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

\AtBeginDocument{{\noindent\small
2005-Oujda International Conference on Nonlinear Analysis.
\newline {\em Electronic Journal of Differential Equations},
Conference 14, 2006, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE/Conf/14 \hfil A solution method]
{A solution method of the Signorini problem}

\author[A. Addou, A. Lidouh, B. Seddoug \hfil EJDE/Conf/14 \hfilneg]
{Ahmed Addou, Abdeluaab Lidouh, Belkassem Seddoug}  % in alphabetical order

\address{Ahmed Addou \newline
Universit\'{e} Mohammed premier, Facult\'{e} des sciences,
 Oujda, Maroc}
\email{addou@sciences.univ-oujda.ac.ma}

\address{Abdeluaab Lidouh \newline
Universit\'{e} Mohammed premier, Facult\'{e} des sciences,
 Oujda, Maroc}
\email{lidouh@sciences.univ-oujda.ac.ma}

\address{Belkassem Seddoug \newline
Universit\'{e} Mohammed premier, Facult\'{e} des sciences,
 Oujda, Maroc}
\email{seddougbelkassem@yahoo.fr}

\date{}
\thanks{Published September 20, 2006.}
\subjclass[2000]{35J85, 35J35}
\keywords{Signorini problem; variational inequality}

\begin{abstract}
 In this note, we are interested in the variational formulation of
 an unilateral contact problem, the so-called Signorini problem. We
 show that the normal derivative of the solution, can be computed
 according to the data of the problem. What permits the
 determination of the solution by solving a Neumann problem.
\end{abstract}

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


\section{Introduction}

In this work, we are interested in the Signorini problem that
consists in finding $u$ such that
\begin{equation}
\begin{gathered}
-\Delta u+u=f\quad \text{in }\Omega , \\
u\geq 0,\frac{\partial u}{\partial n}\geq 0,\quad
u\frac{\partial u}{\partial n}
=0\quad\text{on }\Gamma ,
\end{gathered} \label{P1}
\end{equation}
where $\Omega $ is a bounded open subset of $\mathbb{R}^2$, which
boundary $\Gamma $ is regular enough, and $f\in L^{2}(\Omega )$.
Another unknown of the problem is the coincidence set on the border of
$\Omega $.

It is known that the variational principle applied to \eqref{P1}
produces the variational inequality that consists in finding $u$
such that
\begin{equation}
a(u,v-u)\geq \int_{\Omega }f.(v-u)\quad\forall
v\in K, \label{P2}
\end{equation}
where $a$ is the scalar product of $H^{1}(\Omega )$ and
$K=\{v\in H^{1}(\Omega ): v\geq 0\text{ on }\Gamma \} $.

As in \cite{Stamp}, we introduce the function $\psi $, a solution of
the Dirichlet problem,
\begin{equation}
\begin{gathered}
-\Delta \psi +\psi =f\quad \text{in }\Omega  \\
\psi =0\quad\text{on }\Gamma,
\end{gathered}  \label{P3}
\end{equation}
and we show that the solution of (\ref{P2}) is the solution of the
variational inequality
\begin{equation}
a(u,v-u)\geq \int_{\Omega }f.(v-u)\quad \forall
v\in K_{\psi },  \label{P4}
\end{equation}
where $K_{\psi }=\{ v\in H^{1}(\Omega ): v\geq \psi \text{ in }
\Omega \} $.

To solve  problem (\ref{P4}), we propose a method that consists
in determining first $\frac{\partial u}{\partial n}$, with the
help of the data of the problem, then solving a Neumann problem to
determine $u$.

\section{Formulation of the problem}

Let $\Omega $ be a bounded open subset of $\mathbb{R}^2$, with boundary
$\Gamma $ is regular enough. The Sobolev
space $H^{1}(\Omega )$ is equipped with its scalar product
\begin{equation*}
a(u,v)=\int_{\Omega }\nabla u.\nabla vdx+\int_{\Omega }u.vdx\text{ \ \ }
\forall u,v\in H^{1}(\Omega ).
\end{equation*}
We denote by $K$ the closed convex cone of $H^{1}(\Omega )$ defined
as
\begin{equation*}
K=\{ v\in H^{1}(\Omega ): v\geq 0\text{ on }\Gamma
\}.
\end{equation*}
Then we consider the variational inequality problem:
Find $u\in K$  such that
\begin{equation}
a(u,v-u)\geq \int_{\Omega }f.(v-u),\quad \forall v\in K,
 \label{ineq1}
\end{equation}
where $f\in L^{2}(\Omega )$.

In the sequel, we  use of the following notations:
\[
v^{+}=\max (v,0),\quad
v^{-}=\min (v,0)
\]
such that $v=v^{+}+v^{-}$  for all $v$  in $L^{2}(\Omega )$
 or in $L^{2}(\Gamma)$.

We will note indifferently a function $v$ of $H^{1}(\Omega )$
and its restriction to $\Gamma $.

\begin{proposition} \label{prop2.1}
Problem \eqref{ineq1} is equivalent to the problem:
Find $u\in K_{\psi }$  such that
\begin{equation}
a(u,v-u)\geq \int_{\Omega }f.(v-u),\quad \forall v\in K_{\psi},  \label{ineq2}
\end{equation}
where $\psi $ is given by \eqref{P3} and
$K_{\psi }=\{ v\in H^{1}(\Omega ): v\geq \psi\text{ in }\Omega \}$.
\end{proposition}

\begin{proof}
It suffices to show that the solution $u$ of \eqref{ineq1} is in
$K_{\psi }$, i.e: $u\geq \psi $.

Let $u$ be the solution of \eqref{ineq1}. With
$v=u-(u-\psi)^{-}\in K$ in (\ref {ineq1}), and while taking account of
(\ref{P3}), one has
\begin{equation*}
a( (u-\psi )^{-},(u-\psi )^{-}) \leq 0.
\end{equation*}
So $(u-\psi )^{-}=0$, which implies $u\geq \psi $ in $\Omega $.
\end{proof}

Using Green's formula, \eqref{ineq1} can be written as:
Find $u\in K_{\psi }$  such that
\begin{equation}
a(u-\psi ,v-u)+\langle \frac{\partial \psi }{\partial n}
,v-u \rangle \geq 0,\quad \forall v\in K_{\psi },  \label{ineq3}
\end{equation}
where $\langle .,.\rangle $ designates the scalar product of
$L^{2}(\Gamma )$, and $\frac{\partial \psi }{\partial n}$ the
normal derivative of $\psi $.

To have $\frac{\partial \psi }{\partial n}$ at the same time as
$\psi $, we propose the resolution of (\ref{P3}) by enforcing the
boundary condition with Lagrange multipliers (see \cite{Babuska}),
then the variational formulation of (\ref{P3}) reads:
Find $(\psi ,\lambda )\in H^{1}(\Omega )\times
H^{-\frac{1}{2}}(\Gamma)$  such that
\begin{equation}
\begin{gathered}
a(\psi ,v)-\int_{\Gamma }\lambda v=\int_{\Omega }fv,\quad \forall v\in
H^{1}(\Omega ), \\
\int_{\Gamma }\mu \psi =0,\quad \forall \mu \in H^{-\frac{1}{2}}(\Gamma ),
\end{gathered}  \label{P5}
\end{equation}
whose solution is the saddle point of the Lagrangian
$\mathcal{J}(v,\mu )=\frac{1}{2}a(v,v)-\int_{\Omega }fv-\int_{\Gamma }\mu v$.

It is known (see \cite{Babuska}) that  problem (\ref{P5}) has
a unique solution $(\psi ,\lambda )$, verifying
\begin{equation}
\begin{gathered}
-\Delta \psi +\psi =f\quad \text{in }\Omega , \\
\lambda =\frac{\partial \psi }{\partial n}\quad \text{on }\Gamma , \\
\psi =0\quad\text{on }\Gamma .
\end{gathered}  \label{P6}
\end{equation}

Note that
$f$ being in $L^{2}(\Omega )$, implies $\psi $ in $H^{2}(\Omega )$
which implies $\frac{\partial \psi }{\partial n}$
in $H^{1/2}(\Gamma )$.


\section{Transformation of the problem}

Hereafter we deal with the problem \eqref{ineq3}, and we consider
the function
$\varphi :L^{2}(\Gamma )\to \mathbb{R}$ defined as
\[
\nu \mapsto \langle \big( \frac{\partial \psi
}{\partial n}\big) ^{+},\nu ^{+}\rangle.
\]
Note that $\varphi $ is convex and continuous on $L^{2}(\Gamma )$.

\begin{proposition} \label{prop3.1}
A function $u$ is a solution of \eqref{ineq3} if and only if $w=u-\psi $ is
the solution of the problem: Find $w\in H^{1}(\Omega )$  such that
\begin{equation}
a(w,v-w)+\varphi (v)-\varphi (w)+\langle \big( \frac{\partial \psi }{
\partial n}\big) ^{-},v-w\rangle \geq 0, \quad \forall v\in
H^{1}(\Omega ). \label{ineq4}
\end{equation}
\end{proposition}

\begin{proof}
Since the two problems \eqref{ineq3} and \eqref{ineq4} have an
unique solution, it suffices to show that the solution $w$ of
\eqref{ineq4} is non negative in $\Omega $.
Let $w$ be the solution of \eqref{ineq4}, with $v=w^{+}$ in
\eqref{ineq4}, we have
\begin{equation*}
a(w,-w^{-})+\langle ( \frac{\partial \psi }{\partial
n}) ^{-},-w^{-}\rangle \geq 0
\end{equation*}
then $a(w^{-},w^{-})\leq \langle \big( \frac{\partial \psi
}{\partial n}\big) ^{-},-w^{-}\rangle \leq 0$, so
$w^{-}=0$.
\end{proof}

We remark that problem \eqref{ineq4} differs from \eqref{ineq3} by the fact
that it is without constraints.

\begin{proposition} \label{prop3.3}
A function $w$ is solution of problem \eqref{ineq4} if and only if $(w,\mu )$
is solution of the problem:
Find  $(w,\mu )\in H^{1}(\Omega )\times L^{2}(\Gamma )$
such that
\begin{equation}
\begin{gathered}
a(w,v)+\langle \mu ,v\rangle +\langle (
\frac{\partial \psi }{\partial n}) ^{-},v\rangle
=0\quad \forall v\in H^{1}(\Omega ), \\
\mu \in \partial \varphi (w),
\end{gathered}  \label{ineq5}
\end{equation}
where $\partial \varphi (w)=\{ \lambda \in L^{2}(\Gamma ):
\forall \nu \in L^{2}(\Gamma ),\langle \lambda ,\nu
-w\rangle \leq
\varphi (\nu )-\varphi (w)\} $ designates the subdifferential of
$\varphi $ at $w$.
\end{proposition}

\begin{proof}
Let $w$ be the solution of \eqref{ineq4}.
On the one hand, for all $v$ in $H_{0}^{1}(\Omega )$, we have
$a(w,v)=0$, hence the mapping
\begin{equation*}
v\mapsto a(w,v)+\langle ( \frac{\partial \psi }{\partial n}
) ^{-},v\rangle
\end{equation*}
is well defined on $\Gamma $.

On the other hand, for all $v$ in $H^{1}(\Omega )$, one has
\begin{align*}
a(w,v)+\langle \big( \frac{\partial \psi }{\partial
n}\big)^{-},v\rangle  &\geq \varphi (w)-\varphi (v+w) \\
&\geq \langle \big( \frac{\partial \psi }{\partial
n}\big)^{+},w^{+}-(v+w)^{+}\rangle  \\
&\geq -\langle \big( \frac{\partial \psi }{\partial n}\big)
^{+},v^{+}\rangle  \\
&\geq -\big\| \big( \frac{\partial \psi }{\partial n}\big)
^{+}\big\| _{L^{2}(\Gamma )}\| v_{\mid \Gamma }\|_{L^{2}(\Gamma
)}.
\end{align*}
Using the same argument, and with $-v$ one has
\begin{equation*}
a(w,-v)+\langle ( \frac{\partial \psi }{\partial
n})
^{-},-v\rangle \geq -\big\| \big( \frac{\partial \psi }{\partial n}
\big) ^{+}\big\| _{L^{2}(\Gamma )}\| v_{\mid \Gamma
}\| _{L^{2}(\Gamma )}.
\end{equation*}
Hence for all $v$ in $H^{1}(\Omega ),$ we have
\begin{equation*}
| a(w,v)+\langle ( \frac{\partial \psi }{\partial
n})
^{-},v\rangle | \leq \| ( \frac{\partial \psi }{
\partial n}) ^{+}\| _{L^{2}(\Gamma )}\| v_{\mid \Gamma
}\| _{L^{2}(\Gamma )}.
\end{equation*}
Therefore,  the linear form $v\mapsto a(w,v)+\langle
( \frac{\partial \psi }{\partial n})^{-},v\rangle $ is continuous
on $H^{1/2}(\Gamma )$ for the norm of $L^{2}(\Gamma )$.
Then because of the density of $H^{1/2}(\Gamma )$ in $L^{2}(\Gamma )$,
 there exists $\mu $ in $L^{2}(\Gamma )$ verifying the equality
of (\ref {ineq5}).

Inversely, it is easy to see that if $(w,\mu )$ is solution of
(\ref{ineq5}) then $w$ is solution of \eqref{ineq4}.
\end{proof}

To characterize $\partial \varphi (w)$, we consider the
closed convex set of $L^{2}(\Gamma )$:
\begin{equation*}
C=\{ \lambda \in L^{2}(\Gamma ): \forall \nu \in
L^{2}(\Gamma ):\langle \lambda ,\nu \rangle \leq
\varphi (\nu )\}.
\end{equation*}
It is easy to show that $C=\{ \lambda \in L^{2}(\Gamma ):
0\leq \lambda \leq ( \frac{\partial \psi }{\partial
n}) ^{+}\text{ a.e in }\Gamma \} $.

\begin{lemma} \label{lemma1}
for all $\mu $ in $L^{2}(\Gamma )$ and $w$ in $H^{1}(\Omega )$, one has
$\mu \in \partial \varphi (w)$ if and only if
\[
\mu \in C\quad \text{and}\quad
\langle \lambda -\mu ,w\rangle \leq 0,\quad \forall
\lambda \in C.
\]
In particular $( \frac{\partial \psi }{\partial n}) ^{+}.\chi _{[
w_{\mid \Gamma }>0]} \in \partial \varphi (w)$.
\end{lemma}

\begin{proof}
Let $\mu \in \partial \varphi (w)$, i.e,
\begin{equation}
\forall v\in H^{1}(\Omega ),\quad \langle \mu ,v-w\rangle
\leq \varphi (v)-\varphi (w).  \label{ineq6}
\end{equation}
With $v=2w$ and $v=0$ in (\ref{ineq6}), and while $\varphi $ is
positively homogeneous, one has
\begin{equation*}
\langle \mu ,w\rangle =\varphi (w).
\end{equation*}
Therefore, while taking back (\ref{ineq6}), one deducts that
$\mu\in C$.
Inversely, let $\mu \in C$ and
$\langle \lambda -\mu ,w\rangle \leq 0$, for all $\lambda \in C$.
 For $\lambda $ in $\partial \varphi (w)\subset C$, we have
\begin{equation*}
\langle \lambda ,w\rangle =\varphi (w)\text{ and
}\langle \lambda -\mu ,w\rangle =0.
\end{equation*}
So that for all $v\in H^{1}(\Omega )$, $
\langle \mu ,v-w\rangle =\langle \mu ,v\rangle -\varphi
(w)\leq \varphi (v)-\varphi (w)$.
\end{proof}

Taking into account  lemma \ref{lemma1}, the problem
(\ref{ineq5}) can be written as:
Find $(w,\mu )\in H^{1}(\Omega )\times C$  such that
\begin{equation}
\begin{gathered}
a(w,v)+\langle \mu ,v\rangle +\langle (
\frac{\partial \psi }{\partial n}) ^{-},v\rangle
=0\quad \forall v\in H^{1}(\Omega ), \\
\langle \lambda -\mu ,w\rangle \leq 0,\quad
\forall \lambda \in C.
\end{gathered} \label{ineq7}
\end{equation}

\begin{remark} \label{rmk35} \rm
The bilinear form $a$ is symmetric, then the problem (\ref{ineq7})
is equivalent to the saddle point problem \cite{Ekeland}:
Find $(w,\mu )\in H^{1}(\Omega )\times C$  such that
\[
\mathcal{L}(w,\lambda )\leq \mathcal{L}(w,\mu )\leq \mathcal{L}(v,\mu )=0
\quad \forall (v,\lambda )\in H^{1}(\Omega )\times C,
\]
where the Lagrangian $\mathcal{L}$ is defined on
$H^{1}(\Omega )\times L^{2}(\Gamma )$ by
\begin{equation*}
\mathcal{L}(v,\lambda )=\frac{1}{2}a(v,v)+\langle \lambda
,v\rangle +\langle ( \frac{\partial \psi }{\partial n}) ^{-},v\rangle
\end{equation*}

The study of the mixed formulation (\ref{ineq7}), is in
preparation by the authors. However, it is clear that the
determination, a priori, of $\mu $, permits to solve the problem
(\ref{ineq7}) as being the Neumann problem
\begin{equation*}
\begin{gathered}
-\Delta w+w=0\quad \text{in }\Omega , \\
\frac{\partial w}{\partial n}=-\mu -( \frac{\partial \psi }{\partial n
}) ^{-}\quad \text{on }\Gamma .
\end{gathered}
\end{equation*}
Which defines $u$ as a solution of the problem
\begin{gather*}
-\Delta u+u=f\quad \text{ in }\Omega , \\
\frac{\partial u}{\partial n}=( \frac{\partial \psi }{\partial n}
) ^{+}-\mu \quad \text{ on }\Gamma,
\end{gather*}
\end{remark}

What we propose in this work is to uncouple the problem
(\ref{ineq7}), by showing that $\mu $ can be computed according to
the data of the initial problem. We consider, then, the linear
mapping
$A :H^{-\frac{1}{2}}(\Gamma )\to H^{1}(\Omega )$ defined
by $g \mapsto v$, where $v$ is a solution to
\begin{gather*}
-\Delta v+v=0\quad \text{in }\Omega , \\
\frac{\partial v}{\partial n}=g\quad \text{on }\Gamma,
\end{gather*}
which is continuous, more precisely, one has the following result
\cite[theorem 2.7]{Babuska}.

\begin{lemma} \label{lemma2}
There exist two positive constants $c_{1}$ and
$c_{2}$, such that
\begin{equation*}
c_{1}\| Ag\| _{H^{1}(\Omega )}\leq \| g\| _{H^{-\frac{1
}{2}}(\Gamma )}\leq c_{2}\| Ag\| _{H^{1}(\Omega )},
\end{equation*}
for all $g$ in $H^{-1/2}(\Gamma )$.
\end{lemma}

Using the set $M=A(C)=\{ v\in H^{1}(\Omega ): -\Delta
v+v=0\text{ in }\Omega \text{ and }\frac{\partial v}{\partial
n}\in C\}$ which is closed, bounded and convex in
$H^{1}(\Omega )$,  problem (\ref{ineq7}), can be written as:
Find $(w,z)\in H^{1}(\Omega )\times M$  such that
\begin{gather*}
a(w+z+t,v)=0\quad \forall v\in H^{1}(\Omega ), \\
a(s-z,w)\leq 0,\quad \forall s\in M,
\end{gather*}
where $t=A\big( ( \frac{\partial \psi }{\partial n})
^{-}\big) $ and $z=A\mu $. What defines $z$ as being the
projection of $-t$ on $M$; i.e,
\begin{equation*}
t=A\big( ( \frac{\partial \psi }{\partial n})
^{-}\big) ,z=\mathop{\rm proj}{}_{M}(-t)\quad
\text{and}\quad w=-z-t.
\end{equation*}

\section{A projection algorithm}

For the determination of $z=\mathop{\rm proj}_{M}(-t)$, and therefore $\mu $,
according to the data, we propose a projection algorithm inspired
of the one of Degueil \cite{Degueil}.

\subsection*{Description of the algorithm}

For an initial guess $\mu _{0}\in C$ (for example $\mu _{0}=0$), we compute $
z_{0}$ such that $z_{0}=A\mu _{0}$ then we construct the sequences
$(\mu _{n})$ and $(z_{n})$, as follows:

\begin{enumerate}
\item  Given $z_{n}$ and $\mu _{n}$ such that $z_{n}=A\mu _{n}$ and $
\mu _{n}\in C$, we compute $v_{n}=Ag_{n}$ where
\begin{equation*}
g_{n}=( \frac{\partial \psi }{\partial n}) ^{+}.\chi _{[ (
-t-z_{n}) _{\mid \Gamma }>0] }.
\end{equation*}
By  lemma \ref{lemma1}, we see that $g_{n}\in \partial \varphi
(-t-z_{n})\subset C,$ and then:
\begin{equation}
a(v-v_{n},z_{n}+t)\geq 0,\text{ }\forall v\in M.  \label{cond1}
\end{equation}

\item  We compute $z_{n+1}=\mathop{\rm proj}_{[ z_{n},v_{n}] }(-t)$
(the projection of $-t$ on the segment $[ z_{n},v_{n}]$), i.e,
\begin{equation}
\begin{gathered}
z_{n+1}=\lambda _{n}v_{n}+(1-\lambda _{n})z_{n}, \\
\mu _{n+1}=\lambda _{n}g_{n}+(1-\lambda _{n})\mu _{n} \\
\lambda _{n}=\min \Big( 1,\frac{a(z_{n}-v_{n},z_{n}+t)}{\|
v_{n}-z_{n}\| _{H^{1}(\Omega )}^{2}}=\frac{\langle \mu
_{n}-g_{n},z_{n}+t\rangle }{\langle \mu
_{n}-g_{n},v_{n}-z_{n}\rangle }\Big)
\end{gathered}  \label{cond2}
\end{equation}
if $v_{n}=z_{n}$ then $z_{n+1}=z_{n}$, the algorithm stops.
\end{enumerate}

\subsection*{Convergence result}
As in \cite{Degueil}, we have the following convergence result.

\begin{theorem} \label{thm4.1}
With the hypothesis and notation above, one has
\begin{equation*}
\lim_{n\to +\infty }\| z_{n}-z\|_{H^{1}(\Omega
)}=\lim_{n\to +\infty }\| \mu _{n}-\mu \| _{H^{-
\frac{1}{2}}(\Gamma )}=0.
\end{equation*}
\end{theorem}

\begin{proof}
Let us, first, show that $\lim_{n\to +\infty
}\| z_{n}-z\| _{H^{1}(\Omega )}=0$. For all $n$, one
has:
\begin{align*}
\| z_{n}+t\| _{H^{1}(\Omega )}^{2}
&=\|z_{n}-z_{n+1}\| _{H^{1}(\Omega )}^{2}+\|z_{n+1}+t\|
_{H^{1}(\Omega )}^{2}+2a( z_{n}-z_{n+1},z_{n+1}+t)  \\
&\geq \| z_{n}-z_{n+1}\| _{H^{1}(\Omega )}^{2}+\|
z_{n+1}+t\| _{H^{1}(\Omega )}^{2}
\end{align*}
Then $\| z_{n}-z_{n+1}\| _{H^{1}(\Omega )}^{2}\leq
\| z_{n}+t\| _{H^{1}(\Omega )}^{2}-\|
z_{n+1}+t\| _{H^{1}(\Omega )}^{2}$, what implies that
\begin{equation}
\lim_{n\to +\infty }\| z_{n}-z_{n+1}\|
_{H^{1}(\Omega )}=0. \label{cond3}
\end{equation}
On the other hand, $z=\mathop{\rm proj}_{M}(-t)$, then
\begin{equation}
a(v-z,z+t)\geq 0,\quad \forall v\in M.  \label{cond4}
\end{equation}
With $v=z_{n+1}$ in (\ref{cond4}) and while taking account of
((\ref{cond1}) and (\ref{cond2})), one has
\begin{align*}
&\| z_{n+1}-z\| _{H^{1}(\Omega )}^{2} \\
&=a(z_{n+1}-z,z_{n+1}-z_{n}) +a( z_{n+1}-z,z_{n}+t)
-a(z_{n+1}-z,z+t)  \\
&\leq a( z_{n+1}-z,z_{n+1}-z_{n}) +a(
z_{n+1}-z,z_{n}+t)  \\
&\leq a( z_{n+1}-z,z_{n+1}-z_{n}) +a(
z_{n+1}-z,z_{n}+t) +a( z-v_{n},z_{n}+t)  \\
&\leq a( z_{n+1}-z,z_{n+1}-z_{n}) +a(
z_{n+1}-v_{n},z_{n}+t)  \\
&\leq a( z_{n+1}-z,z_{n+1}-z_{n}) +(1-\lambda
_{n})a(
z_{n}-v_{n},z_{n}+t)  \\
&\leq a( z_{n+1}-z,z_{n+1}-z_{n}) +(1-\lambda
_{n})\lambda
_{n}\| z_{n}-v_{n}\| _{H^{1}(\Omega )}^{2} \\
&\leq a( z_{n+1}-z,z_{n+1}-z_{n}) +(1-\lambda
_{n})\| z_{n+1}-z_{n}\| _{H^{1}(\Omega )}\|
z_{n}-v_{n}\|
_{H^{1}(\Omega )} \\
&\leq \| z_{n+1}-z_{n}\| _{H^{1}(\Omega )}\big\{
\| z_{n+1}-z\| _{H^{1}(\Omega )}+\|
z_{n}-v_{n}\| _{H^{1}(\Omega )}\big\}.
\end{align*}

We conclude using (\ref{cond3}) and the fact that $M$ is bounded.
To show that $\lim_{n\to +\infty }\| \mu _{n}-\mu \|
_{H^{-\frac{1}{2}}(\Gamma )}=0$, it suffices to use Lemma \ref
{lemma2}.
\end{proof}

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\end{document}