the existence of solutions is listed below.

${\bf(H_{0})}$ The functionals $\phi,J,L,H$ and the operators
$\gamma_{1},\gamma_{2}$ satisfy the following conditions:

  1. (i)

$\phi\colon X\to(-\infty,\infty]$ is a proper, convex and lower semicontinuous
functional such that $D(\phi)\cap K\neq\varnothing$;

  2. (ii)

$J\colon Z_{1}\to\mathbb{R}$ and $L\colon Z_{2}\to\mathbb{R}$ are locally
Lipschitz functionals;

  3. (iii)

$\gamma_{1}\colon X\to Z_{1}$ and $\gamma_{2}\colon Y\to Z_{2}$ are linear and
compact operators;

  4. (iv)

$H\colon Y\to(0,m_{H})$ is weakly continuous, i.e., $H(\sigma_{\alpha})\to
H(\sigma)$ whenever $\sigma_{\alpha}\rightharpoonup\sigma$ in $Y$, with some
$m_{H}>0$.

${\bf(H_{B})}$ $B\colon X\times Y\to\mathbb{\mathbb{R}}$ is convex and lower
semicontinuous in $X\times Y$.

${\bf(H_{F}^{1})}$ $F\colon X\to X^{\ast}$ is a nonlinear operator such that
the mapping $u\mapsto\langle F(u),v-u\rangle$ is weakly lower semicontinuous
for each $v\in X$, i.e.,

$\liminf_{n\to\infty}\langle F(u_{n}),v-u_{n}\rangle\geq\langle
F(u),v-u\rangle,$

whenever $u_{n}\rightharpoonup u$ in $X$ as $n\to\infty$ and $v\in X$.

${\bf(H_{G}^{1})}$ $G\colon Y\to Y^{\ast}$ is a nonlinear operator such that
the mapping $\sigma\mapsto\langle G(\sigma),\mu-\sigma\rangle$ is weakly lower
semicontinuous for each $\mu\in Y$, i.e.,

$\displaystyle\liminf_{n\to\infty}\langle
G(\sigma_{n}),\mu-\sigma_{n}\rangle\geq\langle G(\sigma),\mu-\sigma\rangle,$

whenever $\sigma_{n}\rightharpoonup\sigma$ in $Y$ as $n\to\infty$ and $\mu\in
Y$.

Note that, if $J\colon Z_{1}\to\mathbb{R}$ is Lipschitz continuous, then
${\bf(H_{0})}$(ii) is automatically fulfilled as $c_{x}$ is the same for every
$x\in Z_{1}$. Moreover, ${\bf(H_{F}^{1})}$ is automatically fulfilled if
$F(u)\equiv f$ for some fixed $f\in X^{\ast}$.

######  Lemma 2.1.

Assume $X$ and $Y$ are real reflexive Banach spaces and $K\subset X$ and
$\Lambda\subset Y$ are nonempty, bounded, closed and convex subsets. If
${\bf(H_{0}),\;(H_{B}),\;(H_{F}^{1})}$ and ${\bf(H_{G}^{1})}$ hold, then
problem (1.3) possesses at least one solution.

######  Proof 2.2.

Let $E:=X\times Y$, $\mathcal{K}:=K\times\Lambda$ and define
$\varphi\colon\mathcal{K}\to(-\infty,\infty]$ by

$\varphi(u,\sigma):=\phi(u)+I_{K}(u)+I_{\Lambda}(\sigma),$

where $I_{K}$ and $I_{\Lambda}$ are the indicator functions of $K$ and
$\Lambda$, respectively, i.e.,

$I_{K}(u):=\left\\{\begin{array}[]{ll}0,&\mbox{ if }u\in K\\\ +\infty,&\mbox{
otherwise}\end{array}\right.,\mbox{ and
}I_{\Lambda}(\sigma):=\left\\{\begin{array}[]{ll}0,&\mbox{ if
}\sigma\in\Lambda\\\ +\infty,&\mbox{ otherwise}\end{array}\right..$

Recall that $K$ and $\Lambda$ are nonempty, bounded, closed and convex subsets
of real reflexive Banach spaces $X$ and $Y$, respectively, it follows at once
that $\mathcal{K}$ is weakly compact, while $\varphi$ is proper, convex and
lower semicontinuous with $D(\varphi)=(K\cap D(\phi))\times\Lambda$.

Let us consider the bifunction
$g\colon\mathcal{K}\times\mathcal{K}\to\mathbb{R}$ by

$\displaystyle g((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(u),v-u\rangle$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\sigma),\mu-\sigma\rangle.$

It is not difficult to observe that $g((u,\sigma),(u,\sigma))=0$ for all
$(u,\sigma)\in\mathcal{K}$. Let $t\in(0,1)$ and
$(v_{1},\mu_{1}),(v_{2},\mu_{2})\in\mathcal{K}$ be arbitrary and
$v_{t}=tv_{1}+(1-t)v_{2}$ and $\mu_{t}=t\mu_{1}+(1-t)\mu_{2}$. From the
convexity of $\mathcal{K}\ni(v,\mu)\mapsto B(v,\mu)\in\mathbb{R}$ and the
positive homogeneity and subadditivity of $v\mapsto
J^{0}(\gamma_{1}(u);\gamma_{1}(v))$ and $\mu\mapsto
L^{0}(\gamma_{2}(\sigma);\gamma_{1}(\mu))$, we have

$\displaystyle g((u,\sigma)$
$\displaystyle,(v_{t},\mu_{t}))=B(v_{t},\mu_{t})-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v_{t}-u))-\langle
F(u),v_{t}-u\rangle$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu_{t}-\sigma))-\langle
G(\sigma),\mu_{t}-\sigma\rangle$ $\displaystyle\leq$ $\displaystyle
tB(v_{1},\mu_{1})+(1-t)B(v_{2},\mu_{2})+J^{0}(\gamma_{1}(u);t\gamma_{1}(v_{1}-u)+(1-t)\gamma_{1}(v_{2}-u))$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);t\gamma_{2}(\mu_{1}-\sigma)+(1-t)\gamma_{2}(\mu_{2}-\sigma))-B(u,\sigma)$
$\displaystyle-\langle F(u),t(v_{1}-u)+(1-t)(v_{2}-u)\rangle-\langle
G(\sigma),t(\mu_{1}-\sigma)+(1-t)(\mu_{2}-\sigma)\rangle$ $\displaystyle\leq$
$\displaystyle
t\bigg{[}B(v_{1},\mu_{1})+J^{0}(\gamma_{1}(u);\gamma_{1}(v_{1}-u))+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu_{1}-\sigma))-B(u,\sigma)$
$\displaystyle-\langle F(u),v_{1}-u\rangle-\langle
G(\sigma),\mu_{1}-\sigma\rangle\bigg{]}+(1-t)\bigg{[}B(v_{2},\mu_{2})+J^{0}(\gamma_{1}(u);\gamma_{1}(v_{2}-u))$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu_{2}-\sigma))-\langle
F(u),v_{2}-u\rangle-\langle
G(\sigma),\mu_{2}-\sigma\rangle-B(u,\sigma)\bigg{]}$ $\displaystyle\leq$
$\displaystyle
tg((u,\sigma),(v_{1},\mu_{1}))+(1-t)g((u,\sigma),(v_{2},\mu_{2})).$

This means that the mapping $(v,\mu)\mapsto g((u,\sigma),(v,\mu))$ is convex.

We claim that for each $(v,\mu)\in\mathcal{K}$ the mapping
$\mathcal{K}\ni(u,\sigma)\mapsto g((u,\sigma),(v,\mu))\in\mathbb{R}$ is weakly
upper semicontinuous. Let sequence
$\\{(u_{n},\sigma_{n})\\}\subset\mathcal{K}$ and $(u,\sigma)\in\mathcal{K}$ be
such that

$(u_{n},\sigma_{n})\rightharpoonup(u,\sigma)\mbox{ in $X\times Y$ as
$n\to\infty$.}$

It follows from the weakly lower semicontinuity of
$\mathcal{K}\ni(u,\sigma)\mapsto B(u,\sigma)\in\mathbb{R}$, the upper
semicontinuity of $Z_{1}\times Z_{1}\ni(u,v)\mapsto J^{0}(u;v)\in\mathbb{R}$
(see Proposition 1.12) that

$\displaystyle\left\\{\begin{array}[]{ll}\liminf_{n\to\infty}B(u_{n},\sigma_{n})\geq
B(u,\sigma),\\\
\limsup_{n\to\infty}J^{0}(\gamma_{1}(u_{n});\gamma_{1}(v-u_{n}))\leq
J^{0}(\gamma_{1}(u);\gamma_{1}(v-u)),\end{array}\right.$ (2.3)

where we have used the compactness of $\gamma_{1}$. Using hypotheses
${\bf(H_{F}^{1})}$ and ${\bf(H_{G}^{1})}$, it yields

$\displaystyle\liminf_{n\to\infty}\langle F(u_{n}),v-u_{n}\rangle\geq\langle
F(u),v-u\rangle$ (2.4)

and

$\displaystyle\liminf_{n\to\infty}\langle
G(\sigma_{n}),\mu-\sigma_{n}\rangle\geq\langle G(\sigma),\mu-\sigma\rangle.$
(2.5)

Employing hypotheses ${\bf(H_{0})}$(iii)-(iv) and the upper semicontinuity of
the mapping $Z_{2}\times Z_{2}\ni(\sigma,\mu)\mapsto
L^{0}(\sigma;\mu)\in\mathbb{R}$ (see Proposition 1.12), one has

$\displaystyle\limsup_{n\to\infty}H(\sigma_{n})L^{0}(\gamma_{2}(\sigma_{n});\gamma_{2}(\mu-\sigma_{n}))\leq$
$\displaystyle\limsup_{n\to\infty}\underbrace{|H(\sigma_{n})-H(\sigma)|}_{\to
0}\underbrace{L^{0}(\gamma_{2}(\sigma_{n});\gamma_{2}(\mu-\sigma_{n}))}_{bounded}$
$\displaystyle+\limsup_{n\to\infty}H(\sigma)L^{0}(\gamma_{2}(\sigma_{n});\gamma_{2}(\mu-\sigma_{n}))$
$\displaystyle\leq$
$\displaystyle\limsup_{n\to\infty}H(\sigma)L^{0}(\gamma_{2}(\sigma_{n});\gamma_{2}(\mu-\sigma_{n}))$
$\displaystyle\leq$ $\displaystyle
H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma)),$ (2.6)

where we have applied the compactness of $\gamma_{2}$ and the following
estimates

$\displaystyle|L^{0}(\gamma_{2}(\sigma_{n});\gamma_{2}(\mu-\sigma_{n}))|$
$\displaystyle=\left|\max_{\zeta\in\partial_{C}J(\gamma_{2}(\sigma_{n}))}\langle\zeta,\gamma_{2}(\mu-\sigma_{n})\rangle\right|=|\langle\zeta_{n},\gamma_{2}(\mu)-\gamma_{2}(\sigma_{n})\rangle|$
$\displaystyle\leq\|\zeta_{n}\|_{Z_{2}^{*}}\|\gamma_{2}(\mu)-\gamma_{2}(\sigma_{n})\|\leq
c_{\mu}\|\gamma_{2}(\mu)-\gamma_{2}(\sigma_{n})\|.$

The last inequality is obtained by using the fact that
$\gamma_{2}(\sigma_{n})\to\gamma_{2}(\sigma)$ as $n\to\infty$ and Proposition
1.10, and $c_{\sigma}>0$ is the Lipschitz constant of $L$ near the point
$\gamma_{2}\sigma\in Z_{2}$. Taking into account (2.3)–(2.2), we infer that

$\displaystyle\limsup_{n\to\infty}g((u_{n},\sigma_{n}),(v,\mu))\leq
g((u,\sigma),(v,\mu)),$

namely, for each $(v,\mu)\in\mathcal{K}$ the mapping
$\mathcal{K}\ni(u,\sigma)\mapsto g((u,\sigma),(v,\mu))\in\mathbb{R}$ is weakly
upper semicontinuous.

Therefore, all conditions of Theorem 1.13 are verified. So, we are now in
position to apply this theorem with $a:=0$, $h:=g$ and $E$ endowed with the
weak topology to conclude that there exists an element
$(u_{0},\sigma_{0})\in\mathcal{K}\cap D(\varphi)$ such that

$g((u_{0},\sigma_{0}),(v,\mu))+\varphi(v,\mu)-\varphi(u_{0},\sigma_{0})\geq
0\mbox{ for all }(v,\mu)\in E.$ (2.7)

Choosing $\mu:=\sigma_{0}$ in (2.7), we get that

$B(v,\sigma_{0})-B(u_{0},\sigma_{0})+J^{0}(\gamma_{1}(u_{0});\gamma_{1}(v-u_{0}))-\langle
F(u_{0}),v-u_{0}\rangle+\phi(v)-\phi(u_{0})\geq 0$

for all $v\in K$. Whereas, if we take $v:=u_{0}$ in (2.7), then

$B(u_{0},\mu)-B(u_{0},\sigma_{0})+H(\sigma_{0})L^{0}(\gamma_{2}(\sigma_{0});\gamma_{2}(\mu-\sigma_{0}))-\langle
G(\sigma_{0}),\mu-\sigma_{0}\rangle\geq 0$

for all $\mu\in\Lambda$. This shows that $(u_{0},\sigma_{0})$ is a solution of
problem (1.3). ∎

In hypotheses ${\bf(H_{F}^{1})}$ and ${\bf(H_{G}^{1})}$, we require that
$u\mapsto\langle F(u),v-u\rangle$ and $\sigma\mapsto\langle
G(\sigma),\mu-\sigma\rangle$ are both weakly lower semicontinuous. However,
this assumption strictly limits the application of our result, Lemma 2.1. In
order to extend the scope of applications to problem (1.3), we, further, make
necessary changes to the assumptions on $F$ and $G$, respectively. Under these
changes, we obtain a wide variety of existence results to problem (1.3).

To this end, we impose the following assumptions to functions $F$ and $G$.

${\bf(H_{F}^{2})}$ $F\colon X\to X^{\ast}$ is hemicontinuous and there exists
a concave functional $\beta_{F}:X\to\mathbb{R}$ with $\beta_{F}(0)\leq 0$ such
that $F$ is $\beta_{F}$-relaxed dissipative.

${\bf(H_{G}^{2})}$ $G\colon Y\to Y^{\ast}$ is hemicontinuous and there exists
a concave functional $\beta_{G}\colon Y\to\mathbb{R}$ with $\beta_{G}(0)\leq
0$ such that $G$ is $\beta_{G}$-relaxed dissipative.

${\bf(H_{F}^{3})}$ $F\colon X\to X^{\ast}$ is hemicontinuous and there exists
a concave functional $\beta_{T}\colon X\to\mathbb{R}$ with $\beta_{T}(0)\leq
0$ such that the set-valued operator $T\colon X\to 2^{X^{\ast}}$ defined by

$T(u):=F(u)-\gamma_{1}^{\ast}\partial_{C}J(\gamma_{1}(u))$

is $\beta_{T}$-relaxed dissipative.

${\bf(H_{G}^{3})}$ $G\colon Y\to Y^{\ast}$ is hemicontinuous and there exists
a concave functional $\beta_{S}\colon Y\to\mathbb{R}$ with $\beta_{S}(0)\leq
0$ such that the set-valued operator $S\colon Y\to 2^{Y^{\ast}}$ defined by

$S(\sigma):=G(\sigma)-H(\sigma)\gamma_{2}^{\ast}\partial_{C}L(\gamma_{2}(\sigma))$

is $\beta_{S}$-relaxed dissipative.

The second result regarding the existence of solutions for system (1.3) in the
case of bounded constraint sets is stated below.

######  Lemma 2.3.

The conclusion of Lemma 2.1 still holds if we replace ${\bf(H_{F}^{1})}$ by
${\bf(H_{F}^{j})}$ or ${\bf(H_{G}^{1})}$ by ${\bf(H_{G}^{k})}$, with
$j,k\in\\{1,2,3\\}$.

######  Proof 2.4.

As in the proof of Lemma 2.1, let $E:=X\times Y$,
$\mathcal{K}:=K\times\Lambda$ and
$\varphi\colon\mathcal{K}\to(-\infty,\infty]$ be defined by

$\varphi(u,\sigma):=\phi(u)+I_{K}(u)+I_{\Lambda}(\sigma).$

Case 1. Assume ${\bf(H_{F}^{2})}$ and ${\bf(H_{G}^{3})}$ hold.

Let us define the functions
$g,h\colon\mathcal{K}\times\mathcal{K}\to\mathbb{R}$ by

$\displaystyle g((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(v),v-u\rangle$
$\displaystyle+H(\mu)L^{0}(\gamma_{2}(\mu);\gamma_{2}(\mu-\sigma))-\langle
G(\mu),\mu-\sigma\rangle$

and

$\displaystyle h((u$
$\displaystyle,\sigma),(v,\mu)):=B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(u),v-u\rangle$
$\displaystyle-\beta_{F}(v-u)+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\sigma),\mu-\sigma\rangle-\beta_{S}(\mu-\sigma).$

Note that $h((u,\sigma),(u,\sigma))=-\beta_{F}(0)-\beta_{S}(0)\geq 0$ for all
$(u,\sigma)\in\mathcal{K}$. Arguing as in the proof of Lemma 2.1, it is
readily seen that $\mathcal{K}\ni(u,\sigma)\mapsto
g((u,\sigma),(v,\mu))\in\mathbb{R}$ is weakly upper semicontinuous and
$\mathcal{K}\ni(v,\mu)\mapsto h((u,\sigma),(v,\mu))\in\mathbb{R}$ is convex.
Moreover, hypothesis ${\bf(H_{G}^{3})}$ and Proposition 1.12 point out that
there exists $\xi_{\sigma}\in\partial_{C}L(\gamma_{2}(\sigma))$ and
$\xi_{\mu}\in\partial_{C}L(\gamma_{2}(\mu))$ such that

$L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))=\langle\xi_{\sigma},\gamma_{2}(\mu-\sigma)\rangle_{Z_{2}^{\ast}\times
Z_{2}}=\langle\gamma_{2}^{\ast}(\xi_{\sigma}),\mu-\sigma\rangle_{Y^{\ast}\times
Y},$

and

$L^{0}(\gamma_{2}(\mu);\gamma_{2}(\mu-\sigma))=\langle\xi_{\mu},\gamma_{2}(\mu-\sigma)\rangle_{Z_{2}^{\ast}\times
Z_{2}}=\langle\gamma_{2}^{\ast}(\xi_{\mu}),\mu-\sigma\rangle_{Y^{\ast}\times
Y}.$

Thus,

$\displaystyle g((u,\sigma),(v,\mu))-h((u,$
$\displaystyle\sigma),(v,\mu))=\beta_{F}(v-u)-\langle
F(v)-F(u),v-u\rangle+\beta_{S}(\mu-\sigma)$ $\displaystyle-\langle
G(\mu)-H(\mu)\gamma_{2}^{\ast}(\xi_{\mu})-G(\sigma)+H(\sigma)\gamma_{2}^{\ast}(\xi_{\sigma}),\mu-\sigma\rangle\geq
0.$

We apply again Theorem 1.13 with $a:=0$ to get the existence of
$(u_{0},\sigma_{0})\in\mathcal{K}\cap D(\varphi)$ such that

$g((u_{0},\sigma_{0}),(w,\lambda))+\varphi(w,\lambda)-\varphi(u_{0},\sigma_{0})\geq
0\mbox{ for all }(w,\lambda)\in E.$ (2.8)

Let $(v,\mu)\in\mathcal{K}$ be fixed. Then for any $t\in(0,1)$, by inserting
$(w,\lambda):=(u_{0}+t(v-u_{0}),\sigma_{0})$ in (2.8) we get

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
B(u_{0}+t(v-u_{0}),\sigma_{0})-B(u_{0},\sigma_{0})+J^{0}(\gamma_{1}(u_{0});t(\gamma_{1}(v-u_{0})))$
$\displaystyle-\langle
F(u_{0}+t(v-u_{0})),t(v-u_{0})\rangle+\phi(u_{0}+t(v-u_{0}))-\phi(u_{0})$
$\displaystyle\leq$ $\displaystyle
t\left[B(v,\sigma_{0})-B(u_{0},\sigma_{0})+J^{0}(\gamma_{1}(u_{0});\gamma_{1}(v-u_{0}))\right.$
$\displaystyle\left.-\langle
F(u_{0}+t(v-u_{0})),v-u_{0}\rangle+\phi(v)-\phi(u_{0})\right],$

where we have utilized the convexity of $B$ and $\phi$ as well as the
positivity homogeneity and subadditivity of $v\mapsto J^{0}(u;v)$. Dividing by
$t>0$ and then letting $t\to 0_{+}$, we apply the hemicontinuity of $F$ to get
the first inequality of (1.3). In order to get the second inequality of (1.3),
we fix $t\in(0,1)$ and choose
$(w,\lambda):=(u_{0},\sigma_{0}+t(\mu-\sigma_{0}))$ in (2.8) to get

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
B(u_{0},\sigma_{0}+t(\mu-\sigma_{0}))-B(u_{0},\sigma_{0})-\langle
G(\sigma_{0}+t(\mu-\sigma_{0})),t(\mu-\sigma_{0})\rangle$
$\displaystyle+H(\sigma_{0}+t(\mu-\sigma_{0}))L^{0}(\gamma_{2}(\sigma_{0}+t(\mu-\sigma_{0}));t\gamma_{2}(\mu-\sigma_{0}))$
$\displaystyle\leq$ $\displaystyle
t\left[B(u_{0},\mu)-B(u_{0},\sigma_{0})-\langle
G(\sigma_{0}+t(\mu-\sigma_{0})),\mu-\sigma_{0}\rangle\right.$
$\displaystyle\left.+H(\sigma_{0}+t(\mu-\sigma_{0}))L^{0}(\gamma_{2}(\sigma_{0}+t(\mu-\sigma_{0}));\gamma_{2}(\mu-\sigma_{0}))\right].$

Dividing again by $t>0$, then taking the $\limsup$ as $t\to 0_{+}$ we get that
$(u_{0},\sigma_{0})$ also satisfies the second inequality of (1.3).

Case 2. Assume that ${\bf(H_{F}^{2})}$ and ${\bf(H_{G}^{2})}$ hold.

We define the functions $g,h\colon\mathcal{K}\times\mathcal{K}\to\mathbb{R}$
by

$\displaystyle g((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(v),v-u\rangle$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\mu),\mu-\sigma\rangle$

and

$\displaystyle h((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(u),v-u\rangle-\beta_{F}(v-u)$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\sigma),\mu-\sigma\rangle-\beta_{G}(\mu-\sigma).$

It is obvious that $h((u,\sigma),(u,\sigma))=-\beta_{F}(0)-\beta_{S}(0)\geq 0$
for all $(u,\sigma)\in\mathcal{K}$. Moreover, the mapping
$\mathcal{K}\ni(u,\sigma)\mapsto g((u,\sigma),(v,\mu))\in\mathbb{R}$ is weakly
upper semicontinuous and $\mathcal{K}\ni(v,\mu)\mapsto
h((u,\sigma),(v,\mu))\in\mathbb{R}$ is convex. Additionally, we apply
hypotheses ${\bf(H_{F}^{2})}$ and ${\bf(H_{G}^{2})}$ to get

$\displaystyle g((u,\sigma),(v,\mu))-h((u,\sigma),(v,\mu))=$
$\displaystyle\beta_{F}(v-u)-\langle
F(v)-F(u),v-u\rangle+\beta_{G}(\mu-\sigma)$ $\displaystyle-\langle
G(\mu)-G(\sigma),\mu-\sigma\rangle$ $\displaystyle\geq$ $\displaystyle 0.$

This allows us to employ Theorem 1.13 with $a:=0$ to find
$(u_{0},\sigma_{0})\in\mathcal{K}\cap D(\varphi)$ such that (2.8) holds. Let
$(v,\mu)\in\mathcal{K}$ be fixed. Inserting
$(w,\lambda):=(u_{0}+t(v-u_{0}),\sigma_{0})$ with $t\in(0,1)$ in (2.8) we get

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
t\left[B(v,\sigma_{0})-B(u_{0},\sigma_{0})+J^{0}(\gamma_{1}(u_{0});\gamma_{1}(v-u_{0}))\right.$
$\displaystyle\left.-\langle
F(u_{0}+t(v-u_{0})),v-u_{0}\rangle+\phi(v)-\phi(u_{0})\right].$

Hence, from the hemicontinuity of $F$, we obtain the first inequality of
(1.3). Let $t\in(0,1)$ be arbitrary and put
$(w,\lambda):=(u_{0},\sigma_{0}+t(\mu-\sigma_{0}))$ in (2.8). We have

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
B(u_{0},\sigma_{0}+t(\mu-\sigma_{0}))-B(u_{0},\sigma_{0})-\langle
G(\sigma_{0}+t(\mu-\sigma_{0})),t(\mu-\sigma_{0})\rangle$
$\displaystyle+H(\sigma_{0})L^{0}(\gamma_{2}(\sigma_{0});t(\gamma_{2}(\mu-\sigma_{0})))$
$\displaystyle\leq$ $\displaystyle
t\left[B(u_{0},\mu)-B(u_{0},\sigma_{0})-\langle
G(\sigma_{0}+t(\mu-\sigma_{0})),\mu-\sigma_{0}\rangle+H(\sigma_{0})L^{0}(\gamma_{2}(\sigma_{0});\gamma_{2}(\mu-\sigma_{0}))\right].$

Invoking the hemicontinuity of $G$, we obtain the second inequality of (1.3).

Case 3. Suppose that ${\bf(H_{F}^{3})}$ and ${\bf(H_{G}^{2})}$ are satisfied.

Let $g,h\colon\mathcal{K}\times\mathcal{K}\to\mathbb{R}$ be defined by

$\displaystyle g((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(v);\gamma_{1}(v-u))-\langle
F(v),v-u\rangle$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\mu),\mu-\sigma\rangle$

and

$\displaystyle h((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(u),v-u\rangle-\beta_{T}(v-u)$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\sigma),\mu-\sigma\rangle-\beta_{G}(\mu-\sigma).$

It is obvious that $h((u,\sigma),(u,\sigma))=-\beta_{T}(0)-\beta_{S}(0)\geq 0$
for all $(u,\sigma)\in\mathcal{K}$. Again, $\mathcal{K}\ni(u,\sigma)\mapsto
g((u,\sigma),(v,\mu))\in\mathbb{R}$ is weakly upper semicontinuous and
$\mathcal{K}\ni(v,\mu)\mapsto h((u,\sigma),(v,\mu))\in\mathbb{R}$ is convex.
Additionally, we use hypotheses ${\bf(H_{F}^{3})}$ and ${\bf(H_{G}^{2})}$ to
get

$\displaystyle g((u,\sigma),(v,\mu))-h((u,\sigma),(v,\mu))=$
$\displaystyle\beta_{T}(v-u)-\langle
F(v)-\gamma_{1}^{*}\xi_{u}-F(u)+\gamma_{1}^{*}\xi_{v},v-u\rangle$
$\displaystyle+\beta_{G}(\mu-\sigma)-\langle
G(\mu)-G(\sigma),\mu-\sigma\rangle$ $\displaystyle\geq$ $\displaystyle 0,$

where $\xi_{u}\in\partial_{C}J(\gamma_{1}u)$ and $\xi_{v}\in\partial
J_{C}(\gamma_{1}v)$ are such that

$J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))=\langle\xi_{u},\gamma_{1}(v-u)\rangle_{Z_{1}^{\ast}\times
Z_{1}}=\langle\gamma_{1}^{\ast}(\xi_{u}),u-v\rangle_{X^{\ast}\times X},$

and

$J^{0}(\gamma_{1}(v);\gamma_{1}(v-u))=\langle\xi_{v},\gamma_{1}(v-u)\rangle_{Z_{1}^{\ast}\times
Z_{1}}=\langle\gamma_{1}^{\ast}(\xi_{v}),u-v\rangle_{X^{\ast}\times X}.$

We are now in a position to apply Theorem 1.13 with $a:=0$ to deduce that
there exists an element $(u_{0},\sigma_{0})\in\mathcal{K}\cap D(\varphi)$ such
that (2.8) holds. Let $(v,\mu)\in\mathcal{K}$ be fixed. Then for any
$t\in(0,1)$, we insert $(w,\lambda):=(u_{0}+t(v-u_{0}),\sigma_{0})$ in (2.8)
to get

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
t\left[B(v,\sigma_{0})-B(u_{0},\sigma_{0})+J^{0}(\gamma_{1}(u_{0}+t(v-u_{0}));\gamma_{1}(v-u_{0}))\right.$
$\displaystyle\left.-\langle
F(u_{0}+t(v-u_{0})),v-u_{0}\rangle+\phi(v)-\phi(u_{0})\right].$

Hence, from the hemicontinuity of $F$ and upper semicontinuity of $Z_{1}\ni
u\mapsto J^{0}(u;v)$, we obtain the first inequality of (1.3). For $t\in(0,1)$
plugging $(w,\lambda):=(u_{0},\sigma_{0}+t(\mu-\sigma_{0}))$ in (2.8) we have

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
t\left[B(u_{0},\mu)-B(u_{0},\sigma_{0})-\langle
G(\sigma_{0}+t(\mu-\sigma_{0})),\mu-\sigma_{0}\rangle+H(\sigma_{0})L^{0}(\gamma_{2}(\sigma_{0});\gamma_{2}(\mu-\sigma_{0}))\right].$

Invoking the hemicontinuity of $G$ the second inequality of (1.3) is obtained.

Case 4. Let ${\bf(H_{F}^{3})}$ and ${\bf(H_{G}^{3})}$ be fulfilled.

Consider the functions $g,h\colon\mathcal{K}\times\mathcal{K}\to\mathbb{R}$
defined by

$\displaystyle g((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(v);\gamma_{1}(v-u))-\langle
F(v),v-u\rangle$
$\displaystyle+H(\mu)L^{0}(\gamma_{2}(\mu);\gamma_{2}(\mu-\sigma))-\langle
G(\mu),\mu-\sigma\rangle$

and

$\displaystyle h((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(u),v-u\rangle-\beta_{T}(v-u)$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\sigma),\mu-\sigma\rangle-\beta_{S}(\mu-\sigma).$

Then, we have $h((u,\sigma),(u,\sigma))=-\beta_{T}(0)-\beta_{S}(0)\geq 0$ for
all $(u,\sigma)\in\mathcal{K}$, the mapping $\mathcal{K}\ni(u,\sigma)\mapsto
g((u,\sigma),(v,\mu))\in\mathbb{R}$ is weakly upper semicontinuous, while the
mapping $\mathcal{K}\ni(v,\mu)\mapsto h((u,\sigma),(v,\mu))\in\mathbb{R}$ is
convex. Let $\xi_{\sigma}\in\partial_{C}L(\gamma_{2}(\sigma))$ and
$\xi_{\mu}\in\partial_{C}L(\gamma_{2}(\mu))$ be such that

$L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))=\langle\xi_{\sigma},\gamma_{2}(\mu-\sigma)\rangle_{Z_{2}^{\ast}\times
Z_{2}}=\langle\gamma_{2}^{\ast}(\xi_{\sigma}),\mu-\sigma\rangle_{Y^{\ast}\times
Y},$

and

$L^{0}(\gamma_{2}(\mu);\gamma_{2}(\mu-\sigma))=\langle\xi_{\mu},\gamma_{2}(\mu-\sigma)\rangle_{Z_{2}^{\ast}\times
Z_{2}}=\langle\gamma_{2}^{\ast}(\xi_{\mu}),\mu-\sigma\rangle_{Y^{\ast}\times
Y}.$

We apply hypotheses ${\bf(H_{F}^{3})}$ and ${\bf(H_{G}^{3})}$ to get

$\displaystyle g((u,\sigma),(v,\mu))$
$\displaystyle-h((u,\sigma),(v,\mu))=\beta_{T}(v-u)-\langle
F(v)-\gamma_{1}^{*}\xi_{u}-F(u)+\gamma_{1}^{*}\xi_{v},v-u\rangle$
$\displaystyle+\beta_{S}(\mu-\sigma)-\langle
G(\mu)-H(\mu)\gamma_{2}^{\ast}(\xi_{\mu})-G(\sigma)+H(\sigma)\gamma_{2}^{\ast}(\xi_{\sigma}),\mu-\sigma\rangle$
$\displaystyle\geq$ $\displaystyle 0,$

where $\xi_{u}\in\partial_{C}J(\gamma_{1}u)$ and $\xi_{v}\in\partial
J_{C}(\gamma_{1}v)$ are such that

$J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))=\langle\xi_{u},\gamma_{1}(v-u)\rangle_{Z_{1}^{\ast}\times
Z_{1}}=\langle\gamma_{1}^{\ast}(\xi_{u}),u-v\rangle_{X^{\ast}\times X},$

and

$J^{0}(\gamma_{1}(v);\gamma_{1}(v-u))=\langle\xi_{v},\gamma_{1}(v-u)\rangle_{Z_{1}^{\ast}\times
Z_{1}}=\langle\gamma_{1}^{\ast}(\xi_{v}),u-v\rangle_{X^{\ast}\times X}.$

We apply Theorem 1.13 with $a:=0$ to find an element
$(u_{0},\sigma_{0})\in\mathcal{K}\cap D(\varphi)$ such that (2.8) is
fulfilled. Let $(v,\mu)\in\mathcal{K}$ be fixed. Then for any $t\in(0,1)$, we
insert $(w,\lambda):=(u_{0}+t(v-u_{0}),\sigma_{0})$ in (2.8) to get

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
t\left[B(v,\sigma_{0})-B(u_{0},\sigma_{0})+J^{0}(\gamma_{1}(u_{0}+t(v-u_{0}));\gamma_{1}(v-u_{0}))\right.$
$\displaystyle\left.-\langle
F(u_{0}+t(v-u_{0})),v-u_{0}\rangle+\phi(v)-\phi(u_{0})\right].$

Hence, from the hemicontinuity of $F$ and upper semicontinuity of $Z_{1}\ni
u\mapsto J^{0}(u;v)$, we obtain the first inequality of (1.3). Let $t\in(0,1)$
be arbitrary and put $(w,\lambda):=(u_{0},\sigma_{0}+t(\mu-\sigma_{0}))$ in
(2.8). We have

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
t\left[B(u_{0},\mu)-B(u_{0},\sigma_{0})-\langle
G(\sigma_{0}+t(\mu-\sigma_{0})),\mu-\sigma_{0}\rangle\right.$
$\displaystyle\left.+H(\sigma_{0}+t(\mu-\sigma_{0}))L^{0}(\gamma_{2}(\sigma_{0}+t(\mu-\sigma_{0}));\gamma_{2}(\mu-\sigma_{0}))\right].$

Invoking the hemicontinuity of $G$ and upper semicontinuity of
$Y\ni\sigma\mapsto H(\sigma)L^{0}(\gamma_{2}\sigma;\mu)$, we get the second
inequality of (1.3).

Case 5. Let ${\bf(H_{F}^{1})}$ and ${\bf(H_{G}^{2})}$ be fulfilled.

Let us introduce the functions
$g,h\colon\mathcal{K}\times\mathcal{K}\to\mathbb{R}$ given by

$\displaystyle g((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v)-\gamma_{1}(u))-\langle
F(u),v-u\rangle$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu)-\gamma_{2}(\sigma))-\langle
G(\mu),\mu-\sigma\rangle$

and

$\displaystyle h((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v)-\gamma_{1}(u))-\langle
F(u),v-u\rangle$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu)-\gamma_{2}(\sigma))-\langle
G(\sigma),\mu-\sigma\rangle-\beta_{G}(\mu-\sigma)$

for all $(u,\sigma),(v,\mu)\in\mathcal{K}$.

Then, we have that $h((u,\sigma),(u,\sigma))=-\beta_{G}(0)\geq 0$ for all
$(u,\sigma)\in\mathcal{K}$, the mapping $\mathcal{K}\ni(u,\sigma)\mapsto
g((u,\sigma),(v,\mu))\in\mathbb{R}$ is weakly upper semicontinuous, while the
mapping $\mathcal{K}\ni(v,\mu)\mapsto h((u,\sigma),(v,\mu))\in\mathbb{R}$ is
convex. It follows from hypothesis ${\bf(H_{G}^{2})}$ that

$\displaystyle
g((u,\sigma),(v,\mu))-h((u,\sigma),(v,\mu))=\beta_{G}(\mu-\sigma)-\langle
G(\mu)-G(\sigma),\mu-\sigma\rangle\geq 0.$

We apply Theorem 1.13 with $a:=0$ to find an element
$(u_{0},\sigma_{0})\in\mathcal{K}\cap D(\varphi)$ such that (2.8) is
fulfilled. Let $(v,\mu)\in\mathcal{K}$ be fixed. Then for any $t\in(0,1)$, we
insert $(w,\lambda):=(v,\sigma_{0})$ and
$(w,\lambda):=(u_{0},\sigma_{0}+t(\mu-\sigma_{0}))$ in (2.8), respectively,
and pass to the upper limit as $t\to 0^{+}$ for the second resulting
inequality to get the (1.3).

Case 6. Let ${\bf(H_{F}^{1})}$ and ${\bf(H_{G}^{3})}$ be fulfilled.

Let us introduce the functions
$g,h\colon\mathcal{K}\times\mathcal{K}\to\mathbb{R}$ given by

$\displaystyle g((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(u),v-u\rangle$
$\displaystyle+H(\mu)L^{0}(\gamma_{2}(\mu);\gamma_{2}(\mu-\sigma))-\langle
G(\mu),\mu-\sigma\rangle$

and

$\displaystyle h((u$
$\displaystyle,\sigma),(v,\mu)):=B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(u),v-u\rangle$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\sigma),\mu-\sigma\rangle-\beta_{S}(\mu-\sigma).$

for all $(u,\sigma),(v,\mu)\in\mathcal{K}$.

Then, we have that $h((u,\sigma),(u,\sigma))=-\beta_{S}(0)\geq 0$ for all
$(u,\sigma)\in\mathcal{K}$, the mapping $\mathcal{K}\ni(u,\sigma)\mapsto
g((u,\sigma),(v,\mu))\in\mathbb{R}$ is weakly upper semicontinuous, while the
mapping $\mathcal{K}\ni(v,\mu)\mapsto h((u,\sigma),(v,\mu))\in\mathbb{R}$ is
convex. It follows from hypothesis ${\bf(H_{G}^{3})}$ that
$g((u,\sigma),(v,\mu))-h((u,\sigma),(v,\mu))\geq 0.$

We apply Theorem 1.13 with $a:=0$ to find an element
$(u_{0},\sigma_{0})\in\mathcal{K}\cap D(\varphi)$ such that (2.8) is
fulfilled. Let $(v,\mu)\in\mathcal{K}$ be fixed. Then for any $t\in(0,1)$, we
insert $(w,\lambda):=(v,\sigma_{0})$ and
$(w,\lambda):=(u_{0},\sigma_{0}+t(\mu-\sigma_{0}))$ in (2.8), respectively,
and pass to the upper limit as $t\to 0^{+}$ for the second resulting
inequality to get the (1.3).

Case 7. Let ${\bf(H_{F}^{2})}$ and ${\bf(H_{G}^{1})}$ be fulfilled.

Let us introduce the functions
$g,h\colon\mathcal{K}\times\mathcal{K}\to\mathbb{R}$ given by

$\displaystyle g((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(v),v-u\rangle$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\sigma),\mu-\sigma\rangle$

and

$\displaystyle h((u$
$\displaystyle,\sigma),(v,\mu)):=B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(u),v-u\rangle$
$\displaystyle-\beta_{F}(v-u)+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\sigma),\mu-\sigma\rangle.$

for all $(u,\sigma),(v,\mu)\in\mathcal{K}$.

Then, we have that $h((u,\sigma),(u,\sigma))=-\beta_{F}(0)\geq 0$ for all
$(u,\sigma)\in\mathcal{K}$, the mapping $\mathcal{K}\ni(u,\sigma)\mapsto
g((u,\sigma),(v,\mu))\in\mathbb{R}$ is weakly upper semicontinuous, while the
mapping $\mathcal{K}\ni(v,\mu)\mapsto h((u,\sigma),(v,\mu))\in\mathbb{R}$ is
convex. It follows from hypothesis ${\bf(H_{G}^{3})}$ that
$g((u,\sigma),(v,\mu))-h((u,\sigma),(v,\mu))\geq 0.$

We apply Theorem 1.13 with $a:=0$ to find an element
$(u_{0},\sigma_{0})\in\mathcal{K}\cap D(\varphi)$ such that (2.8) is
fulfilled. Let $(v,\mu)\in\mathcal{K}$ be fixed. Then for any $t\in(0,1)$, we
insert $(w,\lambda):=(tv+(1-t)u_{0},\sigma_{0})$ and
$(w,\lambda):=(u_{0},\mu)$ in (2.8), respectively, and pass to the upper limit
as $t\to 0^{+}$ for the first resulting inequality to get the (1.3).

Case 8. Let ${\bf(H_{F}^{3})}$ and ${\bf(H_{G}^{1})}$ be fulfilled.

Consider the functions $g,h\colon\mathcal{K}\times\mathcal{K}\to\mathbb{R}$
given by

$\displaystyle g((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(v);\gamma_{1}(v-u))-\langle
F(v),v-u\rangle$
$\displaystyle+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\sigma),\mu-\sigma\rangle$

and

$\displaystyle h((u,\sigma),(v,\mu)):=$ $\displaystyle
B(v,\mu)-B(u,\sigma)+J^{0}(\gamma_{1}(u);\gamma_{1}(v-u))-\langle
F(u),v-u\rangle$
$\displaystyle-\beta_{T}(v-u)+H(\sigma)L^{0}(\gamma_{2}(\sigma);\gamma_{2}(\mu-\sigma))-\langle
G(\sigma),\mu-\sigma\rangle$

for all $(u,\sigma),(v,\mu)\in\mathcal{K}$.

Then, we have that $h((u,\sigma),(u,\sigma))=-\beta_{T}(0)\geq 0$ for all
$(u,\sigma)\in\mathcal{K}$, the mapping $\mathcal{K}\ni(u,\sigma)\mapsto
g((u,\sigma),(v,\mu))\in\mathbb{R}$ is weakly upper semicontinuous, while the
mapping $\mathcal{K}\ni(v,\mu)\mapsto h((u,\sigma),(v,\mu))\in\mathbb{R}$ is
convex. It follows from hypothesis ${\bf(H_{G}^{3})}$ that
$g((u,\sigma),(v,\mu))-h((u,\sigma),(v,\mu))\geq 0.$

We apply Theorem 1.13 with $a:=0$ to find an element
$(u_{0},\sigma_{0})\in\mathcal{K}\cap D(\varphi)$ such that (2.8) is
fulfilled. Let $(v,\mu)\in\mathcal{K}$ be fixed. Then for any $t\in(0,1)$, we
insert $(w,\lambda):=(tv+(1-t)u_{0},\sigma_{0})$ and
$(w,\lambda):=(u_{0},\mu)$ in (2.8), respectively, and pass to the upper limit
as $t\to 0^{+}$ for the first resulting inequality to get the (1.3).

This completes the proof of the lemma.∎

We end this section with an existence result when at least one of the
constraint sets is unbounded. To this end, we need a coercivity condition.
Here, and hereafter, we consider the space $X\times Y$ is endowed with the
norm $\|(u,\sigma)\|:=\sqrt{\|u\|^{2}_{X}+\|\sigma\|^{2}_{Y}}$. We consider
the following coercivity condition:

${\bf(C)}$ There exist $\bar{w}_{0}\in K\cap D(\phi)$ and
$\bar{\tau}_{0}\in\Lambda$ such that

${\displaystyle\frac{2B(u,\sigma)-B(\bar{w}_{0},\sigma)-B(u,\bar{\tau}_{0})+\langle
F(u),\bar{w}_{0}-u\rangle+\langle
G(\sigma),\bar{\tau}_{0}-\sigma\rangle}{\|(u,\sigma)\|}\to\infty},\mbox{ as
}\|(u,\sigma)\|\to\infty.$

We also consider ${\bf(H_{0}^{\prime})}$ to be set of assumptions obtained
from ${\bf(H_{0})}$ by replacing (ii) with the slightly stronger condition:

(ii’) $J\colon Z_{1}\to\mathbb{R}$ and $L:Z_{2}\to\mathbb{R}$ are locally
Lipschitz functionals and there exist $\alpha_{J}>0$ and $\alpha_{L}>0$ such
that

$\sup_{x\in Z_{1}}c_{x}\leq\alpha_{J}\mbox{ and }\sup_{y\in
Z_{2}}c_{y}\leq\alpha_{L},$

where $c_{x}>0$ (resp. $c_{y}>0$) denotes the Lipschitz constant of $J$ (resp.
$L$) near the point $x\in Z_{1}$ (resp. $y\in Z_{2}$).

######  Theorem 2.5.

Let $X,Y$ be real reflexive Banach spaces. Assume that $K\subseteq X$,
$\Lambda\subseteq Y$ are nonempty closed convex sets such that either
$K\times\Lambda$ is bounded or ${\bf(C)}$ is satisfied. If, in addition,
${\bf(H_{0}^{\prime})}$, ${\bf(H_{B})}$, ${\bf(H_{F}^{j})}$,
${\bf(H_{G}^{k})}$ with $j,k\in\\{1,2,3\\}$, then problem (1.3) possesses at
least one solution in $K\times\Lambda$.

######  Proof 2.6.

If $K\times\Lambda$ is bounded, then the desired conclusion can be obtained
directly by using Lemmas 2.7 and 2.3.

So, we assume now that $K\times\Lambda$ is unbounded. Let $r_{0}>0$ be
sufficiently large such that $K_{r}:=K\cap\bar{B}_{X}(\bar{w}_{0},r)$ and
$\Lambda_{r}:=\Lambda\cap\bar{B}_{Y}(\bar{\tau}_{0},r)$ are both nonempty for
all $r\geq r_{0}$. Let $r\geq r_{0}$ be arbitrary fixed. Next, we consider the
following intermediate problem:

Find $(u,\sigma)\in\left(K_{r}\cap D(\phi)\right)\times\Lambda_{r}$ such that

$\displaystyle\left\\{\begin{array}[]{ll}B(v,\sigma)-B(u,\sigma)+\phi(v)-\phi(u)+J^{0}(\gamma_{1}u;\gamma_{1}(v-u))\geq\langle
F(u),v-u\rangle,\\\
B(u,\mu)-B(u,\sigma)+H(\sigma)L^{0}(\gamma_{2}\sigma;\gamma_{2}(\mu-\
\sigma))\geq\langle G(\sigma),\mu-\sigma\rangle,\end{array}\right.$ (2.11)

for all $v\in K_{r}$ and all $\mu\in\Lambda_{r}$.

In virtue of Lemmas 2.7 and 2.3, we can see that problem (2.11) has at least
one solution, say $(u_{r},\sigma_{r})\in K_{r}\times\Lambda_{r}$.

We claim that there exists a constant $r_{1}\geq r_{0}$ such that each
solution of problem (2.11) with $r=r_{1}$ satisfies the following inequality

$\displaystyle\max\\{\|u_{r_{1}}-\bar{w}_{0}\|_{X},\|\sigma_{r_{1}}-\bar{\tau}_{0}\|_{Y}\\}<r_{1}.$
(2.12)

Arguing by contradiction, we assume that for each $r\geq r_{0}$ it holds

$\displaystyle\max\\{\|u_{r}-\bar{w}_{0}\|_{X},\|\sigma_{r}-\bar{\tau_{0}}\|_{Y}\\}=r.$

Then, plugging $(v,\mu):=(\bar{w}_{0},\bar{\tau}_{0})$ into (2.11) we find
that

$\left\\{\begin{array}[]{l}B(\bar{w}_{0},\sigma_{r})-B(u_{r},\sigma_{r})+\phi(\bar{w}_{0})-\phi(u_{r})+J^{0}(\gamma_{1}(u_{r});\gamma_{1}(\bar{w}_{0}-u_{r}))\geq\langle
F(u_{r}),\bar{w}_{0}-u_{r}\rangle,\\\
B(u_{r},\bar{\tau}_{0})-B(u_{r},\sigma_{r})+H(\sigma_{r})L^{0}(\gamma_{2}(\sigma_{r});\gamma_{2}(\bar{\tau}_{0}-\sigma_{r}))\geq\langle
G(\sigma_{r}),\bar{\tau}-\sigma_{r}\rangle.\end{array}\right.$

Summing up the last two inequalities and we get

$\displaystyle\phi(\bar{w}_{0})$
$\displaystyle-\phi(u_{r})-\langle\zeta_{u_{r}},\gamma_{1}(\bar{w}_{0}-u_{r})\rangle-H(\sigma_{r})\langle\zeta_{\sigma_{r}}\gamma_{2}(\bar{\tau}_{0}-\sigma_{r})\rangle\geq
2B(u_{r},\sigma_{r})-B(u_{r},\bar{\tau}_{0})$
$\displaystyle-B(\bar{w}_{0},\sigma_{r})+\langle
F(u_{r}),\bar{w}_{0}-u_{r}\rangle-\langle
G(\sigma_{r}),\bar{\tau}_{0}-\sigma_{r}\rangle,$ (2.13)

where $\zeta_{u_{r}}\in\partial_{C}J(\gamma_{1}(u_{r}))$ and
$\zeta_{\sigma_{r}}\in\partial_{C}L(\gamma_{2}(\sigma_{r}))$ are such that

$\displaystyle\langle\zeta_{u_{r}},\gamma_{1}(\bar{w_{0}}-u_{r})\rangle_{Z_{1}^{*}\times
Z_{1}}=\max_{\zeta\in\partial_{C}J(\gamma_{1}(u_{r}))}\langle\zeta,\gamma_{1}(\bar{w}_{0}-u_{r})\rangle_{Z_{1}^{*}\times
Z_{1}},$
$\displaystyle\langle\zeta_{\sigma_{r}},\gamma_{2}(\bar{\tau}_{0}-\sigma_{r})\rangle_{Z_{2}^{*}\times
Z_{2}}=\max_{\zeta\in\partial_{C}L(\gamma_{2}(\sigma_{r}))}\langle\zeta,\gamma_{2}(\bar{\tau}_{0}-\sigma_{r})\rangle_{Z_{2}^{*}\times
Z_{2}}.$

Since any convex and l.s.c. functional is bounded below (see, e.g,
(Brezis2011, , Proposition 1.10)) there exist constants
$\alpha_{\phi},\beta_{\phi}\geq 0$ such that

$\displaystyle\phi(u)\geq-\alpha_{\phi}\|u\|_{X}-\beta_{\phi},\ \forall u\in
X.$ (2.14)

Inserting (2.14) into (2.6), we have

$\displaystyle 2B(u_{r},\sigma_{r})$
$\displaystyle-B(u_{r},\bar{\tau}_{0})-B(\bar{w}_{0},\sigma_{r})\langle
F(u_{r}),\bar{w}_{0}-u_{r}\rangle-\langle
G(\sigma_{r}),\bar{\tau}_{0}-\sigma_{r}\rangle\leq\phi(\bar{w}_{0})+\alpha_{\phi}\|u\|_{X}$
$\displaystyle+\beta_{\phi}+|\langle\zeta_{u_{r}},\gamma_{1}(\bar{w}_{0}-u_{r})\rangle|+H(\sigma_{r})|\langle\zeta_{\sigma_{r}},\gamma_{2}(\bar{\tau}_{0}-\sigma_{r})\rangle|.$
(2.15)

Dividing (2.6) by $\sqrt{\|u_{r}\|^{2}_{X}+\|\sigma_{r}\|^{2}_{Y}}$ and
letting $r\to\infty$, it gives

$\displaystyle\infty=$
$\displaystyle\lim_{r\to\infty}\frac{2B(u_{r},\sigma_{r})-B(u_{r},\bar{\tau}_{0})-B(\bar{w}_{0},\sigma_{r})+\langle
F(u_{r}),\bar{w}_{0}-u_{r}\rangle+\langle
G(\sigma_{r}),\bar{\tau}_{0}-\sigma_{r}\rangle}{\|(u_{r},\sigma_{r})\|}$
$\displaystyle\leq$
$\displaystyle\lim_{r\to\infty}\frac{\phi(\bar{w}_{0})+\alpha_{\phi}\|u\|_{X}+\beta_{\phi}+|\langle\zeta_{u_{r}},\gamma_{1}(\bar{w}_{0}-u_{r})\rangle|+H(\sigma_{r})|\langle\zeta_{\sigma_{r}},\gamma_{2}(\bar{\tau}_{0}-\sigma_{r})\rangle|}{\|(u_{r},\sigma_{r})\|}$
$\displaystyle\leq$
$\displaystyle\alpha_{\phi}+\alpha_{J}\|\gamma_{1}\|+\alpha_{L}m_{H}\|\gamma_{2}\|,$

which is obviously a contradiction. Therefore, we conclude that there exists a
constant $r_{1}>0$ such that inequality (2.12) is satisfied.

Let $(u_{r_{1}},\sigma_{r_{1}})\in K_{r_{1}}\times\Lambda_{r_{1}}$ be a
solution to problem (2.11) with $r=r_{1}$. Furthermore, we are going to show
that $(u_{r_{1}},\sigma_{r_{1}})$ solves problem (1.3) as well. Let
$(v,\mu)\in K\times\Lambda$ be arbitrarily fixed. It follows from (2.12)
(i.e., $(u_{r_{1}},\sigma_{r_{1}})\in B_{X}(\bar{w}_{0},r_{1})\times
B_{Y}(\bar{\tau}_{0},r_{1})$) that there exists a constant $t\in(0,1)$ such
that

$(w,\lambda):=(u_{r_{1}},\sigma_{r_{1}})+t(v-u_{r_{1}},\mu-\sigma_{r_{1}})\in
K_{r_{1}}\times\Lambda_{r_{1}}.$

Inserting $(u,\sigma)=(u_{r_{1}},\sigma_{r_{1}})$ and $(v,\mu)=(w,\lambda)$
into the first inequality of (2.11), we have

$\displaystyle t\langle F(u_{r_{1}},v-u_{r_{1}})\rangle\leq$ $\displaystyle
B(w,\sigma_{r_{1}})-B((u_{r_{1}},\sigma_{r_{1}}))+\phi(w)-\phi(u_{r_{1}})+J^{0}(\gamma_{1}(u_{r_{1}});\gamma_{1}(w-u_{r_{1}}))$
$\displaystyle\leq$ $\displaystyle
t\left[B(v,\sigma_{r_{1}})-B(u_{r_{1}},\sigma_{r_{1}})+\phi(v)-\phi(u_{r_{1}})+J^{0}(\gamma_{1}(u_{r});\gamma_{1}(v-u_{r}))\right].$

Dividing by $t>0$ we infer directly that $(u_{r_{1}},\sigma_{r_{1}})$
satisfies the first inequality of (1.3). Likewise, we also can prove that
$(u_{r_{1}},\sigma_{r_{1}})$ satisfies the second inequality of (1.3). This
means that $(u_{r_{1}},\sigma_{r_{1}})\in K_{r_{1}}\times\Lambda_{r_{1}}$ is a
solution to problem (1.3), hence the proof is now complete. ∎

##  3 Applications to Contact Mechanics

Throughout this section ${\cal S}^{m}$ denotes the linear space of second
order symmetric tensors on ${\mathbb{R}}^{m}$, i.e., ${\cal
S}^{m}={\mathbb{R}}^{m\times m}_{s}$. The inner products and the corresponding
norms on $\mathbb{R}^{m}$ and ${\cal S}^{m}$ are defined by

$\mbox{\boldmath{$u$}}\cdot\mbox{\boldmath{$v$}}=u_{i}v_{i},\quad|\mbox{\boldmath{$v$}}|=(\mbox{\boldmath{$v$}}\cdot\mbox{\boldmath{$v$}})^{1/2},\quad\mbox{\boldmath{$u$}}:=(u_{i}),\
\mbox{\boldmath{$v$}}:=(v_{i})\in{\mathbb{R}}^{m},$

and, respectively,

$\mbox{\boldmath{$\tau$}}\cdot\mbox{\boldmath{$\sigma$}}=\tau_{ij}\sigma_{ij},\quad|\mbox{\boldmath{$\tau$}}|=(\mbox{\boldmath{$\tau$}}\cdot\mbox{\boldmath{$\tau$}})^{1/2},\quad\mbox{\boldmath{$\tau$}}:=(\tau_{ij}),\
\mbox{\boldmath{$\sigma$}}:=(\sigma_{ij})\in{\cal S}^{m}.$

Here and hereafter, $m$ is a positive integer which stands for the dimension
of the spatial variable, indices $i$ and $j$ run from $1$ to $m$ and the
summation convention of the repeated indices is adopted. For a bounded open
set $\Omega\subset{\mathbb{R}}^{m}$ with sufficiently smooth boundary $\Gamma$
(normally, we assume that $\Gamma$ is Lipschitz continuous) we denote by $\nu$
the outward unit vector on $\Gamma$ and we introduce the following function
spaces which will play a key role in applications to nonsmooth mechanics
problems

$H:=L^{2}\left(\Omega;{\mathbb{R}}^{m}\right),\quad{\cal
H}:=\left\\{\mbox{\boldmath{$\tau$}}=(\tau_{ij}):\ \ \tau_{ij}=\tau_{ji}\in
L^{2}\left(\Omega\right)\right\\}=L^{2}\left(\Omega;{\cal S}^{m}\right),$
$H_{1}:=\left\\{\mbox{\boldmath{$u$}}\in H:\ \
\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}})\in{\cal
H}\right\\}=H^{1}\left(\Omega;{\mathbb{R}}^{m}\right),\quad{\cal
H}_{1}:=\left\\{\mbox{\boldmath{$\tau$}}\in{\cal H}:\ \ {\rm
Div\;}\mbox{\boldmath{$\tau$}}\in H\right\\},$

where $\varepsilon$ and ${\rm Div}$ are the deformation operator and the
divergence operator, respectively, and are defined in the following way

$\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}):=(\varepsilon_{ij}(\mbox{\boldmath{$u$}})),\quad\varepsilon_{ij}(\mbox{\boldmath{$u$}}):=\frac{1}{2}(u_{i,j}+u_{j,i}),\qquad{\rm
Div\;}\mbox{\boldmath{$\tau$}}=(\tau_{ij,j}),$

accordingly. The index following a comma represents the partial derivative
with respect to the corresponding component of $x\in\Omega$, i.e.,
$u_{i,j}=\partial u_{i}/\partial x_{j}$. Keep in mind that the above
meantioned spaces are Hilbert spaces endowed with the following corresponding
inner products

$(\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}})_{H}:=\int_{\Omega}\mbox{\boldmath{$u$}}\cdot\mbox{\boldmath{$v$}}\;dx,\quad(\mbox{\boldmath{$\tau$}},\mbox{\boldmath{$\sigma$}})_{\cal
H}:=\int_{\Omega}\mbox{\boldmath{$\tau$}}\cdot\mbox{\boldmath{$\sigma$}}\;dx,\quad(\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}})_{H_{1}}:=(\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}})_{H}+(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}))_{\cal
H},$ $(\mbox{\boldmath{$\tau$}},\mbox{\boldmath{$\sigma$}})_{{\cal
H}_{1}}:=(\mbox{\boldmath{$\tau$}},\mbox{\boldmath{$\sigma$}})_{\cal H}+({\rm
Div\;}\mbox{\boldmath{$\tau$}},{\rm Div\;}\mbox{\boldmath{$\sigma$}})_{H}.$

We recall that the trace operator
$\gamma:H^{1}(\Omega;{\mathbb{R}}^{m})\rightarrow
H^{1/2}(\Gamma;{\mathbb{R}}^{m})\subset L^{2}(\Gamma;{\mathbb{R}}^{m})$ is
linear and compact. Sometimes, for simplicity, we will omit to write $\gamma$
to indicate the Sobolev trace on the boundary, i.e., writing $v$ instead of
$\gamma\mbox{\boldmath{$v$}}$. Also, for a given $\mbox{\boldmath{$v$}}\in
H^{1/2}(\Gamma;{\mathbb{R}}^{m})$ we denote by $v_{\nu}$ and
$\mbox{\boldmath{$v$}}_{\tau}$ the normal and the tangential components of $v$
on the boundary, i.e.,
$v_{\nu}:=\mbox{\boldmath{$v$}}\cdot\mbox{\boldmath{$\nu$}}$ and
$\mbox{\boldmath{$v$}}_{\tau}:=\mbox{\boldmath{$v$}}-v_{\nu}\mbox{\boldmath{$\nu$}}$,
respectively. Similarly, for a tensor field $\sigma$, we define $\sigma_{\nu}$
and $\mbox{\boldmath{$\sigma$}}_{\tau}$ to be the normal and the tangential
components of the Cauchy vector field $\sigma$$\nu$, that is
$\sigma_{\nu}:=(\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}})\cdot\mbox{\boldmath{$\nu$}}$
and
$\mbox{\boldmath{$\sigma$}}_{\tau}:=\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}-\sigma_{\nu}\mbox{\boldmath{$\nu$}}$,
respectively. Recall that the following Green formula holds

$(\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}))_{\cal
H}+({\rm
Div\;}\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$v$}})_{H}=\int_{\Gamma}(\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}})\cdot\mbox{\boldmath{$v$}}\;d\Gamma,\mbox{
for all }\mbox{\boldmath{$v$}}\in H_{1}.$ (3.1)

We consider the following contact model involving a deformable body which
occupies a bounded domain $\Omega\subset\mathbb{R}^{m}$ ($m\geq 2$) with
Lipschitz boundary $\Gamma$, which is partitioned into four disjoint
measurable parts $\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{3}$.

${\bf(P)}$ Find a displacement
$\mbox{\boldmath{$u$}}:\Omega\rightarrow{\mathbb{R}}^{m}$ and a stress tensor
$\mbox{\boldmath{$\sigma$}}:\Omega\rightarrow{\cal S}^{m}$ such that

$\displaystyle-{\rm Div\;}\mbox{\boldmath{$\sigma$}}=\mathbf{f}_{0}$ in
$\displaystyle\Omega,$ (3.2)
$\displaystyle\mbox{\boldmath{$\sigma$}}\in\partial\chi(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}))$
in $\displaystyle\Omega,$ (3.3) $\displaystyle\mbox{\boldmath{$u$}}=0$ on
$\displaystyle\Gamma_{1},$ (3.4)
$\displaystyle\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}=\mathbf{f}_{2}$
on $\displaystyle\Gamma_{2},$ (3.5) $\displaystyle
u_{\nu}=0,-\mbox{\boldmath{$\sigma$}}_{\tau}\in\partial_{C}j(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau})$
on $\displaystyle\Gamma_{3}^{a},$ (3.6)
$\displaystyle\mbox{\boldmath{$\sigma$}}_{\tau}=0,\sigma_{\nu}\leq
0,u_{\nu}\leq 0,\sigma_{\nu}u_{\nu}=0$ on $\displaystyle\Gamma_{3}^{b}.$ (3.7)

Problem ${\bf(P)}$ describes the contact between a deformable body and a
foundation. Relation (3.2) represents the equilibrium equation, showing that
the body is subjected to volume forces of density $\mathbf{f}_{0}$. The
behaviour of the material is described by a nonlinear constitutive law
expressed as a (convex) subdifferential inclusion, namely, (3.3). On
$\Gamma_{1}$ the body is clamped, therefore the displacement vanishes here.
Equation (3.5) shows that surface tractions of density $\mathbf{f}_{2}$ act on
$\Gamma_{2}$. On $\Gamma_{3}$ the body may come in frictional contact with the
foundation and this contact is modelled considering there are two potential
contact zones: $\Gamma_{3}^{a}$ where a nonmonotone friction law is
considered; and $\Gamma_{3}^{b}$ where the contact is frictionless and
satisfies Signorini unilateral contact model. Note that we allow the case of
only one contact zone, i.e., $meas(\Gamma_{3}^{a})=0$ or
$meas(\Gamma_{3}^{b})=0$.

For examples of nonlinear constitutive laws of the form $\eqref{P1Eqn2}$ and
nonmonotone friction laws of the form (3.6) we refer the reader to Cos-Csi-
Var15 ; Cos-Mat12 ; Nan-Pan ; Pan93 ; Sof-Mat12 .

We assume that the constitutive function $\chi$ and the initial data
$\mathbf{f}_{0},\mathbf{f}_{2}$ satisfy the following conditions:

  1. $(\mathcal{H}_{1})$

$\mathbf{f}_{0}\in L^{2}(\Omega;\mathbb{R}^{m})$ and $\mathbf{f}_{2}\in
L^{2}(\Gamma_{2};\mathbb{R}^{m})$;

  2. $(\mathcal{H}_{2})$

$\chi:\mathcal{S}^{m}\to\mathbb{R}$ is a convex and l.s.c. functional with the
property that there exist $\alpha,\beta\in(0,1)$ such that

$\alpha|\mbox{\boldmath{$\mu$}}|^{2}\leq\chi(\mbox{\boldmath{$\mu$}})\leq\beta|\mbox{\boldmath{$\mu$}}|^{2},\
\forall\mbox{\boldmath{$\mu$}}\in\mathcal{S}^{m};$

  3. $(\mathcal{H}_{3})$

If $meas(\Gamma_{3}^{a})>0$, then
$j:\Gamma_{3}^{a}\times\mathbb{R}\to\mathbb{R}$ is functional enjoying the
following properties:

    1. (i)

$\mbox{\boldmath{$x$}}\mapsto j(\mbox{\boldmath{$x$}},t)$ is measurable on
$\Gamma_{3}^{a}$ for all $t\in\mathbb{R}$;

    2. (ii)

$j(\mbox{\boldmath{$x$}},0)\in L^{1}(\Gamma_{3}^{a})$;

    3. (iii)

there exists $p\in L^{2}(\Gamma_{3}^{a})$ such that

$|j(\mbox{\boldmath{$x$}},t_{1})-j(\mbox{\boldmath{$x$}},t_{2})|\leq
p(x)|t_{1}-t_{2}|,\mbox{ for a.e. $\mbox{\boldmath{$x$}}\in\Gamma_{3}^{a}$ and
all $t_{1},t_{2}\in\mathbb{R}$}.$

In order to derive the variational formulation of problem ${\bf(P)}$ assume
$u$ and $\sigma$ are regular functions that satisfy (3.2)-(3.7). Multiplying
(3.2) by $\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}$ and integrating over
$\Omega$ we apply Green’s formula to find that

$(\mathbf{f}_{0},\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})_{H}=-(\mbox{\rm
Div\,}\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})_{H}=(\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}))_{\mathcal{H}}-\int_{\Gamma}(\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}})\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})d\Gamma,$

for all $\mbox{\boldmath{$v$}}\in X:=\\{\mbox{\boldmath{$w$}}\in H_{1}:\
\mbox{\boldmath{$w$}}=0\mbox{ a.e. on }\Gamma_{1}\\}$. It is well known that
$X$ is a closed subspace of $H_{1}$, therefore it is a Hilbert space endowed
with the inner product

$(\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}})_{X}:=(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}),\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}))_{\mathcal{H}}.$

Keeping in mind the boundary conditions on $\Gamma_{3}$ we define the set of
admissible displacements

$\mathcal{K}_{0}:=\\{\mbox{\boldmath{$w$}}\in X:\ w_{\nu}=0\mbox{ on
}\Gamma_{3}^{a}\mbox{ and }w_{\nu}\leq 0\mbox{ on }\Gamma_{3}^{b}\\},$

which is nonempty, unbounded, closed and convex subset of $X$. Keeping in mind
that

$(\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}})\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})=\sigma_{\nu}(v_{\nu}-u_{\nu})+\mbox{\boldmath{$\sigma$}}_{\tau}\cdot(\mbox{\boldmath{$v$}}_{\tau}-\mbox{\boldmath{$u$}}_{\tau})$

for all $\mbox{\boldmath{$v$}}\in\mathcal{K}_{0}$ we have

$-\int_{\Gamma_{1}}\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})\;d\Gamma=0,$
$-\int_{\Gamma_{2}}\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})\;d\Gamma=-\int_{\Gamma_{2}}\mathbf{f}_{2}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})\;d\Gamma,$
$-\int_{\Gamma_{3}^{a}}\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})\;d\Gamma=\int_{\Gamma_{3}^{a}}-\mbox{\boldmath{$\sigma$}}_{\tau}\cdot(\mbox{\boldmath{$v$}}_{\tau}-\mbox{\boldmath{$u$}}_{\tau})\;d\Gamma\leq\int_{\Gamma_{3}^{a}}j^{0}_{,2}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};\mbox{\boldmath{$v$}}_{\tau}-\mbox{\boldmath{$u$}}_{\tau})\;d\Gamma,$

and

$-\int_{\Gamma_{3}^{b}}\mbox{\boldmath{$\sigma$}}\mbox{\boldmath{$\nu$}}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})\;d\Gamma=-\int_{\Gamma_{3}^{b}}\sigma_{\nu}(v_{\nu}-u_{\nu})\;d\Gamma=-\int_{\Gamma_{3}^{b}}\sigma_{\nu}v_{\nu}\;d\Gamma\leq
0.$

Thus,

$(\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}))_{\mathcal{H}}+\int_{\Gamma_{3}^{a}}j^{0}_{,2}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};\mbox{\boldmath{$v$}}_{\tau}-\mbox{\boldmath{$u$}}_{\tau})\;d\Gamma\geq(\mathbf{f}_{0},\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})_{H}+\int_{\Gamma_{2}}\mathbf{f}_{2}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})\;d\Gamma,\quad\forall\mbox{\boldmath{$v$}}\in\mathcal{K}_{0}.$
(3.8)

Since the stress tensor $\sigma$ satisfies (3.8) it is natural to define now
the set of admissible stress tensors with respect to a displacement
$\mbox{\boldmath{$w$}}\in\mathcal{K}_{0}$ to be the following subset of
$\mathcal{H}$

$\Theta(\mbox{\boldmath{$w$}}):=\left\\{\mbox{\boldmath{$\mu$}}\in\mathcal{H}:\
(\mbox{\boldmath{$\mu$}},\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}))_{\mathcal{H}}+\int_{\Gamma_{3}^{a}}j^{0}_{,2}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$w$}}_{\tau};\mbox{\boldmath{$v$}}_{\tau})\;d\Gamma\geq(\mathbf{f}_{0},\mbox{\boldmath{$v$}})_{H}+\int_{\Gamma_{2}}\mathbf{f}_{2}\cdot\mbox{\boldmath{$v$}}\;d\Gamma,\
\forall\mbox{\boldmath{$v$}}\in\mathcal{K}_{0}\right\\}.$

We define the separable bipotential
$b:\mathcal{S}^{m}\times\mathcal{S}^{m}\to(-\infty,+\infty]$ by

$b(\mbox{\boldmath{$\tau$}},\mbox{\boldmath{$\mu$}}):=\chi(\mbox{\boldmath{$\tau$}})+\chi^{\ast}(\mbox{\boldmath{$\mu$}}).$

We point out the fact that $b$ connects the constitutive law (3.3), the
function $\chi$ and its Fenchel conjugate $\chi^{\ast}$ as

$b(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}),\mbox{\boldmath{$\sigma$}})=\mbox{\boldmath{$\sigma$}}\cdot\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}})$

and

$b(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}),\mbox{\boldmath{$\mu$}})\geq\mbox{\boldmath{$\mu$}}\cdot\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}),\quad\forall\mbox{\boldmath{$v$}}\in
X,\forall\mbox{\boldmath{$\mu$}}\in\mathcal{H}.$

Note that, if $(\mathcal{H}_{3})$ holds, then according to (Mat13, , Lemma 1)
one has

$(1-\beta)|\mbox{\boldmath{$\mu$}}|^{2}\leq\chi^{\ast}(\mbox{\boldmath{$\mu$}})\leq\frac{1}{4\alpha}|\mbox{\boldmath{$\mu$}}|^{2},\
\forall\mbox{\boldmath{$\mu$}}\in\mathcal{S}^{m}.$

Therefore, we have

$\chi(\mbox{\boldmath{$\mu$}}(\cdot))\in L^{1}(\Omega)\mbox{ and
}\chi^{\ast}(\mbox{\boldmath{$\mu$}}(\cdot))\in L^{1}(\Omega),\
\forall\mbox{\boldmath{$\mu$}}\in\mathcal{H}.$

Thus, the coupling function $B:X\times\mathcal{H}\to\mathbb{R}$ defined via
the bipotential $b$

$B(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\mu$}}):=\int_{\Omega}b(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}),\mbox{\boldmath{$\mu$}})\;dx,$

is well defined and the following estimates hold

$B(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})=(\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}))_{\mathcal{H}}\mbox{
and
}B(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\mu$}})\geq(\mbox{\boldmath{$\mu$}},\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}))_{\mathcal{H}},\forall\mbox{\boldmath{$v$}}\in
X,\forall\mbox{\boldmath{$\mu$}}\in\mathcal{H}.$ (3.9)

Moreover, there exists a positive constant $C=C(\alpha,\beta)$ such that

$B(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\mu$}})\geq
C(\|\mbox{\boldmath{$v$}}\|^{2}_{X}+\|\mbox{\boldmath{$\mu$}}\|_{\mathcal{H}}^{2}),\
\forall\mbox{\boldmath{$v$}}\in
X,\forall\mbox{\boldmath{$\mu$}}\in\mathcal{H}.$ (3.10)

Choosing $\mbox{\boldmath{$v$}}=\mbox{\boldmath{$u$}}+\mbox{\boldmath{$w$}}$
in (3.8) we deduce that
$\mbox{\boldmath{$\sigma$}}\in\Theta(\mbox{\boldmath{$u$}})$, therefore
$\Theta(\mbox{\boldmath{$u$}})\neq\varnothing$. Keeping in mind the definition
of $\Theta(\mbox{\boldmath{$u$}})$ and (3.9) we get

$B(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\mu$}})+\int_{\Gamma_{3}^{a}}j_{,2}^{0}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};\mbox{\boldmath{$u$}}_{\tau})\;d\Gamma\geq(\mathbf{f}_{0},\mbox{\boldmath{$u$}})_{H}+\int_{\Gamma_{2}}\mathbf{f}_{2}\cdot\mbox{\boldmath{$u$}}\;d\Gamma.$
(3.11)

On the other hand, taking $\mbox{\boldmath{$v$}}:=\mathbf{0}_{X}$ in (3.8)

$-B(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})+\int_{\Gamma_{3}^{a}}j_{,2}^{0}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};-\mbox{\boldmath{$u$}}_{\tau})\;d\Gamma\geq-(\mathbf{f}_{0},\mbox{\boldmath{$u$}})_{H}-\int_{\Gamma_{2}}\mathbf{f}_{2}\cdot\mbox{\boldmath{$u$}}\;d\Gamma.$
(3.12)

Adding (3.11) and (3.12) we get

$B(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\mu$}})-B(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})\geq-\int_{\Gamma_{3}^{a}}[j^{0}_{,2}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};\mbox{\boldmath{$u$}}_{\tau})+j^{0}_{,2}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};-\mbox{\boldmath{$u$}}_{\tau})]\;d\Gamma,\
\forall\mbox{\boldmath{$\mu$}}\in\Theta(\mbox{\boldmath{$u$}}).$ (3.13)

Using relations (3.8), (3.9) and (3.13) we derive the following variational
formulation in terms of bipotentials of problem ${\bf(P)}$:

${\bf(P)}$: Find $\mbox{\boldmath{$u$}}\in\mathcal{K}_{0}$ and
$\mbox{\boldmath{$\sigma$}}\in\Theta(\mbox{\boldmath{$u$}})$ such that

$\left\\{\begin{array}[]{l}{\displaystyle
B(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\sigma$}})-B(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})+\int_{\Gamma_{3}^{a}}j_{,2}^{0}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};\mbox{\boldmath{$v$}}_{\tau}-\mbox{\boldmath{$u$}}_{\tau})\;d\Gamma\geq(\mathbf{f}_{0},\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})_{H}+\int_{\Gamma_{2}}\mathbf{f}_{2}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})\;d\Gamma}\\\
\vskip 6.0pt plus 2.0pt minus
2.0ptB(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\mu$}})-B(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})\geq
0,\end{array}\right.$ (3.14)

for all
$(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\mu$}})\in\mathcal{K}_{0}\times\Theta(\mbox{\boldmath{$u$}})$.

Note that, if $(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})$ is a weak
solution of problem ${\bf(P)}$, i.e., it solves (3.14), then (3.8) and (3.13)
are automatically fulfilled as

$B(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\sigma$}})-B(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})\geq(\mbox{\boldmath{$\sigma$}},\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}))_{\mathcal{H}},\
\forall\mbox{\boldmath{$v$}}\in\mathcal{K}_{0},$

and

$0=j_{,2}^{0}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};\mathbf{0}_{X})=j_{,2}^{0}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};\mbox{\boldmath{$u$}}_{\tau}-\mbox{\boldmath{$u$}}_{\tau})\leq
j_{,2}^{0}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};\mbox{\boldmath{$u$}}_{\tau})+j_{,2}^{0}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};-\mbox{\boldmath{$u$}}_{\tau}).$

######  Theorem 3.1.

Assume $(\mathcal{H}_{1}),(\mathcal{H}_{2})$ and $(\mathcal{H}_{3})$ hold.
Then problem ${\bf(P)}$ possesses at least one weak solution.

######  Proof 3.2.

Let us define the function spaces $Y:=\mathcal{H}$,
$Z_{1}:=L^{2}(\Gamma_{3}^{a};\mathbb{R}^{m})$,
$Z_{2}:=L^{2}(\Omega;\mathcal{S}^{m})$. Moreover, assume that $\gamma_{2}:Y\to
Z_{2}$ is the embedding operator and $\gamma_{1}:X\to Z_{1}$ is defined by

$\gamma_{1}(\mbox{\boldmath{$u$}}):=(\gamma(\mbox{\boldmath{$u$}}_{\tau}))|_{\Gamma_{3}^{a}},$

where $\gamma:X\to H^{1/2}(\Gamma;\mathbb{R}^{m})$ is the trace operator. Also
define $J:Z_{1}\to\mathbb{R}$, $\phi:X\to(-\infty,\infty]$,
$H:Y\to(0,\infty)$, $L:Z_{2}\to\mathbb{R}$, $F:X\to X^{\ast}$ and $G:Y\to
Y^{\ast}$ by the following instructions

$J(\mbox{\boldmath{$y$}}):=\int_{\Gamma_{3}^{a}}j(\mbox{\boldmath{$x$}},\mbox{\boldmath{$y$}}(\mbox{\boldmath{$x$}}))d\Gamma,\
\phi:=I_{\mathcal{K}_{0}},\ H\equiv 1,\ L\equiv 0,\ G\equiv 0_{Y^{\ast}}$

and

$\langle
F(\mbox{\boldmath{$u$}}),\mbox{\boldmath{$v$}}\rangle:=(\mathbf{f}_{0},\mbox{\boldmath{$v$}})_{\mathcal{H}}+\int_{\Gamma_{2}}\mathbf{f}_{2}\cdot\mbox{\boldmath{$v$}}d\Gamma,$

respectively.

By virtue of definition of $B\colon X\times Y\to\mathbb{R}$, it is easy to
check that ${\bf(H_{B})}$, ${\bf(H_{F}^{1})}$ and ${\bf(H_{G}^{1})}$ are
fulfilled. Furthermore, the Aubin-Clarke Theorem (see, e.g., Clarke (Clarke, ,
Theorem 2.7.5)) ensures $J$ is Lipschitz continuous of constant
$\|p\|_{L^{2}}$ and

$J^{0}(\mbox{\boldmath{$y$}};\mbox{\boldmath{$z$}})\leq\int_{\Gamma_{3}^{a}}j^{0}_{,2}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$y$}};\mbox{\boldmath{$z$}})\;d\Gamma,\quad\forall\mbox{\boldmath{$y$}},\mbox{\boldmath{$z$}}\in
Z_{1}.$

It follows immediately that

$J^{0}(\gamma_{1}(\mbox{\boldmath{$u$}});\gamma_{1}(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}))\leq\int_{\Gamma_{3}^{a}}j^{0}_{,2}(\mbox{\boldmath{$x$}},\mbox{\boldmath{$u$}}_{\tau};\mbox{\boldmath{$v$}}_{\tau}-\mbox{\boldmath{$u$}}_{\tau})d\Gamma,\quad\forall\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}}\in
X.$ (3.15)

Thus, ${\bf(H_{0}^{\prime})}$ holds with $\alpha_{J}:=\|p\|_{L^{2}}$ and
$\alpha_{L}:=1$. Let $\bar{w}_{0}:=\mathbf{0}_{X}$ and
$\bar{\mbox{\boldmath{$\tau$}}}_{0}\in Y$ be arbitrarily fixed. Then

$\displaystyle\frac{2B(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})-B(\mathbf{0}_{X},\mbox{\boldmath{$\sigma$}})-B(\mbox{\boldmath{$u$}},\bar{\mbox{\boldmath{$\tau$}}}_{0})+\langle
F(\mbox{\boldmath{$u$}}),-\mbox{\boldmath{$u$}}\rangle+\langle
G(\mbox{\boldmath{$\sigma$}}),\bar{\mbox{\boldmath{$\tau$}}}_{0}-\mbox{\boldmath{$\sigma$}}\rangle}{\|(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})\|}$
$\displaystyle=\frac{B(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})-{\displaystyle\int_{\Omega}[\chi(\mathbf{0}_{X})+\chi^{\ast}(\bar{\mbox{\boldmath{$\tau$}}}_{0})]\;dx-(\mathbf{f}_{0},\mbox{\boldmath{$u$}})_{\mathcal{H}}-\int_{\Gamma_{2}}\mathbf{f}_{2}\cdot\mbox{\boldmath{$u$}}d\Gamma}}{\sqrt{\|\mbox{\boldmath{$u$}}\|_{X}^{2}+\|\mbox{\boldmath{$\sigma$}}\|_{Y}^{2}}}$
$\displaystyle\geq\frac{C(\|\mbox{\boldmath{$u$}}\|_{X}^{2}+\|\mbox{\boldmath{$\sigma$}}\|_{Y}^{2})-c_{1}\|u\|_{X}-c_{2}}{\sqrt{\|\mbox{\boldmath{$u$}}\|_{X}^{2}+\|\mbox{\boldmath{$\sigma$}}\|_{Y}^{2}}}\to\infty\mbox{
as }\|(\mbox{\boldmath{$u$}},\mbox{\boldmath{$\sigma$}})\|\to\infty,$

for some suitable constants $c_{1},c_{2}>0$. Therefore, the coercivity
condition ${\bf(C)}$ also holds. Consequently, problem (1.3) possesses at
least one solution for any constraint sets $K\subseteq X$ and
$\Lambda\subseteq Y$. Choosing $K:=\mathcal{K}_{0}$ and $\Lambda:=Y$, Theorem
2.5 ensures the existence of a pair
$(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\sigma$}}_{1})\in\mathcal{K}_{0}\times
Y$ such that

$\left\\{\begin{array}[]{l}B(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\sigma$}}_{1})-B(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\sigma$}}_{1})+J^{0}(\gamma_{1}(\mbox{\boldmath{$u$}}_{1});\gamma_{1}(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_{1}))\geq\langle
F(\mbox{\boldmath{$u$}}_{1}),\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_{1}\rangle,\
\forall v\in\mathcal{K}_{0},\\\
B(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\mu$}})-B(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\sigma$}}_{1})\geq
0,\ \forall\mbox{\boldmath{$\mu$}}\in Y.\end{array}\right.$ (3.16)

Now, applying again Theorem 2.5 with $K:=\mathcal{K}_{0}$ and
$\Lambda:=\Theta(\mbox{\boldmath{$u$}}_{1})$ we infer there exist
$\mbox{\boldmath{$u$}}_{2}\in\mathcal{K}_{2}$ and
$\mbox{\boldmath{$\sigma$}}_{2}\in\Theta(\mbox{\boldmath{$u$}}_{1})$ such that

$\left\\{\begin{array}[]{l}B(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\sigma$}}_{2})-B(\mbox{\boldmath{$u$}}_{2},\mbox{\boldmath{$\sigma$}}_{2})+J^{0}(\gamma_{1}(\mbox{\boldmath{$u$}}_{2});\gamma_{1}(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_{2}))\geq\langle
F(\mbox{\boldmath{$u$}}_{2}),\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_{2}\rangle,\
\forall v\in\mathcal{K}_{0},\\\
B(\mbox{\boldmath{$u$}}_{2},\mbox{\boldmath{$\mu$}})-B(\mbox{\boldmath{$u$}}_{2},\mbox{\boldmath{$\sigma$}}_{2})\geq
0,\
\forall\mbox{\boldmath{$\mu$}}\in\Theta(\mbox{\boldmath{$u$}}_{1}).\end{array}\right.$
(3.17)

But, from the definition of the coupling function $B$ we have

$B(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\sigma$}}_{1})-B(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\sigma$}}_{1})=B(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\sigma$}}_{2})-B(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\sigma$}}_{2}),\
\forall\mbox{\boldmath{$v$}}\in X,$

and

$B(\mbox{\boldmath{$u$}}_{2},\mbox{\boldmath{$\mu$}})-B(\mbox{\boldmath{$u$}}_{2},\mbox{\boldmath{$\sigma$}}_{2})=B(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\mu$}})-B(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\sigma$}}_{2}),\
\forall\mbox{\boldmath{$\mu$}}\in Y,$

which combined with (3.16)-(3.17) shows that
$(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\sigma$}}_{2})$ solves the
following system

$\left\\{\begin{array}[]{l}B(\mbox{\boldmath{$v$}},\mbox{\boldmath{$\sigma$}}_{2})-B(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\sigma$}}_{2})+J^{0}(\gamma_{1}(\mbox{\boldmath{$u$}}_{1});\gamma_{1}(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_{1}))\geq\langle
F(\mbox{\boldmath{$u$}}_{1}),\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_{1}\rangle,\
\forall v\in\mathcal{K}_{0},\\\
B(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\mu$}})-B(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\sigma$}}_{2})\geq
0,\
\forall\mbox{\boldmath{$\mu$}}\in\Theta(\mbox{\boldmath{$u$}}_{1}).\end{array}\right.$
(3.18)

It is readily seen that, due to (3.15), the pair
$(\mbox{\boldmath{$u$}}_{1},\mbox{\boldmath{$\sigma$}}_{2})$ also solves
(3.14), i.e., it is a weak solution for problem ${\bf(P)}$.

## Conflict of interests

There is no conflict of interests.

## Data availability statements

Data sharing not applicable to this article as no data sets were generated or
analysed during the current study.

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