\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 50, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2011/50\hfil Generalized frameworks]
{Generalized frameworks for first-order evolution inclusions
based on Yosida approximations}
\author[R. U. Verma\hfil EJDE-2011/50\hfilneg]
{Ram U. Verma}
\address{Ram U. Verma \newline
Texas A\&M University\\
Department of Mathematics\\
Kingsville, TX 78363, USA}
\email{verma99@msn.com}
\thanks{Submitted December 25, 2010. Published April 11, 2011.}
\subjclass[2000]{49J40, 65B05}
\keywords{Generalized first-order evolution inclusions;
variational problems; \hfill\break\indent
maximal monotone mapping; $A$-maximal relaxed monotone mapping;
\hfill\break\indent generalized resolvent operator;
general Yosida approximations}
\begin{abstract}
First, general frameworks for the first-order evolution
inclusions are developed based on the $A$-maximal relaxed
monotonicity, and then using the Yosida approximation the
solvability of a general class of first-order nonlinear evolution
inclusions is investigated. The role the $A$-maximal relaxed
monotonicity is significant in the sense that it not only empowers
the first-order nonlinear evolution inclusions but also
generalizes the existing Yosida approximations and its
characterizations in the current literature.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\section{Preliminaries}
The notion of the $A$-maximal relaxed monotonicity \cite{v2} is not
only limited to the first-order evolution equations/inclusions in
conjunction with Yosida approximations, but goes way beyond,
including the fields of optimization and control theory,
variational inequality and variational inclusion problems, and
unify a greater degree of investigations relating to other fields
as well. The obtained results seem to be general in nature, and
have a greater potential for applications. For more details, we
refer the reader to the references in this article.
Consider a real separable Hilbert space with the norm $\|\cdot\|$
and the inner product $\langle \cdot , \cdot \rangle$.
We study a general class of first-order nonlinear evolution
inhomogeneous inclusions of the form
\begin{equation} \label{e1.1}
\begin{gathered}
u'(t)+Mu(t) \ni f(t) \quad\text{for almost all }t\in(0,T),\\
u(0)=u_{0},
\end{gathered}
\end{equation}
where $M: X\to 2^X$ is a multivalued mapping on $X$,
$f\in W_{2}^1(0,T; X)$, $T$ is fixed, $00$.
Furthermore, $M$ is $m$-accretive (or $A$-maximal accretive)
if $M$ is accretive and $R_{\rho, A}^M$ exists for every
$\rho > 0$ on $X$, where $R_{\rho, A}^M$ is the resolvent of $M$.
\end{definition}
\begin{lemma} \label{lem1.1}
Let $M: X\to 2^X$ be a set-valued
mapping on a real Hilbert space $X$. Then following properties are
equivalent:
\begin{itemize}
\item[(i)] $M$ is monotone.
\item[(ii)] $M$ is accretive.
\end{itemize}
\end{lemma}
\begin{lemma} \label{lem1.2}
Let $M: X\to 2^X$ be a set-valued
mapping on a real Hilbert space $X$. Then we have the following
implications equivalent:
\begin{itemize}
\item[(i)] $M$ is $A$-maximal relaxed monotone.
\item[(ii)] $M$ is monotone and $R(A+\rho M)=X$.
\item[(iii)] $M$ is $m$-accretive.
\end{itemize}
\end{lemma}
We plan to explore the solvability of the inclusion problem \eqref{e1.1}
based on the notion of the $A$-maximal relaxed monotonicity \cite{v2}
and the generalized Yosida approximations. The generalized Yosida
approximation turns out to be Lipschitz continuous, while we
explore the solvability of the inclusion problem \eqref{e1.1}. The
obtained results seem to be application-enhanced to
problems arising from other fields, including optimization theory,
decision and management sciences, engineering science, variational
inequality and variational inclusion problems. There are also some
detailed results that are investigated on the generalized Yosida
approximations to the context of the
$A$-maximal relaxed monotonicity frameworks. For more details,
we refer the reader to the references in this article.
\section{Auxiliary results}
\begin{definition} \label{def2.1} \rm
Let $A: X \to X$ be an $(r)$-strongly monotone single-valued
mapping and $M: X \to 2^X$ be a set-valued mappings. The map
$M:X\to 2^X$ is said to be $A$-maximal relaxed monotone if
\begin{itemize}
\item[(i)] $M$ is $(m)$-relaxed monotone; i.e.,
$$
\langle u^*-v^*, u-v\rangle \geq -m\|u-v\|^2\quad
\forall (u,u^*),(v,v^*)\in M,
$$
\item[(ii)] $R(A+\rho M)=X$ for $\rho > 0$.
\end{itemize}
\end{definition}
\begin{definition} \label{def2.2}\rm
Let $A: X \to X$ be a single-valued mapping and $M: X \to 2^X$ be a set-valued mapping. Let $A$ be $(r)$-strongly monotone. The map $M$ is said to be accretive iff
$(A+\rho M)^{-1}$ is single-valued and $(A+\rho M)^{-1}$ is
$(\frac{1}{r-\rho m})$-Lipschitz continuous for all $\rho >0$ and
$r-\rho m>0$.
\end{definition}
\begin{proposition}[\cite{v2}] \label{prop2.1}
Let $A: X \to X$ be a
single-valued mapping, and $M: X \to 2^X$ be a set-valued mapping
such that $D(A)\cap D(M)\not=\emptyset$. Let $A$ be
$(r)$-strongly monotone, and let $M$ be an $A$-maximal relaxed monotone
mapping. Then the generalized resolvent operator associated with
$M$ and defined by
$$
R_{\rho,{\it A}}^{M}(u)=({\it A}+\rho M)^{-1}(u)\quad \forall \, u\in X,
$$
is $(\frac{1}{r-\rho m})$-Lipschitz continuous.
\end{proposition}
Next, we generalize the Yosida approximation $M_{\rho}$ by
$M_\rho =\rho^{-1}(I-AR_{\rho,A}^{M})$, where
$A:X\to X$ is an $(r)$-strongly monotone mapping on $X$
for $\rho>0$, and for $R_{\rho,A}^{M}= (A+\rho M)^{-1}$,
which reduces to the Yosida approximation of $M$ for $A=I$:
$$
M_\rho =\rho^{-1}(I- R_{\rho}^{M}),
$$
where $I$ is the identity and $R_{\rho}^{M}=(I+\rho M)^{-1}$.
\begin{lemma} \label{lem2.1}
Let $A: X \to X$ and $M: X \to 2^X$ be
mappings such that $D(A)\cap D(M)\not=\emptyset$. Let $A$ be
$(r)$-strongly monotone and $AoR_{\rho,A}^{M}$ be cocoercive, and
let $M$ be an $A$-maximal relaxed monotone mapping. Then the
generalized Yosida approximation $M_\rho$ of $M$ defined by
$$
M_{\rho} = \rho^{-1}(I-A R_{\rho,A}^{M}),
$$
where
$$
R_{\rho,{\it A}}^{M}(u)=({\it A}+\rho M)^{-1}(u)\quad
\forall \, u\in D(A)\cap D(M),
$$
is $(\frac{1}{\rho})$-Lipschitz continuous.
\end{lemma}
\begin{proof}
For any $u,v\in X$, we have
\begin{align*}
& \langle M_\rho (u)-M_\rho (v),u-v\rangle\\
&= \langle M_\rho(u)-M_\rho(v), \rho[M_\rho(u)
-M_\rho(v)-(M_\rho(u)-M_\rho(v))]+u-v\rangle\\
&= \rho\|M_\rho (u)-M_\rho (v)\|^2\\
&\quad -\langle \rho^{-1}[u-v-(AR_{\rho,A}^{M}(u)-AR_{\rho,A}^{M}(v))],
-(AR_{\rho,A}^{M}(u)-AR_{\rho,A}^{M}(v))\rangle\\
&= \rho\|M_\rho (u)-M_\rho (v)\|^2+ \rho^{-1}\langle u-v,
AR_{\rho,A}^{M}(u)-AR_{\rho,A}^{M}(v)\rangle\\
&\quad -\rho^{-1}\langle AR_{\rho,A}^{M}(u)-AR_{\rho,A}^{M}(v),
AR_{\rho,A}^{M}(u)-AR_{\rho,A}^{M}(v)\rangle\\
&\geq \rho\|M_\rho (u)-M_\rho (v)\|^2 +\rho^{-1}
\|AR_{\rho,A}^{M}(u)-AR_{\rho,A}^{M}(v)\|^2 \\
&\quad -\rho^{-1}\|AR_{\rho,A}^{M}(u)-AR_{\rho,A}^{M}(v)\|^2\\
&\geq \rho\|M_\rho (u)-M_\rho (v)\|^2.
\end{align*}
\end{proof}
For $A=I$, Lemma \ref{lem2.1} reduces to
\begin{lemma} \label{lem2.2} \rm
Let $M: X \to 2^X$ be a set-valued mapping. Let $M$ be a
maximal monotone mapping. Then the Yosida approximation
$M_\rho$ of $M$ defined by
$$
M_\rho = \rho^{-1}(I-R_{\rho}^{M}),
$$
where
$$
R_{\rho}^{M}(u)=(I+\rho M)^{-1}(u)\quad \forall \, u\in D(M),
$$
is $(\frac{1}{\rho })$-Lipschitz continuous.
\end{lemma}
\begin{proposition} \label{prop2.2}
Let $A: X \to X$ and $M: X \to 2^X$ be mappings such that
$D(A)\cap D(M)\not=\emptyset$. Let $A$
be $(r)$-strongly monotone and $AoR_{\rho,A}^{M}$ be cocoercive,
and let $M$ be an $A$-maximal relaxed monotone mapping. Then the
generalized Yosida approximation $M_\rho$ of $M$ defined by
$M_{\rho} = \rho^{-1}(I-AR_{\rho,A}^{M})$, satisfies:
\begin{itemize}
\item[(i)] $M_\rho(u)\in MR_{\rho,A}^{M}(u)$.
\item[(ii)] $M_{\rho}$ is $A$-maximal relaxed monotone.
\item[(iii)] $(M_{\lambda})_\mu=M_{\lambda+\mu}$.
\end{itemize}
\end{proposition}
\begin{proof} To prove (i), consider
\[
w=R_{\rho,A}^{M}(u)\Rightarrow u\in (A+\rho M)(w)
\Rightarrow \rho M_{\rho}(u)=u-A(w)\in \rho M( w).
\]
The proofs of (ii) and (iii) follow, respectively,
from the Lipschitz continuity of the generalized resolvent
$R_{\rho,A}^{M}(u)$ and the definition of $M_{\rho}$.
\end{proof}
\begin{proposition} \label{prop2.3}
Let $A: X \to X$ and $M: X \to 2^X$ be mappings such that
$D(A)\cap D(M)\not=\emptyset$. Let $A$
be $(r)$-strongly monotone and $AoR_{\rho,A}^{M}$ be cocoercive,
and let $M$ be an $A$-maximal relaxed monotone mapping. Then the
generalized Yosida approximation $M_\rho$ of $M$ defined by
$M_{\rho} = \rho^{-1}(I-AR_{\rho,A}^{M})$, satisfies: for all
$u\in D(M)$,
\begin{gather*}
M_\rho (u) \rightarrow M_0 (u),\quad
\|M_\rho(u)\|\uparrow \|M_0 (u)\| \quad\text{as } \rho\downarrow 0,\\
\|M_{\rho}(u) -M_0 (u)\|^2 \leq \|M_0(u)\|^2 -\|M_\rho (u)\|^2
\quad\text{for all } \rho >0.
\end{gather*}
\end{proposition}
\section{Generalized First-Order Evolution Inclusions}
Let $A: X \to X$ be a single-valued mapping, and
$M: X \to 2^X$ be a multivalued mapping.
In this section, we consider the solvability of first-order
nonlinear evolution inclusions of the form
\begin{equation} \label{e3.1}
\begin{gathered}
u'(t)+Mu(t) \ni f(t) \quad\text{for almost all } t\in(0,T)\\
u(0)=u_{0},
\end{gathered}
\end{equation}
where $M: X\to 2^X$ is a multivalued mapping on $X$,
$f\in W_{2}^1(0,T; X)$, $T$ is fixed, $00$.
Suppose that the given
$$
u_0\in D(M), \quad f\in W_{2}^1(0,T;X)
$$
are fixed. Then \eqref{e3.1} has exactly one solution
$u\in W_{2}^1(0,T;X)$ such that $M:X\to 2^X$ is $A$-maximal
relaxed monotone.
\end{theorem}
\begin{proof}
The proof is based on the results from Section 2,
especially Lemma \ref{lem2.1} and Proposition \ref{prop2.1}. First, we consider
the regularized problems
\begin{equation} \label{e3.2}
u_{\rho}'(t)+M_{\rho} u_\rho (t)=f(t),\quad
u_\rho(0)=u_0, \rho >0.
\end{equation}
As the function $f$ is continuous on [0,T] by the hypotheses,
and $M_\rho$ is $(\frac{1}{\rho})$-Lipschitz continuous by
Lemma \ref{lem2.1}, problems \eqref{e3.2} can be solved as for
first-order evolution equations. To achieve that goal,
we need to arrive at \emph{ a priori} estimate
\begin{equation} \label{e3.3}
\|u_{\rho}'(t)\|\leq C\quad \forall \, \rho >0,\; t\in[0,T].
\end{equation}
Now we differentiate \eqref{e3.2} by setting
$g_{\rho}(t) = M_\rho$o$u_\rho(t)$ as follows:
\begin{equation} \label{e3.4}
u_{\rho}''(t)+g_{\rho}'(t)=f{'}(t)\quad\text{for almost all } t.
\end{equation}
Under the hypotheses all the derivatives exist.
Since $A\circ R_{\rho,A}^{M}$ is cocoercive (and hence
$A\circ R_{\rho,A}^{M}$ is nonexpansive), it implies
$$
\langle A(R_{\rho,A}^{M}(u))- A(R_{\rho,A}^{M}(u)),u-v\rangle
\leq \|u-v\|^2.
$$
It follows that $M_\rho$ is monotone, and thus we have
$$
\langle M_\rho u_\rho(t+h)-M_\rho u_\rho(t), u_\rho(t+h)
-u_\rho(t)\rangle \geq 0.
$$
It follows that
$$
\langle g_{\rho}'(t), u_{\rho}'(t)\rangle \geq 0.
$$
Therefore,
\begin{align*}
\langle u_{\rho}''(t),u_{\rho}'(t)\rangle
&\leq -\langle g_{\rho}'(t), u_{\rho}'(t)\rangle +\langle f'(t),u_{\rho}'(t)\rangle\\
&\leq \langle f'(t),u_{\rho}'(t)\rangle\\
&\leq \| f'(t)\|\,\|u_{\rho}'(t)\|\\
&\leq \frac{1}{2}\|f'(t)\|^2 + \frac{1}{2}\|u_{\rho}'(t)\|^2.
\end{align*}
Applying integration by parts to
$\langle u_{\rho}''(t),u_{\rho}'(t)\rangle$, we have
\begin{align*}
2\int_{0}^t \langle u_{\rho}''(s),u_{\rho}'(s)\rangle ds
&= \|u_{\rho}'(t)\|^2 -\|u_{\rho}'(0)\|^2\\
&\leq \|f\|_{Y}^2 +\int_{0}^t \|u_{\rho}'(s)\|^2 ds,
\end{align*}
where $Y=W_{2}^1 (0,T;X)$.
This is equivalent to
$$
\|u_{\rho}'(t)\|^2 -\|u_{\rho}'(0)\|^2
\leq \|f\|_{Y}^2 +\int_{0}^t \|u_{\rho}'(s)\|^2 ds,
$$
where $Y=W_{2}^1 (0,T;X)$.
Now by Gronwall lemma,
$$
\|u_{\rho}'(t)\|^2 \leq c (\|u_{\rho}'(0)\|^2 + \|f\|_{Y}^2).
$$
Finally, using \eqref{e3.2}, we have
$u_{\rho}'(0) = -M_{\rho}(u_0)+f(0)$ and
$\|M_\rho u_0\|\leq \|M_0 u_0\|$, and thus,
it follows that \eqref{e3.3} holds.
\end{proof}
\begin{corollary} \label{coro3.1}
Let $M: X \to 2^X$ be maximal monotone on a separable Hilbert
space $X$. Suppose that the given
$$
u_0\in D(M), \quad f\in W_{2}^1(0,T;X)
$$
are fixed. Then \eqref{e3.1} has exactly one solution
$u\in W_{2}^1(0,T;X)$ such that $M:X\to 2^X$ is maximal monotone.
\end{corollary}
\subsection*{Concluding Remarks}
The obtained results on the first-order evolution inclusions
can further be generalized to the case of a real Banach space
setting in terms of accretivity and $m$-accretivity.
More importantly, the solution concept is also changed as
an integral solution based on the difference method belonging
to \eqref{e3.1} as backward differences.
The uniqueness proof assures that each classical solution
of \eqref{e3.1} is also an integral solution.
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\end{document}