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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 56, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}


\begin{document}
\title[\hfilneg EJDE-2007/56\hfil Boundary-value problems]
{Boundary-value problems for second-order differential operators 
with nonlocal boundary conditions}

\author[M. Denche, A. Meziani\hfil EJDE-2007/56\hfilneg]
{Mohamed Denche, Abderrahmane Meziani}

\address{Mohamed Denche \newline
Laboratoire Equations Differentielles \\
Departement de Mathematiques \\
Faculte des Sciences, Universite Mentouri \\
25000 Constantine, Algeria}
\email{denech@wissal.dz}

\address{Abderrahmane Meziani \newline
Laboratoire Equations Differentielles \\
Departement de Mathematiques \\
Faculte des Sciences, Universite Mentouri \\
25000 Constantine, Algeria}
\email{mezianiar@yahoo.fr}

\thanks{Submitted May 10, 2006. Published April 17, 2007.}
\subjclass[2000]{47E05, 35K20}
\keywords{Green's function; regular and non regular boundary conditions;
\hfill\break\indent  semi group with singularities; weighted mixed boundary
conditions}

\begin{abstract}
 In this paper, we study a second-order differential operator combining
 weighting integral boundary condition with another two-point boundary
 condition. Under certain conditions on the weighting functions, called
 regular and non regular cases, we prove that the resolvent decreases with
 respect to the spectral parameter in $L^{p}(0,1)$, but there is no maximal
 decrease at infinity for $p>1$. Furthermore, the studied operator generates
 in $L^{p}(0,1) $, an analytic semi group for $p=1$ in the regular case, and
 an analytic semi group with singularities for $p>1$, in both cases, and for 
 $p=1$, in the non regular case only. The obtained results are then used to
 show the correct solvability of a mixed problem for parabolic partial
 differential equation with non regular boundary conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section} 
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition} 
\newtheorem{remark}[theorem]{Remark} 
\allowdisplaybreaks

\section{Introduction}

In space $L^{p}(0,1) $ we consider the boundary-value problem
\begin{equation}
\begin{gathered} L(u):=u''=f(x), \\ 
\begin{aligned}
B_{i}(u)&:=a_{i}u(0)+b_{i}u'(0)+c_{i}u(1)+d_{i}u'(1) \\ 
&\quad
+\int_{0}^{1}R_{i}(t)u(t)dt +\int_{0}^{1}S_{i}(t)u'(t)dt=0, \end{aligned}
\end{gathered}  \label{eq1}
\end{equation}
where $i=\overline{1,2}$ and the functions $R_{i},S_{i}$ belong to 
$C([0,1],\mathbb{C})$. To problem \eqref{eq1} in $L^{p}(0,1)$ we associate the
operator
\begin{equation*}
L_{p}(u)=u'',
\end{equation*}
with domain $D(L_{p})=\{u\in W^{2,p}(0,1) :B_{i}(u)=0,i=\overline{1,2}\}$.

Many papers and books give the full spectral theory of Birkhoff regular
differential operators with two point linearly independent boundary
conditions, in terms of coefficients of boundary conditions. The reader
should refer to \cite{dun, gal5, naim, ras1, ras2, tam, yak2, yak3} and
references therein. Few works were devoted to the study of a non regular
situation. The case of separated non regular boundary conditions was studied
by, Eberhard, Hopkins, Jakson, Keldysh, Khromov, Seifert, Stone, Ward (see
Yakubov and Yakubov \cite{yak3} for exact references). A situation of non
regular non-separated boundary conditions was considered by Benzinger
 \cite{ben}, Denche \cite{den1, denk1, denk2}, Eberhard and Freiling \cite{eber},
Gasumov and Magerramov \cite{gas1, gas2}, Khromov \cite{khr}, Mamedov 
\cite{mam}, Shkalikov \cite{shk1}, Silchenko \cite{sil4}, Tretter \cite{tret},
Vagabov \cite{vag}, Yakubov \cite{yak5} and Yakubov \cite{yak6}.

A mathematical model with integral boundary conditions was derived by 
\cite{far, ros} in the context of optical physics. The importance of 
this kind of
problems has been also pointed out by Samarskii \cite{sam}.

In this paper, we study a problem for second order ordinary differential
equation with mixed nonlocal boundary conditions combined weighting integral
boundary conditions with another two point boundary conditions. The regular
case was studied in the space $L_{1}(0,1) $ by Gallardo \cite{gal7}. The
Particular case where $S_{i}(t)=0$, and nonregular boundary conditions is
studied by Silchenko \cite{sil4}. A situation of a variable coefficient of 
$u''(x)$ in the equation has been treated in \cite{galak, skub}.
The integral boundary conditions are again non regular but they assume less
restrictions on the functions $R_{i}(t)$ (here again $S_{i}(t) =0$, 
$a_{i}=b_{i}=c_{i}=d_{i}=0$). In particular the corresponding estimate in 
$L_{2}(0,1) $ has been established.

Following the technique in 
\cite{denk1, denk2, gal6, gal7, naim, ras1, ras2}, we should bound the 
resolvent in the space $L^{p}(0,1) $ by means of a
suitable formula for Green's function. Under certain conditions on the
weighting functions and on the coefficients in the boundary conditions,
called non regular boundary conditions, we prove that the resolvent
decreases with respect to the spectral parameter in $L^{p}(0,1) $, but there
is no maximal decreasing at infinity for $p\geq 1$. In contrast to the
regular case this decreasing is maximal for $p=1$ as shown in \cite{gal6,
gal7}. Furthermore, the studied operator generates in$\ L^{p}(0,1) $ an
analytic semi group with singularities for $p\geq 1$. The obtained results
are then used to show the correct solvability of a mixed problem for a
parabolic partial differential equation with non regular non local boundary
conditions.

\section{Green's Function}

Let $\lambda \in \mathbb{C}$, $u_{1}(x)=u_{1}(x,\lambda )$ and 
$u_{2}(x)=u_{2}(x,\lambda )$ be a fundamental system of solutions of equation
\begin{equation*}
L(u)-\lambda u=0.
\end{equation*}
Following \cite{naim} , the Green's function of problem (\ref{eq1}) is
\begin{equation}
G(x,s,\lambda )=\frac{N(x,s,\lambda )}{\Delta (\lambda )},
\label{eq2}
\end{equation}
where $\Delta (\lambda )$ is the characteristic determinant of the
considered problem defined by
\begin{equation}
\Delta (\lambda )=\left|
\begin{matrix}
B_{1}(u_{1}) & B_{1}(u_{2}) \\
B_{2}(u_{1}) & B_{2}(u_{2})
\end{matrix}
\right| ,  \label{eq3}
\end{equation}
and
\begin{equation}
N(x,s,\lambda )=\left|
\begin{matrix}
u_{1}(x) & u_{2}(x) & g(x,s,\lambda) \\
B_{1}(u_{1}) & B_{1}(u_{2}) & B_{1}(g)_{x} \\
B_{2}(u_{1}) & B_{2}(u_{2}) & B_{2}(g)_{x}
\end{matrix}
\right| ,  \label{eq4}
\end{equation}
for $x,s\in [0,1]$. The function $g(x,s,\lambda )$ is defined as follows
\begin{equation}
g(x,s,\lambda )=\pm \frac{1}{2}\frac{u_{2}(s) u_{1}(x)-u_{2}(x)u_{1}(s)}{
u_{1}'(s)u_{2}(s)-u_{1}(s) u_{2}'(s)},  \label{eq5}
\end{equation}
where it takes the plus sign for $x>s$ and the minus for $x<s$. Given an
arbitrary $\delta \in (0,\frac{\pi }{2})$, we consider the sector
\begin{equation*}
\sum\nolimits_{\delta }{}=\{\lambda \in \mathbb{C}; |\arg (\lambda )| \leq
\frac{\pi }{2} +\delta ,\lambda \neq 0\}.
\end{equation*}
For $\lambda \in \sum_{\delta }$, define $\rho $ as the square root of $%
\lambda $ with positive real part. For $\lambda \neq 0$, we can consider a
fundamental system of solutions of equation $u''=\lambda
u=\rho^2u$ given by $u_{1}(t)=e^{-\rho t}$ and $u_{2}(t)=e^{\rho t}$. In the
following we are going to deduce an adequate formulae for $\Delta (\lambda )$
and $G(x,s,\lambda )$. First of all, for $i,j=\overline{1,2}$, we have
\begin{align*}
B_{i}(u_{j})&=a_{i}+(-1)^{j}\rho b_{i}+c_{i}e^{(-1)^{j}\rho }+(-1)^{j}\rho
d_{i}e^{(-1)^{j}\rho } \\
&\quad+\int_{0}^{1}(R_{i}(t)+(-1)^{j}\rho S_{i}(t))e^{(-1)^{j}\rho t}dt.
\end{align*}
So we obtain from (\ref{eq3}),
\begin{align}
\Delta (\lambda )&=(a_{1}-\rho b_{1}+c_{1}e^{-\rho }-\rho d_{1}e^{-\rho
}+\int_{0}^{1}(R_{1}(t)-\rho S_{1}(t))e^{-\rho t}dt)  \notag \\
&\quad \times (a_{2}+\rho b_{2}+c_{2}e^{\rho }+\rho d_{2}e^{\rho
}+\int_{0}^{1}(R_{2}(t)+\rho S_{2}(t)) e^{\rho t}dt)  \notag \\
&\quad -(a_{2}-\rho b_{2}+c_{2}e^{-\rho }-\rho d_{2}e^{-\rho
}+\int_{0}^{1}(R_{2}(t)-\rho S_{2}(t)) e^{-\rho t}dt)  \notag \\
&\quad \times (a_{1}+\rho b_{1}+c_{1}e^{\rho }+\rho d_{1}e^{\rho
}+\int_{0}^{1}(R_{1}(t)+\rho S_{1}(t)) e^{\rho t}dt),  \label{eq6}
\end{align}
and $g(x,s,\lambda )$ has the form
\begin{equation*}
g(x,s,\lambda )=
\begin{cases}
\frac{1}{4\rho }(e^{\rho (x-s)}-e^{\rho (s-x)}) & \text{if }x>s, \\
\frac{1}{4\rho }(e^{\rho (s-x)}-e^{\rho (x-s)}) & \text{if }x<s.
\end{cases}%
\end{equation*}
Thus we have
\begin{align*}
B_{i}(g)_{x}&=\frac{e^{\rho s}}{4\rho }\Big[a_{i}-\rho b_{i}-c_{i}e^{-\rho
}+\rho d_{i}e^{-\rho } \\
&\quad +\int_{0}^{s}(R_{i}(t)-\rho S_{i}(t) )e^{-\rho t}dt
+\int_{s}^{1}(-R_{i}(t)+\rho S_{i}(t))e^{-\rho t}dt] \\
&\quad +\frac{e^{-\rho s}}{4\rho }[-a_{i}-\rho b_{i}+c_{i}e^{\rho }+\rho
d_{i}e^{\rho } \\
&\quad -\int_{0}^{s}(R_{i}(t)+\rho S_{i}(t) )e^{\rho t}dt
+\int_{s}^{1}(R_{i}(t)+\rho S_{i}(t))e^{\rho t}dt\Big],
\end{align*}
where $i=\overline{1,2}$. For $x,y\in \{a_{i},b_{i},c_{i},d_{i}\}$ and 
$F,G\in \{ R,S\}$, we introduce $\Delta _{xy}=x_{1}y_{2}-x_{2}y_{1},\Delta
_{xF}(t)=x_{1}F_{2}(t)-x_{2}F_{1}(t) ,\Delta _{F}(t,\xi )=F_{1}(t)F_{2}(\xi
)-F_{2}(t)F_{1}(\xi )$, and $\Delta _{FG}(t,\xi )=F_{1}(t)G_{2}(\xi )
-F_{2}(t)G_{1}(\xi )$. After a long calculation, formula (\ref{eq4}) can be
written as
\begin{equation}
N(x,s,\lambda )=\varphi (x,s;\lambda )+\varphi _{i}(x,s,\lambda ),
\label{eq7}
\end{equation}
where
\begin{align}
&\varphi (x,s,\lambda )  \notag \\
& =\frac{1}{2\rho }\Big[e^{\rho (x+s)}\{(\Delta _{ac}-\rho (\Delta
_{ad}+\Delta _{bc})+\rho ^2 \Delta _{bd})e^{-\rho }  \notag \\
&\quad -\int_{0}^{s}(\Delta _{cR}(t)-\rho (\Delta _{dR}(t)+\Delta
_{cS}(t))+\rho^2 \Delta _{dS}(t))e^{-\rho (t+1)}dt  \notag \\
&\quad +\int_{s}^{1}(\Delta _{aR}(t)-\rho (\Delta _{aS}(t)+\Delta
_{bR}(t))+\rho^2 \Delta _{bS}(t))e^{-\rho t}dt  \notag \\
&\quad +\int_{s}^{1}\int_{0}^{s}(\Delta _{R}(\xi ,t) +\rho (\Delta
_{RS}(t,\xi )-\Delta _{RS}(\xi ,t))+\rho^2 \Delta _{S}(\xi ,t))e^{-\rho (\xi
+t)}d\xi dt\}  \notag \\
&\quad +e^{-\rho (x+s)}\{(\Delta _{ac}+\rho ( \Delta _{ad}+\Delta
_{bc})+\rho^2 \Delta _{bd})e^{\rho }  \notag \\
&\quad -\int_{0}^{s}(\Delta _{cR}(t)+\rho (\Delta _{dR}(t)+\Delta
_{cS}(t))+\rho^2 \Delta _{dS}(t))e^{\rho (t+1)}dt  \notag \\
&\quad +\int_{s}^{1}(\Delta _{aR}(t)+\rho (\Delta _{bR}(t)+\Delta
_{aS}(t))+\rho^2 \Delta _{bS}(t))e^{\rho t}dt  \notag \\
&\quad +\int_{s}^{1}\int_{0}^{s} (\Delta _{R}(\xi ,t)+\rho (\Delta _{RS}(\xi
,t)-\Delta _{RS}( t,\xi ))+\rho^2 \Delta _{S}(\xi ,t))e^{\rho (\xi +t)}d\xi
dt\}\Big],  \label{eq8}
\end{align}
and the function $\varphi _{i}(x,s,\lambda )$ is given by
\begin{equation}
\varphi _{i}(x,s,\lambda )=%
\begin{cases}
\varphi _{1}(x,s,\lambda ) & \text{if }x>s, \\
\varphi _{2}(x,s,\lambda ) & \text{if }x<s,
\end{cases}
\label{eq9}
\end{equation}
with
\begin{align}
&\varphi _{1}(x,s,\lambda )  \notag \\
&=\frac{1}{2\rho }\Big[e^{\rho (x-s)}\{(-\Delta _{ac}+\rho (\Delta
_{ad}-\Delta _{bc})+\rho^2\Delta _{bd})e^{-\rho }  \notag \\
&\quad +2\rho \Delta _{ab}+\int_{0}^{s}(\Delta _{aR}(t)+\rho (\Delta
_{aS}(t)+\Delta _{bR}(t))-\rho^2\Delta _{bS}(t))e^{\rho t}dt  \notag \\
&\quad +\int_{0}^{s}(\Delta _{cR}(t)+\rho (\Delta _{cS}(t)-\Delta
_{dR}(t))-\rho^2\Delta _{dS}(t))e^{\rho (t-1)}dt  \notag \\
&\quad -\int_{0}^{1}(\Delta _{aR}(t)+\rho (\Delta _{bR}(t)-\Delta
_{aS}(t))-\rho^2\Delta _{bS}(t))e^{-\rho t}dt  \notag \\
&\quad +\int_{0}^{1}\int_{0}^{s}(\Delta _{R}(t,\xi )+\rho (\Delta _{RS}(\xi
,t)-\Delta _{RS}(t,\xi ) )-\rho^2\Delta _{S}(t,\xi ))e^{\rho (\xi -t)}d\xi
dt\}  \notag \\
&\quad +e^{\rho (s-x)}\{(-\Delta _{ac}+\rho (\Delta _{bc}-\Delta
_{ad})+\rho^2\Delta _{bd})e^{\rho }-2\rho \Delta _{ab}  \notag \\
&\quad +\int_{0}^{s}(\Delta _{aR}(t)+\rho (\Delta _{bR}(t)-\Delta
_{aS}(t))-\rho^2\Delta _{bS}(t))e^{-\rho t}dt  \notag \\
&\quad +\int_{0}^{s}(\Delta _{cR}(t)+\rho (\Delta _{dR}(t)-\Delta
_{cS}(t))-\rho^2\Delta _{dS}(t))e^{\rho (1-t)}dt  \notag \\
&\quad -\int_{0}^{1}(\Delta _{aR}(t)+\rho (\Delta _{aS}(t)-\Delta
_{bR}(t))-\rho^2\Delta _{bS}(t))e^{\rho t}dt  \notag \\
&\quad +\int_{0}^{1}\int_{0}^{s} (\Delta _{R}(t,\xi )-\rho (\Delta _{RS}(\xi
,t)+\Delta _{RS}( t,\xi ))-\rho^2\Delta _{S}(t,\xi ))e^{\rho (t-\xi )}d\xi
dt\}\Big],  \label{eq10}
\end{align}
and
\begin{align}
&\varphi _{2}(x,s,\lambda )  \notag \\
&=\frac{1}{2\rho }\Big[e^{\rho (x-s)}\{(-\Delta _{ac}+\rho (\Delta
_{bc}-\Delta _{ad})+\rho^2 \Delta _{bd})e^{\rho }-2\rho \Delta _{ad}  \notag
\\
&\quad -\int_{s}^{1}(\Delta _{aR}(t)+\rho (\Delta _{aS}(t)-\Delta
_{bR}(t))-\rho^2 \Delta _{bS}(t))e^{\rho t}dt  \notag \\
&\quad -\int_{s}^{1}(\Delta _{cR}(t)+\rho (\Delta _{cS}(t)-\Delta
_{dR}(t))-\rho^2 \Delta _{dS}(t))e^{\rho (t-1)}dt  \notag \\
&\quad +\int_{0}^{1}(\Delta _{cR}(t)+\rho (\Delta _{dR}(t)-\Delta
_{cS}(t))-\rho^2 \Delta _{dS}(t))e^{\rho (1-t)}dt  \notag \\
&\quad +\int_{0}^{1}\int_{s}^{1}(\Delta _{R}(\xi ,t)-\rho (\Delta
_{RS}(t,\xi )+\Delta _{RS}(\xi ,t) )-\rho^2 \Delta _{S}(\xi ,t))e^{\rho (\xi
-t)}d\xi dt\}  \notag \\
&\quad +e^{\rho (s-x)}\{(-\Delta _{ac}+\rho (\Delta _{ad}-\Delta
_{bc})+\rho^2 \Delta _{bd})e^{-\rho }+2\rho \Delta _{cd}  \notag \\
&\quad -\int_{s}^{1}(\Delta _{aR}(t)+\rho (\Delta _{bR}(t)-\Delta
_{aS}(t))-\rho^2 \Delta _{bS}(t))e^{-\rho t}dt  \notag \\
&\quad -\int_{s}^{1}(\Delta _{cR}(t)+\rho (\Delta _{dR}(t)-\Delta
_{cS}(t))-\rho^2 \Delta _{dS}(t))e^{\rho (1-t)}dt  \notag \\
&\quad +\int_{0}^{1}(\Delta _{cR}(t)+\rho (\Delta _{cS}(t)-\Delta
_{dR}(t))-\rho^2 \Delta _{dS}(t))e^{\rho (t-1)}dt  \notag \\
&\quad +\int_{0}^{1}\int_{s}^{1} (\Delta _{R}(\xi ,t)+\rho (\Delta
_{RS}(t,\xi )+\Delta _{RS}(\xi ,t))-\rho^2 \Delta _{S}(\xi ,t))e^{\rho
(t-\xi )}d\xi dt\}\Big]  \label{eq11}
\end{align}

\section{Bounds on the Resolvent}

Every $\lambda \in\mathbb{C}$ such that $\Delta (\lambda )\neq 0$ belongs to
$\rho (L_{p})$, and the associated resolvent operator $R(\lambda,L_{p})$ can
be expressed as a Hilbert Schmidt operator
\begin{equation}
(\lambda I-L_{p})^{-1}f=R(\lambda ;L_{p}) f=-\int_{0}^{1}G(.,s;\lambda
)f(s)ds,\text{ }f\in L^{p}(0,1) .  \label{eq12}
\end{equation}
Then, for every $f\in L^{p}(0,1) $ we estimate (\ref{eq12}),
\begin{equation*}
\|R(\lambda ;L_{p})f\|_{L_{p}(0,1) }\leq \Big(\sup_{0\leq s\leq 1}
\int_{0}^{1}|G(x,s;\lambda )| ^{p}dx\Big)^{1/p}\|f\|_{L_{p}(0,1)},
\end{equation*}
and so we need to bound
\begin{equation}
(\sup_{0\leq s\leq 1} \int_{0}^{1}|G( x,s;\lambda )| ^{p}dx)^{1/p}
=\frac{1 }{|\Delta (\lambda )| }(\sup_{0\leq s\leq 1} \int_{0}^{1}|N(x,s; 
\lambda )|^{p}dx)^{1/p}.  \label{eq13}
\end{equation}

\subsection{Estimation of N$(x,s,\protect\lambda)$}

We will denote by $\|\cdot\|$ the supremum norm for functions in one and two
variables. Since
\begin{equation*}
N(x,s,\lambda )=
\begin{cases}
\varphi (x,s,\lambda )+\varphi _{1}(x,s,\lambda ) & \text{if }x>s, \\
\varphi (x,s,\lambda )+\varphi _{2}(x,s,\lambda ) & \text{if } x<s,
\end{cases}
\end{equation*}
it follows that
\begin{equation*}
|N(x,s,\lambda )| ^{p}\leq 2^{p-1}( |\varphi (x,s,\lambda )| ^{p}+| \varphi
_{i}(x,s,\lambda )| ^{p}).
\end{equation*}
Form (\ref{eq9}), we have
\begin{equation}  \label{eq13'}
\begin{aligned} &\int_{0}^{1}|N(x,s,\lambda )| ^{p}dx\\ &\leq
2^{p-1}\Big(\int_{0}^{1}|\varphi (x,s,\lambda ) |
^{p}dx+\int_{s}^{1}|\varphi _{1}( x,s,\lambda )| ^{p}dx +
\int_{0}^{s}|\varphi _{1}(x,s,\lambda )| ^{p}dx\Big). \end{aligned}
\end{equation}
From (\ref{eq8}), we have
\begin{align*}
&\int_{0}^{1}|\varphi (x,s,\lambda )|^{p}dx \\
&\leq \frac{2^{2p-2}}{2p|\rho | ^{p}\mathop{\rm Re}(\rho )}\Big[(|\rho |^2
|\Delta _{bd}| +|\rho | ( |\Delta _{ad}| +|\Delta _{bc}| )+|\Delta
_{ac}|)^{p} \\
&\quad \times \Big((e^{ps\mathop{\rm Re}(\rho )}-e^{p(s-1)\mathop{\rm Re}
(\rho )})+(e^{p(1-s)\mathop{\rm Re}(\rho )}-e^{-ps\mathop{\rm Re}( \rho )})
\Big) \\
&\quad +\Big(\frac{1}{\mathop{\rm Re}(\rho )} (|\rho |^2 \|\Delta _{bS}\|
+|\rho | (\|\Delta _{aS}\|+\| \Delta _{bR}\|) +\|\Delta _{aR}\|)\Big)^{p} \\
&\quad \times ((e^{p\mathop{\rm Re}(\rho )}-1)\times (1-e^{( s-1)\mathop{\rm
Re}(\rho )})^{p} +(1-e^{-p\mathop{\rm Re}(\rho )})\times (e^{(1-s) 
\mathop{\rm Re}(\rho )}-1)^{p}) \\
&\quad +\Big(\frac{1}{\mathop{\rm Re}(\rho )} (|\rho |^2\|\Delta
_{dS}\|+|\rho | (\|\Delta _{dR}\| + \|\Delta _{cS}\|) +\|\Delta _{cR}\|)\Big)
^{p} \times \big((e^{p\mathop{\rm Re}(\rho ) }-1) \\
&\quad\times (e^{(s-1)\mathop{\rm Re}(\rho )}-e^{-\mathop{\rm Re} (\rho
)})^{p} +(1-e^{-p\mathop{\rm Re}(\rho )})\times (e^{\mathop{\rm Re}(\rho
)}-e^{(1-s)\mathop{\rm Re}(\rho )})^{p}\big) \\
&\quad +\Big(\frac{1}{(\mathop{\rm Re}(\rho ))^{2}} ( \|\Delta _{R}\|
+2|\rho | \|\Delta _{RS}\|+ |\rho |^2 \|\Delta _{S}\|)\Big)^{p} \times ((e^{p
\mathop{\rm Re}(\rho )}-1) \\
&\quad \times (e^{s\mathop{\rm Re}(\rho )}-1)^{p} \times (e^{-s\mathop{\rm
Re}(\rho )}-e^{-\mathop{\rm Re}(\rho ) })^{p}+(1-e^{-p\mathop{\rm Re}(\rho
)}) \\
&\quad \times (e^{\mathop{\rm Re}( \rho )}-e^{s\mathop{\rm Re}(\rho )})^{p}
(1-e^{-s\mathop{\rm Re}(\rho )})^{p})\Big],
\end{align*}
from (\ref{eq10}), we have
\begin{align*}
&\int_{s}^{1}|\varphi _{1}(x,s,\lambda )|^{p}dx \\
&\leq \frac{5^{p-1}}{2p|\rho | ^{p}\mathop{\rm Re}(\rho)}[(|\rho |^2 |\Delta
_{bd}| +|\rho | (|\Delta _{ad}| +|\Delta _{bc}| ) \\
&\quad +|\Delta _{ac}|)^{p}\times ((e^{p\mathop{\rm Re}(\rho ) }-e^{-ps
\mathop{\rm Re}(\rho )})+(e^{-ps\mathop{\rm Re}(\rho )}-e^{-p\mathop{\rm Re}
(\rho )}))+(2|\rho | |\Delta _{ab}| )^{p} \\
&\quad \times \big(e^{p(1-s)\mathop{\rm Re}(\rho )}-e^{p( s-1)\mathop{\rm Re}
(\rho )}\big)+(\frac{1}{\mathop{\rm Re}(\rho )} (|\rho |^2 \|\Delta
_{bS}\|+|\rho | (\| \Delta _{aS}\|+\|\Delta _{bR}\|) \\
&\quad + \|\Delta _{aR}\|))^{p}\times ((e^{-ps\mathop{\rm Re}(\rho )}-e^{-p
\mathop{\rm Re}(\rho )})\times (e^{(1+s)\mathop{\rm Re}(\rho )}-1)^{p} +(e^{p
\mathop{\rm Re}(\rho )} \\
&\quad -e^{ps\mathop{\rm Re}(\rho )}) \times (1-e^{-(1+s) \mathop{\rm Re}
(\rho )})^{p})+(\frac{1}{\mathop{\rm Re}(\rho )} (|\rho |^2 \|\Delta
_{dS}\|+|\rho | ( \|\Delta _{dR}\| \\
&\quad + \|\Delta _{cS}\|)+ \| \Delta _{cR}\|))^{p}\times ((e^{-ps
\mathop{\rm Re}( \rho )}-e^{-p\mathop{\rm Re}(\rho )})\times (e^{p( s+1)
\mathop{\rm Re}(\rho )}-e^{p\mathop{\rm Re}(\rho )})^{p} \\
&\quad +(e^{p\mathop{\rm Re}(\rho )}-e^{ps\mathop{\rm Re}(\rho )})\times
(e^{-\mathop{\rm Re}(\rho )}-e^{-(1+s)\mathop{\rm Re}(\rho )})^{p})+(\frac{1
}{(\mathop{\rm Re}(\rho ) )^{2}} (\|\Delta _{R}\| \\
&\quad +2|\rho | \|\Delta _{RS}\| +|\rho |^2 \|\Delta _{S}\|))^{p}\times
((e^{-ps \mathop{\rm Re}(\rho )}-e^{-p\mathop{\rm Re}(\rho )})\times (e^{
\mathop{\rm Re} (\rho )}-1)^{p} \\
&\quad \times (e^{s\mathop{\rm Re}(\rho )}-1)^{p} +(e^{p \mathop{\rm Re}
(\rho )}-e^{ps\mathop{\rm Re}(\rho )})\times (1-e^{- \mathop{\rm Re}(\rho
)})^{p}\times (1-e^{-s\mathop{\rm Re}(\rho)})^{p})\Big].
\end{align*}
From (\ref{eq11}), we have
\begin{align*}
&\int_{0}^{s}|\varphi _{2}(x,s,\lambda )|^{p}dx \\
&\leq \frac{5^{p-1}}{2p|\rho | ^{p}\mathop{\rm Re}(\rho)}\Big[(|\rho |^2
|\Delta _{bd}| +|\rho | (|\Delta _{ad}| +|\Delta _{bc}|) \\
&\quad +|\Delta _{ac}|)^{p}\times ((e^{p\mathop{\rm Re}(\rho )}-e^{p(1-s)
\mathop{\rm Re}(\rho )})+( e^{p(s-1)\mathop{\rm Re}(\rho )}-e^{-p\mathop{\rm
Re}(\rho )})) \\
&\quad +(2|\rho | |\Delta _{cd}| )^{p}\times (e^{ps\mathop{\rm Re}(\rho
)}-e^{-ps\mathop{\rm Re}(\rho )})+(\frac{1}{\mathop{\rm Re}(\rho )}(|\rho
|^2 \|\Delta _{bS}\|+|\rho | (\|\Delta_{aS}\| \\
&\quad +\|\Delta _{bR}\|)+\|\Delta _{aR}\|))^{p}\times ((e^{ps\mathop{\rm Re}
(\rho )}-1)\times (e^{(1-s)\mathop{\rm Re}(\rho )}-1)^{p} \\
&\quad +(1-e^{-ps\mathop{\rm Re}(\rho )}) \times ( 1-e^{(s-1)\mathop{\rm Re}
(\rho )})^{p})+(\frac{1}{\mathop{\rm Re}(\rho )}(|\rho |^2 \|\Delta
_{dS}\|+|\rho |(\|\Delta _{dR}\| \\
&\quad + \|\Delta _{cS}\|)+\|\Delta _{cR}\|))^{p}\times ((e^{ps\mathop{\rm
Re}(\rho )}-1)\times (e^{(1-s)\mathop{\rm Re}(\rho ) }-e^{-\mathop{\rm Re}
(\rho )}) ^{p} \\
&\quad + (1-e^{-ps\mathop{\rm Re}(\rho )})\times (e^{\mathop{\rm Re} (\rho
)}-e^{(s-1)\mathop{\rm Re}(\rho )}) ^{p})+(\frac{1}{(\mathop{\rm Re}(\rho
))^{2}} (\|\Delta _{R}\|+2|\rho | \| \Delta _{RS}\| \\
&\quad +|\rho |^2 \|\Delta _{S}\|))^{p}\times (( e^{ps\mathop{\rm Re}(\rho
)}-1)\times (e^{(1-s)\mathop{\rm Re} (\rho )}-1)^{p}\times (1-e^{-
\mathop{\rm Re}(\rho)})^{p} \\
&\quad +(1-e^{-ps\mathop{\rm Re}(\rho )})\times ( e^{\mathop{\rm Re}(\rho
)}-1)^{p}\times (1-e^{( s-1)\mathop{\rm Re}(\rho )})^{p})\Big].
\end{align*}
So that
\begin{align*}
&\int_{0}^{1}|N(x,s,\lambda )| ^{p}dx \\
&\leq \frac{2^{p-1}\times 5^{p-1}}{2p|\rho | ^{p}\mathop{\rm Re}( \rho )}
[(|\rho |^2|\Delta _{bd}| +|\rho | ( |\Delta _{ad}| +|\Delta _{bc}|)+|\Delta
_{ac}|)^{p} \\
&\quad \times 2(e^{p\mathop{\rm Re}(\rho )}-e^{-p\mathop{\rm Re}(\rho )})+(
2|\rho | |\Delta _{ab}| )^{p}\times (e^{p(1-s)\mathop{\rm Re}(\rho )}-e^{p(
s-1)\mathop{\rm Re}(\rho )}) \\
&\quad +(2|\rho | |\Delta _{cd}| )^{p}\times (e^{ps\mathop{\rm Re}(\rho
)}-e^{-ps\mathop{\rm Re}(\rho )})+(\frac{1}{\mathop{\rm Re}(\rho )}(|\rho
|^2 \|\Delta _{bS}\| +|\rho | (\|\Delta_{aS}\| \\
&\quad +\|\Delta _{bR}\|)+\|\Delta _{aR}\|))^{p}\times ((e^{p\mathop{\rm Re}
(\rho ) }-1)\times (1-e^{(s-1)\mathop{\rm Re}(\rho )})^{p} \\
&\quad +(1-e^{-p\mathop{\rm Re}(\rho )}) (e^{(1-s)\mathop{\rm Re} (\rho
)}-1)^{p}+(e^{-ps\mathop{\rm Re}(\rho )}-e^{-p\mathop{\rm Re} (\rho
)})\times (e^{(1+s)\mathop{\rm Re}(\rho )}-1)^{p} \\
&\quad +(e^{p\mathop{\rm Re}(\rho )}-e^{ps\mathop{\rm Re}(\rho )}) (
1-e^{-(1+s)\mathop{\rm Re}(\rho )})^{p}+(e^{ps\mathop{\rm Re} (\rho
)}-1)\times (e^{(1-s)\mathop{\rm Re}( \rho )}-1)^{p} \\
&\quad +(1-e^{-ps\mathop{\rm Re}(\rho )}) ( 1-e^{(s-1)\mathop{\rm Re}(\rho
)})^{p})+( \frac{1}{\mathop{\rm Re}(\rho )}(|\rho |^2 \|\Delta _{dS}\|+|\rho
| (\| \Delta _{dR}\| \\
&\quad + \|\Delta _{cS}\|)+\|\Delta _{cR}\|))^{p}((e^{p\mathop{\rm Re}(\rho
) }-1)\times (e^{(s-1)\mathop{\rm Re}(\rho )}-e^{-\mathop{\rm Re} (\rho
)})^{p}+(1-e^{-p\mathop{\rm Re}(\rho )}) \\
&\quad \times (e^{\mathop{\rm Re}(\rho )}-e^{(1-s)\mathop{\rm Re}( \rho
)})^{p}+(e^{-ps\mathop{\rm Re}(\rho )}-e^{-p\mathop{\rm Re}( \rho )})\times
(e^{p(s+1)\mathop{\rm Re}(\rho ) }-e^{p\mathop{\rm Re}(\rho )})^{p} \\
&\quad +(e^{p\mathop{\rm Re}(\rho )}-e^{ps\mathop{\rm Re}(\rho )})\times (
e^{-\mathop{\rm Re}(\rho )}-e^{-(1+s)\mathop{\rm Re}(\rho ) })^{p}+(e^{ps
\mathop{\rm Re}(\rho )}-1) \\
&\quad \times (e^{(1-s)\mathop{\rm Re}(\rho )}-e^{-\mathop{\rm Re}( \rho
)})^{p}+ (1-e^{-ps\mathop{\rm Re}(\rho ) })\times (e^{\mathop{\rm Re}(\rho
)}-e^{(s-1)\mathop{\rm Re} (\rho )})^{p}) \\
&\quad +(\frac{1}{(\mathop{\rm Re}(\rho ))^{2}}(\| \Delta _{R}\|+2|\rho |
\|\Delta _{RS}\|+|\rho |^2 \|\Delta _{S}\|))^{p}\times ((e^{p\mathop{\rm Re}
( \rho )}-1) \\
&\quad \times (e^{s\mathop{\rm Re}(\rho )}-1)^{p}\times (e^{-s \mathop{\rm
Re}(\rho )}-e^{-\mathop{\rm Re}(\rho )})^{p}+(1-e^{-p\mathop{\rm Re} (\rho
)})\times (e^{\mathop{\rm Re}(\rho )}-e^{s\mathop{\rm Re} (\rho )})^{p} \\
&\quad \times (1-e^{-s\mathop{\rm Re}(\rho )})^{p}+(e^{-ps\mathop{\rm Re}(
\rho )}-e^{-p\mathop{\rm Re}(\rho )})\times (e^{\mathop{\rm Re}(\rho
)}-1)^{p}\times (e^{s\mathop{\rm Re}(\rho )}-1)^{p} \\
&\quad +(e^{p\mathop{\rm Re}(\rho )}-e^{ps\mathop{\rm Re}(\rho )})\times (
1-e^{-\mathop{\rm Re}(\rho )})^{p}\times (1-e^{-s\mathop{\rm Re}( \rho
)})^{p}+(e^{ps\mathop{\rm Re}(\rho )}-1) \\
&\quad \times (e^{(1-s)\mathop{\rm Re}(\rho )}-1) ^{p}\times (1-e^{-%
\mathop{\rm Re}(\rho )})^{p}+(1-e^{-ps \mathop{\rm Re}(\rho )})\times (e^{
\mathop{\rm Re}(\rho ) }-1)^{p} \\
&\quad \times (1-e^{(s-1)\mathop{\rm Re}(\rho ) })^{p})\Big].
\end{align*}
Since $\mathop{\rm Re}(\rho )>0$, we obtain
\begin{align}
&\sup_{0\leq s \leq 1} \Big(\int_{0}^{1}|N( x,s,\lambda )| ^{p}dx\Big)^{1/p}
\notag \\
&\leq \frac{2^{(2-\frac{2}{p})}\times 5^{(1-\frac{1}{p})}}{p^{ \frac{1}{p}
}|\rho | (\mathop{\rm Re}(\rho ) )^{1/p}}e^{\mathop{\rm Re}(\rho )}\Big[
|\rho |^2|\Delta _{bd}|  \notag \\
&\quad +|\rho | (|\Delta _{ad}| +|\Delta _{bc}| +|\Delta _{cd}| )+|\Delta
_{ac}|+\frac{3}{\mathop{\rm Re}(\rho )}(|\rho | ^2\|\Delta _{bS}\|+|\rho |
(\|\Delta_{aS}\|  \notag \\
&\quad +\|\Delta _{bR}\|)+\|\Delta _{aR}\|)+ \frac{3}{\mathop{\rm Re}(\rho )}
(|\rho |^2 \|\Delta _{dS}\|+|\rho | (\|\Delta _{dR}\|+\|\Delta _{cS}\|)
\notag \\
&\quad +\|\Delta _{cR}\|)+ \frac{3}{(\mathop{\rm Re}(\rho ))^{2}}(\|\Delta
_{R}\|+2|\rho | \|\Delta _{RS}\|+|\rho |^2\|\Delta _{S}\|)\Big].
\label{eq17}
\end{align}
From the above inequality, (\ref{eq13}) and (\ref{eq13'}), we obtain
\begin{align*}
\|R(\lambda ,L_{p})\| & \leq \frac{2^{(2-\frac{2}{p})}\times 5^{(1-\frac{1}{p
})}}{| \Delta (\rho^2)| |\rho | (\mathop{\rm Re}(\rho ))^{1/p}p^{1/p}}
e^{\mathop{\rm Re}(\rho )} \Big[(|\rho |^2|\Delta _{bd}| +|\rho| (|\Delta
_{ad}| +|\Delta_{bc}| \\
&\quad +|\Delta _{ab}|+|\Delta _{cd}|)+|\Delta _{ac}|+\frac{3}{\mathop{\rm
Re} (\rho )}(|\rho |^2 (\|\Delta _{bS}\|+\|\Delta _{dS}\|) \\
&\quad +|\rho | (\|\Delta _{dR}\|+\|\Delta _{cS}\|+\|\Delta _{aS}\|+\|\Delta
_{bR}\|)+\| \Delta _{cR}\|+\|\Delta _{aR}\|) \\
&\quad + \frac{3}{(\mathop{\rm Re}(\rho ))^{2}}( \|\Delta _{R}\|+2|\rho | \|
\Delta _{RS}\|+|\rho |^2 \|\Delta _{S}\|)\Big],
\end{align*}
for $\rho \in \sum_{\frac{\delta }{2}}=\{\rho \in\mathbb{C} :|\arg \rho |
\leq \frac{\pi }{4}+\frac{\delta }{2},\rho \neq 0\}$, we have 
$(\mathop{\rm
Re}(\rho ))^{-1}<\frac{ 1}{|\rho | \cos (\frac{\pi }{4}+\frac{\delta }{2} )}$. 
Then
\begin{align*}
&\|R(\lambda ,L_{p})\| \\
&\leq \frac{2^{(2-\frac{2}{p})}\times 5^{(1-\frac{1}{p})}}{| \Delta
(\rho^2)| |\rho | ^{1+\frac{1}{p}}(\cos (\frac{\pi }{4}+\frac{\delta }{2}
))^{1/p}p^{ \frac{1}{p}}}e^{\mathop{\rm Re}(\rho )}\Big[(|\rho |^2|\Delta
_{bd}| +|\rho | (|\Delta _{ad}| \\
&\quad + |\Delta _{bc}| +|\Delta _{bc}| +|\Delta _{ab}|+|\Delta
_{cd}|)+|\Delta _{ac}|+\frac{3}{ |\rho | \cos (\frac{\pi }{4}
+\frac{\delta }{2} )}(|\rho |^2(\|\Delta _{bS}\|+\|\Delta _{dS}\|) \\
&\quad +|\rho | (\|\Delta _{dR}\|+\|\Delta _{cS}\|+\|\Delta _{aS}\|+\|\Delta
_{bR}\|)+\| \Delta _{cR}\|+\|\Delta _{aR}\|) \\
&\quad + \frac{3}{(|\rho | \cos (\frac{\pi }{4 }+\frac{\delta }{2}))^{2}}
(\|\Delta _{R}\|+2|\rho | \|\Delta_{RS}\|+|\rho |^2 \|\Delta _{S}\|)\Big].
\end{align*}
Finally, we obtain, for $\lambda =\rho^2\in \sum_{\delta }$,
\begin{align}
&\|R(\lambda ,L_{p})\|  \notag \\
&\leq \frac{c}{|\Delta (\rho^2)| |\rho | ^{1+\frac{1}{p}}}e^{\mathop{\rm Re}
(\rho )}\Big[(|\rho |^2|\Delta _{bd}| +|\rho | (|\Delta _{ab}| +|\Delta_{ad}|
\notag \\
&\quad +|\Delta _{bc}|+|\Delta _{cd}|+\|\Delta _{bS}\|+\|\Delta
_{dS}\|)+|\Delta _{ac}|+\|\Delta _{aS}\| +\|\Delta _{bR}\|+\|\Delta _{dR}\|
\notag \\
&\quad +\|\Delta _{cS}\|+\|\Delta _{S}\| + \frac{1}{|\rho | }(\|\Delta
_{aR}\|+\|\Delta _{cR}\|+2\|\Delta_{RS}\|)+\frac{\|\Delta _{R}\|}{ |\rho |^2}
\Big]  \notag \\
&\leq c\frac{H(\rho )}{|\rho | ^{1+\frac{1}{p}}},  \label{eq17'}
\end{align}
where
\begin{align}
H(\rho )&=\frac{e^{\mathop{\rm Re}(\rho )}}{|\Delta (\rho^2)| }\Big[(|\rho
|^2 |\Delta _{bd}| +|\rho| (|\Delta _{ab}| +|\Delta_{ad}|  \notag \\
&\quad +|\Delta _{bc}|+|\Delta _{cd}|+\|\Delta _{bS}\|+\|\Delta
_{dS}\|)+|\Delta _{ac}|+\|\Delta _{aS}\| +\|\Delta _{bR}\|+\|\Delta _{dR}\|
\notag \\
&\quad +\|\Delta _{cS}\|+\|\Delta _{S}\| + \frac{1}{|\rho | }(\|\Delta
_{aR}\|+\|\Delta _{cR}\|+2\|\Delta _{RS}\|)+\frac{\|\Delta _{R}\|}{|\rho |^2}
\Big],  \label{eq18}
\end{align}
and
\begin{equation*}
c=\frac{2^{(2-\frac{2}{p})}\times 5^{(1-\frac{1}{p} )}}{p^{1/p}}\max 
\Big(\frac{1}{(\cos (\frac{ \delta }{2}))^{1/p}},\frac{3}{(\cos 
(\frac{ \delta }{2}))^{1+\frac{1}{p}}},\frac{3}
{(\cos ( \frac{\delta }{2}))^{2+\frac{1}{p}}}
\Big).
\end{equation*}
The following step is to analyze the function $H(\rho )$ in order to
determine the cases for which it is bounded in the sector $\sum_{\delta/2}$.

\subsection{Estimation of the characteristic determinant, regular case}

The next step is to determine the cases for which $|\Delta (\lambda )|$
remains bounded below. It will then be necessary to bound $|\Delta (\lambda
)|$ appropriately. However, formula (\ref{eq6}) is not useful for this
purpose, it will be then necessary to make some additional regularity
hypotheses on the functions $R_{i}$ and $S_{i}$, and so we assume that the
functions $R_{i}$ and $S_{i}$, $i=\overline{1,2}$, are in 
$C^{1}([0,1],\mathbb{C})$. Integrating twice by parts in (\ref{eq6}), we obtain
\begin{equation}
\begin{aligned} \Delta (\lambda )&=e^{\rho }\Big[-\rho^2\Delta _{bd} +\rho
(\Delta _{ad}-\Delta _{bc}+\Delta _{dS}( 0)-\Delta _{bS}(1))+\Delta _{ac} \\
&\quad +\Delta _{aS}(1)+\Delta _{cS}(0)-\Delta _{bR}(1)-\Delta
_{dR}(0)+\Delta _{S}( 1,0)+\frac{1}{\rho }(\Delta _{aR}(1) \\ &\quad -\Delta
_{cR}(0)+\Delta _{RS}(0,1) +\Delta _{RS}(1,0)) +\frac{1}{\rho^2}(\Delta
_{R}(0,1) )+\Phi (\rho ) \Big], \end{aligned}  \label{eq19}
\end{equation}%
where
\begin{align*}
\Phi (\rho )& =2e^{-\rho }[\rho (\Delta _{ab}+\Delta _{cd}+\Delta
_{dS}(0)-\Delta _{bS}(1))+\frac{1}{\rho }(\Delta _{cR}(1) \\
& \quad -\Delta _{aR}(0)-\Delta _{RS}(0,0)-\Delta _{RS}(1,1))] \\
& \quad \times e^{-2\rho }[\rho ^{2}\Delta _{bd}+\rho (\Delta _{ad}-\Delta
_{bc}+\Delta _{dS}(0)-\Delta _{bS}(1))-\Delta _{ac}-\Delta _{aS}(1) \\
& \quad -\Delta _{cS}(0)+\Delta _{bR}(1)+\Delta _{dR}(0)-\Delta _{S}(1,0)+%
\frac{1}{\rho }(\Delta _{aR}(1)-\Delta _{cR}(0) \\
& \quad +\Delta _{RS}(0,1)+\Delta _{RS}(1,0))+\frac{1}{\rho ^{2}}(\Delta
_{R}(1,0))] \\
& \quad -\frac{1}{\rho }[\int_{0}^{1}(\Delta _{aR'}(t)+\rho (\Delta
_{aS'}(t)-\Delta _{bR'}(t))-\rho ^{2}\Delta _{bS^{\prime
}}(t))e^{\rho (t-1)}dt \\
& \quad +\int_{0}^{1}(\Delta _{aR'}(t)+\rho (\Delta _{bR^{\prime
}}(t)-\Delta _{aS'}(t))-\rho ^{2}\Delta _{bS'}(t))e^{-\rho
(t+1)}dt \\
& \quad +\int_{0}^{1}(\Delta _{cR'}(t)+\rho (\Delta _{cS^{\prime
}}(t)-\Delta _{dR'}(t))-\rho ^{2}\Delta _{dS'}(t))e^{\rho
(t-2)}dt \\
& \quad +\int_{0}^{1}(\Delta _{cR'}(t)+\rho (\Delta _{dR^{\prime
}}(t)-\Delta _{cS'}(t))-\rho ^{2}\Delta _{dS'}(t))e^{-\rho
t}dt] \\
& \quad +\frac{1}{\rho ^{2}}[\int_{0}^{1}(\Delta _{R'R}(t,0)-\rho
(\Delta _{R'S}(t,0)+\Delta _{RS'}(0,t))-\rho ^{2}\Delta
_{S'S}(t,0))e^{\rho (t-1)}dt \\
& \quad +\int_{0}^{1}(\Delta _{R'R}(0,t)-\rho (\Delta _{R'S}(0,t)
+\Delta _{RS'}(t,0))+\rho ^{2}\Delta _{S^{\prime
}S}(1,t))e^{-\rho (t+1)}dt \\
& \quad +\int_{0}^{1}(\Delta _{R'R}(1,t)+\rho (\Delta _{R'S}(1,t)
+\Delta _{RS'}(t,1))+\rho ^{2}\Delta _{S^{\prime
}S}(t,1))e^{\rho (t-2)}dt \\
& \quad +\int_{0}^{1}(\Delta _{R'R}(t,1)+\rho (\Delta _{R'S}(t,1)
+\Delta _{RS'}(1,t))+\rho ^{2}\Delta _{S^{\prime
}S}(1,t))e^{-\rho t}dt \\
& \quad +\int_{0}^{1}\int_{0}^{1}(\Delta _{R'}(\xi ,t)+\rho (\Delta
_{R'S'}(t,\xi )+\Delta _{R'S'}(\xi ,t))
\\
& \quad -\rho ^{2}\Delta _{S'}(\xi t))e^{\rho (\xi -t-1)}d\xi dt].
\end{align*}% \label{eq20'}
Suppose first that $\Delta _{bd}\neq 0$. From (\ref{eq19}) we can write the
characteristic determinant in the form
\begin{equation*}
\Delta (\lambda )=\rho ^{2}\Delta _{bd}e^{\rho }(-1+F(\rho ))
\end{equation*}
for a certain function $F(\rho )$, It is not difficult to see that a
constant $r_{0}$ can be chosen in order that $|F(\rho )|\leq \frac{1}{2}$
for $|\rho |>r_{0}$. Thus, for $\rho \in r_{0}$ $+\sum_{\frac{\delta }{2}}$,
we have
\begin{equation*}
|\Delta (\lambda )|\geq |\rho |^{2}|\Delta _{bd}|e^{\mathop{\rm Re}(\rho
)}(1-|F(\rho )|)\geq \frac{1}{2}|\rho |^{2}|\Delta _{bd}|e^{\mathop{\rm Re}%
(\rho )}.
\end{equation*}
Finally, from (\ref{eq18}) we obtain
\begin{align*}
H(\rho )& \leq \frac{2}{|\Delta _{bd}|}\Big[|\Delta _{bd}|+\frac{1}{r_{0}}
(|\Delta _{ab}|+|\Delta _{ad}|+|\Delta _{bc}|+|\Delta _{cd}|+\Vert \Delta
_{bS}\Vert +\Vert \Delta _{dS}\Vert ) \\
& \quad +\frac{1}{r_{0}^{2}}(|\Delta _{ac}|+\Vert \Delta _{aS}\Vert +\Vert
\Delta _{bR}\Vert +\Vert \Delta _{dR}\Vert +\Vert \Delta _{cS}\Vert +\Vert
\Delta _{S}\Vert ) \\
& \quad +\frac{1}{r_{0}^{3}}(\Vert \Delta _{aR}\Vert +\Vert \Delta
_{cR}\Vert +2\Vert \Delta _{RS}\Vert )+\frac{\Vert \Delta _{R}\Vert }{
r_{0}^{4}}\Big]\equiv H_{0},
\end{align*}
which proves that $H(\rho )$ is bounded by a constant $H_{0}$ in the sector 
$%
r_{0}$ $+\sum_{\frac{\delta }{2}}$. The other cases can be treated in a
similar way \cite{denk1, denk2, gal5, gal7}, and we do not include them here
for lack of space. After doing the complete analysis of cases, we obtain
that $H(\rho )$ is bounded by a constant $H_{0}>0$ in a sector of the form 
$r_{0}$ $+\sum_{\frac{\delta }{2}}$, only in the following five cases

\begin{enumerate}
\item $\Delta _{bd}\neq 0$

\item $\Delta _{bd}=0$ and $\Delta_{ad}-\Delta _{bc}-\Delta _{bS}(1)+\Delta
_{dS}(0) \neq 0$

\item $\Delta _{ab}=\Delta _{ad}=\Delta _{bc}=\Delta _{bd}=\Delta _{cd}=0$, 
$\Delta _{bS}\equiv 0$, $\Delta _{dS}\equiv 0$ and $\Delta _{ac}+\Delta
_{aS}(1)+\Delta _{cS}(0) -\Delta _{bR}(1)-\Delta _{dR}(0)+\Delta _{S}(
1,0)\neq 0$

\item $\Delta _{ab}=\Delta _{ac}=\Delta _{ad}=\Delta _{bc}=\Delta
_{bd}=\Delta _{cd}=0$, $\Delta _{bR}\equiv 0,\Delta _{dR}\equiv 0,\Delta
_{aS}\equiv 0,\Delta _{bS}\equiv 0$, $\Delta _{cS}\equiv 0$, $\Delta
_{dS}\equiv 0$, $\Delta _{S}\equiv 0$ and $\Delta_{aR}(1)-\Delta
_{cR}(0)+\Delta _{RS}(0,1) +\Delta _{RS}(1,0)\neq 0$

\item $a_{i}=b_{i}=c_{i}=d_{i}=0,S_{i}\equiv 0$, $i=\overline{1,2}$ and $%
\Delta _{R}(0,1) \neq 0$.
\end{enumerate}

\begin{definition} \label{def1} \rm
Suppose that $R_{i},S_{i}\in C^{1}([0,1],\mathbb{C})$,
$i=\overline{1,2}$. The boundary conditions in (\ref{eq1}) are
called regular if they verify one of the conditions above.
\end{definition}

The above arguments prove the following theorem.

\begin{theorem} \label{thm1}
If the boundary value conditions in \eqref{eq1} are regular,
then $\sum_{\delta }\subset \rho (L_{p})$ for sufficiently
large $|\lambda | $ and there exists $c>0$ such that
\begin{equation*}
\|R(\lambda ,L_{p})\|\leq \frac{c}{
|\lambda | ^{\frac{1}{2}+\frac{1}{2p}}},|
\lambda | \to +\infty .
\end{equation*}
\end{theorem}

\begin{remark} \label{rmk1} \rm
 From theorem \ref{thm1} results that the operator $L_{p}$ for
$p\neq \infty$, generates an analytic semi group with singularities
\cite{sil3} of type $A(\frac{p-1}{p+1},\frac{3p-1}{p+1})$.
\end{remark}

\begin{remark} \label{rmk2} \rm
For $p=1$, the decrease of the resolvent is maximal and the
operator $L_{1}$ generates an analytic semi group \cite{gal7}.
\end{remark}

\subsection{Estimation of the characteristic determinant, non regular case}

As in the regular case Formula (\ref{eq19}) is not useful for the estimation
of the characteristic determinant, it will be then necessary to make some
additional hypotheses on the functions $R_{i\text{ }}$and $S_{i}$, and so we
assume that the functions $R_{i}$ and $S_{i}$ are in
 $C^{2}([0,1],\mathbb{C}) $, $i=\overline{1,2}$. Integrating twice by parts
  in (\ref{eq19}), we
obtain
\begin{align}
\Delta (\lambda )&=e^{\rho }\Big[-\rho^2\Delta _{bd} +\rho (\Delta
_{ad}-\Delta _{bc}+\Delta _{dS}( 0)-\Delta _{bS}(1))+\Delta _{ac}  \notag \\
&\quad +\Delta _{aS}(1)+\Delta _{cS}(0)-\Delta _{bR}(1)-\Delta
_{dR}(0)+\Delta _{S}( 1,0)+\Delta _{bS'}(1)  \notag \\
&\quad +\Delta _{dS'}(0)+\frac{1}{\rho }(\Delta _{aR}(1)-\Delta
_{cR}(0)+\Delta _{bR'}(1)-\Delta _{aS'}(1) +\Delta
_{cS'}(0)  \notag \\
&\quad -\Delta _{dR'}(0)+\Delta _{RS}(0,1) +\Delta _{RS}(1,0)+\Delta
_{SS'}(1,0) +\Delta _{SS'}(0,1) )  \notag \\
&\quad +\frac{1}{\rho^2}(\Delta _{R}(0,1) -\Delta _{RS'}(0,1)
+\Delta _{RS'}(1,0)-\Delta _{R'S}( 1,0)+\Delta
_{R'S}(0,1)  \notag \\
&\quad -\Delta _{dR'}(1)-\Delta _{cR'}(0)) +\frac{1}{\rho
^{3}}(\Delta _{R'R}(1,0)+\Delta _{R'R}(0,1) )+\Phi (\rho )],
\end{align}
\label{eq21} where
\begin{align*}
\Phi (\rho )&=2e^{-\rho }\Big[\rho (\Delta _{ab}+\Delta _{cd}+\Delta
_{bS}(0)-\Delta _{dS}(1)) +\frac{1}{\rho }(\Delta _{cR}(1) \\
&\quad -\Delta _{aR}(0)+\Delta _{dR'}(1)-\Delta
_{cS'}(1)+\Delta _{aS'}(0) -\Delta _{bR'}(0)-\Delta
_{RS}(0,0) \\
&\quad -\Delta _{RS}(1,1)+\Delta _{S'S}(1,1) +\Delta
_{S'S}(0,0))+\frac{1}{\rho ^{3}}(\Delta _{RR'}(1,1)+\Delta
_{RR'}(0,0))] \\
&\quad +e^{-2\rho }[\rho^2 \Delta _{bd}+\rho (\Delta _{ad}-\Delta
_{bc}+\Delta _{dS}( 0)-\Delta _{bS}(1))-\Delta _{ac}-\Delta _{aS}(1) \\
&\quad -\Delta _{cS}(0)+\Delta _{bR}(1)+\Delta _{dR}(0)-\Delta
_{S}(1,0)-\Delta _{bS'}(1)-\Delta _{dS'}(0) \\
&\quad +\frac{1}{\rho }(\Delta _{aR}(1)-\Delta _{cR}(0) +\Delta
_{bR'}(1)-\Delta _{aS'}( 1)+\Delta _{cS'}(0)-\Delta
_{dR'}(0) \\
&\quad +\Delta _{RS}(0,1) +\Delta _{RS}(1,0)+\Delta
_{SS'}(1,0)+\Delta _{SS'}(0,1) ) \\
&\quad +\frac{1}{\rho^2}(\Delta _{R}(0,1) -\Delta _{RS'}(0,1)
+\Delta _{RS'}(1,0)-\Delta _{R'S}(1,0)+\Delta
_{R'S}(0,1) \\
&\quad -\Delta _{dR'}(1)-\Delta _{cR'}( 0)) +\frac{1}{\rho
^{3}}(\Delta _{R'R}(1,0)+\Delta _{R'R}(0,1) )] \\
&\quad +\frac{1}{\rho^2}[\int_{0}^{1}(\Delta _{aR''}(t) +\rho
(\Delta _{aS''}(t)-\Delta _{bR''}(t))-\rho ^2
\Delta _{bS''}(t))e^{\rho (t-1)}dt \\
&\quad +\int_{0}^{1}(\Delta _{cR''}(t)+\rho (\Delta
_{cS''}(t)-\Delta _{dR''}(t))-\rho^2 \Delta
_{dS''}(t))e^{\rho (t-2)}dt \\
&\quad -\int_{0}^{1}(\Delta _{aR''}(t)+\rho (\Delta
_{bR''}(t)-\Delta _{aS''}(t))-\rho^2 \Delta
_{bS''}(t))e^{-\rho (t+1)}dt \\
&\quad -\int_{0}^{1}(\Delta _{cR''}(t)+\rho (\Delta
_{dR''}(t)-\Delta _{cS''}(t))-\rho^2 \Delta
_{dS''}(t))e^{-\rho t}dt \\
&\quad +\{(\int_{0}^{1}(R_{2}'(t)-\rho S_{2}'(t))e^{-\rho
t}dt)\times ( \int_{0}^{1}(R_{1}'(t)+\rho S_{1}'(t))e^{\rho
(t-1)}dt) \\
&\quad - (\int_{0}^{1}(R_{1}'(t) -\rho S_{1}'(t))e^{-\rho
t}dt)\times (\int_{0}^{1}(R_{2}'(t)+\rho S_{2}'(t))e^{\rho
(t-1)}dt)\} \Big] \\
&\quad +\frac{1}{\rho ^{3}}\Big[-\int_{0}^{1}(\Delta
_{RR''}(1,t) +\rho (\Delta _{RS''}(1,t)+\Delta
_{R''S}(t,1)) \\
&\quad +\rho^2 \Delta _{S''S}(t,1))e^{\rho (t-2)}dt \\
&\quad +\int_{0}^{1}(\Delta _{RR''}(0,t)+\rho (\Delta
_{S''R}(t,0)+\Delta _{SR''}(0,t))+\rho^2 \Delta
_{S''S}(t,0))e^{-\rho (t+1)}dt \\
&\quad +\int_{0}^{1}(\Delta _{R''R}(t,1)+\rho (\Delta
_{R''S}(1,t)+\Delta _{RS''}(1,t))+\rho^2 \Delta
_{SS''}(1,t))e^{-\rho t}dt \\
&\quad -\int_{0}^{1} (\Delta _{R''R}(t,0) +\rho (\Delta
_{R''S}(t,0)+\Delta_{RS''}(0,t))+\rho^2 \Delta
_{SS''}(0,t))e^{\rho ( t-1)}dt\Big].
\end{align*}
After a straightforward calculation, we obtain the following inequality
valid for $\rho \in \sum_{\frac{\delta }{2}}$, with $|\rho | $ sufficiently
large,
\begin{align*}
&|\Phi (\rho )| \\
&\leq 2e^{-\mathop{\rm Re}(\rho )}\Big[|\rho | (|\Delta _{ab}| +|\Delta
_{cd}| +\|\Delta _{bS}\|+\|\Delta_{dS}\|) \\
&\quad +\frac{1}{|\rho |}(\|\Delta _{cR}\|+\|\Delta _{aR}\| +\|\Delta
_{dR'}\|+\|\Delta _{cS'}\|+\|\Delta
_{aS'}\|+\|\Delta _{bR'}\| \\
&\quad +2\|\Delta _{SR}\|+2\|\Delta _{S'S}\|)\frac{1}{|\rho | ^{3}}
(\|\Delta _{RR'}\|+\|\Delta _{RR'}\|) \\
&\quad +e^{-\mathop{\rm Re}(\rho )}[|\rho |^2 |\Delta _{bd}|+|\rho |(|\Delta
_{ad}|+|\Delta _{bc}|+\|\Delta _{dS}\|+\|\Delta _{bS}\|) \\
&\quad +|\Delta _{ac}|+\|\Delta _{bR}\|+\|\Delta _{aS}\| +\|\Delta
_{dR}\|+\|\Delta_{cS}\|+\|\Delta _{dS'}\| +\|\Delta _{bS'}\|
\\
&\quad +\|\Delta _{S}\|+\frac{1}{|\rho |}(\|\Delta _{aR}\|+\|\Delta
_{cR}\|+\|\Delta _{bR'}\|+\|\Delta _{aS'}\|+\|\Delta
_{cS'}\| \\
&\quad +\|\Delta _{dR'}\| +2\|\Delta _{RS'}\|+2\|\Delta
_{S'S}\|)+\frac{1}{|\rho |^2 }(\|\Delta _{aR'}\|+\|\Delta
_{cR'}\| \\
&\quad +\|\Delta _{R}\|+2\|\Delta _{RS'}\| +2\|\Delta
_{R'S}\|+\frac{2}{|\rho |^{3}}\|\Delta _{R'R}\|\Big] \\
&\quad +\frac{1}{|\rho |^2 \mathop{\rm Re}(\rho )}[(\|\Delta
|_{aR''}|+|\rho |(\|\Delta _{aS''}\| +\|\Delta
_{bR''}\|)+|\rho |^2 \|\Delta _{bS''}\|)\times
(1-e^{-\mathop{\rm Re}(\rho)}) \\
&\quad +(\|\Delta _{cR''}\|+|\rho |(\|\Delta
_{cS''}\|+\|\Delta _{dR''}\|)+|\rho |^2 \|\Delta
_{dS''}\|)\times (e^{-\mathop{\rm Re}(\rho)}-e^{-2\mathop{\rm
Re}(\rho )}) \\
&\quad +(\|\Delta _{aR''}\|+|\rho |(\|\Delta
_{aS''}\|+\|\Delta _{bR''}\|)+|\rho |^2 \|\Delta
_{bS''}\|)\times (e^{-\mathop{\rm Re}(\rho )}-e^{-2\mathop{\rm
Re}(\rho )}) \\
&\quad +(\|\Delta _{cR''}\|+|\rho |(\|\Delta
_{cS''}\|+\|\Delta _{dR''}\|)+|\rho |^2 \|\Delta
_{dS''}\|)\times (1-e^{-\mathop{\rm Re}(\rho)}) \\
&\quad +\frac{1}{|\rho |^2(\mathop{\rm Re}(\rho ))^2 }(\|\Delta
_{R'}\|+2|\rho \||\Delta _{R'S'}\| +|\rho
|^2\|\Delta _{S'}\|)\times (1-e^{-\mathop{\rm Re}(\rho )})^2 
+\frac{2}{|\rho |^{3}\mathop{\rm Re}(\rho )} \\
&\quad \times (\|\Delta _{R''R}\|+|\rho |(\|\Delta
_{R''S}\|+\|\Delta _{RS''}\|)+|\rho |^2 \|\Delta
_{SS''}\|)\times (e^{-\mathop{\rm Re}(\rho )}-e^{-2\mathop{\rm
Re}(\rho )}) \\
&\quad +\frac{2}{|\rho |^{3}\mathop{\rm Re}(\rho )}(\|\Delta
_{R''R}\| +|\rho |(\|\Delta _{R''S}\|+\|\Delta
_{RS''}\|)+|\rho |^2 \|\Delta _{SS''}\|)\times
(1-e^{-\mathop{\rm Re}(\rho)}).
\end{align*}
Then
\begin{align}
|\Phi (\rho )| & \leq \frac{4}{|\rho |^{2}(\cos (\frac{\pi }{4}+\frac{\delta
}{2}))^{2}} \Big[|\rho | (|\Delta _{ab}| +|\Delta _{cd}|+\|\Delta
_{bS}\|+\|\Delta _{dS}\|)  \notag \\
&\quad +\frac{1}{|\rho |}(\|\Delta _{cR}\|+\|\Delta _{aR}\| +\|\Delta
_{dR'}\|+\|\Delta _{cS'}\|+\|\Delta
_{aS'}\|+\|\Delta _{bR'}\|  \notag \\
&\quad +2\|\Delta _{SR}\|+2\|\Delta _{S'S}\|)\frac{1}{|\rho | ^{3}}
(\|\Delta _{RR'}\| +\|\Delta _{RR'}\|)  \notag \\
&\quad +\frac{1}{2|\rho |^{2}(\cos (\frac{\pi }{4}+\frac{\delta }{2} ))^{2}}
[|\rho |^2 |\Delta _{bd}|+|\rho |(|\Delta _{ad}|+|\Delta _{bc}|+\|\Delta
_{dS}\|+\|\Delta _{bS}\|)  \notag \\
&\quad +|\Delta _{ac}|+\|\Delta _{bR}\|+\|\Delta _{aS}\|+\|\Delta
_{dR}\|+\|\Delta _{cS}\|+\|\Delta _{dS'}\|+\|\Delta _{bS'}\|
\notag \\
&\quad +\|\Delta _{S}\|+\frac{1}{|\rho |}(\|\Delta _{aR}\|+\|\Delta
_{cR}\|+\|\Delta _{bR'}\|+\|\Delta _{aS'}\|+\|\Delta
_{cS'}\|  \notag \\
&\quad +\|\Delta _{dR'}\| +2\|\Delta _{RS'}\|+2\|\Delta
_{S'S}\|)+\frac{1}{|\rho |^2 }(\|\Delta _{aR'}\|+\|\Delta
_{cR'}\|  \notag \\
&\quad +\|\Delta _{R}\|+2\|\Delta _{RS'}\|+2\|\Delta
_{R'S}\| +\frac{2}{|\rho |^{3}}\|\Delta _{R'R}\|]  \notag \\
&\quad +\frac{2}{|\rho |^{3}\cos (\frac{\pi }{4}+\frac{\delta }{2})}
[(\|\Delta _{aR''}\|+|\rho |(\|\Delta
_{aS''}\|+\|\Delta _{bR''}\|)+|\rho |^2 \|\Delta
_{bS''}\|)  \notag \\
&\quad +(\|\Delta _{cR''}\|+|\rho
|(\|\Delta_{cS''}\| +\|\Delta _{dR''}\|)+|\rho |^2
\|\Delta _{dS''}\|)\Big]  \notag \\
&\quad +\frac{1}{|\rho |^{4}(\cos (\frac{\pi }{4}+\frac{\delta }{2} ))^2 }
(\|\Delta _{R'}\|+2|\rho \||\Delta _{R'S'}\|+|\rho
|^2 \|\Delta _{S'}\|)  \notag \\
&\quad +\frac{4}{|\rho |^{4}\cos (\frac{\pi }{4}+\frac{\delta }{2})}
(\|\Delta _{R''R}\|+|\rho |(\|\Delta
_{R''S}\|+\|\Delta _{RS''}\|)+|\rho | ^2 \|\Delta
_{SS''}\|).  \label{eq23}
\end{align}
Where we have used that $\mathop{\rm Re}(\rho )>|\rho | \cos (\frac{\pi }{4}+%
\frac{\delta }{2})$, $1-e^{-\mathop{\rm Re}(\rho )}<1$, $1-e^{-2
\mathop{\rm
Re}(\rho )}<1$, $|\rho|^2 e^{-\mathop{\rm Re}(\rho )}\leq 2(\cos (\frac{\pi
}{4}+\frac{ \delta }{2}))^{-2}$ and $2|\rho |^2 e^{-2\mathop{\rm Re}(\rho
)}\leq (\cos (\frac{\pi }{4}+\frac{ \delta }{2}))^{-2}$. There are several
cases to analyze depending on the functions $R_{i}$ and $S_{i}$, 
$i=\overline{1,2}$.

\subsection*{Case 1.}

Suppose that $\Delta _{bd}=0$, $\Delta _{ad}-\Delta _{bc}+\Delta
_{dS}(0)-\Delta _{bS}(1)=0$,
\begin{equation*}
\max (|\Delta _{ab}|,|\Delta _{ad}|,|\Delta _{bc}|,|\Delta _{cd}|,\Vert
\Delta _{bS}\Vert ,\Vert \Delta _{dS}\Vert )\neq 0
\end{equation*}%
and $k_{1}=\Delta _{ac}+\Delta _{aS}(1)+\Delta _{cS}(0)-\Delta
_{bR}(1)-\Delta _{dR}(0)+\Delta _{S}(1,0)+\Delta _{bS'}(1)+\Delta
_{dS'}(0)\neq 0$ From (\ref{eq21}), we have for $|\rho |$
sufficiently large
\begin{align*}
|\Delta (\lambda )|& \geq e^{\mathop{\rm Re}(\rho )}\Big[|\Delta
_{ac}+\Delta _{aS}(1)+\Delta _{cS}(0)-\Delta _{bR}(1)-\Delta _{dR}(0)+\Delta
_{S}(1,0) \\
& \quad +\Delta _{bS'}(1)+\Delta _{dS'}(0)|-\frac{1}{|\rho
|}|\Delta _{aR}(1)-\Delta _{cR}(0)+\Delta _{bR'}(1)-\Delta
_{aS'}(1) \\
& \quad +\Delta _{cS'}(0)-\Delta _{dR'}(0)+\Delta
_{RS}(0,1)+\Delta _{RS}(1,0)+\Delta _{SS'}(1,0)+\Delta _{SS'}(0,1)| \\
& \quad -\frac{1}{|\rho |^{2}}|\Delta _{R}(0,1)-\Delta _{RS'}(0,1)
+\Delta _{RS'}(1,0)-\Delta _{R'S}(1,0)+\Delta
_{R'S}(0,1) \\
& \quad -\Delta _{dR'}(1)-\Delta _{cR'}(0)|-\frac{1}{|\rho
|^{3}}|\Delta _{R'R}(1,0)+\Delta _{R'R}(0,1)|-\Phi (\rho )
\Big].
\end{align*}
From (\ref{eq23}) we deduce for $\rho \in \sum_{\frac{\delta }{2}},|\rho
|\geq r_{0}>0$.
\begin{equation*}
|\Phi (\rho )|\leq \frac{c(r_{0})}{|\rho |}.
\end{equation*}%
Then, we have
\begin{align*}
|\Delta (\lambda )|& \geq e^{\mathop{\rm Re}(\rho )}[|\Delta _{ac}+\Delta
_{aS}(1)+\Delta _{cS}(0)-\Delta _{bR}(1)-\Delta _{dR}(0)+\Delta _{S}(1,0) \\
& \quad +\Delta _{bS'}(1)+\Delta _{dS'}(0)|-\frac{1}{|\rho
|}|\Delta _{aR}(1)-\Delta _{cR}(0)+\Delta _{bR'}(1)-\Delta
_{aS'}(1) \\
& \quad +\Delta _{cS'}(0)-\Delta _{dR'}(0)+\Delta
_{RS}(0,1)+\Delta _{RS}(1,0)+\Delta _{SS'}(1,0)+\Delta _{SS'}(0,1)| \\
& \quad -\frac{1}{|\rho |^{2}}|\Delta _{R}(0,1)-\Delta _{RS'}(0,1)
+\Delta _{RS'}(1,0)-\Delta _{R'S}(1,0)+\Delta
_{R'S}(0,1) \\
& \quad -\Delta _{dR'}(1)-\Delta _{cR'}(0)|-\frac{1}{|\rho
|^{3}}|\Delta _{R'R}(1,0)+\Delta _{R'R}(0,1)|-\frac{c(r_{0})}{|\rho |}\Big],
\end{align*}%
we can now choose $r_{0}>0$, such that
\begin{align*}
& \frac{1}{r_{0}}|\Delta _{aR}(1)-\Delta _{cR}(0)+\Delta _{bR'}(1)-\Delta _{aS'}(1)
+\Delta _{cS'}(0)-\Delta
_{dR'}(0) \\
& +\Delta _{RS}(0,1)+\Delta _{RS}(1,0)+\Delta _{SS'}(1,0)+\Delta
_{SS'}(0,1)|+\frac{1}{r_{0^{2}}}|\Delta _{R}(0,1) \\
& -\Delta _{RS'}(0,1)+\Delta _{RS'}(1,0)-\Delta
_{R'S}(1,0)+\Delta _{R'S}(0,1)-\Delta _{dR^{\prime
}}(1)-\Delta _{cR'}(0)| \\
& +\frac{1}{r_{0}^{3}}|\Delta _{R'R}(1,0)+\Delta _{R'R}(0,1)|+\frac{c(r_{0})}{|\rho |} \\
& \leq \frac{1}{2}|\Delta _{ac}+\Delta _{aS}(1)+\Delta _{cS}(0) \\
& \quad -\Delta _{bR}(1)-\Delta _{dR}(0)+\Delta _{S}(1,0)+\Delta
_{bS'}(1)+\Delta _{dS'}(0)|,
\end{align*}%
then, for $\rho \in \sum_{\frac{\delta }{2}},|\rho |\geq r_{0}>0$, we get
\begin{equation*}
|\Delta (\rho )|\geq \frac{e^{\mathop{\rm Re}(\rho )}}{2}|k_{1}|.
\end{equation*}%
From (\ref{eq17'}), we deduce the following bound, valid for every $|\arg
\rho |\leq \frac{\pi }{4}+\frac{\delta }{2}$, $\rho \neq 0$,
\begin{equation*}
\Vert R(\lambda ,L_{p})\Vert \leq \frac{cH(\rho )}{|\rho |^{1+\frac{1}{p}}}
\leq \frac{1}{|\rho |^{1/p}}\times \frac{cH(\rho )}{|\rho |},
\end{equation*}%
where
\begin{align*}
\frac{H(\rho )}{|\rho |}& =\frac{e^{\mathop{\rm Re}(\rho )}}{|\Delta (\rho
^{2})||\rho |}[|\rho |(|\Delta _{ab}|+|\Delta _{ad}|+|\Delta _{bc}|+|\Delta
_{cd}|+\Vert \Delta _{bS}\Vert \\
& \quad +\Vert \Delta _{dS}\Vert )+(|\Delta _{ac}|+\Vert \Delta _{aS}\Vert
+\Vert \Delta _{bR}\Vert +\Vert \Delta _{dR}\Vert +\Vert \Delta _{cS}\Vert \\
& \quad +\Vert \Delta _{S}\Vert )+\frac{1}{|\rho |}(\Vert \Delta _{aR}\Vert
+\Vert \Delta _{cR}\Vert +2\Vert \Delta _{RS}\Vert )+\frac{\Vert \Delta
_{R}\Vert }{|\rho |^{2}}] \\
& =\frac{e^{\mathop{\rm Re}(\rho )}}{|\Delta (\rho ^{2})|}[(|\Delta
_{ab}|+|\Delta _{ad}|+|\Delta _{bc}|+|\Delta _{cd}|+\Vert \Delta _{bS}\Vert
+\Vert \Delta _{dS}\Vert ) \\
& \quad +\frac{1}{|\rho |}(|\Delta _{ac}|+\Vert \Delta _{aS}\Vert +\Vert
\Delta _{bR}\Vert +\Vert \Delta _{dR}\Vert +\Vert \Delta _{cS}\Vert +\Vert
\Delta _{S}\Vert ) \\
& \quad +\frac{1}{|\rho |^{2}}(\Vert \Delta _{aR}\Vert +\Vert \Delta
_{cR}\Vert +2\Vert \Delta _{RS}\Vert )+\frac{\Vert \Delta _{R}\Vert }{|\rho
|^{3}}] \\
& \leq \frac{2}{\left\vert k_{1}\right\vert }[(|\Delta _{ab}|+|\Delta
_{ad}|+|\Delta _{bc}|+|\Delta _{cd}|+\Vert \Delta _{bS}\Vert +\Vert \Delta
_{dS}\Vert ) \\
& \quad +\frac{1}{|\rho |}(|\Delta _{ac}|+\Vert \Delta _{aS}\Vert +\Vert
\Delta _{bR}\Vert +\Vert \Delta _{dR}\Vert +\Vert \Delta _{cS}\Vert +\Vert
\Delta _{S}\Vert ) \\
& \quad +\frac{1}{|\rho |^{2}}(\Vert \Delta _{aR}\Vert +\Vert \Delta
_{cR}\Vert +2\Vert \Delta _{RS}\Vert )+\frac{\Vert \Delta _{R}\Vert }{|\rho
|^{3}}],
\end{align*}
as $|\lambda |\rightarrow +\infty $, where
\begin{equation*}
\frac{cH(\rho )}{|\rho |}\leq \frac{2c}{\left\vert k_{1}\right\vert }
(|\Delta _{ab}|+|\Delta _{ad}|+|\Delta _{bc}|+|\Delta _{cd}|+\Vert \Delta
_{bS}\Vert +\Vert \Delta _{dS}\Vert )=c_{1}
\end{equation*}%
then
\begin{equation*}
\Vert R(\lambda ,L_{p})\Vert \leq \frac{c_{1}}{|\lambda |^{\frac{1}{2p}}}.
\end{equation*}

\subsection*{Case 2.}

If $\Delta _{ab}=\Delta _{ad}=\Delta _{bc}=\Delta _{bd}=\Delta _{cd}=0$, 
$\Delta _{bS}\equiv 0$, $\Delta _{dS}\equiv 0$, $\Delta _{ac}+\Delta
_{aS}(1)+\Delta _{cS}(0)-\Delta _{bR}(1)-\Delta _{dR}(0)+\Delta _{S}(1,0)=0$,
\begin{equation*}
\max (|\Delta _{ac}|,\Vert \Delta _{aS}\Vert ,\Vert \Delta _{cS}\Vert ,\Vert
\Delta _{bR}\Vert ,\Vert \Delta _{dR}\Vert ,\Vert \Delta _{S}\Vert )\neq 0
\end{equation*}
and $k_{2}=\Delta _{aR}(1)-\Delta _{cR}(0)+\Delta _{bR'}(1)-\Delta
_{aS'}(1)+\Delta _{cS'}(0)-\Delta _{dR^{\prime
}}(0)+\Delta _{RS}(0,1)+\Delta _{RS}(1,0)+\Delta _{SS'}(1,0)+\Delta
_{SS'}(0,1)\neq 0$, we have the following bound, valid for $\lambda
\in \sum_{\delta }$ and $|\lambda |$ big enough,
\begin{equation*}
\Vert R(\lambda ,L_{p})\Vert \leq \frac{cH(\rho )}{|\rho |^{1+\frac{1}{p}}}
\leq \frac{1}{|\rho |^{1/p}}\times \frac{cH(\rho )}{|\rho |},
\end{equation*}%
where
\begin{align*}
\frac{H(\rho )}{|\rho |}& =\frac{e^{\mathop{\rm Re}(\rho )}}{|\rho ||\Delta
(\rho ^{2})|}[(|\Delta _{ac}|+\Vert \Delta _{aS}\Vert +\Vert \Delta
_{bR}\Vert +\Vert \Delta _{dR}\Vert +\Vert \Delta _{cS}\Vert \\
& \quad +\Vert \Delta _{S}\Vert )+\frac{1}{|\rho |}(\Vert \Delta _{aR}\Vert
+\Vert \Delta _{cR}\Vert +2\Vert \Delta _{RS}\Vert )+\frac{\Vert \Delta
_{R}\Vert }{|\rho |^{2}}] \\
& \leq \frac{2}{\left\vert k_{2}\right\vert }[(|\Delta _{ac}|+\Vert \Delta
_{aS}\Vert +\Vert \Delta _{bR}\Vert +\Vert \Delta _{dR}\Vert +\Vert \Delta
_{cS}\Vert +\Vert \Delta _{S}\Vert ) \\
& \quad +\frac{1}{|\rho |}(\Vert \Delta _{aR}\Vert +\Vert \Delta _{cR}\Vert
+2\Vert \Delta _{RS}\Vert )+\frac{\Vert \Delta _{R}\Vert }{|\rho |^{2}}],
\end{align*}%
as $|\rho |\rightarrow +\infty $, we have
\begin{equation*}
\frac{cH(\rho )}{|\rho |}\leq \frac{2c}{\left\vert k_{2}\right\vert }
(|\Delta _{ac}|+\Vert \Delta _{aS}\Vert +\Vert \Delta _{bR}\Vert +\Vert
\Delta _{dR}\Vert +\Vert \Delta _{cS}\Vert +\Vert \Delta _{S}\Vert )=c_{2}
\end{equation*}
then
\begin{equation*}
\Vert R(\lambda ,L_{p})\Vert \leq \frac{c_{2}}{|\lambda |^{\frac{1}{2p}}}.
\end{equation*}

\subsection*{Case 3.}

If $\Delta _{ab}=\Delta _{ac}=\Delta _{ad}=\Delta _{bc}=\Delta _{db}=\Delta
_{cd}=0$, $\Delta _{bR}\equiv 0,\Delta _{dR}\equiv 0$, $\Delta _{aS}\equiv 0$, 
$\Delta _{bS}\equiv 0$, $\Delta _{cS}\equiv 0$, $\Delta _{dS}\equiv 0$,
$\Delta _{aR}(1)-\Delta _{cR}(0)+\Delta _{RS}(0,1)+\Delta _{RS}(1,0)=0$, 
$\max (\Vert \Delta _{aR}\Vert ,\Vert \Delta _{cR}\Vert ,\Vert \Delta
_{RS}\Vert )\neq 0$ and $k_{3}=\Delta _{R}(0,1)-\Delta _{RS'}(0,1)
+\Delta _{RS'}(1,0)-\Delta _{R'S}(1,0)+\Delta
_{R'S}(0,1)-\Delta _{dR'}(1)-\Delta _{cR'}(0)\neq
0$. Similarly, we get
\begin{equation*}
\Vert R(\lambda ,L_{p})\Vert \leq \frac{cH(\rho )}{|\rho |^{1+\frac{1}{p}}}
\leq \frac{1}{|\rho |^{1/p}}\times \frac{cH(\rho )}{|\rho |},
\end{equation*}%
where
\begin{align*}
\frac{H(\rho )}{|\rho |}& =\frac{e^{\mathop{\rm Re}(\rho )}}{|\rho ||\Delta
(\rho ^{2})|}\Big[\frac{1}{|\rho |}(\Vert \Delta _{aR}\Vert +\Vert \Delta
_{cR}\Vert +2\Vert \Delta _{RS}\Vert )+\frac{\Vert \Delta _{R}\Vert }{|\rho
|^{2}}\Big] \\
& =\frac{e^{\mathop{\rm Re}(\rho )}}{|\rho |^{2}|\Delta (\rho ^{2})|}\Big[
(\Vert \Delta _{aR}\Vert +\Vert \Delta _{cR}\Vert +2\Vert \Delta _{RS}\Vert
)+\frac{\Vert \Delta _{R}\Vert }{|\rho |}\Big] \\
& \leq \frac{2}{\left\vert k_{3}\right\vert }\big[(\Vert \Delta _{aR}\Vert
+\Vert \Delta _{cR}\Vert +2\Vert \Delta _{RS}\Vert )+\frac{\Vert \Delta
_{R}\Vert }{|\rho |}\Big],
\end{align*}%
as $|\rho |\rightarrow +\infty $, we have
\begin{equation*}
\frac{cH(\rho )}{|\rho |}\leq \frac{2c}{\left\vert k_{3}\right\vert }(\Vert
\Delta _{aR}\Vert +\Vert \Delta _{cR}\Vert +2\Vert \Delta _{RS}\Vert )=c_{3}
\end{equation*}%
then, we have
\begin{equation*}
\Vert R(\lambda ,L_{p})\Vert \leq \frac{c_{3}}{|\lambda |^{\frac{1}{2p}}}.
\end{equation*}

\subsection*{Case 4.}

If $a_{i}=b_{i}=c_{i}=d_{i}=0$, $S_{i}\equiv 0$ where $i=\overline{1,2}$, 
$\Delta _{R}(0,1)=0$, $\Vert \Delta _{R}\Vert \neq 0$ and $k_{4}=\Delta
_{R'R}(1,0)+\Delta _{R'R}(0,1)\neq 0$, again in this case,
we have
\begin{equation*}
\Vert R(\lambda ,L_{p})\Vert \leq \frac{cH(\rho )}{|\rho |^{1+\frac{1}{p}}}
\leq \frac{1}{|\rho |^{1/p}}\times \frac{cH(\rho )}{|\rho |},
\end{equation*}%
where
\begin{equation*}
\frac{H(\rho )}{|\rho |}=\frac{e^{\mathop{\rm Re}(\rho )}}{|\rho ||\Delta
(\rho ^{2})|}\times \frac{\Vert \Delta _{R}\Vert }{|\rho |^{2}}\newline
=\frac{e^{\mathop{\rm Re}(\rho )}}{|\rho |^{2}|\Delta (\rho ^{2})|}\times
\frac{\Vert \Delta _{R}\Vert }{|\rho |}\leq \frac{2}{k_{4}}\Vert \Delta
_{R}\Vert ,
\end{equation*}
as $|\rho |\rightarrow +\infty $, we have
\begin{equation*}
\frac{cH(\rho )}{|\rho |}\leq \frac{2c}{\left\vert k_{4}\right\vert }\Vert
\Delta _{R}\Vert =c_{4}
\end{equation*}
then
\begin{equation*}
\Vert R(\lambda ,L_{p})\Vert \leq \frac{c_{4}}{|\lambda |^{\frac{1}{2p}}}.
\end{equation*}

\begin{definition} \label{def2} \rm
The boundary conditions in (\ref{eq1}) are called non regular if the
functions $R_{i}$, $S_{i}\in C^{2}([0,1],\mathbb{C})$,
$i=\overline{1,2}$ and if and only if one of the following
conditions holds.

\begin{enumerate}
\item $\Delta _{bd}=0$,
$\Delta _{ad}-\Delta _{bc}+\Delta _{dS}(0)-\Delta _{bS}(1)=0$,
$$
\max (|\Delta _{ab}|,|\Delta _{ad}|,|\Delta _{bc}|,|\Delta
_{cd}|,\|\Delta _{bS}\|,\|\Delta _{dS}\|)\neq 0
$$
and $\Delta _{ac}+\Delta _{aS}(1)+\Delta _{cS}(0)-\Delta
_{bR}(1)-\Delta _{dR}(0)+\Delta _{S}(
1,0)+\Delta _{bS'}(1)+\Delta _{dS'}(0)\neq 0$

\item $\Delta _{ab}=\Delta _{ad}=\Delta _{bc}=\Delta _{bd}=\Delta
_{cd}=0$, $\Delta _{bS}\equiv 0,\Delta _{dS}\equiv 0$,
$\Delta _{ac}$ $+\Delta_{aS}(1)+\Delta _{cS}(0)-\Delta _{bR}(
1)-\Delta _{dR}(0)+\Delta _{S}(1,0)=0$,
$$
\max (|\Delta _{ac}|,\|\Delta _{aS}\|,\|\Delta _{cS}\|,\|\Delta
_{bR}\|,\|\Delta _{dR}\|,\|\Delta _{S}\|)\neq 0
$$
and $\Delta_{aR}(1)-\Delta _{cR}(0)+\Delta _{bR'}(1)-\Delta _{aS'}(1)+\Delta
_{cS'}(0)-\Delta _{dR'}(0)
+\Delta _{RS}(0,1) +\Delta _{RS}(1,0)+\Delta
_{SS'}(1,0)+\Delta _{SS'}(0,1)
\neq 0$

\item $\Delta _{ab}=\Delta _{ac}=\Delta _{ad}=\Delta _{bc}=\Delta
_{db}=\Delta _{cd}=0$,
$\Delta _{bR}\equiv 0$, $\Delta _{dR}\equiv 0$,
$\Delta _{aS}\equiv 0$, $\Delta _{bS}\equiv 0$,
$\Delta _{cS}\equiv 0$, $\Delta _{dS}\equiv 0$,
$\Delta _{aR}(1)-\Delta _{cR}(0)+\Delta _{RS}(0,1) +\Delta _{RS}(1,0)=0$,
$\max (\|\Delta _{aR}\|,\|\Delta _{cR}\|,\|\Delta _{RS}\|)\neq 0$
and $\Delta _{R}(0,1) -\Delta _{RS'}(0,1)+\Delta _{RS'}(1,0)-\Delta _{R'S}(
1,0)+\Delta _{R'S}(0,1) -\Delta _{dR'}(1)-\Delta _{cR'}(0)\neq 0$

\item If $a_{i}=b_{i}=c_{i}=d_{i}=0$, $S_{i}\equiv 0$ where
$i=\overline{1,2}$, $\Delta _{R}(0,1) =0$,
$\|\Delta _{R}\|\neq 0$ and $\Delta_{R'R}(1,0)+\Delta _{R'R}(0,1)\neq 0$.
\end{enumerate}
\end{definition}

The above arguments prove the following theorem.

\begin{theorem} \label{thm2}
If the boundary value conditions in \eqref{eq1} are non regular,
then $\sum_{\delta }\subset \rho (L_{p})$ for sufficiently
large $|\lambda | $ and there exists $c>0$ such that
\begin{equation*}
\|R(\lambda ,L_{p})\|\leq \frac{c}{|\lambda | ^{\frac{1}{2p}}}.
\end{equation*}
\end{theorem}

\begin{remark} \label{rmk3} \rm
 From theorem \ref{thm2} it follows that the operator $L_{p}$, for
$p\neq \infty $, generates an analytic semi group with singularities
\cite{sil3} of type $A(2p-1,4p-1)$.
\end{remark}

\begin{remark} \label{rmk4} \rm
As particular cases, we obtain the results of \cite{denk1, denk2}.
\end{remark}

\begin{remark} \label{rmk5} \rm
Contrarily to the regular case, we have a loss of $\frac{1}{2}$ in the
resolvent estimate.
\end{remark}

\begin{remark} \label{rmk6} \rm
It is not difficult to see that the above definitions of
regularity and non regularity of boundary conditions do not depend
on possible elementary simplifications on the boundary conditions
or integrations by parts.
\end{remark}

\subsection{Applications}

In the following, we apply the results obtained to study of a class of a
mixed problem for a parabolic equation with weighted integral boundary
condition combined with anther two point boundary condition of the form
\begin{equation}
\begin{gathered} \frac{\partial u(t,x)}{\partial t}-a\frac{\partial ^{2}u(
t,x)}{\partial x^{2}}=f(t,x) \\ \begin{aligned}
L_{1}(u)&:=a_{1}u(0,t)+b_{1}u'(0,t)+c_{1}u(1,t)+d_{1}u'(1,t) \\ 
&\quad
+\int_{0}^{1}R_{1}(\xi )u(t,\xi )d\xi +\int_{0}^{1}S_{1}(\xi )u'(t,\xi )d\xi
=0, \end{aligned} \\ \begin{aligned}
L_{2}(u)&:=a_{2}u(0,t)+b_{2}u'(0,t)+c_{2}u(1,t)+d_{2}u'(1,t) \\ 
&\quad
+\int_{0}^{1}R_{2}(\xi )u(t,\xi )d\xi +\int_{0}^{1}S_{2}(\xi )u'(t,\xi )d\xi
=0, \end{aligned} \\ u(0,x)=u_{0}(x), \end{gathered}  \label{eq24}
\end{equation}
where $(t,\xi )\in [0,T]\times [0,1]$. Boundary-value problems for parabolic
equations with integral boundary conditions are studied by \cite{beil, can,
den2, ion1, ion2, kam, sil3} using various methods. For instance, the
potential method in \cite{can} and \cite{kam}, Fourier method in \cite{beil,
ion1, ion2, ion3} and the energy inequalities method has been used in 
\cite{den2, den3, yur}. In our case, we apply the method of operator differential
equation. The study of the problem is then reduced to a Cauchy problem for a
parabolic abstract differential equation, where the operator coefficients
has been previously studied. For this purpose, let $E,E_{1}$ and $E_{2}$ be
Banach spaces. Introduce two Banach spaces
\begin{equation*}
C_{\mu }((0,T],E)=\big\{f\in C((0,T],E):\|f\| =\sup_{t\in (0,T]} \|t^{\mu
}f(t)\|<+\infty \big\},\quad \mu \geq 0,
\end{equation*}
\begin{align*}
C_{\mu }^{\gamma }((0,T],E)&=\Big\{f\in C((0,T],E):\|f\|=\sup_{t\in
(0,T]}\|t^{\mu }f(t)\| \\
&\quad +\sup_{0<t<t+h\leq T} \|f(t+h) -f(t)\|h^{-\gamma }t^{\mu }<+\infty %
\Big\}, \quad \mu \geq 0,\; \gamma \in (0,1],
\end{align*}
and a linear space
\begin{equation*}
C^{1}((0,T],E_{1},E_{2})=\big\{f\in C((0,T],E_{1})\cap C^{1}((0,T] ,E_{2})%
\big\},\quad E_{1}\subset E_{2},
\end{equation*}
where $C((0,T],E)$ and $C^{1}((0,T],E)$ are spaces of continuous and
continuously differentiable, respectively, vector-functions from $(0,T]$
into $E$. We denote, for a linear operator $A$ in a Banach space $E$, by
\begin{gather*}
E(A)=\{u\in D(A):\| u\|_{E(A)}=(\|Au\| ^{2}+\|u\|^{2})^{\frac{1}{2}}\}, \\
C^{1}((0,T],E(A),E)=\{f\in C((0,T],E(A)): f'\in C((0,T],E)\}.
\end{gather*}
Let us derive a theorem which was proved by various methods in \cite{sil3}
and \cite{yak3}. Consider, in a Banach space $E$, the Cauchy problem
\begin{equation}
\begin{gathered} u'(t)=Au(t)+f(t),\quad t\in [0,T], \\ u(0)=u_{0},
\end{gathered}  \label{eq25}
\end{equation}
where $A$ is, generally speaking, unbounded linear operator in $E,u_{0}$ is
a given element of $E$, $f(t)$ is a given vector-function and $u(t)$ is an
unknown vector-function in $E$.

\begin{theorem} \label{thm3}
Let the following conditions be satisfied:
\begin{enumerate}
\item $A$ is a closed linear operator in a Banach space $E$ and for some $
\beta \in (0,1],\alpha >0$
\begin{equation*}
\|R(\lambda ,A)\|\leq C|\lambda
| ^{-\beta },\quad |\arg \lambda | \leq \frac{\pi }{2
}+\alpha ,|\lambda | \to +\infty ;
\end{equation*}

\item $f\in C_{\mu }^{\gamma }((0,T],E)$ for some
$\gamma \in (1-\beta ,1],\mu \in [0,\beta );$

\item $u_{0}\in D(A)$.
\end{enumerate}
Then the Cauchy problem \eqref{eq25} has a unique solution
\begin{equation*}
u\in C((0,T],E)\cap C^{1}((0,T] ,E(A),E),
\end{equation*}
and for the solution $u$,
\begin{gather*}
\|u(t)\|\leq C(\|Au_{0}\|+\|u_{0}\|+\|f\|_{C_{\mu }^{\gamma }((0,T],E)}),
t\in (0,T],
\\
\|u'(t)\|+\|Au(t)\|\leq C(t^{\beta -1}(\|
Au_{0}\|+\|u_{0}\|)+t^{\beta -\mu
-1}\|f\|_{C_{\mu }^{\gamma }((0,t]
,E)}),t\in (0,T].
\end{gather*}
\end{theorem}

As a result of this we get the following theorem.

\begin{theorem} \label{thm4}
Let the following conditions be satisfied:
\begin{enumerate}
\item $a\neq 0$, $|\arg a| <\pi/2$,

\item The functions $R(t),S(t)\in C^{2}([0,1],\mathbb{C})$,
$i=\overline{1,2}$, and one of the following conditions is
satisfied $\Delta _{bd}=0$,
$\Delta _{ad}-\Delta _{bc}+\Delta _{dS}(0)-\Delta _{bS}(1)=0$,
$\max (|\Delta _{ab}|,|\Delta _{ad}|,|\Delta _{bc}|,|\Delta
_{cd}|,\|\Delta _{bS}\|,\|\Delta _{dS}\|)\neq 0$ and
$\Delta_{ac}+\Delta _{aS}(1)+\Delta _{cS}(0)-\Delta
_{bR}(1)-\Delta _{dR}(0)+\Delta _{S}(
1,0)+\Delta _{bS'}(1)+\Delta _{dS'}(0)\neq 0$ or
$\Delta _{ab}=\Delta _{ad}=\Delta _{bc}=\Delta
_{bd}=\Delta _{cd}=0$, $\Delta _{bS}\equiv 0$,
$\Delta _{dS}\equiv 0$,
$\Delta _{ac}+\Delta _{aS}(1)+\Delta _{cS}(0)
-\Delta _{bR}(1)-\Delta _{dR}(0)+\Delta _{S}(1,0)=0$,
$$
\max (|\Delta _{ac}|,\|\Delta _{aS}\|,\|\Delta
_{cS}\|,\|\Delta _{bR}\|,\|\Delta _{dR}\|,\|\Delta _{S}\|)\neq 0
$$
and $\Delta _{aR}(1)-\Delta _{cR}(0)+\Delta_{bR'}(1)-\Delta _{aS'}(1)
+\Delta _{cS'}(0)-\Delta _{dR'}(
0)+\Delta _{RS}(0,1) +\Delta _{RS}(1,0)
+\Delta _{SS'}(1,0)+\Delta _{SS'}(0,1) \neq 0$ or
$\Delta _{ab}=\Delta _{ac}=\Delta _{ad}=\Delta
_{bc}=\Delta _{db}=\Delta _{cd}=0$, $\Delta _{bR}\equiv 0$,
$\Delta_{dR}\equiv 0,\Delta _{aS}\equiv 0$, $\Delta _{bS}\equiv 0$,
$\Delta_{cS}\equiv 0$, $\Delta _{dS}\equiv 0$, $\Delta _{aR}(1)-\Delta
_{cR}(0)+\Delta _{RS}(0,1) +\Delta _{RS}(1,0)=0$,
$$
\max (\|\Delta _{aR}\|,\|\Delta _{cR}\|,\|\Delta_{RS}\|)\neq 0
$$
and $\Delta _{R}(0,1) -\Delta_{RS'}(0,1) +\Delta _{RS'}(1,0)
-\Delta _{R'S}(1,0)+\Delta _{R'S}(0,1) -\Delta _{dR'}(1)
-\Delta _{cR'}(0)\neq 0$ or If: $a_{i}=b_{i}=c_{i}=d_{i}=0,S_{i}\equiv 0$
where $i=\overline{1,2}$, $\Delta _{R}(0,1) =0$,
$\|\Delta_{R}\|\neq 0$ and $\Delta _{R'R}(1,0)+\Delta
_{R'R}(0,1) \neq 0$

\item $f\in C_{\mu }^{\gamma }((0,T],L^{q}(0,1) )$ for some
 $\gamma \in (1-\frac{1}{2q},1]$
and some $\mu \in [0,\frac{1}{2q})$,

\item $u_{0}\in W^{2,q}((0,1),L_{i}u=0,i=\overline{1,2})$.

\end{enumerate}
Then problem (\ref{eq24}) has a unique solution
\begin{equation*}
u\in C((0,T],L^{q}(0,1) )\cap
C^{1}((0,T],W^{2,q}(0,1) ,L^{q}(0,1) )
\end{equation*}
and for this solution we have the estimates:
\begin{gather}
\|u(t,.)\|_{L^{q}(0,1) }\leq
c(\|u_{0}\|_{W^{2,q}(0,1) }+\|
f\|_{C_{\mu }((0,t],L^{q}(0,1)
)}),t\in (0,T],  \label{eq26}
\\  \label{eq27}
\begin{aligned}
&\|u''(t,.)\|_{L^{q}(0,1) }+\|u'(t,.)\|_{L^{q}(0,1) }   \\
&\leq c(t^{\frac{1}{2q}-1}\|u_{0}\|_{W^{2,q}(0,1) }
+t^{\frac{1}{2q}-\mu -1}\|f\|_{C_{\mu}^{\gamma }((0,t],L^{q}(0,1) )})
,t\in (0,T].
\end{aligned}
\end{gather}
\end{theorem}

\begin{proof}
In the space $L^{q}(0,1) $, $1\leq q<+\infty $, we consider the operator $A$
defined by
\begin{equation*}
A(u)=au''(x),\quad D(A)=\{u\in W^{2,q}(0,1) ,\;L_{i}(u)=0, 
i=\overline{1,2}\}.
\end{equation*}
Then problem (\ref{eq24}) can be written as
\begin{gather*}
u'(t)=Au(t)+f(t), \\
u(0)=u_{0},
\end{gather*}
where $u(t)=u(t,.),f(t)=f(t,.)$, and $u_{0}=u_{0}(.)$ are functions with
values in the Banach space $L^{q}(0,1) $. From Theorem \ref{thm2} we
conclude that $\|R(\lambda ,A)\|\leq c|\lambda | ^{-\frac{1}{2q}}$, for
 $|\arg \lambda | \leq \frac{\pi }{2}+\alpha $, as $|\lambda | \to +\infty $.
Then , from Theorem \ref{thm3} the problem (\ref{eq24}) has a unique
solution
\begin{equation*}
u\in C((0,T],L^{q}(0,1) )\cap C^{1}((0,T],W^{2,q}(0,1) ,L^{q}(0,1) )
\end{equation*}
and we have the following estimates
\begin{gather}
\|u(t,.)\|_{L^{q}(0,1) }\leq c(\|Au_{0}\|_{L^{q}(0,1) }+\|
u_{0}\|_{L^{q}(0,1) }+\|f\|_{C_{\mu }((0,t],L^{q}(0,1) )})\,,  \label{eq28'}
\\[4pt]
\begin{aligned} &\|u'(t,.)\|_{L^{q}(0,1) }+\|Au(t,.)\|_{L^{q}(0,1) } \\
&\leq c(t^{\frac{1}{2q}-1}(\|Au_{0}\| _{L^{q}(0,1) }+\|u_{0}\|_{L^{q}(0,1)
})+t^{\frac{1}{2q}-\mu -1}\|f\|_{C_{\mu }^{\gamma }((0,t],L^{q}(0,1) )}),
\end{aligned}  \label{eq29}
\end{gather}
where $t\in [0,T]$, from (\ref{eq28'}) we get
\begin{align*}
\|u(t,.)\|_{L^{q}(0,1) } &\leq c(\|u_{0}''\|_{L^{q}(0,1)
}+\|u_{0}\|_{L^{q}(0,1) }+\|f\|_{C_{\mu }((0,t],L^{q}(0,1) )}) \\
&\leq c(\|u_{0}\|_{W^{2,q}(0,1) }+\|f\|_{C_{\mu }((0,t],L^{q}(0,1) )}),\quad
t\in [0,T].
\end{align*}
And from (\ref{eq29}) we get
\begin{align*}
&\|u'(t,.)\|_{L^{q}(0,1) }+\|u''(t,.)\| _{L^{q}(0,1) }
\\
&\leq c\Big(t^{\frac{1}{2q}-1}(\|u_{0}\| _{L^{q}(0,1)
}+\|u_{0}''\| _{L^{q}(0,1) })+t^{\frac{1}{2q}-\mu -1}\|
f\|_{C_{\mu }^{\gamma }((0,t],L^{q}(0,1) )}\Big) \\
&\leq c\Big(t^{\frac{1}{2q}-1}\|u_{0}\|_{W^{2,q}(0,1) } +t^{\frac{1}{2q}-\mu
-1}\|f\|_{C_{\mu}^{\gamma }((0,t],L^{q}(0,1) )}\Big) ,\quad t\in [0,T].
\end{align*}
\end{proof}

\subsection*{Acknowledgements}

The authors gratefully acknowledge the useful suggestions of the anonymous
referee.

\begin{thebibliography}{99}
\bibitem{beil} Beilin S. A.: \textit{Existence of solutions for
one-dimensional wave equations with nonlocal conditions,} Electronic Journal
of Differential Equations, vol. 2001 (2001), No. 76, 1-8.

\bibitem{ben} Benzinger H. E.: \textit{Green's function for ordinary
differential operators}, J. Differen. Equations, \textbf{7} (1970), 478-496.

\bibitem{can} Cannon J. R., Lin Y. and Van Der Hook J.: \textit{A
quasi-linear parabolic equation with nonlocal boundary conditions, } Rend.
Mat. Appl. (7), \textbf{9} (1989), 239-264.

\bibitem{den1} Denche M.: \textit{Defect-Coerciveness for non regular
transmission problem}. Advances in Mathematical Sciences and Applications,
\textbf{9 }(1999), 1 , 229-241.

\bibitem{denk1} Denche M., Kourta A. : \textit{Boundary value problem for
second-order differential operators with mixed nonlocal boundary conditions,
}Journal of Inequalities in Pure and Applied Mathematics, \textbf{5}, 2
(2004), Article 38, 16 pp.

\bibitem{denk2} Denche M., Kourta A. : \textit{Boundary value problem for
second-order differential operators with nonregular integral boundary
conditions, }Rocky Mountain Journal of Mathematics, \textbf{36}, 3 (2006),
893-913.

\bibitem{den2} Denche, M., Marhoune A. L.: \textit{Mixed problem with
nonlocal boundary conditions for a third-order partial differential equation
of mixed type, }IJMMS, \textbf{26}, 7 (2001), 417-426.

\bibitem{den3} Denche M., Marhoune A. L.: \textit{Mixed problem with
integral boundary condition for a high order mixed type partial differential
equation, }Journal of Applied Mathematics and Stochastic Analysis, \textbf{16%
}, 1 (2003), 69-79.

\bibitem{dun} Dunford N., Schwartz J. T.:\emph{Linear Operators. Part III.
Spectral operators}, Interscience, New York, 1971.

\bibitem{eber} Eberhard W., Freiling G.: \textit{Ston regul\"{a}re Eigenwert
problem}. Math.Zeit., \textbf{160}, 2 (1978), 139-161.

\bibitem{far} Farjas J., Rosell J. I., Herrero R., Pons R., Pi F. and
Orriols G.: \textit{Equivalent low-order model for a nonlinear diffusion
equation, }Phys. D., \textbf{95} (1996), 107-127.

\bibitem{galak} Galakhov E. I., Skubachevskii A. L.: \textit{A nonlocal
spectral problem, }Differential Equations, \textbf{33} (1997), 24-31.

\bibitem{gal5} Gallardo J. M.: \textit{Generation of analytic semi -groups
by second-order differential operators with nonseparated boundary conditions}
, Rocky Mountain Journal of Mathematics, \textbf{30 }(2000), 2, 869-899.

\bibitem{gal6} Gallardo J. M.: \textit{second-order differential operators
with integral boundary conditions and generation of analytic semigroups},
Rocky Mountain Journal of Mathematics, \textbf{30} (2000), 4, 1265-1291.

\bibitem{gal7} Gallardo J. M .: \textit{Differential operators with mixed
boundary conditions: generation of analytic semigroups, }Nonlinear Analysis,
\textbf{47} (2001), 1333-1344.

\bibitem{gas1} Gasumov M. G., Magerramov A. M.: \textit{On fold-completeness
of a system of eigenfunctions and associated functions of a class of
differential operators, }DAN Azerb.SSR, \textbf{30} (1974), 3, 9-12.

\bibitem{gas2} Gasumov M. G., Magerramov A. M.: \textit{Investigation of a
class of differential operator pencils of even order, }DAN SSR, \textbf{265}
(1982), 2, 277-280.

\bibitem{ion1} Ionkin N. I.: \textit{Solution of boundary value problem in
heat conduction theory with nonlocal boundary conditions, }Differ. Uravn.,
\textbf{13} (1977), 294-304.

\bibitem{ion2} Ionkin N. I.: \textit{Stability of a problem in heat
conduction theory with nonlocal boundary conditions, }Differ. Uravn.,
\textbf{15} (1979), 7, 1279-1283.

\bibitem{ion3} Ionkin N. I., Moiseev E. I.: \textit{A problem for the heat
conduction equation with two-point boundary condition, }Differ. Uravn.,
\textbf{15} (1979), 7, 1284-1295.

\bibitem{kam} Kamynin N. I.: \textit{A boundary value problem in the theory
of heat conduction with non classical boundary condition, }Theoret. Vychisl.
Mat. Fiz., \textbf{4} (1964), 6 1006-1024.

\bibitem{khr} Khromov A. P.: \textit{Eigenfunction expansions of ordinary
linear differential operators in a finite interval}, Soviet Math. Dokl.,
\textbf{3} (1962), 1510-1514.

\bibitem{mam} Mamedov Yu. A.:\textit{\ On functions expansion to the series
of solutions residue of irregular boundary value problems}, in ``
Investigations on differential equations'', AGU named after S.M. Kirov
(1984), 94-98.

\bibitem{naim} Naimark M.A.: \textit{Linear differential operators}, vols.
\textbf{1-2}, Ungar Publishing, (1967).

\bibitem{ras1} Rasulov M. L.: \textit{Methods of Contour Integration, }
North-Holland, Amsterdam, 1967.

\bibitem{ras2} Rasulov M. L.: \textit{Applications of the contour Integral
Method, }Nauka, Moscow, 1975.

\bibitem{ros} Rosell J. I., Farjas J., Herrero R., Pi F. and Orriols G.:
\textit{Homoclinic phenomena in opto-thermal bistability with localized
absorption, }Phys. D, \textbf{85} (1995), 509-547.

\bibitem{sam} Samarskii A. A.: \textit{Some problems in differential
equations theory}, Differ. Uravn. \textbf{16}, (1980), 11, 1925-1935.

\bibitem{shk1} Shkalikov A. A.: \textit{On completenes of eigenfunctions and
associated function of an ordinary differential operator with separated
irregular boundary conditions}, Func. Anal. i evo Prilozh, \textbf{10}
(1976), 4, 69-80.

\bibitem{sil3} Silchenko J. T.: \textit{Differential equations with
non-densely defined operator coefficients, generating semigroups with
singularities}, Nonlinear Anal., \textbf{36} (1999), 3, Ser. A: Theory
Methods, 345--352.

\bibitem{sil4} Silchenko Yu. T.: \textit{An ordinary differential operator
with irregular boundary conditions} (Russian), Sibirsk. Mat. Zh. \textbf{40}
(1999), 1, 183--190, iv; translation in Siberian Math. J. \textbf{40}
(1999), 1, 158--164.

\bibitem{skub} Skubachevskii A. L., Steblov G. M.: \textit{On spectrum of
differential operators with domain non-dense in }$L_{2}(0,1) $, DAN SSSR
\textbf{321} (1991), 1158-1163.

\bibitem{tam} Tamarkin J. D.: \textit{About Certain General Problems of
Theory of Ordinary Linear Differential Equations and about Expansions of
Derivative Functions into series, }Petrograd, 1917.

\bibitem{tret} Tretter C.: \textit{On $\lambda $-Nonlinear Boundary
Eigenvalue Problems}, Akademic Verlag, Berlin, 1993.

\bibitem{vag} Vagabov A. I.: \textit{On eigenvectors of irregular
differential pencils with non-separated boundary conditions, }DAN SSSR,
\textbf{261} (1981), 2, 268-271.

\bibitem{yak2} Yakubov S.: \textit{Completness of Root Functions of Regular
Differential Operators}, Longman Scientific Technical, New York, (1994).

\bibitem{yak3} Yakubov S., Yakubov Y.: \textit{Differential-operator
equations. Ordinary and partial differential equations}, Chapman \& Hall/CRC
Monographs and Surveys in Pure and Applied Mathematics, 103. Chapman \&
Hall/CRC, Boca Raton, FL, (2000).

\bibitem{yak5} Yakubov S.: \textit{On a new method for solving irregular
problems}, J. Math. Anal. Appl. \textbf{220} (1998), 1, 224--249.

\bibitem{yak6} Yakubov Y: \textit{Irregular boundary value problems for
ordinary differential equations}, Analysis (Munich) \textbf{18} (1998), 4,
359--402.

\bibitem{yur} Yurchuk N. I.:\textit{Mixed problem with an integral condition
for certain parabolic equations, }Differ. Uravn., \textbf{22} (12) (1986),
2117-2126.
\end{thebibliography}

\end{document}