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\begin{document}
\setcounter{page}{33}
\noindent
\parbox{2.85cm}{\includegraphics*[keepaspectratio=true,scale=1.75]{BJMA.jpg}}
\noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent
Banach J. Math. Anal. 2 (2008), no. 1, 33--39\\
$\frac{\rule{4.55in}{0.05in}}{{}}$\\
{\footnotesize
\textcolor[rgb]{0.65,0.00,0.95}{\textsc{\textbf{\large{B}}anach
\textbf{\large{J}}ournal of \textbf{\large{M}}athematical
\textbf{\large{A}}nalysis}}\\
ISSN: 1735-8787 (electronic)\\
\textcolor[rgb]{0.00,0.00,0.84}{\textbf{http://www.math-analysis.org }}\\
$\frac{{}}{\rule{4.55in}{0.05in}}$}\\[.5in]}

\title[Extended general variational inequalities]{ Auxiliary Principle
Technique for Extended General Variational  Inequalities }

\author[M. Aslam Noor ]{Muhammad Aslam Noor$^1$}

\address{$^1$ Department of Mathematics, COMSATS Institute of
Information Technology, Islamabad, Pakistan }

\email{\textcolor[rgb]{0.00,0.00,0.84}{noormaslam@hotmail.com }}

\dedicatory{{\rm  Submitted by Th. M. Rassias}}

\subjclass[2000]{Primary 49J40; Secondary 90C33}

\keywords{Variational inequalities, nonconvex functions,
fixed-point problem, convergence, auxiliary principle.}

\date{Received: 10  January  2008; Accepted: 22 February 2008.}


\begin{abstract}
In this paper,  we use the auxiliary
principle technique to study the existence of a solution of the extended
general variational inequalities, which were introduced and studied by
the author. Several special cases are also discussed.
\end{abstract}

\maketitle


\section{Introduction }

\noindent Variational inequalities, which were introduced in
1960's, represent the  optimality conditions for the
differentiable convex functions on the  convex sets in normed space.
It is known that the properties of the solutions of the variational
inequalities may not hold, in general, when the convex set is nonconvex.
In the recent years, the concept of convexity has been generalized in
several directions, see, for example, \cite{2} and the references therein.
Noor \cite{15, 16} introduced the nonconvex set, which is   called the
$(h,g)$-convex set, using  the idea of  a segmental type of non-connected
convexity for sets by taking into account only convex combinations of
special types of points, see also \cite{2}.
Noor \cite{15} has shown  that the minimum of a differentiable  $(h,g)$-convex
function on a $(h,g)$-convex set can be   characterized by a class of
variational inequalities. This fact has motivated  Noor \cite{13}  to
introduce  and consider a new class of  variational inequalities, which is
called the  {\it extended general  nonlinear variational inequality }
involving three
operators. It has been shown \cite{15} that the extended general variational
inequalities are equivalent to the fixed point problems. This
alternative equivalence has been used to suggest a wide class of
iterative methods for solving the extended general variational
inequalities. It is known that it is very difficult to find the
projection of the operator except in very special cases. To overcome
this drawback, on uses the auxiliary principle technique. This technique
is mainly due to Glowinski, Lions and Tremolieres \cite{4}. This technique is
more flexible and has been used to develop several
numerical methods for solving the  variational inequalities and the
equilibrium problems.  In this paper, we
 again use the auxiliary principle technique to study the existence of a
solution  of the extended general  variational inequalities.
 Since the extended general variational
inequalities include  various classes of variational inequalities
and complementarity problems as special cases, results proved in
this paper continue to hold for these problems. Results proved in this
paper may be viewed  as important and significant improvement of the
previously known results. It is interesting to explore the applications of
these extended general variational inequalities in mathematical and
engineering sciences with new and novel aspects.

\bigskip

\section{Preliminaries }


Let $H$ be a real Hilbert space whose inner product and norm are
denoted by $\la \cdot, \cdot \ra$ and $\| . \|,$ respectively.  Let $K$ be
a nonempty closed  and convex  set in $H.$

 For given nonlinear operators $T,g, h:H \rightarrow H$, consider the
problem of finding $u \in H, h(u) \in K$ such that
\begin{equation}\label{2.1}
\la Tu, g(v) - h(u)\ra      \geq 0, \qquad \forall v\in H: g(v) \in K.
\end{equation}
Inequality of type \eqref{2.1} is called the {\it extended general
 variational inequality involving three operators,} which was introduced
and studied by Noor \cite{15}.

\medskip

  We  now show that  the
minimum of a class of differentiable   nonconvex functions on $(h,
g)$-convex set $K$ in $H$ can be characterized  by extended general
variational inequality \eqref{2.1}. For this purpose, we recall the following
well known concepts, see \cite{2}.

\begin{definition} Let $K$ be any set in $H$.  The set $K$
is said to be $(h,g)$-convex, if there exist  functions $g, h: H
\longrightarrow H $ such that
\begin{eqnarray*}
h(u)+t(g(v)-h(u)) \in K, \quad  \forall u,v \in H: h(u), g(v) \in K, \quad
t\in [0,1].
\end{eqnarray*}
\end{definition}

Note that every convex set is $(h,g)$-convex, but the converse is not
true, see \cite{2}. If $g = h, $ then the $(h,g)$-convex set $K $ is called the
$g$-convex set, which was introduced by Youness \cite{21}. See also Cristescu
and Lupsa \cite{2} for its various extensions and generalization.

\vskip .2pc

\begin{definition} The function $F: K \longrightarrow H$ is
said to be $(h,g)$-convex on the $(g,h)$-convex set $K,$ if there exist
two functions $h, g$ such that
\begin{eqnarray*}
F(h(u)+t(g(v)-h(u))) & \leq & (1-t)F(h(u))+tF(g(v))\,, \\
&& \quad  \forall u,v \in H: h(u),g(v) \in K, \quad t\in [0,1].
\end{eqnarray*}
\end{definition}

Clearly every convex function is $(h,g)$-convex, but the converse is not
true. For $g =h, $ Definition 2.2  is due to Youness \cite{21}.

It is known \cite{15} that the minimum of a differentiable $(h,g)$-convex
function  on a $(h,g)$-convex set $K$ in $H$  can be characterized by
the extended general variational inequality \eqref{2.1}. For the sake of
completeness and to convey an idea of the technique, we include its proof.

\vskip .2pc

\begin{lemma} Let $F: K \longrightarrow H$ be a
differentiable $(h,g)$-convex function on the $(g,h)$-convex set $K. $
Then $u \in H:h(u) \in K $ is the
minimum of
$(h,g)$-convex function $F$ on $K$ if and only if $u \in H: h(u) \in K $
satisfies the  inequality
\begin{eqnarray}\label{2.2}
\la F^{\prime }(h(u)), g(v)-h(u)\ra  \geq 0, \quad \forall v\in H: g(v)
\in K,
\end{eqnarray}
where $F^{\prime }(u) $ is the Frechet differential of $F$ at $h(u) \in
K.$
\end{lemma}
\begin{proof} Let $u \in H : h(u) \in K $ be a minimum of
$hg$-convex function $F$ on $K.$ Then
\begin{eqnarray}\label{2.3}
F(h(u)) \leq F(g(v))\,, \quad  \forall v\in H: g(v) \in K.
\end{eqnarray}
Since $K$ is a $hg$-convex set, so, for all $u,v \in H: h(u),g(v) \in K, t
\in [0,1],
g(v_t)=h(u)+t(g(v)-h(u)) \in K.$ Setting $g(v)= g(v_t)$ in \eqref{2.3}, we have
\begin{eqnarray*}
F(h(u)) \leq F(h(u)+ t(g(v)-h(u)) \leq F(h(u))+t(F(g(v))-F(h(u))).
\end{eqnarray*}
Dividing the above inequality by $t$ and taking $t \longrightarrow 0,$ we
have
\begin{eqnarray*}
\la F^{\prime }(h(u)), g(v)-h(u)\ra  \geq 0, \quad \forall v\in H: g(v)
\in K,
\end{eqnarray*}
which is the required result \eqref{2.2}.

Conversely, let $u \in H: h(u) \in K$ satisfy the inequality \eqref{2.2}. Since
$F$ is a $hg$-convex function, $\quad \forall u,v \in H: h(u),g(v) \in
K, \quad t \in [0,1], \quad h(u)+t(g(v)-h(u)) \in K $ and
\noindent
\begin{eqnarray*}
F(h(u)+t(g(v)-h(u))) \leq (1-t)F(h(u))+tF(g(v))\,,
\end{eqnarray*}
which implies that
\noindent
\begin{eqnarray*}
F(g(v))-F(h(u)) \geq \frac{F(h(u)+t(g(v)-g(u)))-F(h(u))}{t}.
\end{eqnarray*}
\begin{eqnarray*}
F(g(v))-F(h(u)) \geq \la F^{\prime }(h(u)),g(v)-h(u)\ra \geq 0, \quad
\mbox{using \eqref{2.2},}
\end{eqnarray*}
which implies that
\begin{eqnarray*}
F(h(u)) \leq F(g(v))\,, \quad \forall v\in H: g(v) \in K
\end{eqnarray*}
showing that $u \in K$ is the minimum of $F$ on $K$ in $H.$

\end{proof}
Lemma 2.3 implies that $hg$-convex programming problem can be studied via
the extended  general variational inequality \eqref{2.1} with $Tu = F'(h(u))$.
In a similar way, one can show that the extended general variational
inequality is the Fritz-John condition of  the inequality constrained
optimization problem.

\vskip .2pc

We now list some special cases of the extended general variational
inequalities.

\medskip

\noindent\textbf{I. } \ If $g =h, $ then problem \eqref{2.1} is equivalent to
finding $u \in H:  g(u) \in K $ such that
\begin{eqnarray}\label{2.4}
\la Tu,g(v)-g(u) \ra \geq 0, \qquad \forall v\in H:g(v) \in K,
\end{eqnarray}
which is known as general variational inequality, introduced and studied
by Noor \cite{5} in 1988. It turned out that odd order and nonsymmetric
obstacle, free, moving, unilateral and equilibrium problems arising in
various branches of pure and applied sciences can be studied via general
variational inequalities.

\medskip


\textbf{II.} \ For $g \equiv I$, the identity operator, the extended  general
variational inequality \eqref{2.1} collapses to:
find $u \in H: h(u) \in  K $ such that
\begin{eqnarray}\label{2.5}
\la Tu,v-h(u) \ra \geq 0, \quad \forall v\in K,
\end{eqnarray}
which is also called the  general variational inequality, see Noor \cite{6}.

\medskip


\textbf{III. } \ For $ h = I, $ the identity operator, the extended general
variational inequality \eqref{2.1} is equivalent to finding $u \in KI $ such
that
\begin{eqnarray}\label{2.6}
\la Tu,g(v)- u \ra \geq 0, \quad \forall v\in H: g(v) \in K,
\end{eqnarray}
which is also called the general variational inequality involving two
nonlinear operators which was introduced and studied by Noor \cite{16, 17}.

\medskip

 We would like to emphasize the fact that
general variational inequalities \eqref{2.4}, \eqref{2.5} and \eqref{2.6} are quite
different from each other and have different applications.

\medskip

 \noindent\textbf{VI. } \ For $ g = h = I, $ the identity operator, the
extended general variational inequality \eqref{2.1} is equivalent to finding $u
\in K $ such that
\begin{eqnarray*}
\la Tu,v-u \ra \geq 0, \qquad \forall v \in K,
\end{eqnarray*}
which is known as the classical variational inequality and was introduced
in 1964 by Stampacchia \cite{22}. For the recent applications, numerical
methods,  sensitivity analysis, dynamical systems and formulations of
variational inequalities, see \cite{1}--\cite{22} and the references therein.

\medskip



\noindent\textbf{V.} \ If $K^{*} = \{ u \in H ; \la u,v \ra \geq 0,  \quad
\forall  v\in K  \} $  is a polar (dual) convex cone of a closed convex
cone $K$ in  $H,$ then problem \eqref{2.1} is equivalent to finding $u \in H$
such that
\begin{eqnarray}\label{2.7}
g(u) \in K, \quad  Tu \in K^{*},  \quad \la g(u), Tu \ra = 0,
\end{eqnarray}
which is known as the general complementarity problem, see \cite {12,18}.
If $g = I,$ the identity operator, then problem \eqref{2.7} is called the
generalized complementarity problem. For $g(u)= u-m(u),$ where $m$ is
a point-to-point mapping, then problem \eqref{2.7} is called the quasi(implicit)
complementarity problem, see \cite{12, 18} and the references therein.

From the above discussion, it is clear that the extended general
variational inequalities \eqref{2.1} is most general and includes several
previously known classes of variational inequalities and related
optimization problems as special cases. These variational inequalities
have important applications in mathematical programming and engineering
sciences.


\medskip

 We also need the following concepts and results.

\medskip

\noindent\begin{definition}  For all $u,v \in H$, an operator  $T: H \rightarrow H$
is said to be: \\

\textbf{(i)}{\it strongly monotone}, if there exists a constant $ \alpha > 0$
such that $$ \la Tu - Tv, u-v \ra  \geq \alpha ||u-v||^2 $$

\textbf{(ii)} {\it Lipschitz continuous}, if there exists a constant
 $\beta  > 0$ such that $$ ||Tu-Tv|| \leq \beta ||u-v||. $$
\end{definition}
From (i) and (ii), it follows that $ \alpha \leq \beta .$

\begin{remark}  It follows from the strongly monotonicity of
the operator $T, $ that
\begin{eqnarray*}
\alpha \|u-v\|^2 \leq \la Tu-Tv,u-v \ra \leq \|Tu-Tv\|\|u-v\|, \quad
\forall u,v \in H,
\end{eqnarray*}
which implies that
\begin{eqnarray*}
\|Tu-Tv\| \geq \alpha \|u-v\|, \quad  \forall u,v \in H.
\end{eqnarray*}
\end{remark}
This observation enables us to define the following concept.

\begin{definition}  The operator $T$ is said  to firmly  expanding if
\begin{eqnarray*}
\|Tu-Tv \| \geq \|u-v\|, \quad \forall u,v \in H.
\end{eqnarray*}
\end{definition}
\bigskip

%\setcounter{equation}{0}
%\setcounter{chapter}{3}

\section{ Main Results}

In this Section, we use the auxiliary principle technique of Glowinski,
Lions and Tremolieres \cite{4} to study the existence of a solution of the
extended general  variational inequality \eqref{2.1}.

\vskip .2pc
\begin{theorem} Let $T$ be a strongly monotone with constant
$\alpha > 0 $ and Lipschitz continuous with constant $\beta > 0. $ Let $g
$ be a strongly monotone and Lipschitz continuous  operator with constants
$ \sigma > 0 $ and $\delta > 0 $ respectively.  If the operator $h$ is
firmly expanding and  there exists a constant  $\rho > 0 $ such that
\begin{eqnarray}\label{3.1}
|\rho - \frac{\alpha }{\beta ^2 }|< \frac{\sqrt{\alpha ^2 - \beta ^2
k(2-k)}}{\beta ^2}, \quad \alpha > \beta \sqrt{k(2-k)}, \quad k < 1,
\end{eqnarray}
where
\begin{eqnarray}\label{3.2}
\theta & = & k +\sqrt{1-2\rho \alpha +\rho ^2 \beta ^2 }\nonumber\\
&&\\
k & = & \sqrt{1-2 \sigma + \delta ^2}\,.\nonumber
\end{eqnarray}
then the extended  general variational inequality \eqref{2.1} has a unique
solution.
\end{theorem}

\medskip

\begin{proof} We use the auxiliary principle technique  to prove
the existence of a solution of \eqref{2.1}. For a given $u \in H: g(u) \in K $
satisfying   the  extended general  variational inequality \eqref{2.1}, we
consider the
problem of finding a solution $ w \in  H:h(w) \in K $ such that
\begin{eqnarray}\label{3.3}
\la \rho Tu + h(w)-g(u), g(v)-h(w) \ra \geq 0, \quad
\forall v \in H:g(v) \in K\,,
\end{eqnarray}
where $\rho > 0 $ is a constant.

The inequality of type \eqref{3.3} is called the auxiliary extended general
variational inequality associated with the problem \eqref{2.1}. It is clear that
the relation \eqref{3.3} defines a mapping $u \mapsto w. $ It is enough
to show that the mapping $ u \mapsto w $ defined by the relation
\eqref{3.3} has a unique fixed point belonging to $H $ satisfying the general
variational inequality \eqref{2.1}.  Let $w_1 \neq  w_2 $ be two solutions of
\eqref{3.3} related to $u_1, u_2 \in H $ respectively. It is sufficient to show
that for a well chosen $ \rho > 0 $,
\begin{eqnarray*}
\|w_1-w_2\| \leq \theta \|u_1-u_2\|\,,
\end{eqnarray*}
with $ 0 < \theta < 1, $ where $\theta $ is independent of $u_1$ and $u_2.
$ Taking $v = w_2 $(respectively $w_1$) in \eqref{3.3} related to
$u_1 $ (respectively $u_2$), adding the resultant,
 we have
\begin{eqnarray*}
\la h(w_1)-h(w_2), h(w_1)-h(w_2)  \ra \leq \la g(u_1)-g(u_2)-\rho
(Tu_1-Tu_2), h(w_1)-h( w_2) \ra\,,
\end{eqnarray*}
from which we have
\begin{eqnarray}\label{3.4}
\|h(w_1)-h(w_2)\|& \leq &\|g(u_1)-g(u_2)-\rho (Tu_1-Tu_2 )\| \nonumber \\
& \leq & \|u_1-u_2-(g(u_1)-g(u_2))| + \|u_1-u_2 - \rho (Tu_1-Tu_2)\|\nonumber \\.
\end{eqnarray}
Since $T$ is both strongly monotone and Lipschitz continuous operator with
constants $\alpha > 0 $ and $\beta > 0 $ respectively, it follows that
\begin{eqnarray}\label{3.5}
\|u_1-u_2 -\rho (Tu_1-Tu_2)\|^2 &\leq & \|u_2-u_2\|^2 - 2 \rho \la
u_1-u_2, Tu_1-Tu_2 \ra \nonumber \\ &&+ \rho ^2\|Tu_1-Tu_2\|^2 \nonumber \\
& \leq & \left(1-2\rho \alpha + \rho ^2\beta ^2 \right)\|u_1-u_2\|^2.
\end{eqnarray}
In a similar way, using the strongly monotonicity with constant $\sigma >
0$ and Lipschitz continuity with constant $\delta > 0, $ we have
\begin{eqnarray}\label{3.6}
\|u_1-u_2-(g(u_1)-g(u_2))\| \leq \sqrt{1-2\sigma + \delta ^2}\|u_1-u_2\|.
\end{eqnarray}
From \eqref{3.4}, \eqref{3.5} and \eqref{3.6} and using the fact that the operator $h $ is
firmly expanding, we have
\begin{eqnarray*}
\|w_1-w_2\| &\leq &\left\{ k + \sqrt{1-2\rho \alpha + \rho ^2\beta
^2} \right\}\|u_1-u_2\| \\
& = & \theta \|u_1-u_2\|,
\end{eqnarray*}
From \eqref{3.1} and \eqref{3.2}, it follows that $\theta < 1 $ showing that the
mapping defined by \eqref{3.3} has a fixed point belonging to $K, $ which is
the solution of \eqref{2.1}, the required result. \hfill \qquad $\Box $
\end{proof}

\bigskip

\textbf{Acknowledgement.} The author  would like to thank Dr. S. M.
Junaid Zaidi, Rector, CIIT, for providing excellent research facilities.
\bigskip


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