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\title{\sc Hartman-Stampacchia results for stably pseudomonotone
operators and nonlinear hemivariational inequalities
\thanks{Correspondence address: Vicen\c tiu R\u adulescu,
Department of Mathematics, University of Craiova, 200585 Craiova,
Romania. E-mail: {\tt vicentiu.radulescu@imar.ro}}}
\author{\sc Nicu\c sor Costea$\,^{a}$ and Vicen\c tiu R\u adulescu$\,^{a,b}$\\
\small $^a\,$Department of Mathematics, University of Craiova,
200585 Craiova,
Romania\\
\small $^b\,$Institute of Mathematics ``Simion Stoilow'' of the Romanian Academy,\\
\small P.O. Box 1-764, 014700 Bucharest, Romania\\
\small E-mail addresses: {\tt nicusorcostea@yahoo.com} \qquad {\tt
vicentiu.radulescu@imar.ro}}

\begin{document}
\baselineskip16pt \maketitle \noindent{\small{\sc Abstract}. We
are concerned with two classes of nonstandard hemivariational
inequalities. In the first case we establish a Hartman-Stampacchia
type existence result in the framework of stably pseudomonotone
operators. Next, we prove an existence result for a class of
nonlinear perturbations of canonical hemivariational inequalities.
Our analysis includes both the cases of compact sets and of closed
convex sets in Banach spaces. Applications to noncoercive
hemivariational and variational-hemivariational inequalities
illustrate the abstract results of this paper.
\\
\small{\bf 2000 Mathematics
Subject Classification:}  35J85, 47H05, 49J53.\\
\small{\bf Key words:} hemivariational inequality; Clarke's
generalized gradient; stably pseudomonotone operator;  fixed point
theorem; set valued mappings.}


\section{Introduction}
Inequality theory plays an important role in many fields, such as
mechanics, engineering sciences, economics, optimal control, etc.
Because of their wide applicability, inequality problems have
become an important area of investigation in the past several
decades, an important part of this research focusing on the
existence of the solutions. Inequality problems can be divided
into two main classes: that of variational inequalities and that
of hemivariational inequalities. The study of variational
inequality problems began in the early sixties with the work of G.
Fichera \cite{Fich}, J. L. Lions and G. Stampacchia
\cite{Lio-Sta}. The most basic result is due to Hartman and
Stampacchia \cite{Har-Sta}, which states that if $X$ is a finite
dimensional Banach space, $K\subset X$ is compact and convex, and
$A$ is a continuous operator, then the variational inequality
problem of finding $u\in K$ such that
\neweq{VI}
\langle Au,v-u\rangle\geq 0,\quad\forall v\in K
\endeq
has a solution. When $K$ is not compact, or $X$ is infinite
dimensional, certain monotonicity properties are required to prove
the existence of solution.
\medskip

By replacing the subdifferential of a convex function by the
generalized gradient (in the sense of F. H. Clarke) of a locally
Lipschitz functional, hemivariational inequalities arise whenever
the energetic functional associated to a concrete problem is
nonconvex. This new type of inequalities appears as a
generalization of the variational inequalities, but
hemivariational inequalities are much more general, in the sense
that they are not equivalent to minimum problems but, give rise to
substationarity problems. The theory of hemivariational
inequalities can be viewed as a new field of Nonsmooth Mechanics
since the main ingredient used in the study of these inequalities
is the notion of Clarke subdifferential of a locally Lipschitz
functional. The mathematical theory hemivariational inequalities,
as well as their applications in Mechanics, Engineering or
Economics were introduced and developed by P. D. Panagiotopoulos
\cite{Pan1}-\cite{Pan5} in the case of nonconvex energy functions.
For a treatment of this theory and further comments we recommend
the monographs by Z. Naniewicz and P. D. Panagiotopoulos
\cite{Nan-Pan}, D. Motreanu and P. D. Panagiotopoulos
\cite{Mot-Pan} and by D. Motreanu and V. R\u adulescu
\cite{Mot-Rad2}.

\medskip

Throughout  this paper $V$ will denote a real  Banach space and
let $T:V\ri L^p(\Omega;\RR^k)$ be a linear and continuous
operator, where $1< p<\infty$, $k\geq 1$, and $\Omega$ is a
bounded open set in $\RR^N$. We shall denote $Tu=\hat u$ and by
$p'$ the conjugated exponent of $p$. Let $K$ be a subset of $V$,
$A:K\ri V^\ast$ a nonlinear operator and
$j=j(x,y):\Omega\times\RR^k\ri \RR$ is a function such that the
mapping
\neweq{eq1}
j(\cdot,y):\Omega\ri\RR \quad\mbox{is measurable, for every }
y\in\RR^k.
\endeq
We assume that at least one of the following conditions hold true:
either there exist $l\in L^{p'}(\Omega;\RR)$ such that
\neweq{eq2}
|j(x,y_1)-j(x,y_2)|\leq l(x)|y_1-y_2|, \quad\forall\, x\in
\Omega,\ \forall\, y_1,y_2\in\RR^k,
\endeq
or
\neweq{eq3}
\mbox{ the mapping $j(x,\cdot)$ is locally Lipschitz, } \forall\,
x\in\Omega,
\endeq
and there exist $C>0$ such that
\neweq{eq4}
|z|\leq C(1+|y|^{p-1}),\quad \forall\, x\in \Omega,\forall\,
z\in\de j(x,y).
\endeq

Recall that $j^{\,0}(x,y;h)$ denotes the Clarke's generalized
directional derivative of the locally Lipschitz mapping
$j(x,\cdot)$ at the point $y\in\RR^k$ with respect to the
direction $h\in\RR^k$, where $x\in\Omega$, while $\de j(x,y)$ is
the Clarke's generalized gradient of this mapping at $y\in\RR^k$,
that is (see e.g. \cite{Cla}, \cite{Mot-Rad2}, \cite{Nan-Pan})
$$
j^{\,0}(x,y;h)=\limsup\limits_{\stackrel{y'\ri y}{t\downarrow
0}}\frac{j(x,y'+th)-j(x,y')}{t}\ ,
$$
$$
\de j(x,y)=\{z\in \RR^k:\  z\cdot h  \leq j^{\,0}(x,y;h), \mbox{
for all } h\in\RR^k\},
$$
where the symbol ``$\cdot$'' means the inner product on $\RR^k$.

The euclidian norm in $\RR^k$, $k\geq 1$, resp. the duality
pairing between a Banach space and its dual will be denoted by
$|\cdot|$, respectively $\langle\cdot,\cdot\rangle.$

\begin{defin}\label{def1}
The operator $A:K\ri V^\ast$ is said to be
\begin{description}
\item (i)  hemicontinuous if $A$ is continuous from line segments
in $K$ to the w$^\ast$ topology of $V^\ast$;

\item (ii) monotone if $\langle Av-Au,v-u\rangle\geq 0$, for all
$u,v\in K$;

\item (iii) pseudomonotone if $\langle Au,v-u\rangle\geq 0$
implies $\langle Av,v-u\rangle\geq 0$, for all $u,v\in K$;

\item (iv) stably pseudomonotone with respect to a set $U\subset
V^\ast$ if, $A$ and $A(\cdot)-z$ are pseudomonotone for every
$z\in U$.
\end{description}
\end{defin}

\begin{rem}\label{rem1}
Clearly a monotone map is pseudomonotone and stably pseudomonotone
with respect to any set $U\subset V^\ast$. Stably pseudomonotone
mappings are introduced and discussed in \cite{He-NG,He1,He2}.
Moreover, Yiran He showed in \cite{He1} (see Example 2, p.460)
that the class of of stably pseudomonotone mappings is strictly
broader than the class of monotone mappings (we point out the fact
that the operator in this example is also hemicontinuous).
\end{rem}
The following fixed point theorem will be used in the proof of the
main results. We denote by $2^K$ the family all nonempty subsets
of a set $K$.
\begin{teo}\label{Th1} (Tarafdar \cite{Tar}) Let $K\neq\emptyset$
be a convex subset of a Hausdorff topological vector space $E$.
Let $F:K\ri 2^K$ be a set valued map such that
\begin{description}
\item (1) for each $u\in K$, $F(u)$ is a nonempty convex subset of
$K$;

\item (2) for each $v\in K$, $F^{-1}(v)=\{\, u\in K:\ v\in F(u)\}$
contains an open set $O_v$ which may be empty;

\item(3) $\bigcup_{v\in K} O_v=K$;

\item (4) there exists a nonempty set $V_0$ contained in a compact
convex subset $V_1$ of $K$ such that $D=\bigcap_{v\in V_0} O_v^c$
is either empty or compact (where $O_v^c$ is the complement of
$O_v$ in $K$).
\end{description}
Then there exist a point $u_0\in K$ such that $u_0\in F(u_0)$.
\end{teo}


\section{Hemivariational inequalities of Hartman-Stampacchia
type involving stably pseudomonotone operators}

This section is devoted to the  study of the {\it hemivariational
inequality problem of  Hartman-Stampacchia type} (as it was called
by Panagiotopoulos, Fundo and R\u adulescu in \cite{Pan-Fun-Rad}
to show that this kind of inequalities are a generalization of
those studied by Hartman and Stampacchia in \cite{Har-Sta}):

(${\bf P}$) Find $u\in K$ such that
$$
\langle Au,v-u\rangle+\int_\Omega\, j^{\,0}(x,\hat u(x);\hat
v(x)-\hat u(x))\, dx\geq 0,\quad \forall\, v\in K.
$$

Throughout this section $V$ will denote a real reflexive Banach
space and $T:V\ri L^p(\Omega;\RR^k)$ will stand for a linear and
compact operator.

Let us consider a Banach space $Y$ such that there  exists a
linear and compact operator $L:V\ri Y$ and let $J:Y\ri\RR$ be an
arbitrary locally Lipschitz functional. For each $u\in V$ there
exists (see e.g. \cite {Cla}, \cite{Mot-Rad2}, \cite{Nan-Pan})
${\bf z}_u\in\de J(Lu)$ such that
\neweq{eq6}
J^0(Lu;\zeta)=\langle {\bf z}_u,\zeta\rangle=\max\left\{\, \langle
w,\zeta\rangle:\ w\in\de J(Lu) \right\}.
\endeq
Denoting by $L^\ast:Y^\ast\ri V^\ast$ the adjoint operator of $L$,
we define the subset $U(J,L)$ of $V^\ast$ as follows
\neweq{eq7}
U(J,L)=\{-{\bf Z}_u:\ u\in K \mbox{ and  } {\bf Z}_u=L^\ast {\bf
z}_u \}.
\endeq



\begin{lemma}\label{lema1}
Let $K$ be a nonempty, closed and convex subset of $V$. Assume
that $A:K\ri V^\ast$ is a hemicontinuous and stably pseudomonotone
map with respect to the set $U(J,L)$ defined in \eq{eq7}. Further
we assume that there exists a nonempty subset $V_0$ contained in a
weakly compact subset $V_1$ of $K$ such that the set
$$
D=\{\, u\in K:\ \langle Av,v-u\rangle+J^{\,0}(Lu;Lv-Lu)\geq
0,\quad \forall\, v\in V_0 \}
$$
is weakly compact or empty. Then the problem

${\bf (P')}$ Find $u\in K$ such that
$$
\langle Au,v-u\rangle+J^{\,0}(Lu;Lv-Lu)\geq0, \quad\forall\, v\in
K,
$$
admits at least one solution.
\end{lemma}

\noindent As we will see problem ($P'$) closely links to the
following problem:

(${\bf P^\star}$) Find $u\in K$ such that
$$
\langle Av,v-u\rangle+J^{\,0}(Lu;Lv-Lu)\geq0, \quad\forall\, v\in
K.
$$
We denote by $S\,'$ and $S^\star$ the solution sets of problem
(${\bf P'}$) and problem (${\bf P^\star}$), respectively. Now we
establish the relationship between problems (${\bf P'}$) and
(${\bf P^\star}$).


\begin{prop}\label{prop1}
Let $K$ be a nonempty closed and convex subset of $V$. If $A:K\ri
V^\ast$ is stably pseudomonotone with respect to the set $U(J,L)$
defined in \eq{eq7}, then $S\,'\subseteq S^\star$. In addition, if
$A$ is hemicontinuous, then $S\,'=S^\star$.

\end{prop}

\proof Let $u_0\in S\,'$. For any $v\in K$ we have
\begin{eqnarray*}
0&\leq &\langle Au_0,v-u_0\rangle+J^0(Lu_0;Lv-Lu_0)\\
&=&\langle Au_0,v-u_0\rangle+\langle{\bf
z}_{u_0},L(v-u_0)\rangle\\
&=&\langle Au_0,v-u_0\rangle+\langle L^\ast{\bf z}_{u_0},v-u_0\rangle\\
&=&\langle Au_0,v-u_0\rangle+\langle{\bf Z}_{u_0},v-u_0\rangle.
\end{eqnarray*}
Using the stably pseudomonotonicity of $A$ we get that
$$ \langle
Av,v-u_0\rangle+\langle {\bf Z}_{u_0},v-u_0\rangle\geq0,
\quad\forall v\in K.
$$
The above inequality is equivalent to
$$
\langle Av,v-u_0\rangle+J^0(Lu_0;Lv-Lu_0)\geq 0, \quad \forall
v\in K,
$$
which means that $u_0\in S^\star$.

 In addition, if $A$ is hemicontinuous, we will show that
$S\,'=S^\star$. Suppose $u_0\in S^\star$. For any $v\in K$, let
$$v_t=tv+(1-t)u_0,\quad t\in (0,1).$$
Then $v_t\in K$ and
$$
\langle Av_t,t(v-u_0)\rangle+J^{\,0}(Lu_0;t(Lv-Lu_0))\geq0.
$$
Using the positive homogeneity of the map $J^{\,0}(\cdot\,;\cdot)$
and dividing by $t$ we obtain that
$$
\langle Av_t,v-u_0\rangle+J^{\,0}(Lu_0;Lv-Lu_0)\geq0.
$$
Letting $t\ri 0$, by the hemicontinuity of $A$, we get
$$
\langle Au_0,v-u_0\rangle+J^{\,0}(Lu_0;Lv-Lu_0)\geq0,\quad \forall
v\in K,
$$
which means $u_0\in S\,'$.
\endproof

\noindent {\sc Proof of Lemma \ref{lema1}}. Arguing by
contradiction, suppose that for every $u\in K$, there exists $v\in
K$ such that
$$
\langle Au,v-u\rangle+J^{\,0}(Lu;Lv-Lu)<0.
$$
This implies, by Proposition \ref{prop1}  that for every $u\in K$
there exists $v\in K$ such that
\neweq{eq8}
\langle Av,v-u\rangle+J^{\,0}(Lu;Lv-Lu)<0.
\endeq
We define the set valued map $F:K\ri 2^K$ as follows
$$
F(u)=\{\, v\in K: \langle Au,v-u\rangle+J^{\,0}(Lu;Lv-Lu)<0\}
$$
We will prove next that $F$ satisfies the conditions of Theorem
\ref{Th1}.

Let $u\in K$ be arbitrary but fixed. Obviously $F(u)$ is nonempty
for each $u\in K$ since we have assumed that the problem (${\bf
P'}$) has no solutions. Let $v_1,v_2\in F(u)$ and $t\in [0,1]$.
Taking into account that $L$ is linear and the application
$Lv\longmapsto J^{\, 0}(Lu;Lv)$ is convex we have
$$
\begin{array}{ll}
\langle
Au,tv_1+(1-t)v_2-u\rangle+J^{\,0}(Lu;L(tv_1+(1-t)v_2)-Lu)\leq\\
t\langle Au,v_1-u\rangle+(1-t)\langle Au,v_2-u\rangle
+tJ^{\,0}(Lu;Lv_1-Lu)+(1-t)J^{\,0}(Lu;Lv_2-Lu)<0,
\end{array}
$$
which shows that $F(u)$ is a convex subset of $K$.

Let, now $v\in K$ be arbitrary but fixed.
\begin{eqnarray*}
F^{-1}(v)&=&\{\, u\in K:\ v\in F(u)\}\\
&=&\{\, u\in K:\ \langle Au,v-u\rangle+J^{\,0}(Lu;Lv-Lu)<0\}\\
&\supseteq&\{\,u\in K:\ \langle
Av,v-u\rangle+J^{\,0}(Lu;Lv-Lu)<0\}:=O_v.
\end{eqnarray*}
We shall prove that $[F^{-1}(v)]^c\subseteq O_v^c$ which implies
 $O_v\subseteq F^{-1}(v)$. Let $u\in [F^{-1}(v)]^c$. Then
$$
\langle Au,v-u\rangle+J^{\,0}(Lu;Lv-Lu)\geq0.
$$
Using the fact that $A$ is stably pseudomonotone with respect to
the set $U(J,L)$ we get that
$$
\langle Av,v-u\rangle+J^{\,0}(Lu;Lv-Lu)\geq0,
$$
which leads to the conclusion that $u\in O_v^c$.

We claim that $O_v$ is weakly open. Indeed, if $O_v\ni u_n\weak u$
then, since $L$ is linear and compact $Lu_n\ri Lu$ and by the
upper semicontinuity of $J^{\,0}(\cdot,\cdot)$, we obtain
\begin{eqnarray*}
\limsup\limits_{n\ri\infty} \left[\langle
Av,v-u_n\rangle+J^{\,0}(Lu_n;Lv-Lu_n)\right]&\leq&\lim\limits_{n\ri\infty}\langle
Av,v-u_n\rangle+\limsup\limits_{n\ri\infty}J^{\,0}(Lu_n;Lv-Lu_n)\\
&\leq&\langle Av,v-u\rangle+J^{\,0}(Lu;Lv-Lu).
\end{eqnarray*}
The above relation implies that the application $u\longmapsto
\langle Av,v-u\rangle+J^{\,0}(Lu;Lv-Lu)$ is weakly upper
semicontinuous. Thus by the definition of the upper
semicontinuity, $O_v$ is an weakly open set.

Obviously $\bigcup_{v\in K}O_v\subseteq K$. Now, let $u\in K$. By
\eq{eq8} there exists $v\in K$ such that $\langle
Av,v-u\rangle+J^{\,0}(Lu;Lv-Lu)<0$ and thus $u\in O_v$. It follows
that $K\subseteq \bigcup_{v\in K}O_v$.

By our hypothesis the set
\begin{eqnarray*}
D&=&\{\, u\in K:\ \langle Av,v-u\rangle+J^{\,0}(Lu;Lv-Lu)\geq 0,
\quad\forall\, v\in V_0\}\\
&=&\bigcap\limits_{v\in V_0}\{\, u\in K:\ \langle
Av,v-u\rangle+J^{\,0}(Lu;Lv-Lu)\geq 0\}\\
&=& \bigcap\limits_{v\in V_0} O_v^c
\end{eqnarray*}
is weakly compact or empty.

Thus the mapping $F$ satisfies the conditions of Theorem \ref{Th1}
in the weak topology, so there exists a point $u_0\in K$ such that
$u_0\in F(u_0)$; that is $ 0=\langle
Au_0,u_0-u_0\rangle+J^{\,0}(Lu_0;Lu_0-Lu_0)<0$ and this
contradiction completes the proof of Lemma \ref{lema1}.




We will derive a result applicable to the inequality problem
(${\bf P}$) which constitutes the main result of this section.

\begin{teo}\label{Th3}
Assume that the hypotheses of Lemma \ref{lema1} are fulfilled for
$Y=L^p(\Omega;\RR^k)$, $L=T$ and $J:L^p(\Omega;\RR^k)\ri \RR$
$$
J(u)=\int_\Omega j(x,u(x))\, dx.
$$
Then the problem ${\bf (P)}$ has at least one solution.
\end{teo}


\proof  Conditions \eq{eq1} and \eq{eq2} or \eq{eq1} and
\eq{eq3}-\eq{eq4} on $j$ ensure that $J$ is locally Lipschitz on
$Y$ (see e.g. Clarke \cite{Cla}, p. 83 or Motreanu-R\u adulescu
\cite{Mot-Rad2} Theorem 1.3) and
$$
\int_\Omega j^{\, 0}(x,u(x);v(x))\, dx\geq J^{\,0}(u;v),\quad
\forall\, u,v\in L^p(\Omega;\RR^k).
$$
It follows that
\neweq{eq9}
\int_\Omega j^{\, 0}(x,\hat u(x);\hat v(x))\, dx\geq J^{\,0}(\hat
u;\hat v),\quad \forall\, u,v\in V.
\endeq
It results that if $u\in K$ is a solution of the problem (${\bf
P'}$) then $u$ solves inequality problem (${\bf P}$), too.
\endproof


If in addition $K$ is bounded it follows that $K$ is a weakly
compact set. Setting $V_0=V_1=K$, we notice that the set $D$ in
Theorem \ref{Th3} is weakly compact as it is the intersection of
weakly closed sets $O_v^c$.  The following result follows.


\begin{cor}\label{cor1}
Let $Y=L^p(\Omega;\RR^k)$, $L=T$ and $J:L^p(\Omega;\RR^k)\ri \RR$
be the function
$$
J(u)=\int_\Omega j(x,u(x))\, dx.
$$
Assume that $K$ is a nonempty, closed, bounded and convex subset
of $V$ and let $A:K\ri V^\ast$ be a hemicontinuous and stably
pseudomonotone map with respect to the set $U(J,L)$ defined in
\eq{eq7}. Then the problem ${\bf (P)}$ has at least one solution.
\end{cor}
\medskip



\section{Existence results for nonlinear hemivariational inequalities and applications}

This section is concerned with the study of the following {\it
nonlinear hemivariational inequality problem}

Find $u\in K$ such that
\neweq{eq10}
\Lambda(u,v)+\int_\Omega j^{\,0}(x,\hat u(x);\hat v(x)-\hat
u(x))\;dx\geq \langle f,v-u\rangle,\quad \forall\;v\in K,
\endeq
where  $\Lambda:K\times K\ri \RR$ is a given function and $f\in
V^\ast$.

We point out the fact that the study of inequality problems
involving nonlinear terms has captured special attention in the
last few years. We just refer to the prototype problem of finding
$u\in K$ such that
\neweq{eq11}
\Lambda(u,v)\geq \langle f,v-u\rangle,\quad\forall\; v\in K.
\endeq
Nonlinear inequality problems of the type \eq{eq11} model some
equilibrium problems drawn from operations research, as well as
some unilateral boundary value problems stemming from mathematical
physics and where introduced by J. Gwinner \cite{Gwi} who
investigated the existence theory and abstract stability analysis
in the setting of reflexive Banach spaces.

The main object of this section is to establish existence results
for the nonlinear hemivariational inequality \eq{eq10} for general
maps, without monotonicity assumptions. As a consequence to our
theorems, we will derive some existence results for
hemivariational inequalities that have been studied in
\cite{Mot-Rad1}, \cite{Nan-Pan} and \cite{Pan-Fun-Rad} as it will
be seen at the end of this section.


\subsection{Existence results}


In this subsection we shall establish two existence results for
the inequality problem \eq{eq10}. The first result is given by the
following theorem.
\begin{teo}\label{Th4}
Let $K$ be a nonempty, closed and convex subset of $V$ and $f\in
V^\ast$. Let $\Lambda:K\times K\ri\RR$ a function vanishing on the
diagonal, that is, $\Lambda(u,u)=0$, for all $u\in K$, which
satisfies the following assumptions:
\begin{description}
\item (1) $\Lambda$ is convex with respect to the  second
variable;

\item (2) $\Lambda$ is upper semicontinuous with respect to the
first variable, that is, $\limsup_{n\ri\infty} \Lambda(u_n,v)\leq
\Lambda (u,v)$, for all $v\in K$ whenever $u_n\ri u$ in $K$;

\item (3) There exists a compact convex subset $V_1$ of $K$ such
that for each $u\in K\setminus V_1$ there is some $v$ in $V_1$ for
which we have
$$
\Lambda(u,v)+\int_\Omega j^{\;0}(x,\hat u(x);\hat v(x)-\hat
u(x))\;dx<\langle f,v-u\rangle.
$$
\end{description}
If $j$ satisfies the conditions \eq{eq1} and \eq{eq3}-\eq{eq4} and
$T:V\ri L^p(\Omega;\RR^k)$ is linear and continuous, then the
nonlinear hemivariational inequality problem \eq{eq10} has at
least one solution in $K$.
\end{teo}

\proof For each $v\in K$ we define the set
$$
N(v):=\left\{u\in K:\ \Lambda(u,v)+\int_\Omega j^{\;0}(x,\hat
u(x);\hat v(x)-\hat u(x))\; dx \geq \langle f,v-u\rangle\right\}.
$$
We point out the fact that the solution set of \eq{eq10} is
$S=\bigcap_{v\in K}N(v)$.

First we prove that for each $v\in K$ the set $N(v)$ is closed.
Let $\{u_n\}\subset N(v)$ be a sequence which converges to $u$ as
$n\ri\infty$. We show that $u\in N(v)$. Since $j$ satisfies the
conditions \eq{eq1} and \eq{eq3}-\eq{eq4}, by part (a) of Lemma 1
in \cite{Pan-Fun-Rad} (pp. 44) the application
$$
(u,v)\longmapsto \intom j^{\; 0}(x,\hat u(x);\hat v(x)-\hat
u(x))\; dx
$$
is upper semicontinuous. Since $T$ is linear and continuous, $\hat
u_n\ri \hat u$ and by the fact that $u_n\in N(v)$ for each $n$, we
have
\begin{eqnarray*}
\langle f,v-u\rangle&=&\limsup\limits_{n\ri\infty}\langle
f,v-u_n\rangle\leq \limsup\limits_{n\ri\infty}\left[\Lambda(u_n,v)
+\intom j^{\; 0}(x,\hat u_n(x);\hat v(x)-\hat u_n(x))\; dx\right]\\
&\leq&\limsup\limits_{n\ri\infty}\Lambda(u_n,v)+\limsup\limits_{n\ri\infty}\intom
j^{\; 0}(x,\hat u_n(x);\hat v(x)-\hat u_n(x))\; dx\\
&\leq&\Lambda(u,v)+\intom j^{\; 0}(x,\hat u(x);\hat v(x)-\hat
u(x))\; dx.
\end{eqnarray*}
This is equivalent to $u\in N(v)$.

Arguing by contradiction, suppose that $S=\emptyset$. Then for
each $u\in K$ there exists $v\in K$ such that
\neweq{eq12}
\Lambda(u,v)+\intom j^{\; 0}(x,\hat u(x);\hat v(x)-\hat u(x))\;
dx< \langle f,v-u\rangle.
\endeq
We define the set valued map $F:K\ri 2^K$ by $u\longmapsto F(u)$
where
$$
F(u)=\left\{v\in K:\ \Lambda(u,v)+\intom j^{\; 0}(x,\hat u(x);\hat
v(x)-\hat u(x))\; dx<\langle f,v-u\rangle\right\}.
$$
Taking \eq{eq12} into account we deduce that $F(u)$ is nonempty
for each $u\in K$. Using the fact that $\Lambda$ is convex with
respect to the second variable, $T$ is linear  and the application
$\hat v\longmapsto j^{\;0}(x,\hat u;\hat v)$ is also convex, we
obtain that $F(u)$ is a convex set.

Now, for each $v\in K$, the set
\begin{eqnarray*}
F^{-1}(v)&=&\{u\in K:\ v\in F(u)\}\\
&=&\left\{u\in K:\ \Lambda(u,v)+\intom j^{\; 0}(x,\hat u(x);\hat
v(x)-\hat u(x))\; dx<\langle f,v-u\rangle\right\}\\
&=&\left\{u\in K:\ \Lambda(u,v)+\intom j^{\; 0}(x,\hat u(x);\hat
v(x)-\hat u(x))\; dx\geq\langle f,v-u\rangle\right\}^c\\
&=&\left[N(v)\right]^c=O_v
\end{eqnarray*}
is open in K. We claim next that $\bigcup_{v\in K}O_v=K$. To prove
that, let $u\in K$. As $F(u)$ is nonempty it follows that there
exists $v\in F(u)$ which implies $u\in F^{-1}(v)$. Thus
$K\subseteq \bigcup_{v\in K}O_v$, the converse inclusion being
obvious.

Finally, from the last conditions of the theorem, for each $u\in
K\setminus V_1$ there exists $v\in V_1$ such that
$$
\Lambda(u,v)+\intom j^{\; 0}(x,\hat u(x);\hat v(x)-\hat u(x))\;
dx<\langle f,v-u\rangle
$$ which means that $u\not\in N(v)$. This implies that the set
$D=\bigcap_{v\in V_1}O_v^c=\bigcap_{v\in V_1}N(v)\subset V_1$ is
empty or compact as a closed subset of the compact set $V_1$.
Taking $V_0=V_1$ we have proved that the set valued map $F$
satisfies the conditions of Theorem \ref{Th1}, hence there exists
$u_0\in K$ such that $u_0\in F(u_0)$, that is,
$$
0=\Lambda(u_0,u_0)+\intom j^{\; 0}(x,\hat u_0(x);\hat u_0(x)-\hat
u_0(x))\; dx<\langle f,u_0-u_0\rangle=0,
$$
which is a contradiction. Hence the solution set $S$ of problem
\eq{eq10} is nonempty.
\endproof

In case $K$ is compact, the last condition of Theorem \ref{Th4} is
automatically fulfilled, since we can set $V_1=K$. Thus the
following Corollary has been proved.


\begin{cor}\label{cor2}
Let $K$ be a compact convex subset of $V$ and $f\in V^\ast$. Let
$\Lambda:K\times K\ri \RR$ be a function which vanishes on the
diagonal such that $\Lambda$ is convex with respect to the second
variable and upper semicontinuous with respect to the first
variable. Then, if $j$ satisfies \eq{eq1} and \eq{eq3}-\eq{eq4}
and $T:V\ri L^p(\Omega;\RR^k)$ is linear and continuous, the
problem \eq{eq10} has a solution in $K$.
\end{cor}


\begin{rem}\label{rem2}
If $V$ is reflexive and the operator $T:V\ri L^p(\Omega;\RR^k)$ is
linear and compact the conclusion of Theorem \ref{Th4} still holds
if $\Lambda$ is weakly upper semicontinuous with respect to the
first variable instead of being upper semicontinuous, because in
these conditions, by part (b) of Lemma 1 in \cite{Pan-Fun-Rad} the
application
$$
(u,v)\longmapsto \intom j^{\; 0}(x,\hat u(x);\hat v(x)-\hat
u(x))\; dx
$$
is weakly upper semicontinuous. The proof is identical to that of
Theorem \ref{Th4}, but the conditions of Theorem \ref{Th1} are
satisfied in the weak topology.
\end{rem}
Taking into account the above remark, we can state a variant of
Corollary \ref{cor2} for reflexive Banach spaces  as follows:
\begin{cor}\label{cor3} Let $K$ be a nonempty, bounded, closed
and convex subset of a real reflexive Banach space $V$ and $f\in
V^\ast$. Let $\Lambda:K\times K\ri \RR$ be a function which
vanishes on the diagonal such that $\Lambda$ is convex with
respect to the second variable and weakly upper semicontinuous
with respect to the first variable. Then, if $j$ satisfies
\eq{eq1} and \eq{eq3}-\eq{eq4} and $T:V\ri L^p(\Omega;\RR^k)$ is
linear and compact, the inequality problem \eq{eq10} has a
solution in $K$.
\end{cor}

In order to establish another result concerning the existence of
solutions for the nonlinear hemivariational problem \eq{eq10} we
need the following result which is due to Mosco (see \cite{Mos}):

\smallskip

\noindent{\bf Mosco's Theorem.} Let $K$ be a nonempty convex and
compact subset of a topological vector space $V$. Let
$\Phi:V\ri\RR\cup\{+\infty\}$ be a proper, convex and lower
semicontinuous function such that $D(\Phi)\cap K\neq \emptyset$.
Let $g,h:V\times V\ri\RR$ be two functions such that
\begin{description}
\item (i) $g(x,y)\leq h(x,y)$, for every $x,y\in V$;

\item(ii) the mapping $h(\cdot,y)$ is concave, for any $y\in V$;

\item (iii) the mapping $g(x,\cdot)$ is lower semicontinuous, for
every $x\in V$.
\end{description}
Let $\lambda$ be an arbitrary real number. Then the following
alternative holds: either

-there exists $y_0\in D(\Phi)\cap K$ such that
$g(x,y_0)+\Phi(y_0)-\Phi(x)\leq \lambda$, for any $x\in V$,

or

-there exists $x_0\in V$ such that $h(x_0,x_0)>\lambda$.

We notice that a particular case of interest for the above result
is if $\lambda=0$ and $h(x,x)\leq 0$, for every $x\in V$.
\begin{lemma}\label{lema2}
Let $K$ be a nonempty, bounded, closed and convex subset of a real
reflexive Banach space $V$ and $f\in V^\ast$ be arbitrary but
fixed. Consider a Banach space $Y$ such that there exists a linear
and compact mapping $L:V\ri Y$ and let $J:Y\ri \RR$ be an
arbitrary locally Lipschitz function. Suppose in addition that
$\Lambda:V\times V\ri\RR$ is a function which satisfies the
following conditions:
\begin{description}
\item (1) $\Lambda(u,u)=0,\quad\forall\; u\in V$;

\item (2) $\Lambda(u,v)+\Lambda(v,u)\geq 0,\quad \forall\; u,v\in
V$;

\item (3) $\Lambda$ is weakly upper semicontinuous and concave
with respect to the first variable.

Then it exists $u\in K$ such that for all $v\in K$ we have
$$
\Lambda(u,v)+J^{\; 0}(Lu;Lv-Lu)\geq \langle f,v-u\rangle.
$$
\end{description}
\end{lemma}

\proof Set
$$
g(v,u)=-\Lambda(u,v)-\langle f,u-v\rangle-J^{\; 0}(Lu;Lv-Lu)
$$
and
$$
h(v,u)=\Lambda(v,u)-\langle f,u-v\rangle-J^{\; 0}(Lu;Lv-Lu).
$$
By condition (2) we have
$$
g(v,u)-h(v,u)=-[\Lambda(u,v)+\Lambda(v,u)]\leq 0,\quad\forall\;
u,v\in V.
$$

The mapping $u\longmapsto g(v,u)$ is weakly lower semicontinuous
for each $v\in V$, while the mapping $v\longmapsto h(v,u)$ is
concave for each $u\in V$. We shall apply Mosco's Theorem with
$\lambda=0$ and $\Phi=I_K$, where $I_K$ denotes the indicator
function of the set $K$, that is $I_K(u)=0$ if $u\in K$ and
$I_K(u)=\infty$ otherwise. Clearly $I_K$ is proper, convex and
lower semicontinuous since $K$ is nonempty, convex and closed. We
obtain that exists $u\in K$ satisfying
$$
g(v,u)+I_K(u)-I_K(v)\leq 0,\quad\forall v\in V;
$$
A simple computation yields that there exists $u\in K$ such that
$$
\Lambda(u,v)+J^{\; 0}(Lu;Lv-Lu)\geq \langle
f,v-u\rangle,\quad\forall v\in K.
$$
\endproof

The second existence result concerning the nonlinear
hemivariational inequality problem can now be stated as follows:

\begin{teo}\label{Th5}
Assume that the hypotheses of Lemma \ref{lema2} are fulfilled for
$Y=L^p(\Omega;\RR^k)$, $L=T$ and $J:L^p(\Omega;\RR^k)\ri \RR$
$$
J(u)=\int_\Omega j(x,u(x))\, dx.
$$
Then the inequality problem \eq{eq10} has at least one solution.
\end{teo}







\subsection{Applications}

\subsubsection{Noncoercive hemivariational and variational-hemivariational inequalities} Let us
consider the noncoercive forms of the coercive and semicoercive
hemivariational inequalities investigated in \cite{Nan-Pan} (pp.
65-77, 80-85). For this, let assume that $V$ is a real Hilbert
space with the property that
$$
V\subset \left[L^2(\Omega;\RR^k)\right]^N\subset V^\ast
$$
and the injections are continuous and dense. Moreover, let $T:V\ri
L^2(\Omega;\RR^k)$, $Tu=\hat u$, $\hat u(x)\in \RR^k$ be linear
and continuos. It is also assumed that $f\in V^\ast$ and
$a:V\times V\ri \RR$ is a bilinear symmetric and continuous form.
We consider the problem:

(${\bf P_1}$) Find $u\in K$ such that
$$
a(u,v-u)+\intom j^{\;0}(x,\hat u(x);\hat v(x)-\hat u(x))\; dx\geq
\langle f,v-u\rangle,\quad \forall\; v\in  K.
$$
We take $\Lambda:V\times V\ri \RR$ defined by
$\Lambda(u,v)=a(u,v-u)$.
\begin{description}
\item {\sc Case 1}. If $K$ is compact and convex it suffices to
apply Corollary \ref{cor2} to conclude that the problem (${\bf
P_1}$) has at least one solution. We point out that this result is
more general from the point of view of the absence of the
coercivity and semicoercivity assumption, but less general from
the point of view of the compactness of $K$.

\item {\sc Case 2}. If $K$ is the entire space $V$ (as in
\cite{Nan-Pan}) we can replace the coercivity and semicoercivity
assumptions by the following assumption:\\
There exists a compact convex subset $V_1$ of $V$ such that for
each $u\in V\setminus V_1$ there is some $v\in V_1$ for which we
have
$$
a(u,v-u)+\intom j^{\;0}(x,\hat u(x);\hat v(x)-\hat
u(x))\;dx<\langle f,v-u\rangle.
$$
It suffices in this case to apply Theorem \ref{Th4} to conclude
that the problem (${\bf P_1}$) admits at least one solution.
\end{description}
\medskip

\noindent Let us consider now a functional $\Phi:V\ri
(-\infty,+\infty]$, which is convex, lower semicontinuous and
proper and formulate the following variational-hemivariational
problem:

(${\bf P_2}$) Find $u\in K$ such that
$$
a(u,v-u)+\Phi(v)-\Phi(u)+\intom j^{\;0}(x,\hat u(x);\hat v(x)-\hat
u(x))\; dx\geq \langle f,v-u\rangle,\quad \forall\; v\in K.
$$
Here $V$, $a$ and $j$ satisfy the same conditions as in the case
of the hemivariational problem (${\bf P_1}$).

Defining $\Lambda:V\times V\ri\RR$ by
$\Lambda(u,v)=a(u,v-u)+\Phi(v)-\Phi(u)$ the conclusions from Case
1 and Case 2 still hold with the specification that the condition
which replaces coercivity and semicoercivity assumptions, when we
take $K=V$, becomes: \\
There exists a compact convex subset $V_1$ of $V$ such that for
each $u\in V\setminus V_1$ there is some $v\in V_1$ for which we
have
$$
a(u,v-u)+\Phi(v)-\Phi(u)+\intom j^{\;0}(x,\hat u(x);\hat v(x)-\hat
u(x))\;dx<\langle f,v-u\rangle.
$$


\subsubsection{Hartman-Stampacchia
 type hemivariational and variational-hemivariational inequalities}

Let us consider we are in the framework of \cite{Pan-Fun-Rad}
where it is studied a hemivariational inequality of
Hartman-Stampacchia type. Let $V$ be a real Banach space and let
$T:V\ri L^p(\Omega;\RR^k)$ be a linear continuous operator, where
$1\leq p<\infty$, $k\geq 1$. $K$ is a subset of $V$, and $A:K\ri
V^\ast$ an operator while $j$ satisfies conditions \eq{eq1} and
\eq{eq3}-\eq{eq4}. We consider the problem:

(${\bf P_3}$) Find $u\in K$ such that
$$
\langle Au,v-u\rangle+\int_\Omega\, j^{\,0}(x,\hat u(x);\hat
v(x)-\hat u(x))\, dx\geq 0,\quad \forall\, v\in K.
$$

If $A$ is $w^*$-demicontinuous, that is, for any sequence
$\{u_n\}\subset K$ converging to $u$, the sequence $\{Au_n\}$
converges to $Au$ for the $w^*$ topology of $V^\ast$, then
accordingly to Remark 3 in \cite{Pan-Fun-Rad}
$$
\lim\limits_{n\ri\infty}\langle Au_n,u_n\rangle=\langle
Au,u\rangle, \quad \mbox{ whenever } u_n\ri u.
$$

We define $\Lambda(u,v)=\langle Au,v-u\rangle$ and assume that $A$
is $w^*$-demicontinuous (or equivalently the application
$u\longmapsto \langle Au,v\rangle$ is weakly upper semicontinuos),
then $\Lambda$ satisfies conditions (1)-(2) of Theorem \ref{Th4}.
\begin{description}
\item {\sc Case 1}. Let $K$ be a compact convex subset of the
infinite dimensional Banach space $V$. Applying Corollary
\ref{cor2} we obtain that the problem (${\bf P_3}$) has a
solution.

\item {\sc Case 2}. Weaking  the hypotheses on $K$ by assuming
that $K$ is a nonempty, closed and convex subset of the Banach
space $V$ we need an extra condition to ``balance'' the lack of
compactness. Theorem \ref{Th4} states that a sufficient
condition such that (${\bf P_3}$) has a solution is: \\
There exists a compact convex subset $V_1$ of $K$ such that for
each $u\in K$ there is some $v\in K$ for which we have
$$
\langle Au,v-u\rangle+\intom j^{\;0}(x,\hat u(x);\hat v(x)-\hat
u(x))\; dx<0.
$$
\end{description}

\medskip

\noindent In \cite{Mot-Rad1} it is studied the following
variational-hemivariational inequality problem:

(${\bf P_4}$) Find $u\in K$ such that
$$
\langle Au-f,v-u\rangle+\Phi(v)-\Phi(u)+\int_\Omega\,
j^{\,0}(x,\hat u(x);\hat v(x)-\hat u(x))\, dx\geq 0,\quad
\forall\, v\in K,
$$
where $f\in V^\ast$, $K$ is a subset of the real reflexive Banach
space $V$, $A:V\ri V^\ast$ is a nonlinear operator and
$\Phi:V\ri\RR\cup\{+\infty\}$ is a proper, convex and lower
semicontinuos function. We also consider that
$T:L^p(\Omega;\RR^k)\ri V$ is a linear and compact operator and
$Tu=\hat u$.

If $A$ is monotone and hemicontinuous, and $K$ is nonempty,
bounded, closed and convex we shall rediscover the result of
Theorem 2 in \cite{Mot-Rad1} by means of Theorem \ref{Th5}. We
take $\Lambda:V\times V\ri\RR$ to be $\Lambda(u,v)=\langle
Av,v-u\rangle+\Phi(v)-\Phi(u)$. It is easy to observe that
$\Lambda(v,u)=-\langle Au,v-u\rangle+\Phi(u)-\Phi(v)$. Then by the
monotonicity of $A$ we have
$$
\Lambda(u,v)+\Lambda(v,u)=\langle Av-Au,v-u\rangle\geq 0.
$$
Clearly the application $\Lambda$ defined as above is concave and
weakly upper semicontinuous with respect to the first variable.
Thus by Theorem \ref{Th5} there exists $u\in K$ such that
\neweq{eq13}
\langle Aw,w-u\rangle+\Phi(w)-\Phi(u)+\intom j^{\; 0}(x,\hat
u(x);\hat w(x)-\hat u(x))\; dx \geq \langle f,w-u\rangle,\quad
\forall\; w\in K.
\endeq

We fix $v\in K$ and set $w=tv+(1-t)u\in K$, for $t\in(0,1)$. So,
by \eq{eq13},
$$
t\langle A(tv+(1-t)u),v-u\rangle+\Phi(tv+(1-t)u)-\Phi(u)+\intom
j^{\; 0}(x,\hat u(x);t(\hat v(x)-\hat u(x)))\; dx \geq t\langle
f,v-u\rangle.
$$
Using the convexity of $\Phi$, the fact that $j^{\;0}(x,u;\cdot)$
is positive homogeneous and dividing by $t$ we find
$$
\langle A(tv+(1-t)u),v-u\rangle+\Phi(v)-\Phi(u)+\intom j^{\;
0}(x,\hat u(x);\hat v(x)-\hat u(x))\; dx \geq \langle
f,v-u\rangle.
$$
Now, letting $t\ri 0$ and using the hemicontinuity  of $A$ we find
that $u$ is a solution of {\bf ($P_4$)}.

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\end{document}