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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 87, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/87\hfil Robin boundary value problems]
{Robin boundary value problems for elliptic operational differential
equations with \\ variable coefficients}

\author[R. Haoua, A. Medeghri \hfil EJDE-2015/??\hfilneg]
{Rabah Haoua, Ahmed Medeghri}

\address{Rabah Haoua \newline
 Laboratoire de Math\'ematiques Pures et Appliqu\'ees,
Universit\'e de Mostaganem, 27000 Mostaganem, Alg\'erie}
\email{Haoua.rabah27@gmail.com}

\address{Ahmed Medeghri \newline
 Laboratoire de Math\'ematiques Pures et Appliqu\'ees,
Universit\'e de Mostaganem, 27000 Mostaganem, Alg\'erie}
\email{medeghri@univ-mosta.dz, ahmed\_medeghri@yahoo.com}

\thanks{Submitted September 19, 2014. Published April 7, 2015.}
\subjclass[2000]{34K10, 34K30, 35J25, 35J40, 47A60}
\keywords{Elliptic equation; Robin boundary conditions; analytic semigroup;
\hfill\break\indent  maximal regularity; Dunford operational calculus}

\begin{abstract}
 In this article we give some new results on abstract second-order differential
 equations of elliptic type with variable operator coefficients and general
 Robin boundary conditions, in the framework of H\"older spaces.
 We assume that the family of variable coefficients verify the well known
 Labbas-Terreni assumption used in the sum theory. We use Dunford calculus,
 interpolation spaces and the semigroup theory to obtain existence,
 uniqueness and maximal regularity results for the solution of the problem.
\end{abstract}

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

\section{Introduction}

In a complex Banach space $E$, we consider the second-order differential
equation
\begin{equation}
u''(x)+A(x)u(x)-\omega u(x)=f(x),\quad x\in ]0,1[ ,\label{Pb1}
\end{equation}
together with the general boundary conditions of Robin's type
\begin{equation}
u'(0)-Hu(0)=d_0,\quad u(1)=u_1.\label{Pb2}
\end{equation}
Here $\omega$ is a positive real number, $f\in C^{\theta}([0,1] ;E)$,
$0<\theta<1$, $d_0$ and $u_1$ are given elements
in $E$, $(A(x))_{x\in[  0,1]  }$ is a
family of closed linear operators whose domains $D(A(x))$
are not necessarily dense in $E$ and $H$ is a closed linear
operator with $D(H)\subset E$. Set for $x\in[ 0,1]$
\[
A_{\omega}(x)=A(x)-\omega I,\quad \omega>0.
\]
For $f\in C^{\theta}([  0,1]  ;E)$, we search for a
strict solution $u$ of problem \eqref{Pb1}, \eqref{Pb2}; that is,
\begin{gather*}
\forall x\in[  0,1]  \quad u(x)\in D(A(x)), \quad u\in C^2([  0,1];E)\\
x\mapsto A(x)u(x)\in C([0,1] ;E),\quad u(0)\in D(H).
\end{gather*}

Our purpose is to establish existence, uniqueness and maximal regularity of
the strict solution of this problem improving the results  in
\cite{ChFavLabMaiMed} and completing the one  in \cite{FavLabLabSad}. The
method is essentially based on Dunford calculus, interpolation spaces, the
semigroup theory and some techniques as in \cite{FavLabLabSad,Labbas}.

Throughout this work we suppose that the family $(A(x))_{x\in[  0,1]  }$
satisfies the following hypotheses:
\begin{itemize}
\item[(H1)] There exist $\omega_0>0$ and  $C>0$ such that for all
 $x\in[  0,1]$ and all $z\geq0$, $(A_{\omega_0}(x)-zI)^{-1}\in L(E)$ and
\begin{equation}
\| (A_{\omega_0}(x)-zI)^{-1} \| _{L(E)}\leq\frac{C}{1+z};\label{Hypo1}
\end{equation}
this estimate holds in some sector
\[
\Pi_{\theta_0,r_0}=\{ z\in\mathbb{C}\backslash\{  0\}  :| \arg(z)|
\leq\theta_0\}  \cup\{  z\in\mathbb{C}:| z| \leq r_0\}  ,
\]
where $\theta_0$ and $r_0$ are small positive numbers.
\end{itemize}

\begin{remark} \label{rmk1.1} \rm
It is well known that assumption \eqref{Hypo1} implies the same properties for
$A_{\omega}(x)$, $\omega\geq\omega_0$.
\end{remark}

On the other hand, it is well known that the square roots
\[
Q_{\omega}(x)=-(-A_{\omega}(x))
^{1/2},\text{ \ }x\in[  0,1]  \text{, \ }\omega\geq
\omega_0,
\]
are well defined and generate analytic semigroups not strongly continuous at
zero, (see Balakrishnan \cite{Balacrishnan} for dense domains and
Martinez-Sanz \cite{MartCarraSans} for non dense domains).
\begin{itemize}
\item[(H2)] There exist $C$, $\alpha$, $\mu>0$ such that for all 
$x, \tau\in[  0,1]$ and rall $\omega\geq\omega_0$:
\begin{equation}
\| Q_{\omega}(x)(Q_{\omega}(x)
-zI)^{-1}(Q_{\omega}(x)^{-1}-Q_{\omega}(\tau)^{-1})\| _{L(E)}
\leq \frac{C| x-\tau| ^{\alpha}}{| z+\omega|^{\mu}} \label{Hypo2}
\end{equation}
with $\alpha+\mu-2>0$.
This hypothesis is  known as the Labbas-Terreni assumption.
\end{itemize}

\begin{remark} \label{rmk1.2} \rm
From \eqref{Hypo2} one can prove that there
exist $C,\alpha, \mu>0$ such that for all $x,\tau\in[0,1]$ and
all $\omega\geq\omega_0$,
\[
\| Q_{\omega}(x)(Q_{\omega}(x)
-zI)^{-1}(Q_{\omega}(x)^{-2}-Q_{\omega}(\tau)^{-2})\| _{L(E)}
\leq \frac{C| x-\tau| ^{\alpha}}{| z+\omega|^{\mu}}
\]
with $\alpha+\mu-2>0$.
\end{remark}

\begin{itemize}
\item[(H3)] There exists $C>0$ such that for all $x\in[0,1]$ and all
$\omega \geq\omega_0$: 
$Q_{\omega}(x)-H$ is closable, $(\overline{Q_{\omega
}(x)-H})$ is boundedly invertible and
\begin{equation}
\| (\overline{Q_{\omega}(x)-H}) ^{-1}\| _{L(E)}\leq C(1+\sqrt{\omega})
\label{Hypo3}
\end{equation}
see  Cheggag et al \cite{ChFavLabMaiMed} for the autonomous case.

\item[(H4)] For all $x\in[  0,1]$ and all $\omega\geq\omega_0$,
\begin{equation}
\begin{gathered}
(\overline{Q_{\omega}(x)-H})^{-1}((D(Q_{\omega}(x)),E)
_{1-\theta,\infty})\subset D(Q_{\omega}(x) )\cap D(H), \\
Q_{\omega}(x)(\overline{Q_{\omega}(x)
-H})^{-1}((D(Q_{\omega}(x)),E)_{1-\theta,\infty})\subset(D(Q_{\omega
}(x)),E)_{1-\theta,\infty}.
\end{gathered} \label{Hypo4}
\end{equation}
\end{itemize}

Recall that for all $x\in[  0,1]$,
\begin{align*}
D_{Q_{\omega}(x)}(\theta,+\infty)
& =\{\phi\in E:\sup_{r>0} \| r^{\theta}Q_{\omega}(x)(Q_{\omega}(x)-zI)^{-1}
\phi\| _E<+\infty\} \\
 &: =(D(Q_{\omega}(x)),E)_{1-\theta,\infty}.
\end{align*}
See Grisvard \cite{Grisvard}.

\begin{itemize}
\item[(H5)]  For all $x\in[  0,1] $ and all $\omega\geq\omega_0$,
\begin{equation}
Q_{\omega}(x)^{-1}(\overline{Q_{\omega}(x)
-H})^{-1}=(\overline{Q_{\omega}(x)-H})
^{-1}Q_{\omega}(x)^{-1}.\label{Hypo5}
\end{equation}


\item[(H6)] There exists $C>0$ such that for all $x,\tau\in[  0,1] $
and all $\omega\geq\omega_0$,
\begin{equation}
\| (\overline{Q_{\omega}(x)-H})^{-1}-(\overline{Q_{\omega}(\tau)-H})
^{-1}\| _{L(E)}\leq C| x-\tau|^{l},\label{Hypo6}
\end{equation}
with some $l\geq2$.
\end{itemize}

Note that, from Remark \ref{rmk1.1} there exists a second sector
\[
\Pi_{\theta_1+\frac{\pi}{2},r_1}
=\{  z\in\mathbb{C}\backslash\{  0\}  :| \arg(z)|
\leq\theta_1+\frac{\pi}{2}\}
\cup\{  z\in\mathbb{C}:| z| \leq r_1\}  ,
\]
with small $\theta_1>0$ and $r_1>0$ such that the resolvent set of
$-(-A_{\omega}(x))^{1/2}$ satisfies
\[
\rho(-(-A_{\omega}(x))^{\frac{1}{2}
})\supset\Pi_{\theta_1+\frac{\pi}{2},r_1}
\]
for every $x\in[0,1]$.  Set
\[
\Gamma=\{  z=\rho e^{\pm i(\theta_1+\frac{\pi}{2})}
:\rho\geq r_1\}  \cup\{  z\in\mathbb{C}:| z| =r_1,\;
| \arg(z)| \geq\theta_1+\frac{\pi}{2}\}  ,
\]
oriented from $\infty e^{-i\theta_1}$ to $\infty e^{i\theta_1}$.

Cheggag et al \cite{ChFavLabMaiMed} studied the same problem with
Robin condition for the autonomous case $(A(x)=A)$ and used semigroups theory.
Assumption \eqref{Hypo2} are used in many other situations. One can cite, for
example, Bouziani et al \cite{Bouziani}.

There exists an other approach different from \eqref{Hypo2} which uses the
differentiability of the resolvent operators, see Boutaous et al
\cite{Boutaous} for Dirichlet problems.

In this work, there are two difficulties: we do not assume the
differentiability of the resolvent operators and one boundary condition
contains an unbounded operator. Our essential result is summarized by 
Theorem \ref{thm4.4}.

This article is organized as follows. In Section 2, some preliminary
technical results are proved. In Section $3$, we obtain some new results
verified by the solution $u$ of  \eqref{Pb1} and \eqref{Pb2} by using
an heuristical reasoning. 
In Section 4, we give necessary and sufficient
compatibility conditions in order to obtain the maximal regularity results of
the solution. 
Section 5 is devoted to prove existence of the solution by the
study of the associated approximating problem. Finally, we will present an
example of a partial differential equation where our abstract results apply.

\section{Technical lemmas}

\begin{lemma} \label{lem2.1}
Assume \eqref{Hypo1}. Then there exists a constant $C>0$, such that for all
$\omega\geq\omega_0$, $z\in\Gamma$ and $x\in[  0,1]  $,
\[
\| (Q_{\omega}(x)-zI)^{-1}\|
_{L(E)}\leq\frac{C}{\sqrt{\omega}+| z| }.
\]
\end{lemma}

For a proof of the above lemma, see Haase \cite[p. 57]{Haase}.

\begin{lemma} \label{lem2.2}
Assume \eqref{Hypo1}. For any $\omega\geq\omega_0$ and $x\in[
0,1]  $, the operator $I-e^{2Q_{\omega}(x)}$ has a bounded
inverse and
\[
(I-e^{2Q_{\omega}(x)})^{-1}=\frac{1}{2\pi i}
\int_{\Gamma}\frac{e^{2z}}{1-e^{2z}}(zI-Q_{\omega}(x))^{-1}dz+I.
\]
\end{lemma}

For a proof of the above lemma, see  Lunardi \cite[p. 60]{Lunardi}.
By using Lemma \ref{lem2.1}, there exists $C>0$ such that
for all $\omega\geq\omega_0$ :
\[
\| (I-e^{2Q_{\omega}(x)})^{-1}\| _{L(E)}\leq C.
\]

\begin{lemma} \label{lem2.3}
There exists $C>0$ such that
\begin{enumerate}
\item for all $x>0$ and $\varphi\in D(Q_{\omega}(0))$,
 $\| \varphi-e^{xQ_{\omega}(0)}\varphi\|
_E\leq Cx\| Q_{\omega}(0)\varphi\| _E$

\item for all $x>0$ and $\varphi\in D(Q_{\omega}(0)^2)$,
 $\| \varphi-e^{xQ_{\omega}(0)}
\varphi\| _E\leq Cx^2\| Q_{\omega}(0)
^2\varphi\| _E$.
\end{enumerate}
\end{lemma}

For a proof of the above lemma, see  Boutaous \cite[p. 8]{Boutaous}.
For $\omega\geq\omega_0$ and $x\in[  0,1]  $, we set
\[
\Pi_{\omega}(x)=I+2(I-e^{2Q_{\omega}(x)
})^{-1}Q_{\omega}(x)e^{2Q_{\omega}(x)
}(\overline{Q_{\omega}(x)-H})^{-1}.
\]

\begin{proposition} \label{prop2.4}
Assume \eqref{Hypo1}, \eqref{Hypo3} and \eqref{Hypo5}. Then there exists
$\omega^{\ast}\geq\omega_0$ such that for all $\omega\geq\omega^{\ast}$ and
$x\in[  0,1]  $, $\Pi_{\omega}(x)$ is boundedly invertible.
\end{proposition}

\begin{proof}
By Dore and Yakubov \cite[p. 103]{Dore-Yaku}, for $\alpha\in \mathbb{R}$ 
there exist constants $C$, $k>0$ (which do not depend on $\omega$) such that
for any $y\geq1$,
\[
\| (-A(x)+\omega I)^{\alpha}e^{-y(
-A(x)+\omega I)^{1/2}}\| _{L(E)}
\leq Ce^{-ky\sqrt{\omega}},
\]
thus, for $\omega$ large enough,
\[
\| Q_{\omega}(x)e^{2Q_{\omega}(x)
}\| _{L(E)}\leq Ce^{-2k\sqrt{\omega}},
\]
from which we obtain
\[
\| 2(I-e^{2Q_{\omega}(x)})^{-1}
Q_{\omega}(x)e^{2Q_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}\| _{L(
E)}\leq C(1+\sqrt{\omega})e^{-2k\sqrt{\omega}}.
\]
Then there exists $\omega^{\ast}\geq\omega_0$ such that for any $\omega
\geq\omega^{\ast}$,
\[
\| 2(I-e^{2Q_{\omega}(x)})^{-1}
Q_{\omega}(x)e^{2Q_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}\| _{L(
E)}<1.
\]
Hence, for $\omega\geq\omega^{\ast}$, $\Pi_{\omega}(x)$ is
boundedly invertible.
\end{proof}

\begin{lemma} \label{lem2.5}
Assume \eqref{Hypo1} and \eqref{Hypo3}--\eqref{Hypo6} and consider, for
$\omega\geq\omega_0$, the linear operator
\[
\Lambda_{\omega}(x)=Q_{\omega}(x)
-H+e^{2Q_{\omega}(x)}(Q_{\omega}(x)
+H),\quad x\in[  0,1],
\]
of domain $D(\Lambda_{\omega}(x))=D(Q_{\omega}(x))\cap D(H)$.  
Then there exists $\omega^{\ast}\geq\omega_0$ such that, for all $\omega\geq
\omega^{\ast}$ and $x\in[  0,1] $, $\Lambda_{\omega}(
x)$ is closable, its closure is invertible with
\begin{gather}
(\overline{\Lambda_{\omega}(x)})^{-1}=(
\overline{Q_{\omega}(x)-H})^{-1}(\Pi_{\omega
}(x))^{-1}(I-e^{2Q_{\omega}(x)})^{-1},\label{Pb3}
\\
(\overline{\Lambda_{\omega}(x)})^{-1}=(
\overline{Q_{\omega}(x)-H})^{-1}+(\overline
{Q_{\omega}(x)-H})^{-1}W(x),\label{Pb4}
\end{gather}
where
\begin{gather*}
W(x)\in L(E), \quad (\overline{Q_{\omega}(x)-H})^{-1}W(x)=W(x)
(\overline{Q_{\omega}(x)-H})^{-1},\\
W(x)(E)\subset
\cap_{k=1}^\infty D(Q_{\omega}(x)^{k}). 
\end{gather*}
\end{lemma}

\begin{proof}
Here $\Lambda_{\omega}(x)=(I-e^{2Q_{\omega}(x)})(Q_{\omega}(x)-H)
+2Q_{\omega}(x)e^{2Q_{\omega}(x)}$; therefore
$\Lambda_{\omega}(x)$ is closable (see Kato \cite{Kato}) and
\begin{align*}
 \overline{\Lambda_{\omega}(x)}
& =(I-e^{2Q_{\omega}(x)})(\overline
{Q_{\omega}(x)-H})+2Q_{\omega}(x)e^{2Q_{\omega}(x)}\\
& =(I-e^{2Q_{\omega}(x)})\Pi_{\omega}(
x)(\overline{Q_{\omega}(x)-H}).
\end{align*}
The above equality together with \eqref{Hypo4} and Proposition \ref{prop2.4} gives
$0\in\rho(\overline{\Lambda_{\omega}(x)})$ and
\eqref{Pb3} holds.
By \eqref{Pb3} we have
\[
(\overline{\Lambda_{\omega}(x)})^{-1}=(
\overline{Q_{\omega}(x)-H})^{-1}(I+M(
x))^{-1}(I+N(x))^{-1},
\]
with
\begin{gather*}
M(x)=2(I-e^{2Q_{\omega}(x)})^{-1}Q_{\omega}(x)e^{2Q_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}\in L(E),\\
N(x)=-e^{2Q_{\omega}(x)}\in L(E),\quad
[ M(x)]  (E)\subset \cap_{k=1}^\infty D(Q_{\omega}(x)^{k}), \\
[ N(x)]  (E)\subset \cap_{k=1}^\infty D(Q_{\omega}(x)^{k}). 
\end{gather*}
Set $U(x)=-M(x)(I+M(x)
)^{-1}\in L(E)$ and $V(x)=-N(x)(I+N(x))^{-1}\in L(E)$.
Then
\[
(\overline{\Lambda_{\omega}(x)})^{-1}=(
\overline{Q_{\omega}(x)-H})^{-1}(I+U(
x))(I+V(x)),
\]
with
\begin{gather*}
(\overline{Q_{\omega}(x)-H})^{-1}U(x)=U(x)(\overline{Q_{\omega}(x)-H})^{-1},\\
(\overline{Q_{\omega}(x)-H})^{-1}V(x)=V(x)(\overline{Q_{\omega}(x)-H})^{-1}, \\
[U(x)]  (E), \quad 
[V(x)]  (E) \supset \cap_{k=1}^\infty D(Q_{\omega}(x)^{k})\,.
\end{gather*}
Setting $W(x)=U(x)+V(x)+U(x)V(x)$ we deduce the result.
\end{proof}

\begin{lemma} \label{lem2.6}
There exists $K>0$ such that for each $z\in\Gamma$ and $r>0$, we have
\begin{gather*}
| z+r| \geq K| z| ,\quad | z+r| \geq K| r| ,\\
| z-r| \geq K| z| ,\quad | z-r| \geq K| r| .
\end{gather*}
There exists $K>0$ depending only on $\Gamma$ such that
for all $\lambda>0$ and all $\nu\in[  0,1]$,
\[
\int_{\Gamma}\frac{| dz| }{| z\pm\lambda| | z| ^{\nu}}\leq\frac{K}{\lambda^{\nu}}.
\]
\end{lemma}

For a proof of the above lemma, see Labbas-Terreni 
\cite[Lemmas 6.1 and 6.2]{Labbas-Terreni}.

\section{Representation of the solution}

\subsection{Heuristical reasoning}

Let us recall briefly the case when $A(x)\equiv A-\omega
I\equiv A_{\omega}$ is a constant operator satisfying the natural ellipticity
hypothesis mentioned above (we will take 
$Q_{\omega}=-(-A_{\omega})^{1/2}$). In this case, the representation 
of the solution $v$ of  \eqref{Pb1}, \eqref{Pb2} is given by the formula (see 
Cheggag \cite{ChFavLabMaiMed}).
\begin{align*}
v(x)& =e^{xQ_{\omega}}[  (\overline{\Lambda
_{\omega}})^{-1}d_0+(Q_{\omega}+H)(
\overline{\Lambda_{\omega}})^{-1}e^{Q_{\omega}}u_1] \\
&\quad +\frac{1}{2}e^{xQ_{\omega}}(Q_{\omega}+H)(
\overline{\Lambda_{\omega}})^{-1}Q_{\omega}^{-1}\int_0
^{1}e^{sQ_{\omega}}f(s)ds\\
&\quad -\frac{1}{2}e^{xQ_{\omega}}(Q_{\omega}+H)(
\overline{\Lambda_{\omega}})^{-1}e^{Q_{\omega}}Q_{\omega}^{-1}\int
_0^{1}e^{(1-s)Q_{\omega}}f(s)ds\\
&\quad +e^{(1-x)Q_{\omega}}[  (I-(Q_{\omega
}+H)(\overline{\Lambda_{\omega}})^{-1}e^{2Q_{\omega}
})u_1-(\overline{\Lambda_{\omega}})^{-1}e^{Q_{\omega
}}d_0] \\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}}(Q_{\omega}+H)
(\overline{\Lambda_{\omega}})^{-1}e^{Q_{\omega}}Q_{\omega}
^{-1}\int_0^{1}e^{sQ_{\omega}}f(s)ds\\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}}[  I-(Q_{\omega
}+H)(\overline{\Lambda_{\omega}})^{-1}e^{2Q_{\omega}
}]  Q_{\omega}^{-1}\int_0^{1}e^{(1-s)Q_{\omega}
}f(s)ds \\
&\quad +\frac{1}{2}Q_{\omega}^{-1}\int_0^{x}e^{(x-s)Q_{\omega}
}f(s)ds+\frac{1}{2}Q_{\omega}^{-1}\int_{x}^{1}e^{(
s-x)Q_{\omega}}f(s)ds.
\end{align*}
Set
\begin{align*}
& L_{Q_{\omega}(x)}(x,f)\\
&=\frac{1}{2}e^{xQ_{\omega}(x)}(Q_{\omega}(
x)+H)(\overline{\Lambda_{\omega}(x)
})^{-1}Q_{\omega}(x)^{-1}\int_0^{1}e^{sQ_{\omega
}(x)}f(s)ds\\
&\quad -\frac{1}{2}e^{xQ_{\omega}(x)}(Q_{\omega}(
x)+H)(\overline{\Lambda_{\omega}(x)
})^{-1}e^{Q_{\omega}(x)}Q_{\omega}(x)
^{-1}\int_0^{1}e^{(1-s)Q_{\omega}(x)}f(
s)ds\\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}(
Q_{\omega}(x)+H)(\overline{\Lambda_{\omega
}(x)})^{-1}e^{Q_{\omega}(x)}Q_{\omega
}(x)^{-1}\int_0^{1}e^{sQ_{\omega}(x)}f(
s)ds\\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}[
I-(Q_{\omega}(x)+H)(\overline
{\Lambda_{\omega}(x)})^{-1}e^{2Q_{\omega}(
x)}]  Q_{\omega}(x)^{-1} \\
&\quad\times \int_0^{1}e^{(1-s) Q_{\omega}(x)}f(s)ds
 +\frac{1}{2}Q_{\omega}(x)^{-1}\int_0^{x}e^{(
x-s)Q_{\omega}(x)}f(s)ds\\
&\quad +\frac{1}{2}Q_{\omega}(x)^{-1}\int_{x}^{1}e^{(
s-x)Q_{\omega}(x)}f(s)ds.
\end{align*}
Our heuristical reasoning is the following. Assume that  \eqref{Pb1}
and \eqref{Pb2} has a strict solution $u$; that is,
\begin{gather*}
\forall x\in[  0,1]  \; u(x)\in D(A_{\omega}(x)), \; 
u\in C^2([0,1]  ;E),\\
x\mapsto A_{\omega}(x)u(x)\in C(
[  0,1]  ;E) \text{ and } u(0)\in D(H). 
\end{gather*}
Then we can write 
\[
L_{Q_{\omega}(x)}(x,f)=L_{Q_{\omega}(
x)}(x,u''(x)-Q_{\omega}(x)^2u(x)).
\]
After two integrations by parts and some formal calculus, one obtains
\begin{align*}
& u(x)+\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega
}(x)\int_0^{1}Q_{\omega}(x)e^{sQ_{\omega
}(x)}(Q_{\omega}(s)^{-2}-Q_{\omega
}(x)^{-2})Q_{\omega}(s)^2u(s)ds \\
& -\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
e^{Q_{\omega}(x)}
 \int_0^{1}Q_{\omega}(x)e^{(1-s)Q_{\omega
}(x)}(Q_{\omega}(s)^{-2}-Q_{\omega
}(x)^{-2})Q_{\omega}(s)^2u(s)ds \\
& -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}T_{\omega
}(x)e^{Q_{\omega}(x)}
 \int_0^{1}Q_{\omega}(x)e^{sQ_{\omega}(x)
}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)
^{-2})Q_{\omega}(s)^2u(s)ds \\
& -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}(
I-T_{\omega}(x)e^{2Q_{\omega}(x)})
 \int_0^{1}Q_{\omega}(x)e^{(1-s)Q_{\omega
}(x)}(Q_{\omega}(s)^{-2}-Q_{\omega }(x)^{-2})\\
&\times Q_{\omega}(s)^2u(s)ds
 +\frac{1}{2}\int_0^{x}Q_{\omega}(x)e^{(x-s)
Q_{\omega}(x)}(Q_{\omega}(s)
^{-2}-Q_{\omega}(x)^{-2})Q_{\omega}(s)^2u(s)ds \\
& +\frac{1}{2}\int_{x}^{1}Q_{\omega}(x)e^{(s-x)
Q_{\omega}(x)}(Q_{\omega}(s)
^{-2}-Q_{\omega}(x)^{-2})Q_{\omega}(s)^2u(s)ds\\
& =L_{Q_{\omega}(x)}(x,f)+e^{xQ_{\omega}(
x)}[  (\overline{\Lambda_{\omega}(x)
})^{-1}d_0+T_{\omega}(x)e^{xQ_{\omega}(x)}u_1] \\
&\quad  +e^{(1-x)Q_{\omega}(x)}[  (I-T_{\omega}(x)e^{2Q_{\omega}(x)})
u_1-(\overline{\Lambda_{\omega}(x)})
^{-1}e^{Q_{\omega}(x)}d_0]  ,
\end{align*}
where
\[
T_{\omega}(x)=(Q_{\omega}(x)+H)
(\overline{\Lambda_{\omega}(x)})^{-1}.
\]
Applying $Q_{\omega}(x)^2$, we obtain
\begin{align*}
& Q_{\omega}(x)^2u(x)\\
& +\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
\int_0^{1}Q_{\omega}(x)^{3}e^{sQ_{\omega}(x)
}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)
^{-2})Q_{\omega}(s)^2u(s)ds\\
& -\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
e^{Q_{\omega}(x)}
 \int_0^{1}Q_{\omega}(x)^{3}e^{(1-s)
Q_{\omega}(x)}(Q_{\omega}(s)
^{-2}-Q_{\omega}(x)^{-2})Q_{\omega}(s)
^2u(s)ds\\
& -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}T_{\omega
}(x)e^{Q_{\omega}(x)}
 \int_0^{1}Q_{\omega}(x)^{3}e^{sQ_{\omega}(x)
}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)
^{-2})Q_{\omega}(s)^2u(s)ds\\
& -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}(
I-T_{\omega}(x)e^{2Q_{\omega}(x)})
 \int_0^{1}Q_{\omega}(x)^{3}e^{(1-s)
Q_{\omega}(x)}\Big(Q_{\omega}(s)^{-2}\\
&\quad -Q_{\omega}(x)^{-2}\Big)Q_{\omega}(s)^2u(s)ds
\\
& +\frac{1}{2}\int_0^{x}Q_{\omega}(x)^{3}e^{(
x-s)Q_{\omega}(x)}(Q_{\omega}(s)
^{-2}-Q_{\omega}(x)^{-2})Q_{\omega}(s)
^2u(s)ds\\
& +\frac{1}{2}\int_{x}^{1}Q_{\omega}(x)^{3}e^{(
s-x)Q_{\omega}(x)}(Q_{\omega}(s)
^{-2}-Q_{\omega}(x)^{-2})Q_{\omega}(s)
^2u(s)ds\\
& =Q_{\omega}(x)^2L_{Q_{\omega}(x)}(
x,f)+Q_{\omega}(x)^2e^{xQ_{\omega}(x)}
[  (\overline{\Lambda_{\omega}(x)})
^{-1}d_0+T_{\omega}(x)e^{xQ_{\omega}(x)}
u_1] \\
&\quad +Q_{\omega}(x)^2e^{(1-x)Q_{\omega}(
x)}[  (I-T_{\omega}(x)e^{2Q_{\omega}(
x)})u_1-(\overline{\Lambda_{\omega}(x)
})^{-1}e^{Q_{\omega}(x)}d_0]  .
\end{align*}
By setting
\[
w(\cdot)=Q_{\omega}(\cdot)^2u(
\cdot),
\]
we obtain the new  equation
\[
w+P_{\omega}w=G_{Q_{\omega}(x)}(d_0,u_1,f),
\]
where
\begin{align*}
& (P_{\omega}w)(x)\\
&=\frac{1}{2}T_{\omega}(x)e^{xQ_{\omega}(x)
}\int_0^{1}Q_{\omega}(x)^{3}e^{sQ_{\omega}(x)
}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)
^{-2})w(s)ds\\
& -\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
e^{Q_{\omega}(x)}\int_0^{1}Q_{\omega}(x)
^{3}e^{(1-s)Q_{\omega}(x)}(Q_{\omega
}(s)^{-2}-Q_{\omega}(x)^{-2})w(
s)ds\\
& -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}T_{\omega
}(x)e^{Q_{\omega}(x)}\int_0^{1}Q_{\omega
}(x)^{3}e^{sQ_{\omega}(x)}(Q_{\omega
}(s)^{-2}-Q_{\omega}(x)^{-2})w(
s)ds\\
& +\frac{1}{2}e^{(1-x)Q_{\omega}(x)}T_{\omega
}(x)e^{2Q_{\omega}(x)}\int_0^{1}Q_{\omega
}(x)^{3}e^{(1-s)Q_{\omega}(x)
}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)
^{-2})w(s)ds\\
& -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}\int_0
^{1}Q_{\omega}(x)^{3}e^{(1-s)Q_{\omega}(
x)}(Q_{\omega}(s)^{-2}-Q_{\omega}(
x)^{-2})w(s)ds\\
& +\frac{1}{2}\int_0^{x}Q_{\omega}(x)^{3}e^{(
x-s)Q_{\omega}(x)}(Q_{\omega}(s)
^{-2}-Q_{\omega}(x)^{-2})w(s)ds\\
& +\frac{1}{2}\int_{x}^{1}Q_{\omega}(x)^{3}e^{(
s-x)Q_{\omega}(x)}(Q_{\omega}(s)
^{-2}-Q_{\omega}(x)^{-2})w(s)ds\\
& =\overset{7}{\underset{i=1}{\sum}}I_i,
\end{align*}
and
\begin{align*}
& G_{Q_{\omega}(x)}(d_0,u_1,f)(x)\\
&=Q_{\omega}(x)^2L_{Q_{\omega}(x)}(x,f)
 +Q_{\omega}(x)^2e^{xQ_{\omega}(x)}[
(\overline{\Lambda_{\omega}(x)})^{-1}
d_0+T_{\omega}(x)e^{xQ_{\omega}(x)}u_1] \\
&\quad +Q_{\omega}(x)^2e^{(1-x)Q_{\omega}(
x)}[  (I-T_{\omega}(x)e^{2Q_{\omega}(
x)})u_1-(\overline{\Lambda_{\omega}(x)
})^{-1}e^{Q_{\omega}(x)}d_0]  .
\end{align*}


\begin{proposition} \label{prop3.1}
Under Hypotheses \eqref{Hypo1}--\eqref{Hypo6}, there exists $\omega^{\ast
}>0$, such that for all $\omega\geq\omega^{\ast}$,
\[
\| P_{\omega}\| _{L(
C[  0,1]  ;E)}\leq\frac{1}{2}.
\]
\end{proposition}

\begin{proof}
We treat for example
\begin{align*}
I_1  & =\frac{1}{2}T_{\omega}(x)e^{xQ_{\omega}(
x)}\int_0^{x}Q_{\omega}(x)^{3}e^{sQ_{\omega}(
x)}(Q_{\omega}(s)^{-2}-Q_{\omega}(
x)^{-2})w(s)ds\\
&\quad +\frac{1}{2}T_{\omega}(x)e^{xQ_{\omega}(x)
}\int_{x}^{1}Q_{\omega}(x)^{3}e^{sQ_{\omega}(x)
}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)
^{-2})w(s)ds\\
& =a+b,
\end{align*}
and thus
\[
\| a\| _E\leq K\int_{\Gamma}| z|
^2\Big(\sup_{x\in[  0,1] } \int_0^{x}e^{-c(
x+s)| z| }(x-s)^{\alpha}ds\Big)
\frac{| dz| }{| z+\omega| ^{\mu}
}\| w\| _{C([  0,1]  ;E)}\,.
\]
Using H\"older's inequality as in \cite{Bouziani}, \cite{Labbas-Terreni}, we
prove that
\[
\underset{x\in[  0,1]  }{\sup}\int_0^{x}e^{-c(x+s)
| z| }(x-s)^{\alpha}ds\leq\frac
{K}{| z| ^{1+\alpha}},
\]
from which we deduce using Lemma \ref{lem2.6}, that
\begin{align*}
\| a\| _E
& \leq K\int_{\Gamma}\frac{|z| ^2| dz| }{| z+\omega|
^{\mu}| z| ^{\alpha+1}}\| w\| _{C([  0,1]  ;E)}\\
& \leq K\int_{\Gamma}\frac{| dz| }{|
z+\omega| ^{\mu}| z| ^{\alpha-1}}\|w\| _{C([  0,1]  ;E)}\\
& \leq\frac{K}{\omega^{\alpha+\mu-2}}\| w\| _{C([  0,1]  ;E)},
\end{align*}
We obtain also a similar estimate for $\| b\| _E$.
Therefore there exists $\omega^{\ast}>0$ such that for all 
$\omega\geq \omega^{\ast}$, 
$\| P_{\omega}\| _{L(C[ 0,1]  ;E)}\leq\frac{1}{2}$ which leads us to invert
$I+P_{\omega} $ for $\omega\geq\omega^{\ast}$ in the space 
$C([0,1]  ;E)$.
\end{proof}

\subsection{Regularity of the second member $G_{Q_{\omega}(  x)
}(  d_0,u_1,f)  $}

In this section we use the following lemmas (see \cite[Lemma 1.8]{AquistapaceTerreni} 
and  \cite[Theorem 6 (4)]{FavLabMangTanYag}).

\begin{lemma} \label{lem3.2}
Fix $x\in[  0,1] $ and $\beta\in]  0,1[  $. Then
\begin{enumerate}
\item $s\to e^{sQ_{\omega}(x)}\varphi\in C([  0,1]  ;E)$ if and only if
 $\varphi\in\overline{D(Q_{\omega}(x))}$.

\item $s\to e^{sQ_{\omega}(x)}\varphi\in C^{\beta}([  0,1]  ;E)$ if and only if
 $\varphi\in D_{Q_{\omega}(x)}(\beta,+\infty)$.
\end{enumerate}
\end{lemma}

\begin{lemma} \label{lem3.3}
Let $f\in C^{\beta}([  0,1]  ;E)$. Then
\[
Q_{\omega}(0)\int_0^{1}e^{sQ_{\omega}(0)}(f(s)-f(0))ds\in(D(
Q_{\omega}(0));E)_{1-\theta,\infty}.
\]
\end{lemma}

The regularity results of $G_{Q_{\omega}(x)}(d_0,u_1,f)$ are as follows:

\begin{proposition} \label{prop3.4}
Assume \eqref{Hypo1}--\eqref{Hypo6} and let $f\in C^{\theta}([  0,1]  ;E)$ 
with $\theta\in[  0,1] $.  Then

(1) $G_{Q_{\omega}(x)}(d_0,u_1,f)\in C([  0,1]  ;E)$ if and only if
\begin{equation}
\begin{gathered}
u_1\in D(Q_{\omega}(1)^2), \quad (\overline{Q_{\omega}(0)-H})^{-1}d_0\in D(
Q_{\omega}(0))\cap D(H), \\
Q_{\omega}(0)(\overline{Q_{\omega}(0)
-H})^{-1}(d_0-Q_{\omega}(0)^{-1}f(
0))\in D(Q_{\omega}(0)), \\
Q_{\omega}(0)^2(\overline{Q_{\omega}(0)
-H})^{-1}(d_0-Q_{\omega}(0)^{-1}f(
0))+f(0)\in\overline{D(Q_{\omega}(0))}, \\
Q_{\omega}(1)^2u_1+f(1)\in\overline{D(Q_{\omega}(1))}.
\end{gathered} \label{Pb5}
\end{equation}

(2) $G_{Q_{\omega}(x)}(d_0,u_1,f)\in
C^{\theta}([  0,1]  ;E)$ if and only if
\begin{equation}
\begin{gathered}
u_1\in D(Q_{\omega}(1)^2),\quad 
(\overline{Q_{\omega}(0)-H})^{-1}d_0\in D(Q_{\omega}(0))\cap D(H), \\
Q_{\omega}(0)(\overline{Q_{\omega}(0)
-H})^{-1}(d_0-Q_{\omega}(0)^{-1}f(0))\in D(Q_{\omega}(0)), \\
Q_{\omega}(0)^2(\overline{Q_{\omega}(0)
-H})^{-1}(d_0-Q_{\omega}(0)^{-1}f(
0))+f(0)\in(D(Q_{\omega}(0));E)_{1-\theta,\infty}, \\
Q_{\omega}(1)^2u_1+f(1)\in(D(Q_{\omega}(1));E)_{1-\theta,\infty}.
\end{gathered} \label{Pb6}
\end{equation}
\end{proposition}

\begin{proof}
Let $x\in[  0,1]  $. Then we have
\begin{align*}
& G_{Q_{\omega}(x)}(d_0,u_1,f)(x)\\
& =Q_{\omega}(x)^2e^{xQ_{\omega}(x)}[
(\overline{\Lambda_{\omega}(x)})^{-1}
d_0+T_{\omega}(x)e^{Q_{\omega}(x)}u_1]
\\
&\quad +Q_{\omega}(x)^2e^{(1-x)Q_{\omega}(
x)}[  (I-T_{\omega}(x)e^{2Q_{\omega}(
x)})u_1-(\overline{\Lambda_{\omega}(x)
})^{-1}e^{Q_{\omega}(x)}d_0] \\
&\quad +\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
\int_0^{1}Q_{\omega}(x)e^{sQ_{\omega}(x)
}f(s)ds\\
&\quad -\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
\int_0^{1}Q_{\omega}(x)e^{(2-s)Q_{\omega
}(x)}f(s)ds\\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}T_{\omega
}(x)\int_0^{1}Q_{\omega}(x)e^{(
1+s)Q_{\omega}(x)}f(s)ds\\
&\quad +\frac{1}{2}e^{(1-x)Q_{\omega}(x)}T_{\omega
}(x)\int_0^{1}Q_{\omega}(x)e^{(
3-s)Q_{\omega}(x)}f(s)ds\\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}\int_0
^{1}Q_{\omega}(x)e^{(1-s)Q_{\omega}(x)}f(s)ds\\
&\quad +\frac{1}{2}\int_0^{x}Q_{\omega}(x)e^{(x-s) Q_{\omega}(x)}f(s)ds
 +\frac{1}{2}\int_{x}^{1}Q_{\omega}(x)e^{(s-x) Q_{\omega}(x)}f(s)ds\\
&=Q_{\omega}(x)^2e^{xQ_{\omega}(x)}[
(\overline{\Lambda_{\omega}(x)})^{-1}
d_0+T_{\omega}(x)e^{Q_{\omega}(x)}u_1]\\
&\quad +Q_{\omega}(x)^2e^{(1-x)Q_{\omega}(
x)}[  (I-T_{\omega}(x)e^{2Q_{\omega}(
x)})u_1-(\overline{\Lambda_{\omega}(x)
})^{-1}e^{Q_{\omega}(x)}d_0] \\
&\quad +\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
\int_0^{1}Q_{\omega}(x)e^{sQ_{\omega}(x)
}(f(s)-f(0))ds\\
&\quad -\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
\int_0^{1}Q_{\omega}(x)e^{(2-s)Q_{\omega
}(x)}f(s)ds\\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}T_{\omega
}(x)\int_0^{1}Q_{\omega}(x)e^{(1+s)Q_{\omega}(x)}f(s)ds\\
&\quad +\frac{1}{2}e^{(1-x)Q_{\omega}(x)}T_{\omega
}(x)\int_0^{1}Q_{\omega}(x)e^{(3-s)Q_{\omega}(x)}f(s)ds\\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}\int_0
^{1}Q_{\omega}(x)e^{(1-s)Q_{\omega}(
x)}(f(s)-f(1))ds\\
&\quad +\frac{1}{2}\int_0^{x}Q_{\omega}(x)e^{(x-s)
Q_{\omega}(x)}(f(s)-f(0))ds\\
&\quad +\frac{1}{2}\int_{x}^{1}Q_{\omega}(x)e^{(s-x)
Q_{\omega}(x)}(f(s)-f(1))ds\\
&\quad +\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
(e^{Q_{\omega}(x)}f(0)-f(0))+\frac{1}{2}(e^{xQ_{\omega}(x)}f(0)-f(0))\\
&\quad +\frac{1}{2}e^{(1-x)Q_{\omega}(x)}(
f(1)-e^{Q_{\omega}(x)}f(1)
)+\frac{1}{2}(e^{Q_{\omega}(x)}f(1)-f(1)).
\end{align*}
Let us write
\begin{align*}
f(0)-T_{\omega}(x)f(0)
& =[I-T_{\omega}(x)]  f(0)\\
& =[  I-(Q_{\omega}(x)+H)(\overline{\Lambda_{\omega}(x)})^{-1}]  f(0)\\
& =[  (\overline{\Lambda_{\omega}(x)})-(Q_{\omega}(x)+H)]  (
\overline{\Lambda_{\omega}(x)})^{-1}f(0)
\\
& =-2H(\overline{\Lambda_{\omega}(x)})
^{-1}f(0)+e^{2Q_{\omega}(x)}T_{\omega}(x)f(0).
\end{align*}
Then we obtain
\begin{align*}
& G_{Q_{\omega}(x)}(d_0,u_1,f)(x)\\
& =-e^{xQ_{\omega}(x)}H(\overline{\Lambda_{\omega
}(x)})^{-1}f(0)+Q_{\omega}(
x)^2e^{xQ_{\omega}(x)}(\overline
{\Lambda_{\omega}(x)})^{-1}d_0\\
&\quad +e^{(1-x)Q_{\omega}(x)}f(1)
+Q_{\omega}(x)^2e^{(1-x)Q_{\omega}(x)}u_1\\
&\quad +\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
\int_0^{1}Q_{\omega}(x)e^{sQ_{\omega}(x)
}(f(s)-f(0))ds\\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}\int_0
^{1}Q_{\omega}(x)e^{(1-s)Q_{\omega}(x)}(f(s)-f(1))ds\\
&\quad +\frac{1}{2}\int_0^{x}Q_{\omega}(x)e^{(x-s)Q_{\omega}(x)}(f(s)-f(0)
)ds-\frac{1}{2}f(0)\\
&\quad +\frac{1}{2}\int_{x}^{1}Q_{\omega}(x)e^{(s-x)
Q_{\omega}(x)}(f(s)-f(1))ds-\frac{1}{2}f(1)\\
&\quad +Q_{\omega}(x)^2e^{xQ_{\omega}(x)}e^{Q_{\omega}(x)}T_{\omega}(x)\\
&\quad\times \Big[  u_1-\frac{1}{2}Q_{\omega}(x)^{-1}\int_0
^{1}e^{(1-s)Q_{\omega}(x)}f(s)
ds+\frac{1}{2}(I+e^{Q_{\omega}(x)})Q_{\omega
}(x)^{-2}f(0)\Big] \\
&\quad -Q_{\omega}(x)^2e^{(1-x)Q_{\omega}(
x)}e^{2Q_{\omega}(x)}T_{\omega}(x)
[  u_1-\frac{1}{2}Q_{\omega}(x)^{-1}\int_0
^{1}e^{(1-s)Q_{\omega}(x)}f(s)ds] \\
&\quad -Q_{\omega}(x)^2e^{(1-x)Q_{\omega}(
x)}e^{Q_{\omega}(x)}\\
& \quad\times \Big[  (\overline{\Lambda_{\omega}(x)})
^{-1}d_0+\frac{1}{2}T_{\omega}(x)Q_{\omega}(x)
^{-1}\int_0^{1}e^{sQ_{\omega}(x)}f(s)
ds+\frac{1}{2}Q_{\omega}(x)^{-2}f(1)\Big] \\
& =[  -e^{xQ_{\omega}(x)}H(\overline{\Lambda_{\omega}(x)})^{-1}f(0)+Q_{\omega
}(x)^2e^{xQ_{\omega}(x)}(\overline
{\Lambda_{\omega}(x)})^{-1}d_0] \\
&\quad +[  e^{(1-x)Q_{\omega}(x)}f(
1)+Q_{\omega}(x)^2e^{(1-x)Q_{\omega
}(x)}u_1] \\
&\quad +\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
\int_0^{1}Q_{\omega}(x)e^{sQ_{\omega}(x)
}(f(s)-f(0))ds\\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}\int_0
^{1}Q_{\omega}(x)e^{(1-s)Q_{\omega}(
x)}(f(s)-f(1))ds\\
&\quad +\frac{1}{2}\int_0^{x}Q_{\omega}(x)e^{(x-s)
Q_{\omega}(x)}(f(s)-f(0)
)ds-\frac{1}{2}f(0)\\
&\quad +\frac{1}{2}\int_{x}^{1}Q_{\omega}(x)e^{(s-x)
Q_{\omega}(x)}(f(s)-f(1)
)ds-\frac{1}{2}f(1)\\
&\quad +A_{\omega}(x)R(x,f,d_0,u_1)
\end{align*}
\begin{align*}
&=\overset{4}{\underset{k=1}{\sum}}I_{k}+[  -e^{xQ_{\omega}(
x)}H(\overline{\Lambda_{\omega}(x)})
^{-1}f(0)+Q_{\omega}(x)^2e^{xQ_{\omega}(
x)}(\overline{\Lambda_{\omega}(x)})^{-1}d_0] \\
&\quad +[  e^{(1-x)Q_{\omega}(x)}f(
1)+Q_{\omega}(x)^2e^{(1-x)Q_{\omega}(x)}u_1] 
 +A_{\omega}(x)R(x,f,d_0,u_1),
\end{align*}
where
\begin{align*}
& R(x,f,d_0,u_1)\\
& =-e^{xQ_{\omega}(x)}e^{Q_{\omega}(x)}
T_{\omega}(x)\\
&\quad\times \Big[  u_1-\frac{1}{2}Q_{\omega}(x)^{-1}\int_0
^{1}e^{(1-s)Q_{\omega}(x)}f(s)
ds+\frac{1}{2}(I+e^{Q_{\omega}(x)})Q_{\omega }(x)^{-2}f(0)\Big] \\
& +e^{(1-x)Q_{\omega}(x)}e^{2Q_{\omega}(
x)}T_{\omega}(x)[  u_1-\frac{1}{2}Q_{\omega
}(x)^{-1}\int_0^{1}e^{(1-s)Q_{\omega}(x)}f(s)ds] \\
&\quad +e^{(1-x)Q_{\omega}(x)}e^{Q_{\omega}(x)}
\Big[  (\overline{\Lambda_{\omega}(x)})
^{-1}d_0\\
&\quad+\frac{1}{2}T_{\omega}(x)Q_{\omega}(x)
^{-1}\int_0^{1}e^{sQ_{\omega}(x)}f(s)
ds+\frac{1}{2}Q_{\omega}(x)^{-2}f(1)\Big]  .
\end{align*}
Each term in $R(\cdot,f,d_0,u_1)$ contains 
$e^{Q_{\omega}(x)}$ and $e^{Q_{\omega}(x)}\in L(
E,D(Q_{\omega}(x)^{m}))$ for any $m\in\mathbb{N}$, so
\begin{gather*}
R(\cdot,f,d_0,u_1)\in C^{\infty}([ 0,1]  ;E),\\
A_{\omega}(\cdot)R(\cdot,f,d_0,u_1)\in
C^{\infty}([  0,1]  ;E). 
\end{gather*}

For $I_1$, let us write
\begin{align*}
I_1  & =\frac{1}{2}[  e^{Q_{\omega}(x)}T_{\omega}(
x)Q_{\omega}(x)-e^{Q_{\omega}(0)}T_{\omega}(x)Q_{\omega}(0)]  
\int_0^{1}e^{sQ_{\omega}(0)}(f(s)-f(0))ds\\
&\quad +e^{Q_{\omega}(0)}T_{\omega}(x)Q_{\omega
}(0)\int_0^{1}e^{sQ_{\omega}(0)}(f(s)-f(0))ds\,.
\end{align*}
From Lemma \ref{lem3.3}, we obtain
\[
Q_{\omega}(0)\int_0^{1}e^{sQ_{\omega}(0)
}(f(s)-f(0))ds\in(D(Q_{\omega}(0)),E)_{1-\theta,\infty},
\]
from which we deduce that
\[
e^{xQ_{\omega}(0)}Q_{\omega}(0)\int_0
^{1}e^{sQ_{\omega}(0)}(f(s)-f(
0))ds\in C^{\theta}([  0,1]  ;E)\,.
\]
For the other terms, we use the same technics as in \cite{AquistapaceTerreni}.

Now, from Lemma \ref{lem2.5}, we have
\begin{align*}
& Q_{\omega}(x)^2e^{xQ_{\omega}(x)}(
\overline{\Lambda_{\omega}(x)})^{-1}d_0
-e^{xQ_{\omega}(x)}H(\overline{\Lambda_{\omega}(
x)})^{-1}f(0)\\
& =Q_{\omega}(x)^2e^{xQ_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}d_0-e^{xQ_{\omega
}(x)}H(\overline{Q_{\omega}(x)-H})
^{-1}f(0)\\
&\quad +Q_{\omega}(x)^2e^{xQ_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}W(x)
d_0-e^{xQ_{\omega}(x)}H(\overline{Q_{\omega}(
x)-H})^{-1}W(x)f(0)\\
& =Q_{\omega}(x)^2e^{xQ_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}d_0+e^{xQ_{\omega
}(x)}(Q_{\omega}(x)-H)(
\overline{Q_{\omega}(x)-H})^{-1}f(0)\\
&\quad -e^{xQ_{\omega}(x)}Q_{\omega}(x)(
\overline{Q_{\omega}(x)-H})^{-1}f(0)\\
&\quad +Q_{\omega}(x)^2e^{xQ_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}W(x)
d_0-e^{xQ_{\omega}(x)}H(\overline{Q_{\omega}(
x)-H})^{-1}W(x)f(0)\\
& =Q_{\omega}(x)^2e^{xQ_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}d_0+e^{xQ_{\omega
}(x)}f(0)\\
&\quad -e^{xQ_{\omega}(x)}Q_{\omega}(x)(
\overline{Q_{\omega}(x)-H})^{-1}f(0)\\
&\quad +Q_{\omega}(x)^2e^{xQ_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}W(x)
d_0-e^{xQ_{\omega}(x)}H(\overline{Q_{\omega}(
x)-H})^{-1}W(x)f(0),
\end{align*}
where $W(x)\in L(E)$ and $R(W(x))\subset \cap_{k=1}^\infty D(Q_{\omega}(x)^{k})$. 
 So
\begin{align*}
& Q_{\omega}(x)^2e^{xQ_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}d_0+e^{xQ_{\omega}(x)}f(0)
 -e^{xQ_{\omega}(x)}Q_{\omega}(x)(
\overline{Q_{\omega}(x)-H})^{-1}f(0)\\
& =[  Q_{\omega}(x)^2e^{xQ_{\omega}(x)
}(\overline{Q_{\omega}(0)-H})^{-1}
d_0-Q_{\omega}(0)^2e^{xQ_{\omega}(0)}(
\overline{Q_{\omega}(0)-H})^{-1}d_0]\\
&\quad +[  Q_{\omega}(x)^2e^{xQ_{\omega}(x)
}(\overline{Q_{\omega}(x)-H})^{-1}
d_0-Q_{\omega}(x)^2e^{xQ_{\omega}(x)}(
\overline{Q_{\omega}(0)-H})^{-1}d_0] \\
&\quad +[  -e^{xQ_{\omega}(x)}Q_{\omega}(0)
(\overline{Q_{\omega}(0)-H})^{-1}f(
0)+e^{xQ_{\omega}(0)}Q_{\omega}(0)
(\overline{Q_{\omega}(0)-H})^{-1}f(
0)] \\
&\quad +[  -e^{xQ_{\omega}(x)}Q_{\omega}(x)
(\overline{Q_{\omega}(x)-H})^{-1}f(
0)+e^{xQ_{\omega}(x)}Q_{\omega}(0)
(\overline{Q_{\omega}(0)-H})^{-1}f(
0)] \\
&\quad +[  e^{xQ_{\omega}(x)}f(0)-e^{xQ_{\omega
}(0)}f(0)] \\
&\quad +e^{xQ_{\omega}(0)}[  Q_{\omega}(0)
^2(\overline{Q_{\omega}(0)-H})^{-1}
d_0-Q_{\omega}(0)(\overline{Q_{\omega}(0)-H})^{-1}f(0)+f(0)]  .
\end{align*}
Using the algebraic identity
\begin{align*}
& Q_{\omega}(x)(Q_{\omega}(x)-zI)
^{-1}-Q_{\omega}(0)(Q_{\omega}(0)-zI)^{-1}\\
&= zQ_{\omega}(x)(Q_{\omega}(x)-zI)
^{-1}[  Q_{\omega}(x)^{-1}-Q_{\omega}(0)
^{-1}]  Q_{\omega}(0)(Q_{\omega}(0)-zI)^{-1},
\end{align*}
\marginpar{I added the equal sign before $zQ$ is it
correct?}
Then we obtain
\begin{align*}
& Q_{\omega}(x)^2e^{xQ_{\omega}(x)}(
\overline{Q_{\omega}(x)-H})^{-1}d_0+e^{xQ_{\omega}(x)}f(0)\\
& -e^{xQ_{\omega}(x)}Q_{\omega}(x)(
\overline{Q_{\omega}(x)-H})^{-1}f(0)\\
& =-\frac{1}{2\pi i}\int_{\Gamma}z^2e^{xz}Q_{\omega}(x)
(Q_{\omega}(x)-zI)^{-1}[  Q_{\omega}(
x)^{-1}-Q_{\omega}(0)^{-1}]  \\
&\quad\times  Q_{\omega}(0)(Q_{\omega}(0)-zI)
^{-1}(\overline{Q_{\omega}(0)-H})^{-1}d_0\,dz \\
&\quad +\frac{1}{2\pi i}\int_{\Gamma}ze^{xz}Q_{\omega}(x)(
Q_{\omega}(x)-zI)^{-1}[  Q_{\omega}(
x)^{-1}-Q_{\omega}(0)^{-1}]  \\
&\quad\times  Q_{\omega}(0)(Q_{\omega}(0)-zI)
^{-1}(\overline{Q_{\omega}(0)-H})^{-1}f(0)dz\\
&\quad -\frac{1}{2\pi i}\int_{\Gamma}z^2e^{xz}Q_{\omega}(x)(
Q_{\omega}(x)-zI)^{-1}[  Q_{\omega}(
x)^{-1}-Q_{\omega}(0)^{-1}]  \\
&\quad\times  Q_{\omega}(0)(Q_{\omega}(0)-zI)
^{-1}[  (\overline{Q_{\omega}(x)-H})
^{-1}-(\overline{Q_{\omega}(0)-H})^{-1}]
d_0dz\\
&\quad -\frac{1}{2\pi i}\int_{\Gamma}ze^{xz}Q_{\omega}(0)(
Q_{\omega}(0)-zI)^{-1}[  (\overline
{Q_{\omega}(x)-H})^{-1}-(\overline{Q_{\omega
}(0)-H})^{-1}]  d_0dz\\
&\quad +\frac{1}{2\pi i}\int_{\Gamma}e^{xz}Q_{\omega}(x)(
Q_{\omega}(x)-zI)^{-1}[  (\overline
{Q_{\omega}(x)-H})^{-1}-(\overline{Q_{\omega
}(0)-H})^{-1}]  f(0)dz\\
&\quad -\frac{1}{2\pi i}\int_{\Gamma}e^{xz}Q_{\omega}(x)(
Q_{\omega}(x)-zI)^{-1}[  Q_{\omega}(
x)^{-1}-Q_{\omega}(0)^{-1}] \\
&\quad\times  Q_{\omega}(0)(Q_{\omega}(0)-zI)^{-1}f(0)dz\\
&\quad +e^{xQ_{\omega}(0)}[  Q_{\omega}(0)
^2(\overline{Q_{\omega}(0)-H})^{-1}
d_0-Q_{\omega}(0)(\overline{Q_{\omega}(0)-H})^{-1}f(0)+f(0)]\\
&= \sum_{i=1}^7 a_i\,.
\end{align*}

For $a_1$ we have
\begin{align*}
\| a_1\| _E
& \leq K\int_{\Gamma}| z| e^{-cx| z| }\frac{x^{\alpha}}{| z|
^{\mu}}| dz| \| Q_{\omega}(0)(
\overline{Q_{\omega}(0)-H})^{-1}d_0\| _E\\
& \leq K\int_{\Gamma}e^{-\sigma}\frac{x^{\alpha}}{(\frac{\sigma}
{x})^{\mu-1}}\frac{d\sigma}{x}\| Q_{\omega}(0)
(\overline{Q_{\omega}(0)-H})^{-1}d_0\| _E\\
& \leq Kx^{\alpha+\mu-2}\| Q_{\omega}(0)(
\overline{Q_{\omega}(0)-H})^{-1}d_0\| _E\\
& \leq K\| Q_{\omega}(0)(\overline{Q_{\omega
}(0)-H})^{-1}d_0\| _E.
\end{align*}
The same technics are used for the other terms $a_i$, $i=1,2,\dots ,6$.
For the last term of $G$. One has
\begin{align*}
& [  e^{(1-x)Q_{\omega}(x)}f(1)+Q_{\omega}(x)^2e^{(1-x)Q_{\omega}(x)}u_1] \\
& =[  e^{(1-x)Q_{\omega}(x)}f(
1)-e^{(1-x)Q_{\omega}(1)}f(1)] 
+[  Q_{\omega}(x)^2e^{(1-x)Q_{\omega}(x)}u_1\\
&\quad -Q_{\omega}(1)^2e^{(1-x)Q_{\omega}(1)}u_1] 
  +e^{(1-x)Q_{\omega}(1)}[  f(1)+Q_{\omega}(1)^2u_1]\\
& =-\frac{1}{2\pi i}\int_{\Gamma}e^{(1-x)z}((
Q_{\omega}(x)-zI)^{-1}-(Q_{\omega}(1)-zI)^{-1})f(1)dz\\
&\quad -\frac{1}{2\pi i}\int_{\Gamma}ze^{(1-x)z}(Q_{\omega
}(x)(Q_{\omega}(x)-zI)^{-1}-Q_{\omega}(1)(Q_{\omega}(1)
-zI)^{-1})u_1dz\\
&\quad +e^{(1-x)Q_{\omega}(1)}[  f(1)+Q_{\omega}(1)^2u_1] \\
& =-\frac{1}{2\pi i}\int_{\Gamma}e^{(1-x)z}Q_{\omega}( x)
 (Q_{\omega}(x)-zI)^{-1}(Q_{\omega}(x)^{-1}-Q_{\omega}(1)^{-1})
 Q_{\omega}(1) \\
&\quad\times (Q_{\omega}(1)-zI)^{-1} f(1)dz 
 -\frac{1}{2\pi i}\int_{\Gamma}z^2e^{(1-x)z}Q_{\omega
}(x)(Q_{\omega}(x)-zI)^{-1}\\
&\quad\times (Q_{\omega}(x)^{-1}-Q_{\omega}(1)^{-1})
  Q_{\omega}(1)(Q_{\omega}(1)-zI)^{-1}u_1dz
 +e^{(1-x)Q_{\omega}(1)}\\
&\quad\times[  f(1)+Q_{\omega}(1)^2u_1] \\
& =b_1+b_{2}+b_3.
\end{align*}
For the term $b_1$, we have
\begin{align*}
\| b_1\| _E
&\leq K\int_{\Gamma}e^{-c(1-x)| z| }\frac{(1-x)^{\alpha}}{|
z| ^{\mu}}| dz| \| f(1)\| _E \\
& \leq K\int_{\Gamma}e^{-\sigma}\frac{(1-x)^{\alpha}}{(
\frac{\sigma}{1-x})^{\mu}}\frac{(1-x)d\sigma}{(1-x)^2}\| f(1)\| _E\\
& \leq K(1-x)^{\alpha+\mu-2}\| f(1)\| _E\\
& \leq K\| f(1)\| _E.
\end{align*}
The same technic is used for the other terms. From $a_{7}$ and $b_3$ using
Lemmas \ref{lem3.2} and \ref{lem3.3} we deduce \eqref{Pb5} and \eqref{Pb6}.
\end{proof}

We can write for $\omega\geq\omega^{\ast}$ and $x\in[  0,1]  $,
\begin{equation}
u(x)=Q_{\omega}(x)^{-2}(I+P_{\omega})^{-1}G_{Q_{\omega}(x)}(d_0,u_1,f).\label{Pb7}
\end{equation}


\section{Regularity of the solution}

Throughout this section we assume that $\omega\geq\omega^{\ast}$.

\subsection{Regularity of $P_{\omega}$}

\begin{proposition} \label{prop4.1}
Under assumptions \eqref{Hypo1}$\sim$\eqref{Hypo6}, we have
\begin{enumerate}
\item $P_{\omega}\in L(C(E),C^{\alpha+\mu-2}(E))$

\item $P_{\omega}\in L(C(E),C(E)\cap
B(D_{A(\cdot)}(\frac{\beta}{2},+\infty)
))$, where $\beta\in]  0,\alpha+\mu-2]  $.
\end{enumerate}
\end{proposition}

\begin{proof}
Let us prove the first statement. Let $0\leq\tau\leq x\leq1$ and

$w\in C([  0,1]  ;E)$, we have
\begin{align*}
& (P_{\omega}w)(x)-(P_{\omega}w)(\tau)\\
& =\frac{1}{2}T_{\omega}(x)e^{xQ_{\omega}(x)
}\int_0^{1}Q_{\omega}(x)^{3}e^{sQ_{\omega}(x)
}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)
^{-2})w(s)ds\\
&\quad -\frac{1}{2}T_{\omega}(\tau)e^{\tau Q_{\omega}(
\tau)}\int_0^{1}Q_{\omega}(\tau)^{3}e^{sQ_{\omega
}(\tau)}(Q_{\omega}(s)^{-2}-Q_{\omega }(\tau)^{-2})w(s)ds\\
&\quad  -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}\int_0
^{1}Q_{\omega}(x)^{3}e^{(1-s)Q_{\omega}(
x)}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)^{-2})w(s)ds\\
&\quad +\frac{1}{2}e^{(1-\tau)Q_{\omega}(\tau)}
\int_0^{1}Q_{\omega}(\tau)^{3}e^{(1-s)
Q_{\omega}(\tau)}(Q_{\omega}(s)
^{-2}-Q_{\omega}(\tau)^{-2})w(s)ds \\
&\quad +\frac{1}{2}\int_0^{x}Q_{\omega}(x)^{3}e^{(
x-s)Q_{\omega}(x)}(Q_{\omega}(s)
^{-2}-Q_{\omega}(x)^{-2})w(s)ds\\
&\quad -\frac{1}{2}\int_0^{\tau}Q_{\omega}(\tau)^{3}e^{(
\tau-s)Q_{\omega}(\tau)}(Q_{\omega}(
s)^{-2}-Q_{\omega}(\tau)^{-2})w(s)ds\\
&\quad +\frac{1}{2}\int_{x}^{1}Q_{\omega}(x)^{3}e^{(
s-x)Q_{\omega}(x)}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)^{-2})w(s)ds\\
&\quad -\frac{1}{2}\int_{\tau}^{1}Q_{\omega}(\tau)^{3}e^{(
s-\tau)Q_{\omega}(\tau)}(Q_{\omega}(
s)^{-2}-Q_{\omega}(\tau)^{-2})w(s)ds\\
&\quad -\frac{1}{2}e^{xQ_{\omega}(x)}T_{\omega}(x)
e^{Q_{\omega}(x)}\int_0^{1}Q_{\omega}(x)
^{3}e^{(1-s)Q_{\omega}(x)}(Q_{\omega
}(s)^{-2}-Q_{\omega}(x)^{-2})w(s)ds\\
&\quad +\frac{1}{2}e^{\tau Q_{\omega}(\tau)}T_{\omega}(
\tau)e^{Q_{\omega}(\tau)}\int_0^{1}Q_{\omega}(
\tau)^{3}e^{(1-s)Q_{\omega}(\tau)
}(Q_{\omega}(s)^{-2}-Q_{\omega}(\tau)^{-2})w(s)ds\\
&\quad -\frac{1}{2}e^{(1-x)Q_{\omega}(x)}T_{\omega
}(x)e^{Q_{\omega}(x)}\int_0^{1}Q_{\omega
}(x)^{3}e^{sQ_{\omega}(x)}(Q_{\omega
}(s)^{-2}-Q_{\omega}(x)^{-2})w(s)ds\\
&\quad +\frac{1}{2}e^{(1-\tau)Q_{\omega}(\tau)
}T_{\omega}(\tau)e^{Q_{\omega}(\tau)}\int
_0^{1}Q_{\omega}(\tau)^{3}e^{sQ_{\omega}(\tau)
}(Q_{\omega}(s)^{-2}-Q_{\omega}(\tau)^{-2})w(s)ds\\
& +\frac{1}{2}e^{(1-x)Q_{\omega}(x)}T_{\omega
}(x)e^{2Q_{\omega}(x)}\int_0^{1}Q_{\omega
}(x)^{3}e^{(1-s)Q_{\omega}(x)
}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)^{-2})w(s)ds\\
&\quad -\frac{1}{2}e^{(1-\tau)Q_{\omega}(\tau)
}T_{\omega}(\tau)e^{2Q_{\omega}(\tau)}\int
_0^{1}Q_{\omega}(\tau)^{3}e^{(1-s)Q_{\omega
}(\tau)}(Q_{\omega}(s)^{-2}-Q_{\omega}(\tau)^{-2})w(s)ds\\
& =\overset{14}{\underset{i=1}{\sum}}I_i.
\end{align*}
One has $I_{9}+I_{10}$, $I_{11}+I_{12}$ and $I_{13}+I_{14}$ are 
$o(| x-\tau| ^{\beta})$ where
$0<\beta\leq\alpha+\mu-2$.

For the other terms, we use the same technics, let us treat for example
$I_{5}+I_{6}$ we can write that
\begin{align*}
& I_{5}+I_{6}\\
& =-\frac{1}{4\pi i}\int_0^{\tau}\int_{\Gamma}(e^{(x-s)
z}-e^{(\tau-s)z})z^2Q_{\omega}(\tau)
(Q_{\omega}(\tau)-zI)^{-1} \\
&\quad\times (Q_{\omega}(s)^{-2}-Q_{\omega}(\tau)
^{-2})w(s)\,dz\,ds\\
& -\frac{1}{4\pi i}\int_0^{\tau}\int_{\Gamma}z^2e^{(x-s)
z}(Q_{\omega}(x)(Q_{\omega}(x)
-zI)^{-1}-Q_{\omega}(\tau)(Q_{\omega}(
\tau)-zI)^{-1})\\
&\quad\times  (Q_{\omega}(s)^{-2}-Q_{\omega}(\tau)
^{-2})w(s)\,dz\,ds\\
& -\frac{1}{4\pi i}\int_0^{\tau}\int_{\Gamma}z^2e^{(x-s)
z}Q_{\omega}(x)(Q_{\omega}(x)-zI)
^{-1}(Q_{\omega}(\tau)^{-2}-Q_{\omega}(x)
^{-2})w(s)\,dz\,ds\\
&\quad  -\frac{1}{4\pi i}\int_{\tau}^{x}\int_{\Gamma}z^2e^{(x-s)
z}Q_{\omega}(x)(Q_{\omega}(x)-zI)
^{-1}(Q_{\omega}(s)^{-2}-Q_{\omega}(x)
^{-2})w(s)\,dz\,ds\\
&  \sum_{i=1}^4 J_i,
\end{align*}
For $J_1$, we have
\begin{align*}
\| J_1\| _E  
& \leq K\int_{\Gamma}\int_0^{\tau}
\int_{\tau}^{x}e^{-c(\xi-s)| z| }|
z| ^{3}\frac{(\tau-s)^{\alpha}}{|
z| ^{\mu}}| dz| d\xi ds\| w\|_{C([  0,1]  ;E)}\\
& \leq K\int_{\Gamma}\int_0^{\tau}\int_{\tau}^{x}e^{-\sigma}\frac{(
\tau-s)^{\alpha}}{(\frac{\sigma}{\xi-s})^{\mu-3}}
\frac{d\sigma}{(\xi-s)}d\xi ds\| w\|
_{C([  0,1]  ;E)}\\
& \leq K\int_0^{\tau}\int_{\tau}^{x}(\tau-s)^{\alpha}(
\xi-s)^{\mu-4}d\xi ds\| w\| _{C([0,1]  ;E)}\\
& \leq K\int_0^{\tau}(x-s)^{2\alpha+2\mu-4}ds\|
w\| _{C([  0,1]  ;E)}\\
& \leq K(x-\tau)^{\alpha+\mu-2}\| w\|_{C([  0,1]  ;E)},
\end{align*}
and
\begin{align*}
\| J_{2}\| _E  
& \leq K\int_0^{\tau}\int_{\Gamma }e^{-c(x-s)| z| }| z|
^{3}\frac{(x-\tau)^{\alpha}}{| z| ^{\mu}
}\frac{(\tau-s)^{\alpha}}{| z| ^{\mu}
}| dz| ds\| w\| _{C([0,1]  ;E)}\\
& \leq K\int_0^{\tau}\int_{\Gamma}e^{-\sigma}\frac{(x-\tau)
^{\alpha}(\tau-s)^{\alpha}}{(\frac{\sigma}{x-s})
^{2\mu-3}}\frac{d\sigma}{(x-s)}ds\| w\|
_{C([  0,1]  ;E)}\\
& \leq K\int_0^{\tau}(x-\tau)^{\alpha}(\tau-s)
^{\alpha}(x-s)^{2\mu-4}ds\| w\| _{C([  0,1]  ;E)}\\
& \leq K(x-\tau)^{\alpha+\mu-2}\| w\|
_{C([  0,1]  ;E)}.
\end{align*}
 $J_3$ and $J_4$ are also treated as $J_1$.
\end{proof}

\begin{corollary} \label{coro4.2}
Under assumptions \eqref{Hypo1}--\eqref{Hypo6} and for all $\omega
\geq\omega^{\ast}$, we have
\begin{enumerate}
\item $(I+P_{\omega})^{-1}\in L(C(E))$.

\item $(I+P_{\omega})^{-1}\in L(C^{\beta}(
E))$, where $\beta\in]  0,\alpha+\mu-2]  $.

\item $(I+P_{\omega})^{-1}\in L(C(E)\cap
B(D_{A(\cdot)}(\frac{\beta}{2},+\infty)
))$, where $\beta\in]  0,\alpha+\mu-2]  $.
\end{enumerate}
\end{corollary}

\subsection{``mixed'' regularity of $G_{Q_{\omega}( x)  }(d_0,u_1,f)  $}

\begin{proposition} \label{prop4.3}
Assume \eqref{Hypo1}--\eqref{Hypo6}. Let $\beta\in]  0,\alpha+\mu-2]  $,
\[
(\overline{Q_{\omega}(0)-H})^{-1}d_0\in D_{Q_{\omega}(0)}\cap D(H), \quad
u_1\in D_{A(1)}, \quad f\in C^{\beta}([  0,1];E).
\]
Then
$G_{Q_{\omega}(x)}(d_0,u_1,f)(
\cdot)+f(\cdot)\in B(D_{A(\cdot)
}(\frac{\beta}{2},+\infty))$ if and only if
\begin{gather*}
Q_{\omega}(0)(\overline{Q_{\omega}(0)-H})^{-1}(d_0-Q_{\omega}(0)^{-1}f(
0))\in D(Q_{\omega}(0)), \\
Q_{\omega}(0)^2(\overline{Q_{\omega}(0)-H})^{-1}
(d_0-Q_{\omega}(0)^{-1}f(0))+f(0)\in D_{A(0)}(
\frac{\beta}{2},+\infty), \\
A_{\omega}(1)u_1-f(1)\in D_{A(
1)}(\frac{\beta}{2},+\infty). 
\end{gather*}
\end{proposition}

Summarizing the above results we obtain the following theorem.

\begin{theorem} \label{thm4.4}
Assume \eqref{Hypo1}--\eqref{Hypo6}. Let $\beta\in]  0,\alpha+\mu-2]  $,
\[
(\overline{Q_{\omega}(0)-H})^{-1}d_0\in
D_{Q_{\omega}(0)}\cap D(H), \quad
u_1\in D_{A(1)}, \quad f\in C^{\beta}([  0,1];E)
\]
such that
\begin{gather*}
Q_{\omega}(0)(\overline{Q_{\omega}(0)-H})^{-1}(d_0-Q_{\omega}(0)^{-1}f(
0))\in D(Q_{\omega}(0)), \\
Q_{\omega}(0)^2(\overline{Q_{\omega}(0)
-H})^{-1}(d_0-Q_{\omega}(0)^{-1}f(
0))+f(0)\in D_{A(0)}(
\frac{\beta}{2},+\infty), \\
A_{\omega}(1)u_1-f(1)\in D_{A(1)}(\frac{\beta}{2},+\infty). 
\end{gather*}
Then there exists $\omega^{\ast}>0$ such that for all 
$\omega\geq\omega^{\ast}$, the equation \eqref{Pb7} has a unique solution 
$w(\cdot) =Q_{\omega}(\cdot)^2u(\cdot)$ satisfies
\begin{enumerate}
\item $Q_{\omega}(\cdot)^2u(\cdot)\in C([  0,1]  ;E)$.

\item $Q_{\omega}(\cdot)^2u(\cdot)\in C^{\beta}([  0,1]  ;E)$.

\item $u''\in C^{\beta}([  0,1]  ;E)$.

\item $u''\in C([  0,1]  ;E)\cap B(D_{A(\cdot)}(\frac{\beta}{2},+\infty))$.
\end{enumerate}
\end{theorem}

\begin{proof}
We have
\begin{align*}
u''(\cdot)& =f(\cdot)+Q_{\omega }(\cdot)^2u(\cdot)\\
& =f(\cdot)+[  G_{Q_{\omega}(x)}(d_0,u_1,f)(\cdot)-(P_{\omega}w)(\cdot)] \\
& =[  f(\cdot)+G_{Q_{\omega}(x)}(d_0,u_1,f)(\cdot)]  -(P_{\omega}w)(\cdot).
\end{align*}
\end{proof}

\section{Approximating problem}

In our heuristical reasoning in section two, one proves that the solution of
\eqref{Pb1}--\eqref{Pb2}, when it exists, is given necessarily by \eqref{Pb7}.

To prove that the representation of $u$ in \eqref{Pb7} is the unique strict
solution of \eqref{Pb1}--\eqref{Pb2}, we consider the family of approximating
problems
\begin{gather*}
u_{n}''(x)+A_{n}(x)u_{n}(x)-\omega u_{n}(x)
=f(x), \quad x\in]  0,1[ , \\
u_{n}'(0)-Hu_{n}(0)=d_0, \\
u_{n}(1)=u_1,
\end{gather*}
where $(A_{n}(x))_{x\in[  0,1]  }$ is
the family of Yosida approximations of $(A(x))_{x\in[  0,1]  }$ defined by
\[
A_{n}(x)=-nA(x)(A(x)
-nI)^{-1},\text{ }n\in \mathbb{N}^{\ast}.
\]
Then we use the same arguments as in Labbas \cite{Labbas} or in Bouziani
\cite{Bouziani}.

\section{A concrete general example}

Consider the complex Banach space $E=C([  0,1]  )$
with its usual sup-norm and define the family of closed linear operators
$Q(x)$ for all $x\in[  0,1]  $ by
\begin{gather*}
D(Q(x))=\{  \varphi\in C^2([  0,1]  ):a(x)\varphi(0)
+b(x)\varphi'(0)=0,\text{ }\varphi(1) =0\}, \\
(Q(x)\varphi)(y)=\varphi ''(y), \quad  y\in(0,1), 
\end{gather*}
from which it is easy to deduce that $A(x)=Q(x)^2 $,
\begin{gather*}
D(A(x))=\{  \varphi\in C^{4}([  0,1]  ):a(x)\varphi(0)
+b(x)\varphi'(0)=0,\varphi(1)=0, \\
 a(x)\varphi''(0)+b(x)
\varphi^{\prime\prime\prime}(0)=0,\quad \varphi''(1)=0\}, \\
(A(x)\varphi)(y)=-\varphi
^{(iv)}(y), \quad y\in(0,1). 
\end{gather*}
We assume that $a$, $b\in C^{1}([  0,1]  )$, $a>0$,
$b>0$ and $\inf_{x\in[  0,1]} (a(x)+b(x))>0$. Let us define the linear 
operator $H$ by
\begin{gather*}
D(H)=\{  \varphi\in C^{1}([  0,1]):\varphi(1)=0\} ,\\
(H\varphi)(y)=\alpha\varphi'(y), \quad y\in(0,1). 
\end{gather*}
Therefore the spectra of $Q(x)$, for every $x\in[0,1]  $, is included in 
$]  -\infty,0]  $,
\begin{gather*}
\overline{D(Q(x))}=\{  \varphi\in C^{1}([  0,1]  ):\varphi(0)
=\varphi(1)=0\} \quad \text{if } b(x)=0, \\
\overline{D(Q(x))}=\{  \varphi\in C^{1}([  0,1]  ):\varphi(0)
=0\}  \quad \text{if } b(x)\neq0,
\end{gather*}
so $D(Q(x))$ is not dense in $E$.  
Let $z\in\mathbb{C}\setminus\mathbb{R}_{-}$ and $\psi\in E$,
\[
Q(x)\varphi-z\varphi=\psi\in E,
\]
which implies
\begin{gather*}
\varphi''(y)-z\varphi(y)=\psi(y) \quad y\in(0,1),\\
a(x)\varphi(0)+b(x) \varphi'(0)=0, \quad \varphi(1)=0. 
\end{gather*}
Then we obtain
\[
((Q(x)-zI)^{-1}\psi)(y)=\int_0^{1}K_{\rho}(y,x,s)\psi(s)ds,
\]
where $\rho=\sqrt{z}$
\[
K_{\rho}(y,x,s)=\begin{cases}
\frac{\sinh\rho(1-y)[  a(x)\sinh\rho
s-b(x)\rho\cosh\rho s]  }{\rho[  a(x)
\sinh\rho-b(x)\rho\cosh\rho]  }, & 0\leq s\leq y\\[4pt]
\frac{\sinh\rho(1-s)[  a(x)\sinh\rho
y-b(x)\rho\cosh\rho y]  }{\rho[  a(x)
\sinh\rho-b(x)\rho\cosh\rho]  }, & y\leq s\leq 1.
\end{cases}
\]
We consider $Q(x)=-(-A(x))^{1/2}$.
One has
\[
Q(x)^{-1}=\int_0^{1}K_0(y,x,s)\psi(s)ds,
\]
where
\[
K_0(y,x,s)=\begin{cases}
\frac{(1-y)[  a(x)s+b(x)
]  }{a(x)-b(x)}, & 0\leq s\leq y, \\[4pt]
\frac{(1-s)[  a(x)y-b(x)
]  }{1+b(x)}, & y\leq s\leq 1.
\end{cases}
\]
By direct calculations, one proves \eqref{Hypo1} and \eqref{Hypo2}, see
\cite[p. 52 first example and Proposition 7.1]{AquistapaceTerreni}. 
Then, all our results apply to the following concrete quasi-elliptic
boundary value problem for a large $\omega>0$,
\begin{gather*}
\frac{\partial^2u}{\partial x^2}(x,y)-\frac{\partial
^{4}u}{\partial y^{4}}(x,y)-\omega u(x,y)
=f(x,y),\quad (x,y)\in[  0,1] \times[  0,1]  , \\
a(x)u(x,0)+b(x)\frac{\partial u}{\partial y}(x,0)=0, \quad x\in[  0,1] \\
\frac{\partial u}{\partial x}(0,y)
-\frac{\partial u}{\partial y}(0,y)=d_0(y),\quad 
u(1,y) =u_1(y),\quad y\in[  0,1] \\
a(x)\frac{\partial^2u}{\partial y^2}(x,0)
-b(x)\frac{\partial^{3}u}{\partial y^{3}}(x,0)
=0,\quad x\in[  0,1] \\
\frac{\partial^2u}{\partial y^2}(x,1)=u(x,1) =0,\quad x\in[  0,1] \\
u(0,y)=\varphi(y),\quad 
u(1,y) =\psi(y),\quad y\in[  0,1]  .
\end{gather*}


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