\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 170, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/170\hfil Controllability of nonlinear inclusions]
{Controllability of nonlinear third-order dispersion inclusions
with infinite delay}
\author[M. Li, X. Wang, H. Wang \hfil EJDE-2013/170\hfilneg]
{Meili Li, Xiaoxia Wang, Haiqing Wang} % in alphabetical order
\address{Meili Li \newline
Department of Applied Mathematics, Donghua University,
Shanghai 201620, China}
\email{stylml@dhu.edu.cn}
\address{Xiaoxia Wang \newline
Department of Applied Mathematics, Donghua University,
Shanghai 201620, China}
\email{772091534@qq.com}
\address{Haiqing Wang \newline
Department of Applied Mathematics, Donghua University,
Shanghai 201620, China}
\email{aqhai123456@163.com}
\thanks{Submitted June 4, 2013. Published July 26, 2013.}
\subjclass[2000]{93B05, 34G20, 93C20, 34K09}
\keywords{Controllability; semigroup theory; nonlinear dispersion inclusions;
\hfill\break\indent Korteweg-de Vries equation; infinite delay}
\begin{abstract}
This article shows the controllability of nonlinear third-order
dispersion inclusions with infinite delay. Sufficient conditions
are obtained by using a fixed-point theorem for multivalued maps.
Particularly, the compactness of the operator
semigroups is not assumed in this article.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
In 1895, Korteweg and de Vries considered the following equation
as a model for propagation of
small amplitude long waves in a uniform channel \cite{Korteweg}
\begin{equation} \label{e1.1}
\eta_t=\frac{3}{2}\sqrt{\frac{g}{l}}\big(\frac{1}{2}\eta^2
+\frac{2}{3}\alpha\eta+\frac{1}{3}\sigma \eta_{\xi\xi}\big)_\xi
\end{equation}
where $\eta$ is the surface elevation above the equilibrium level
$l$, $\alpha$ is a small constant related to the uniform motion of
the liquid, $g$ is the gravitational constant, and
$\sigma=\frac{l^3}{3}-\frac{Tl}{\rho g}$ with surface capillary
tension $T$ and density $\rho$. When posed on the whole real line $R$
or on a periodic domain, \eqref{e1.1} can always be reduced by certain
variable transformations to its standard form
\[
x_t+x x_{\xi}+x_{\xi\xi\xi}=0
\]
where $x\equiv x(\xi,t)$ is a real valued function of two real
variables $\xi$ and $t$ and subscript is the corresponding
partial derivatives. It is well known that many physical phenomena
can be described by the KDV equation. This equation arises in many
physical contexts as a model equation incorporating
the effects of dispersion, dissipation and nonlinearity. In particular, the
equation is now a fundamental model of the weakly nonlinear waves in
the weakly dispersive media and has been studied extensively by researchers
in various aspects (see \cite{Lin, Sakthivel-1} and references
cited therein).
As one of the fundamental concepts in mathematical control theory,
controllability plays an important role in control theory and engineering.
Roughly speaking, controllability generally means that it is possible to
steer a dynamical control system from an arbitrary initial state to an
arbitrary final state using the set of admissible controls.
For the controllability problem,
there are different methods for various types of nonlinear systems and
the details can be found in various papers
\cite{ Dauer, Klamka, Sakthivel-3, Sakthivel-2}.
Many authors have studied on the controllability problems of third-order
dispersion equation. In 1993, Russell and Zhang \cite{Russell}
discussed the controllability and stabilizability of the
third-order linear dispersion equation on a periodic domain. They
discussed the exponential decay rates with distributed controls of
restricted form and for the equation with boundary dissipation.
Later on, George, Chalishajar and Nandakumaran \cite{George}
discussed the exact controllability of nonlinear third-order
dispersion equation. They established the controllability results
using two standard types of nonlinearities, namely, Lipschitzian
and monotone. Chalishajar \cite{Chalishajar} studied the exact
controllability of nonlinear integro-differential third-order
dispersion system by using the Schaefer fixed-point theorem.
Recently, Sakthivel, Mahmudov and Ren \cite{Sakthivel} focused on
the approximate controllability for the nonlinear third-order
dispersion equation. They discussed the approximate
controllability under the assumption that the corresponding
linear control system is approximately controllable. More
recently, Muthukumar and Rajivganthi \cite{Muthukumar} studied the
approximate controllability of stochastic nonlinear third-order
dispersion equation by using fixed-point theory, infinite
dimensional semigroup properties, stochastic analysis techniques.
It has been widely argued and accepted \cite{Hale1, Wu} that for
various reasons, time delay should be taken into consideration in
modeling. Obviously, the KDV equation with time delay has more
actual significance. Zhao and Xu \cite{Zhao} have studied the existence
of solitary waves for KDV equation with time delay. Li and Wang
\cite{Li} have discussed the controllability of nonlinear third-order
dispersion equation with infinite distributed delay.
In recent years the corresponding parts of multivalued analysis
were applied to obtain various controllability results for systems
governed by semilinear differential and functional differential
inclusions in infinite dimensional Banach spaces (refer to
\cite{Abada, Chang, Obukhovski, Rykaczewski} and
others). The attention of researchers to such systems is caused by
the fact that many control processes arising in mathematical
physics may be naturally presented in this form (refer to
\cite{Kamenskii}). Specially, it should be point out that
Obukhovski and Zecca \cite{Obukhovski} investigated the
controllability problems for a system governed by a semilinear differential
inclusion in a Banach space with a noncompact semigroup and as
application they considered the controllability for a system
governed by a perturbed wave equation.
In this paper, we establish sufficient conditions for the
controllability of nonlinear third-order dispersion inclusions
with infinite delay by using a fixed-point theorem for multivalued
maps combined with a noncompact operator semigroup. To the best
of the author's knowledge, the controllability of nonlinear
third-order dispersion inclusions has not been studied yet in the
literature.
\section{Preliminaries}
The purpose of this paper is to study the controllability of the
nonlinear third-order dispersion inclusions with infinite delay
\begin{equation} \label{e2.1}
\frac{\partial x}{\partial t}(\xi,t)+ \frac{\partial ^3
x}{\partial \xi^3}(\xi,t)\in(Gu)(\xi,t) + F(t, x_{t}(\xi,\cdot))
\end{equation}
on the domain $ t\in J$, $0\leq \xi \leq 2\pi$,with periodic
boundary conditions
\begin{equation} \label{e2.2}
\frac{\partial^k x}{\partial \xi^k}(0,t)=\frac{\partial^k
x}{\partial \xi^k}(2\pi,t), \quad k=0,1,2,
\end{equation}
and initial condition
\begin{equation} \label{e2.3}
x(\xi,\theta)=x_0(\xi,\theta), \quad -\infty<\theta\leq 0, \;
0\leq \xi \leq 2\pi,
\end{equation}
where $J=[0,b]$, $F$ is a multivalued continuous function.
$x_0: [0,2\pi] \times (-\infty,0]\to R$ are continuous
functions. $x_t(\xi,\theta)=x(\xi,t+\theta)$,
$-\infty<\theta\leq 0$. $u$ is the control function and the operator
$G$ is defined by
\begin{equation} \label{e2.4}
(Gu)(\xi,t)=g_1(\xi)\big\{u(\xi,t)-\int_{0}^{2\pi}g_1(s)u(s,t)ds\big\}.
\end{equation}
Then $G$ is a bounded linear operator and $g_1(\xi)$ is a piece-wise
continuous nonnegative function on $[0,2\pi]$ such that
\[
[g_1]:=\int_{0}^{2\pi}g_1(s)ds=1.
\]
The state $x(\cdot,t)$ takes values in a Banach space
$X=L^2(0,2\pi)$ with the norm $\|\cdot\|$ and inner product
$\langle\cdot, \cdot\rangle$. The control function $u(\cdot,t)$
is given in $L^2(J,U)$, a Banach space of all admissible control
functions, with $U=L^2(0,2\pi)$ as a Banach space. Define an
operator $A$ on $X$ with domain $D=D(A)$ given by
\[
D(A)=\big\{x\in H^3(0,2\pi):\frac{\partial ^k x}{\partial
\xi^k}(0)=\frac{\partial^k x}{\partial \xi^k}(2\pi); \,
k=0,1,2\big\},
\]
such that
\[
Ax=-\frac{\partial^3 x}{\partial \xi^3}.
\]
It follows from Lemma 8.5.2 and Korteweg-de Vries equation of Pazy
\cite{Pazy} that $A$ is the infinitesimal generator of a
$C_0$-group of isometry on $X$. Let $\{T(t)\}_{t\geq 0}$ be the
$C_0$-group generated by $A$. Obviously, one can show for all
$x\in D(A)$,
\[
\langle Ax,x \rangle_{L^2(0,2\pi)}=0.
\]
Also, there exists a constant $M>0$ such that
\[
\sup\{\|T(t)\|:t\in J\}\leq M.
\]
To study system \eqref{e2.1}-\eqref{e2.3}, we assume that the histories
$x_t: (-\infty,0]\to X$, $x_t(\theta)=x(t+\theta)$ belong
to some abstract phase space $\mathcal{B}$, which is defined
axiomatically. In this article, we will employ an axiomatic
definition of the phase space introduced by Hale and
Kato \cite{Hale} and follow the terminology used in \cite{Hino}.
Thus, $\mathcal{B}$ will be a linear space of functions mapping
$(-\infty,0]$ into $X$ endowed with a seminorm
$\|\cdot\|_\mathcal{B}$. We will assume that $\mathcal{B}$
satisfies the following axioms:
\begin{itemize}
\item[(A1)] If $x: (-\infty,\sigma+a)\to X$, $a>0$, is
continuous on $[\sigma,\sigma+a)$ and $x_\sigma\in \mathcal{B}$,
then for every $t\in [\sigma,\sigma+a)$ the following conditions
hold:
\begin{itemize}
\item[(i)] $x_t$ is in $\mathcal{B}$;
\item[(ii)] $\|x(t)\|\leq H\|x_t\|_\mathcal{B}$;
\item[(iii)] $\|x_t\|_\mathcal{B} \leq K(t-\sigma)\sup\{\|x(s)\|:
\sigma\leq s\leq t\}+M(t-\sigma)\|x_\sigma\|_\mathcal{B}$.
\end{itemize}
Here $H\geq 0$ is a constant, $K,M:[0,+\infty) \to
[1,+\infty)$, $K$ is continuous, $M$ is locally bounded, and
$H,K,M$ are independent of $x(\cdot)$.
\item[(A2)] For the function $x(\cdot)$ in (A1), $x_t$ is a
$\mathcal{B}$-valued continuous function on $[\sigma,\sigma+a]$.
\item[(A3)] The space $\mathcal{B}$ is complete.
\end{itemize}
\begin{example} \label{examp2.1} \rm
The phase space $C_r \times L^p(\rho_1,X)$.
Let $r\geq 0, 1\leq p<\infty$ and let $\rho_1: (-\infty,-r)\to
R$ be a non-negative measurable function which satisfies the
conditions (g-5), (g-6) in the terminology of \cite{Hino}. In
other words, this means that $\rho_1$ is locally integrable and there
exists a non-negative, locally bounded function $\delta$ on
$(-\infty,0]$ such that $\rho_1(\mu+\nu)\leq \delta(\mu)\rho_1(\nu)$,
for all $\mu\leq 0$ and $\nu \in (-\infty,-r)\setminus N_{\mu}$,
where $N_{\mu} \subseteq(-\infty,-r)$ is a set with Lebesgue measure
zero. The space $C_r \times L^p(\rho_1,X)$ consists of all classes of
functions $\phi: (-\infty,0]\to X$ such that $\phi$ is
continuous on $[-r,0]$, Lebesgue-measurable, and $\rho_1\|\phi\|^p$ is
Lebesgue integrable on $(-\infty,-r)$. The seminorm in $C_r \times
L^p(\rho_1,X)$ is defined by
\[
\|\phi\|_\mathcal{B}=\sup\{\|\phi(\nu)\|: -r\leq \nu \leq
0\}+\big(\int_{-\infty}^{-r}\rho_1(\nu)\|\phi(\nu)\|^pd\nu\big)^{1/p}.
\]
The space $C_r \times L^p(\rho_1,X)$ satisfies axioms
(A1), (A2), (A3). Moreover, if $r=0$ and $p=2$,the phase space
$C_r \times L^p(\rho_1,X)$ is reduced to
$\mathcal{B}=C_0 \times L^2(\rho_1,X)$. We can take
$H=1$, $M(t)=\delta (-t)^{1/2}$, and
$K(t)=1+\big(\int_{-t}^{0}\rho_1(\nu)d\nu\big)^{1/2}$ for $t\geq
0$. We refer the reader to \cite{Hino} for details.
\end{example}
Next, we introduce definitions, notation and preliminary facts
from multivalued analysis which are used throughout this paper.
Let $C(J,X)$ be the Banach space of continuous functions from $J$
to $X$ with the norm $\|x\|_J=\sup\{\|x(t)\|: t\in J\}$. $B(X)$
denotes the Banach space of bounded linear operators from $X$ into
itself. A measurable function $x: J \to X$ is Bochner
integrable if and only if $\|x\|$ is Lebesgue integrable (For
properties of the Bochner integral see Yosida \cite{Yosida}).
$L^1(J,X)$ denotes the Banach space of Bochner integrable
functions $x: J\to X$ with norm
$\|x\|_{L^1}=\int_{0}^{b}\|x(t)\|dt$ for all $x\in L^1(J,X)$.
For a metric space $(X,d)$, we introduce the following symbols:
\begin{gather*}
P(X)=\{y \in 2^X, Y\neq \emptyset\}, \quad
P_{cl}(X)=\{y \in P(X) :y\text{ is closed}\},\\
P_{b}(X)=\{y \in P(X) : y\text{ is bounded}\}, \quad
P_{cp}(X)=\{y \in P(X) : y\text{ is compact}\},\\
P_{b,cl}(X)=\{y \in P(X) : y\text{ is bounded and closed}\}.
\end{gather*}
We define $H_d: P(X) \times P(X) \to R_+ \cup \{\infty\}$ by
\[
H_d(A,B)=\max\{\sup_{a \in A}d(a,B), \quad
\sup_{b \in B}d(A,b)\},
\]
where
\[
d(A, b) = \inf\limits_{a \in A}d(a, b), \ d(a, B) = \inf\limits_{b
\in B} d(a, b).
\]
Then, $(P_{b,cl}(X), H_d)$ is a metric space and $(P_{cl}(X),
H_d)$ is a generalized (complete) metric space.
In what follows, we briefly introduce some facts on multivalued
analysis. For more details, one can see \cite{Deimling, Hu}.
$\bullet$ $\Gamma$ has a fixed point if there is $x \in X$ such
that $x \in \Gamma(x)$. The set of fixed points of the
multivalued operator $\Gamma$ will be denoted by Fix$\Gamma$.
$\bullet$ A multivalued map $\Gamma : J \to P_{cl}(X)$ is
said to be measurable, if for each $x \in X$, the function $Y: J
\to \mathbb{R}$, defined by
\[
Y(t)=d(x,\Gamma(t))=\inf\{\|x-z\|: z\in \Gamma(t)\},
\]
belongs to $L^1(J,R)$.
\begin{definition} \label{def2.1} \rm
A multivalued operator $\Gamma: X \to P_{cl}(X)$ is called:
\begin{itemize}
\item[(a)] $\gamma$-Lipschitz if there exists $ \gamma>0$
such that
\[
H_d(\Gamma(x),\Gamma(y))\leq \gamma d(x,y), \ \text{for each} \
x,y \in X;
\]
\item[(b)] a contraction if it is $\gamma$-Lipschitz
with $\gamma<1$.
\end{itemize}
\end{definition}
Our main results are based on the following lemma.
\begin{lemma}[\cite{Covitz}] \label{lem2.1}
Let $(X,d)$ be a complete metric space. If $\Gamma: X\to P_{cl}(X)$
is a contraction, then $\operatorname{Fix}\Gamma\neq \emptyset$.
\end{lemma}
By the variation of constant formula, we can write a mild solution
of \eqref{e2.1}-\eqref{e2.3} as
\begin{equation} \label{e2.5}
x(\xi,t)=T(t)x(\xi,0)+\int_{0}^{t}T(t-s)(Gu)(\xi,s)ds
+\int_{0}^{t}T(t-s)f(s)(\xi) ds,
\end{equation}
where $f \in S_{F,x} = \{f \in L^{1}(J,X) : f(t)(\xi) \in
F(t, x_t(\xi,\cdot)), \, \text{for a.e. } t \in J, \, \xi \in
[0,2\pi] \}$.
\begin{definition} \label{def2.2} \rm
System \eqref{e2.1}-\eqref{e2.3} is said to be exactly
controllable on the interval $J$, if for any given $x_{b} \in X$ with
$[x_{b}]=0$, there exists a control $u\in L^2(0,b;
L^2(0,2\pi))=L^2(J,U)$ such that the mild solution
$x(.,t)$ of \eqref{e2.1}-\eqref{e2.3} satisfies $x(.,b)=x_{b}$.
\end{definition}
For $\theta\leq 0$, $\xi \in [0,2\pi]$ and $\phi
\in \mathcal{B}$, we define
\begin{equation} \label{e2.6}
x(t)(\xi)=x(\xi,t), \quad
F(t, \phi)(\xi)=F(t, \phi(\xi, \cdot)), \quad
\phi(\theta)(\xi)=\phi(\xi,\theta)=x_0(\xi,\theta).
\end{equation}
Russell and Zhang \cite{Russell} studied the exact controllability
of a corresponding linear system (i.e. with $F\equiv 0$ in
\eqref{e2.1}-\eqref{e2.3}). In their analysis, they considered controls which
conserve the quantity $[x(\cdot,t)]$, which corresponds to the
\emph{volume}. The following is their controllability result for
the linear system.
\begin{theorem}[\cite{Russell}] \label{thm2.1}
Let $b>0$ be given and let $g_1\in C^0[0,2\pi]$ be associated with
$G$ in \eqref{e2.4}.
Given any final state $x_b\in X$ with $[x_b]=0$, there exists a
control $u\in L^2(J,U)$ such that the solution $x$ of
\begin{equation} \label{e2.7}
\frac{\partial x}{\partial t}(\xi,t)
+ \frac{\partial ^3 x}{\partial \xi^3}(\xi,t)=(Gu)(\xi,t)
\end{equation}
together with boundary conditions
\begin{equation} \label{e2.8}
\frac{\partial^k x}{\partial \xi^k}(0,t)=\frac{\partial^k
x}{\partial \xi^k}(2\pi,t), \quad k=0,1,2,
\end{equation}
and initial condition
\begin{equation} \label{e2.9}
x(\xi,0)=0
\end{equation}
satisfies the terminal condition $x(\cdot,b)=x_b$ in $X$.
Moreover, there exist a positive constant $C_1$ independent of
$x_b$ such that
\begin{equation} \label{e2.10}
\|x\|_{L^2(J,X)}\leq C_1\|x_b\|_X.
\end{equation}
\end{theorem}
The main purpose of this paper is to obtain sufficient conditions on
the perturbed nonlinear term $F$ which will preserve the exact
controllability. Usually authors assume the compactness of
semigroup while studying the controllability.
Here we drop this assumption and prove the controllability result.
\section{Controllability}
We assume the following conditions hold:
\begin{itemize}
\item[(H1)] $F : J \times \mathcal{B} \to P_{cp}(X):(\cdot, \phi)
\to F(\cdot, \phi)$ is measurable for each $\phi \in
\mathcal{B}$.
\item[(H2)] $H_d(F(t, \phi_1),F(t, \phi_2))\leq
l(t)\|\phi_1-\phi_2\|_\mathcal{B}$,for each $t \in J$ and
$\phi_1,\phi_2\in \mathcal{B}$,
where $l \in L^1(J,R_+)$ and $d(0, F(t, 0))\leq l(t)$,for a.e. $t\in
J$.
\end{itemize}
Denote
\[
\Gamma_0^b=\int_{0}^{b}T(b-s)GG^*T^*(b-s)ds.
\]
Note that the linear system \eqref{e2.7}-\eqref{e2.9}
is exactly controllable if and
only if there exists a $\zeta>0$ such that
\[
\langle \Gamma_0^b x,x \rangle \geq \zeta \|x\|^2, \quad
\text{for all } x\in X,
\]
Then $\Gamma_0^b$ is invertible and
\[
\|(\Gamma_0^b)^{-1}\|\leq \frac{1}{\zeta}.
\]
\begin{theorem} \label{thm3.1}
Assume that conditions {\rm (H1)--(H2)} and
$[x_b]=0$ are satisfied. Then the nonlinear third-order dispersion
inclusions \eqref{e2.1}-\eqref{e2.3} is controllable on $J$ provided
\begin{equation} \label{e3.1}
(1+\frac{1}{\zeta}M^2M_G^2b)MLK_b<1,
\end{equation}
where
$M_G=\|G\|$, $L=\int_{0}^{b}l(s)ds$, $K_b=\sup\{K(t): t\in J\}$.
\end{theorem}
\begin{proof}
Define the control function
\begin{equation} \label{e3.2}
u(\xi,t)=G^*T^*(b-t)(\Gamma_0^b)^{-1}\big(x_b-T(b)x(\xi,0)-
\int_{0}^{b}T(b-s)f(s)(\xi) ds\big),
\end{equation}
where $f\in S_{F,x}$.
Let $Z_b=\{x(\xi,t)\in C((-\infty,b]; X):
x_0(\xi,\theta)=\phi(\xi,\theta), \ \phi\in \mathcal{B}\}$. Set
$\|\cdot\|_b$ be a seminorm
in $Z_b$ defined by
\[
\|x(\xi,t)\|_b=\|x_0(\xi,t)\|_\mathcal{B}+\sup\limits_{s\in
J}\|x(\xi,s)\|, \quad x(\xi,t)\in Z_b.
\]
Now, we shall show that, when using the control \eqref{e3.2}, the
operators $\Gamma: Z_b \to 2^{Z_b}$ defined by
\begin{align*}
(\Gamma x)(\xi,t)
= \Big\{&w(\xi,t)\in Z_b: w(\xi,t)=
T(t)x(\xi,0)+ \int_{0}^{t}T(t-s)(Gu)(\xi,s)ds\\
& +\int_{0}^{t}T(t-s)f(s)(\xi) ds, \, t\in J, \, f \in S_{F,x}\Big\}
\end{align*}
has a fixed point. This fixed point is then a mild solution of
\eqref{e2.1}-\eqref{e2.3}. Obviously, $x_b\in (\Gamma x)(\cdot,b)$.
Let $\widehat{x}(\xi,t)\in C((-\infty,b],X)$ be the function
defined by
\[
\widehat{x}(\xi,t)= \begin{cases}
x_0(\xi,t), & t \in (-\infty,0],\\
T(t)x(\xi,0), & t\in J.
\end{cases}
\]
Set $x(\xi,t)=y(\xi,t)+\widehat{x}(\xi,t)$, $t\in (-\infty,b]$.
It is easy to see that $y$ satisfies
\begin{gather*}
y(\xi,t)=0, \quad t\in (-\infty,0], \\
y(\xi,t) = \int_{0}^{t}T(t-s)(Gu)(\xi,s)ds +\int_{0}^{t}T(t-s)f(s)(\xi) ds, \quad
t\in J,
\end{gather*}
where $f \in S_{F,y} = \{f \in L^{1}(J,X) : f(t)(\xi) \in
F(t, y_t(\xi,\cdot)+\hat{x}_t(\xi,\cdot)), \text{ for a.e. } t \in J, \,
\xi \in [0,2\pi] \}$.
Let $Z_b^0=\{y(\xi,t)\in Z_b: y(\xi,t)=0, \, t\in (-\infty,0]\}$.
For each $y(\xi,t)\in Z_{b}^0$, let
$\|y(\xi,t)\|_b=\sup_{s\in J}\|y(\xi,s)\|$, thus $(Z_b^0, \|\cdot\|_b)$
is a Banach space.
Consider the operator $\Gamma_1: Z_b^0 \to 2^{Z_b^0}$
defined by
\begin{align*}
(\Gamma_1 y)(\xi,t)
= \Big\{&{v}(\xi,t)\in Z_b^0:
{v}(\xi,t)= \int_{0}^{t}T(t-s)(Gu)(\xi,s)ds\\
& +\int_{0}^{t}T(t-s)f(s)(\xi) ds, \ t\in J, \ f \in S_{F,y}\Big\}.
\end{align*}
Next, we shall show that $\Gamma_1$ satisfy the hypotheses of
Lemma \ref{lem2.1}. The proof will be given in two steps.
\smallskip
\noindent\textbf{Step 1.}
We show that $(\Gamma_1 y)(\xi,t) \in P_{cl}(Z_b^0)$.
Indeed, let $y^{(n)}(\xi,t)\to y^*(\xi,t)$,
$\big({v}_n(\xi,t)\big)_{n\geq 0} \in (\Gamma_1 y)(\xi,t)$ such
that ${v}_n(\xi,t) \to {v}_*(\xi,t)$ in
$Z_b^0$. Then ${v}_*(\xi,t) \in Z_b^0$ and there exists
$f_n\in S_{F,y^{(n)}}$ such that, for each $t\in J$,
\[
{v}_n(\xi,t)= \int_{0}^{t}T(t-s)(Gu_{y^{(n)}})(\xi,s)ds
+\int_{0}^{t}T(t-s)f_n(s)(\xi) ds, \quad t\in J,
\]
where
\[
u_{y^{(n)}}(\xi,t)=G^*T^*(b-t)(\Gamma_0^b)^{-1}\big(x_b-T(b)x(\xi,0)-
\int_{0}^{b}T(b-s)f_n(s)(\xi) ds\big).
\]
Using the fact that $F$ has compact values and (H2) holds, we may
pass to a subsequence if necessary to obtain that $f_n$ converges
to $f_*$ in $L^1(J,X)$; hence, $f_*\in S_{F,y^*}$.
Then, for each $t\in J$,
\[
{v}_n(\xi,t)\to {v}_*(\xi,t)=
\int_{0}^{t}T(t-s)(Gu_{y^{*}})(\xi,s)ds
+\int_{0}^{t}T(t-s)f_*(s)(\xi) ds, \quad t\in J,
\]
where
\[
u_{y^{\ast}}(\xi,t)=G^*T^*(b-t)(\Gamma_0^b)^{-1}\big(x_b-T(b)x(\xi,0)-
\int_{0}^{b}T(b-s)f_{\ast}(s)(\xi) ds\big).
\]
So, ${v}_*(\xi,t)\in (\Gamma_1 y)(\xi, t)$ and, in particular,
$(\Gamma_1 y)(\xi, t) \in P_{cl}(Z_b^0)$.
\smallskip
\noindent\textbf{Step 2.} We show that $(\Gamma_1 y)(\xi,t)$ is a contractive
multivalued map for each $y(\xi,t)\in Z_b^0$.
Let $y(\xi,t), \overline{y}(\xi,t)\in Z_b^0$ and let
${v}(\xi,t) \in (\Gamma_1 y)(\xi,t)$. Then there exists
$f\in S_{F,y}$ such that
\begin{align*}
{v}(\xi,t)
&= \int_{0}^{t}T(t-s)(Gu)(\xi,s)ds
+\int_{0}^{t}T(t-s)f(s)(\xi) ds\\
&=\int_{0}^{t}T(t-\eta)GG^*T^*(b-\eta)(\Gamma_0^b)^{-1}\big(x_b-T(b)x(\xi,0)\\
&\quad-\int_{0}^{b}T(b-s)f(s)(\xi) ds\big)d\eta
+\int_{0}^{t}T(t-s)f(s)(\xi)ds.
\end{align*}
From (H2), it follows that, for each $t\in J$,
\[
H_d(F(\phi_1),F(\phi_2))\leq l(t)\|\phi_1-\phi_2\|_\mathcal{B}, \quad
\phi_1, \phi_2 \in \mathcal{B}.
\]
Hence, there exists $\omega(t)(\xi) \in F(t, \overline{y}_t(\xi,\cdot)
+\widehat{x}_t(\xi,\cdot))$ such that
\[
\|f(t)-\omega(t)\| \leq l(t) \|y_t-\overline{y}_t\|_\mathcal{B}.
\]
Consider $\Omega: J\to 2^X$,given by
\[
\Omega(t)=\{\omega(t) \in X: \|f(t)-\omega(t)\|\leq
l(t)\|y_t-\overline{y}_t\|_{\mathcal{B}}\}.
\]
Since the multivalued operator
$W(t)=\Omega(t)\cap F(t, \overline{y}_t+\widehat{x}_t)$ is measurable
\cite[Proposition III.4]{Castaing}, there exists a function
$\overline{f}(t)$, which is a measurable selection for $W$. So,
$\overline{f}(t)(\xi) \in F(\overline{y}_t(\xi,\cdot)+\widehat{x}_t(\xi,\cdot)) $
and
\[
\|f(t)-\overline{f}(t)\| \leq l(t)
\|y_t-\overline{y}_t\|_\mathcal{B}, \quad \text{for each } t\in J.
\]
For each $t\in J$, we define
\begin{align*}
{\overline{v}}(\xi,t)
&= \int_{0}^{t}T(t-s)(Gu)(\xi,s)ds
+\int_{0}^{t}T(t-s)\overline{f}(s)(\xi) ds\\
&=\int_{0}^{t}T(t-\eta)GG^*T^*(b-\eta)(\Gamma_0^b)^{-1}\big(x_b-T(b)x(\xi,0)\\
&\quad -\int_{0}^{b}T(b-s)\overline{f}(s)(\xi) ds\big)d\eta
+\int_{0}^{t}T(t-s)\overline{f}(s)(\xi)ds.
\end{align*}
Then, for $t\in J$, we obtain
\begin{align*}
&\|{v}(\xi,t)-{\overline{v}}(\xi,t)\|\\
&=\|\big[\int_{0}^{t}T(t-\eta)GG^*T^*(b-\eta)(\Gamma_0^b)^{-1}
\big(x_b-T(b)x(\xi,0)\\
&\quad - \int_{0}^{b}T(b-s)f(s)(\xi) ds\big)d\eta
+\int_{0}^{t}T(t-s)f(s)(\xi)ds\big]\\
& \quad -\big[\int_{0}^{t}T(t-\eta)GG^*T^*(b-\eta)(\Gamma_0^b)^{-1}
\big(x_b-T(b)x(\xi,0)\\
&\quad -\int_{0}^{b}T(b-s)\overline{f}(s)(\xi) ds\big)d\eta
+\int_{0}^{t}T(t-s)\overline{f}(s)(\xi)ds\big]\\
&\leq \|\int_{0}^{t}T(t-\eta)GG^*T^*(b-\eta)(\Gamma_0^b)^{-1}
\int_{0}^{b}T(b-s)[f(s)(\xi)-\overline{f}(s)(\xi)] dsd\eta\|\\
&\quad+\|\int_{0}^{t}T(t-s)[f(s)(\xi)-\overline{f}(s)(\xi)]ds\|\\
& \leq (1+\frac{1}{\zeta}M^2M_G^2b)M\int_{0}^{b}l(s)
\|y_s-\overline{y}_s\|_{\mathcal{B}} ds\\
& \leq (1+\frac{1}{\zeta}M^2M_G^2b)MLK_b\|y-\overline{y}\|_b
\end{align*}
Then
\[
\|{v}-{\overline{v}}\|_b\leq
(1+\frac{1}{\zeta}M^2M_G^2b)MLK_b\|y-\overline{y}\|_b.
\]
By an analogous relation, obtained by interchanging the roles of
${v}$ and ${\overline{v}}$, it follows that
\[
H_d((\Gamma_1 y)(\xi,t),(\Gamma_1 \overline{y})(\xi,t))\leq
(1+\frac{1}{\zeta}M^2M_G^2b)MLK_b\|y-\overline{y}\|_b.
\]
In view of \eqref{e3.1}, we conclude that $\Gamma_1$ is contractive. As a
consequence of Lemma \ref{lem2.1}, we deduce that $\Gamma_1$ have a fixed
point $y^*(\xi,t)\in Z_b^0$. Let
$x(\xi,t)=y^*(\xi,t)+\widehat{x}(\xi,t), \ t\in (-\infty,b]$. Then
$x$ is a fixed point of the operator $\Gamma$ which is a mild
solution of problem \eqref{e2.1}-\eqref{e2.3}.
\end{proof}
\begin{remark} \label{rmk3.1}\rm
We say system \eqref{e2.1}-\eqref{e2.3} is approximately controllable on
$J$ if for any given $x_b \in X$ and $\epsilon >0$, there exists a
control $u\in L^2(J,U)$ such that the mild solution $x(\cdot,t)$
of \eqref{e2.1}-\eqref{e2.3} satisfies $\|x(\cdot,b)-x_b\|<\epsilon$.
Actually we may also discuss the approximate controllability for system
\eqref{e2.1}-\eqref{e2.3} under weaker conditions,
more precisely, it is possible to formulate and prove sufficient conditions
for approximate controllability of nonlinear third-order dispersion inclusions
with infinite delay by suitably using techniques similar to those presented
in \cite{Hassane,Rykaczewski,Sakthivel}. We will go on to do it as a
subsequent work.
\end{remark}
\subsection*{Conclusion}
We have considered controllability problems of nonlinear third-order
dispersion inclusions with infinite delay.
By using a fixed-point theorem for contraction multivalued maps due
to Covitz and Nadler, sufficient conditions have been
given without compactness condition for the semigroup generated
by the linear part of the system.
In the future research, the controllability of stochastic nonlinear
third-order dispersion inclusions may be considered.
In addition, it is interesting to investigate the case with both
delays and impulsive effects.
\subsection*{Acknowledgments}
The authors are grateful with the anonymous referees for their
careful reading of the original manuscript and for sending us
their helpful comments that helped us this article.
This research was supported by grants 12ZR1400100, 11ZR1400200 from the
National Science Foundation of Shanghai.
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\end{document}