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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 14, pp. 1--9.\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-2007/14\hfil Boundary-value problems]
{Boundary-value problems for ordinary differential equations with
matrix coefficients containing a spectral parameter}

\author[M. Denche, A. Guerfi\hfil EJDE-2007/14\hfilneg]
{Mohamed Denche, Amara Guerfi}  % in alphabetical order

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

\address{Amara Guerfi \newline
Department of Mathematics and Computer engineering\\
Faculty of Science and Engineering\\
University of Ouargla \\
30000 Ouargla, Algeria}
\email{amaraguerfi@yahoo.fr}

\thanks{Submitted August 15, 2006. Published January 8, 2007.}
\subjclass[2000]{34L10, 34E05, 47E05}
\keywords{Characteristic determinant; expansion formula;
 Green matrix; \hfill\break\indent regularity conditions}

\begin{abstract}
 In the present work, we study a multi-point boundary-value problem
 for an ordinary differential equation with matrix coefficients
 containing a spectral parameter in the boundary conditions.
 Assuming some regularity conditions, we show that the characteristic
 determinant has an infinite number of zeros, and specify their
 asymptotic behavior. Using the asymptotic behavior of Green matrix
 on contours expending at infinity, we establish the series expansion
 formula of sufficiently smooth functions in terms of residuals
 solutions to the given problem. This formula actually gives the
 completeness of root functions as well as the possibility of
 calculating the coefficients of the series.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

We study a multi-point boundary-value problem
\begin{gather}
y'-\lambda A(x,\lambda )y=f(x),\quad -\infty <a\leq x\leq b<\infty, \label{eq1}\\
L(y)=\sum_{k=0}^{P}\lambda ^{k}\big( \alpha
^{(k)}y(a,\lambda )+\beta ^{(k)}y(b,\lambda )\big) =0,
\label{eq2}
\end{gather}
with
\[
A(x,\lambda )=\sum_{j=0}^{\infty }\lambda ^{-j}A_{j}(x)\,,
\]
where $\lambda \gg 1$,
$A_{j}(x)$, ($j=0,1,\dots $), $\alpha ^{(k)},\beta ^{(k)}$ are
matrices of order
$n\times n$, $f(x)$ is a vector function of order $n$, which is continuous
(or integrable bounded) in $[ a,b]$.

The study of the boundary-value problem \eqref{eq1}--\eqref{eq2} in the case of
ordinary differential equations originates in the papers by
Birkhoff \cite{birkh1}, \cite{birkh2}. Later,
Tamarkin  \cite{tam} considered the same problem, under
more general hypothesis, and introduced  the
classes of regular and strongly regular problems.

We note that boundary-value problems with a parameter in the boundary
conditions have interesting applications, since many concrete problems of
mathematical physics (e.g., \cite{tikh}) lead to problems of this form.
This happens whenever one applies the method of separation of variables to
solve the corresponding partial differential equation with boundary
conditions, which contain a directional derivative.

In general, the spectral properties of \eqref{eq1}--\eqref{eq2} are mainly
determined not only by the boundary conditions, but also by the highest
coefficients of all the polynomials in $\lambda $, $A(x,\lambda )$. Hence,
for the same boundary conditions, but different matrix functions
$A(x,\lambda )$, the problems can be both regular and non regular.

Various questions connected with the theory of ordinary differential
operators have been studied intensively; see for example
\cite{dan,naim,khrom,ras2,ras1,shk,shk1,vag2,vag3,vag4,vag5,yak}.
Problems of the form \eqref{eq1}--\eqref{eq2}, for the case of first
order systems, have been  studied in \cite{ras2},
where the results by Tamarkin  \cite{tam} were generalized.
In this paper, we consider more general problems of the
form \eqref{eq1}--\eqref{eq2}, where
regularity conditions are taken to be more  general than those
in \cite{naim,ras2}, and coincide with the one in \cite{ras1},
 when the coefficients of the equation are independent of
$\lambda $. Using the asymptotic behavior of the system of
fundamental solutions of equation (\ref{eq1}) given in \cite{vag1},
we formulate the regularity notion of the problem
\eqref{eq1}--\eqref{eq2}. For the introduced regular problems, we
show that the characteristic determinant $\Delta (\lambda )$ has an
infinite number of zeros. We establish that in the exterior of
$\delta $-neighboring of those zeros the elements of the Green
matrix have the uniform estimate
$G_{pq}(x,\xi ,\lambda )=0( 1) $. Using this estimate on the contours
which expand at infinity, we obtain
the series expansion formula of sufficiently smooth functions in terms of
solutions residuals to the given problem. In fact, this formula gives the
completeness of root functions as well as the possibility of calculating the
coefficients of the series.

\section{Preliminaries}\label{sec2}

 Suppose that:
\begin{enumerate}
\item \label{cond1}
$A_{j}(x)$ belongs to $C[ a,b]$ for $j=0,1,\dots$.

\item \label{cond2} For $x\in [ a,b]$, the roots
$\varphi _{1}(x),\dots ,\varphi _{n}(x)$ of the characteristic equation
in the sense of Birkhoff \cite{ras2}
\begin{equation}
\det (A_{0}(x)-\varphi E)=0,\,  \label{eq3}
\end{equation}
are distinct, not identically zero, their arguments and the
arguments of their differences are independent of $x$.

Let $M(x)$ be a matrix which transforms $A_{0}(x)$ to the diagonal matrix
$D(x)$ i.e.
\begin{equation*}
M^{-1}(x)A_{0}(x)M(x)=D(x)=\mathop{\rm diag}(\varphi_{1}(x),\dots ,\varphi _{n}(x)).
\end{equation*}
We require that at least one of the matrix $M'(x)$, $A_{1}(x)$
belong to the Holder space $H_{\alpha }$

\item \label{cond3} For $|\lambda| $ sufficiently large,
the following matrix has rank $n\times 2np$:
\[
\begin{pmatrix}
\alpha _{11}^{(1)}  \dots\;  \alpha _{1n}^{(1)} & \dots &
\alpha_{11}^{(P)}  \dots\;  \alpha _{1n}^{(P)}
& \beta _{11}^{(1)} \dots\; \beta _{1n}^{(1)} & \dots
& \beta _{11}^{(P)} \dots\;  \alpha _{1n}^{(P)}\\
\vdots &  &  &  &  &  \vdots \\
\alpha _{n1}^{(1)}  \dots\;  \alpha _{nn}^{(1)} & \dots &
\alpha_{n1}^{(P)}  \dots\; \alpha _{nn}^{(P)}
& \beta _{n1}^{(1)} \dots\; \beta _{nn}^{(1)} & \dots
& \beta _{n1}^{(P)} \dots\; \beta _{nn}^{(P)}
\end{pmatrix}
\end{equation*}

\end{enumerate}
We first start by giving the notion of sectors that we need later on. For
this purpose, we consider the set of values $\lambda $ that satisfy
\begin{equation}
\mathop{\rm Re}\lambda \varphi _{k}(x)=\mathop{\rm Re}\lambda \varphi _{s}(x)\quad k\neq
s\; x\in [ a,b].  \label{eq4}
\end{equation}
This equality  determines a finite number of sectors $(\Sigma_{j})$
for which by a convenable numeration of zeros of  (\ref{eq3}), we
have the inequalities
\begin{equation*}
\mathop{\rm Re}\lambda \varphi _{1}(x)\leq \mathop{\rm Re}\lambda \varphi _{2}(x)\leq \dots \leq
\mathop{\rm Re}\lambda \varphi _{n}(x).
\end{equation*}
Consider now the set of values $\lambda $ satisfying
\begin{equation}
\mathop{\rm Re}\lambda \omega _{s}=0~,\quad s=\overline{1,n}~,
\label{eq5}
\end{equation}
where $\omega _{s}=\int_{a}^{b}\varphi _{s}(t)dt$. By condition
\ref{cond2} the equalities (\ref{eq5}) define a certain number of
straight lines coming through the origin of $\lambda $-plane, and
each is applied by the origin into two straight-half lines through
this origin. We denote them by $d_{1},d_{2},\dots ,d_{2\mu }$, and the
argument of $d_{j}$ by $-\alpha _{j}+\frac{\pi }{2}$, where
$\alpha_{j}$ are numerated as follows:
\begin{equation*}
0\leq \alpha _{1}<\alpha _{2}<\dots <\alpha _{2\mu }<2\pi .
\end{equation*}
Consider a second set of straight half lines $d_{j}'$ $(j=\overline{
1,2\mu })$ distributed as
\begin{equation*}
d_{1}', d_{1}, d_{2}', d_{2},\dots , d_{2\mu}',
d_{2\mu }, d_{1}'.
\end{equation*}
The rays $d_{j}'$ divided the $\lambda $-plane into $2\mu$ sectors
$T_{1},T_{2},\dots,T_{2\mu }$. Let us consider an arbitrary
$T_{j}$.
Let $\omega _{1j},\dots,\omega _{\upsilon _{j}j}$ be taken from the
numbers $\omega _{1},\dots ,\omega _{p}$, which are situated on a straight
line issued from the origin and making an angle $\alpha _{j}$
with the real axis:
\begin{equation*}
\omega _{sj}=\mu _{sj}e^{\alpha _{j}\sqrt{-1}},\quad
s=\overline{1,\nu _{j}}.
\end{equation*}
In addition, we can always choose a numeration of the numbers
$\omega _{sj}$ such that we have the following inequalities hold:
\begin{equation*}
\mu _{1j}<\mu _{2j}<\dots <\mu _{s_{j}j}<0<\mu _{s_{j}+1j}<\dots
<\mu _{\nu_{j}j}.
\end{equation*}
If all $\mu _{sj}$ are strictly positive, then we put $s_{j}=0$. Otherwise,
if $\mu _{sj}$ are strictly negative, then $s_{j}=\nu _{j}$.

After excluding $\omega _{sj}\,(\overline{1,\nu _{j}})$ from the set
$\{ \omega _{1},\dots ,\omega _{n}\} $, the remaining $\omega _{s}$
can be divided into two groups: $(\omega _{s}^{(1)}),(\omega _{s}^{(2)})$.
The first group is formed by those one for which
$\mathop{\rm Re}\lambda \omega _{s}\to -\infty $, whereas the second group
those for $\mathop{\rm Re}\lambda \omega _{s}\to +\infty $. Hence, in each
$(T_{j})$ the roots of equation (\ref{eq3}) are numerated as
\begin{equation*}
\omega _{1}^{(1)},\dots ,\omega _{\kappa _{j}}^{(1)},\omega _{1j},\dots ,\omega
_{\nu _{j}j},\omega _{\kappa _{j}+\nu _{j}+1}^{(2)},\dots ,\omega _{n}^{(2)}.
\end{equation*}

The boundaries of the sectors $(\Sigma_{j})$ and $(T_{j})$ divide the whole
$\lambda $-plane into a finite number of sectors $(R_{j})$, where each of
those is simultaneously situated in one of the sectors $(T_{j})$ and in one
of the sectors $(\Sigma_j )$. So, in $(R_{j})$ we have
\begin{align*}
\mathop{\rm Re}\lambda \varphi _{1}(x)
&\leq \mathop{\rm Re}\lambda \varphi _{2}(x)\leq \dots \leq
\mathop{\rm Re}\lambda \varphi _{\tau _{j}}(x)\leq 0 \\
&\leq \mathop{\rm Re}\lambda \varphi _{\tau _{j+1}}(x)\leq \dots
\leq \mathop{\rm Re}\lambda \varphi_{n}(x),
\end{align*}
where $\tau _{j}=\kappa _{j}+s_{j}$.

\begin{definition} \label{def1}\rm
A sequence of curves $\Gamma _{\nu }$ in the
$\lambda $-plane is called an expanding sequence, if there is a
 constant $K$ such that, for $\lambda \in $ $\Gamma _{\nu }$ and
all positive integer $\nu $, the inequalities
$\mathop{\rm meas} \Gamma _{\nu }\leq Kr_{\nu }$, and
$|d\lambda| \leq r_{\nu }'d\theta $ hold, where
$r_{\nu }$ is the distance from the origin of $\lambda $-plane to the
nearest point of the $\Gamma _{\nu }$, $r_{\nu }'$ is the largest
distance between points of curve $\Gamma _{\nu }$, and $d\theta $  is the
angle subtended by the chord $d\lambda $ at the origin.
\end{definition}

\begin{lemma}[\cite{vag2}] \label{le1}
Let $\xi (\lambda ,z,x)$ be a continuous
function defined in the half-plane $\mathop{\rm Re}{c}\lambda \leq 0$,
with  $c$ constant not equal to zero, $x\in [ a,b]$,
 $z\in (0,Z)$. Suppose that
\begin{equation*}
|\xi (\lambda ,z,x)| \leq c/|\lambda
| ^{\alpha }~,\text{ \ }\alpha >\frac{1}{2},
\quad |\lambda| \gg 1.
\end{equation*}
Let $\psi (z)$ be a bounded function. Then
\begin{equation*}
J(\psi )=\int_{0}^{Z}\psi (z)dz\int_{\Gamma _{\nu
}}\xi (\lambda ,z,x)e^{c\lambda Z}d\lambda
\end{equation*}
tends to zero uniformly with respect to $x\in [ a,b]$,
as $\nu $ approaches infinity on the contour
$\Gamma _{\nu}$(where $\Gamma _{\nu }$ is an expanding sequence
situated in the half-plane $\mathop{\rm Re} {c}\lambda \leq 0 $).
\end{lemma}

\section{Main Results}

\subsection*{Construction of the Green Matrix}
The Green matrix of problem \eqref{eq1}--\eqref{eq2} is
\begin{equation*}
G(x,\xi ,\lambda )=g(x,\xi ,\lambda )-y^{0}(x,\lambda )U^{-1}(\lambda
)L(g(x,\xi ,\lambda ))\,,
\end{equation*}
where
\begin{gather*}
G(x,\xi ,\lambda )=\big( G_{pq}(x,\xi ,\lambda )\big)_{p,q=1}^{n},
\quad U(\lambda )=L(y^{0}(x,\lambda )=\big( U_{pq}(\lambda
)\big) _{p,q=1}^{n}, \\
 L(g(x,\xi ,\lambda ))=\big( L_{pq}(g(x,\xi ,\lambda ))\big)_{p,q=1}^{n},
\end{gather*}
$y^{0}(x,\lambda )$ is the solution of the homogeneous
equation (\ref{eq1}), and
\begin{equation*}
G_{pq}(x,\xi ,\lambda )=\frac{\Delta _{pq}
(x,\xi ,\lambda )}{\Delta (\lambda)},
\end{equation*}
where
\begin{gather*}
\Delta _{pq}(x,\xi ,\lambda )
=\det \begin{pmatrix}
g_{pq}(x,\xi ,\lambda ) & y_{p1}^{0}(x,\lambda ) & \dots &
y_{pn}^{0}(x,\lambda ) \\
L_{1q}(g) & U_{11}(\lambda ) & \dots & U_{1n}(\lambda ) \\
\vdots & \vdots &  & \vdots \\
L_{nq}(g) & U_{n1}(\lambda ) & \ldots & U_{nn}(\lambda )
\end{pmatrix}, \\
g_{pq}(x,\xi ,\lambda )=\begin{cases}
\frac{1}{2}\sum_{s=1}^{n}y_{pq}^{0}(x,\lambda )Z_{sq}(\xi
,\lambda )
& \text{if }  a\leq \xi \leq x\leq b \\
-\frac{1}{2}\sum_{s=1}^{n}y_{pq}^{0}(x,\lambda )Z_{sq}(\xi
,\lambda )
& \text{if }  a\leq x\leq \xi \leq b,
\end{cases}
\end{gather*}
$Z(x,\lambda )=T(x,\lambda )/W(x,\lambda )$, where
$T(x,\lambda )$ is the matrix of order $n\times n$ when we take the
transposed of the matrix made up using the co-factors of the
elements of the matrix $y^{0}(x,\lambda )$, and
$W(x,\lambda )=\det y^{0}(x,\lambda )$,
\begin{gather*}
L_{pq}(g(x,\xi ,\lambda
))=\sum_{s=1}^{n}\sum_{k=0}^{P}\lambda ^{k}\big(
\alpha _{ps}^{(k)}g_{sq}(a,\xi ,\lambda )+\beta
_{ps}^{(k)}g_{sq}(b,\xi ,\lambda )\big) ,
\\
U_{pq}(\lambda )=\sum_{s=1}^{n}\sum_{k=0}^{P}\lambda
^{k}\big( \alpha _{ps}^{(k)}y_{sq}^{0}(a,\lambda )+\beta
_{ps}^{(k)}y_{sq}^{0}(b,\lambda )\big) ,\,
\end{gather*}
where
\begin{equation}
\Delta (\lambda )=\det U(\lambda )   \label{eq7}
\end{equation}
is the characteristic determinant of problem \eqref{eq1}--\eqref{eq2}.
Thus, the general solution of problem \eqref{eq1}--\eqref{eq2}
is
\begin{equation*}
y(x,\lambda ,f)=\int_{a}^{b}G(x,\xi ,\lambda )f(\xi )d\xi,
\end{equation*}
for $x\in \lbrack a,b]$.

\subsection*{Asymptotic Representation of the Zeros of the Characteristic
Determinant}

According to the Vagabov theorem \cite{vag1}, the fundamental system of
solutions for the homogeneous equation corresponding to (\ref{eq1}), have in
each sector $(\Sigma_j )$ the asymptotic behavior
\begin{equation}
y^{0}(x,\lambda )=\Big( M(x)+0\big( \frac{1}{|\lambda
| ^{\alpha }}\big) \Big) \exp \Big( \lambda
\int_{a}^{x}D(\xi )d\xi \Big) ,  \label{eq8}
\end{equation}
where $0<\alpha \leq 1$, $x\in \lbrack a,b]$, and
$M(x)=(M_{pq}(x)) _{p,q=1}^{n}$ is one of the matrix indicated in condition
\ref{cond2}.
Using the notation
\begin{equation*}
\widehat{\Phi }(x)=\Phi (x)+0\big( \frac{1}{|\lambda|
^{\alpha }}\big) ,
\end{equation*}
and substituting (\ref{eq8}) from the boundary conditions (\ref{eq2}), we
obtain
\begin{equation}
U_{pq}(\lambda )=A_{pq}(\lambda )+B_{pq}(\lambda )e^{\lambda \omega
_{q}}\,,\quad p,q=\overline{1,n}, \label{eq9}
\end{equation}
where
\begin{equation}
A_{pq}(\lambda )=\sum_{s=1}^{n}\sum_{k=0}^{P}\lambda
^{k}\alpha _{ps}^{(k)}\widehat{M}_{sq}(a)\,,  \label{eq10}
\end{equation}
and
\begin{equation}
B_{pq}(\lambda )=\sum_{s=1}^{n}\sum_{k=0}^{P}\lambda
^{k}\beta _{ps}^{(k)}\widehat{M}_{sq}(b).  \label{eq10'}
\end{equation}
On the other hand, if we denote
\begin{equation*}
A^{(q)}=\begin{pmatrix}
A_{1q} \\
\vdots \\
A_{nq} \end{pmatrix},
\quad
B^{(q)}=\begin{pmatrix}
B_{1q} \\
\vdots \\
B_{nq} \end{pmatrix},
\end{equation*}
then $\Delta (\lambda )$ can be written in the form
\begin{equation}
\Delta (\lambda )=\det\begin{pmatrix}
A^{(1)}+B^{(1)}e^{\lambda \omega _{1}} & \dots & A^{(n)}+B^{(n)}e^{\lambda
\omega _{n}}
\end{pmatrix}.  \label{eq11}
\end{equation}
Using (\ref{eq7}), (\ref{eq9}), (\ref{eq10}), and (\ref{eq10'}) we conclude
from (\ref{eq11}) that the following asymptotic relations hold:
\begin{equation}
\Delta (\lambda )e^{-\lambda \sum_{s=\kappa _{j}+\nu
_{j}+1}^{n}\omega _{s}^{(2)}}=\widehat{M}_{1j}(\lambda )e^{m_{1j}Z}+\dots +
\widehat{M}_{\sigma _{j}j}(\lambda )e^{m_{\sigma _{j}j}Z}\,,
\label{eq12}
\end{equation}
where $m_{1j}<m_{2j}<\dots <m_{\sigma _{j}j}$,
$Z=\lambda e^{\exp(\alpha _{j}\sqrt{-1})}$, and
\begin{gather*}
m_{1j}= \begin{cases}
\sum_{s=1}^{s_{j}}\mu _{sj} & \text{for }  s_{j}>0 \\
0 & \text{for }  s_{j}=0,
\end{cases}
\quad
 m_{\sigma _{j}j}=\begin{cases}
\sum_{s=s_{j}+1}^{\nu _{j}}\mu _{sj} & \text{for }
s_{j}<\nu _{j}
\\
0 & \text{for }  s_{j}=\nu _{j},
\end{cases}
\\
\begin{aligned}M_{1j}(\lambda )= \det
\Big(& A^{(1)}\dots A^{(\kappa _{j})}B^{(\kappa _{j}+1)}\dots 
B^{(\kappa_{j}+s_{j})}A^{(\kappa _{j}+s_{j}+1)}\dots \\
&A^{(\kappa _{j}+\nu _{j})}B^{(\kappa _{j}+\nu_{j}+1)}\dots B^{(n)}
\Big),
\end{aligned}\\
M_{\sigma _{j}j}=\det\begin{pmatrix}
A^{(1)}&\dots& A^{(\kappa _{j}+s_{j})}B^{(\kappa
_{j}+s_{j}+1)}&\dots& B^{(n)}\end{pmatrix} .
\end{gather*}

\begin{definition}  \label{def2} \rm
 A function $f(\lambda )$ $\,$is called an
asymptotic power function of degree $\kappa $, if there exist
$a\in \mathbb{C}\backslash \{ 0\} $, $0<\alpha \leq 1$ and
$\kappa \in \mathbb{Z}$ such that
\begin{equation*}
f(\lambda )=\lambda ^{\kappa }\Big( a+0\big( \frac{1}{|\lambda
| ^{\alpha }}\big) \Big) ,\quad |\lambda| \to \infty .
\end{equation*}
\end{definition}

A similar definition is given in \cite{benz} and \cite{eber}
 for $\alpha =1 $.

\begin{definition}[Regularity]  \label{def3} \rm
 The boundary-value problem \eqref{eq1}--\eqref{eq2} is said to be regular
 if in all sectors $R_{j}$, the
functions $M_{1j}(\lambda )$ are asymptotic power functions of degree
$\kappa $ where\ $\kappa $ is a positive integer, and all the other
determinants built by different columns of the matrix\
$(A^{(1)}\dots A^{(n)}B^{(1)}\dots B^{(n)})$ are asymptotic power functions of
degree $\leq \kappa $.
\end{definition}

\begin{theorem}\label{th1}
Suppose that the boundary-value problem
\eqref{eq1}--\eqref{eq2} is regular, and the
conditions \ref{cond1}, \ref{cond2}, \ref{cond3},
of section \ref{sec2} are satisfied, then in each sector $(T_{j})$ we have
\begin{enumerate}
\item $\Delta (\lambda )$ admits an infinite number of zeros which can be
divided into $2\mu $ groups. The values of $j^{th}-$ group are contained in
the strip ($D_{j}$) of finite width and parallel to rays $d_{j}$ which is
inside ($D_{j}$).

\item  If the interiors of circles of sufficiently small radius $\delta $ with
centers at zeros of \thinspace $\Delta (\lambda )$ are removed, then in the
remained plane, we get
\begin{equation*}
\big|\lambda ^{-\kappa }\Delta (\lambda )\exp\big(-\lambda
\sum_{s=\kappa _{j}+\nu _{j}+1}^{n}\omega
_{s}^{(2)}\big)\big| \geq k_{\delta }\,,
\end{equation*}
where $k_{\delta }$ is a positive number depending only on
$\delta$.

\item The number of zeros of $\Delta (\lambda )$ which are near to the origin
is finite. The zeros $\lambda _{N}^{(j)}$ of $j^{th}$-group have the
asymptotic representation
\begin{equation*}
|\lambda _{N}^{(j)}| =\frac{2N\pi }{m_{\sigma
_{j}j}-m_{1j}}\big( 1+0\big( \frac{1}{N}\big) \big) .
\end{equation*}

\item Each zero of $\Delta (\lambda )$ is a pole of the solution of
problem \eqref{eq1})--\eqref{eq2}.
\end{enumerate}
\end{theorem}

The proof of this theorem can be done as in  \cite[Theorem 4, page 205]{ras1}.

\subsection*{Asymptotic Representation of a Solution of
Boundary Value Problem \eqref{eq1}--\eqref{eq2}}
According to condition \ref{cond2} of section \ref{sec2}, the root
arguments of the characteristic equation (\ref{eq3}) are independent of $x$.
So, we can write
\begin{equation*}
\varphi _{s}(x)=\pi _{s}q_{s}(x),\quad
x\in [ a,b],\; s=\overline{1,n}\,,
\end{equation*}
where $\pi _{s}$is in general a complex constant, $q_{s}(x)>0$,
hence from (\ref{eq8}) it results
\begin{equation}
\mathop{\rm Re}\lambda \pi _{1}\leq \mathop{\rm Re}\lambda \pi _{2}\leq \dots
\leq \mathop{\rm Re}\lambda \pi _{\tau_{j}}\leq 0\leq \mathop{\rm Re}\lambda \pi _{\tau _{j+1}}
\leq \dots \leq \mathop{\rm Re}\lambda \pi_{n}\,.   \label{eq13}
\end{equation}
Let us set
\begin{equation*}
x_{s}=\int_{a}^{x}q_{s}(t)dt,\quad
\xi_{s}=\int_{a}^{\xi }q_{s}(t)dt,\quad
x_{0s}=\int_{a}^{b}q_{s}(x)dt.
\end{equation*}
By appropriate transformations, the matrix $G(x,\xi ,\lambda )$ can
be written, in each sector $R_{j}(\delta)$
(where $R_{j}(\delta)$ denotes the remaining part of sector $R_{j}$ after
removing the interior of the circle of sufficiently small rays
$\delta $ centered in the zeros of $\Delta (\lambda )$), in the
following form
\begin{equation}
\begin{aligned}
G_{pq}(x,\xi ,\lambda )
&=g_{pq}^{0}(x,\xi ,\lambda )
+\Big( \sum_{l=1}^{\tau _{j}}\sum_{s=\tau
_{j}+1}^{n}P_{ls}(\lambda )\widehat{M}_{pl}(x)\widehat{V}_{sq}(\xi
)e^{\lambda \pi _{l}x_{l}-\lambda
\pi _{s}\xi _{s}} \\
&\quad +\sum_{l=\tau _{j}+1}^{n}\sum_{s=\tau
_{j}+1}^{n}P_{ls}(\lambda )\widehat{M}_{pl}(x)\widehat{V}_{sq}(\xi
)e^{\lambda \pi _{l}(x_{l}-x_{0l})-\lambda \pi _{s}\xi _{s}} \\
&\quad +\sum_{l=1}^{\tau _{j}}\sum_{s=1}^{\tau _{j}}Q_{ls}(\lambda )
\widehat{M}_{pl}(x)\widehat{V}_{sq}(\xi )e^{\lambda \pi _{l}x_{l}-\lambda
\pi _{s}(\xi _{s}-x_{0s})} \\
&\quad +\sum_{l=\tau _{j}+1}^{n}\sum_{s=1}^{\tau
_{j}}Q_{ls}(\lambda )\widehat{M}_{pl}(x)\widehat{V}_{sq}(\xi
)e^{\lambda \pi_{l}(x_{l}-x_{0l})
-\lambda \pi _{s}(\xi _{s}-x_{0s})}\Big) ,
\end{aligned}   \label{eq14}
\end{equation}
where
\begin{equation}
P_{ls}(\lambda )=\begin{cases}
\dfrac{\lambda ^{-\kappa }e^{-\lambda
W\sum_{m=1}^{n}A_{ms}(\lambda )\Delta _{ml}(\lambda
)}}{\lambda ^{-\kappa }e^{-\lambda W}\Delta (\lambda )}
& \text{if }  l\leq \tau _{j} \\[8pt]
\dfrac{\lambda ^{-\kappa }e^{-\lambda W+\lambda \pi
_{l}x_{0l}\sum_{m=1}^{n}A_{ms}(\lambda )\Delta _{ml}(\lambda )}}{
\lambda ^{-\kappa }e^{-\lambda W}\Delta (\lambda )} &\text{if } l\geq
\tau _{j}+1
\end{cases}  \label{eq15}
\end{equation}
\begin{equation}
Q_{ls}(\lambda )=\begin{cases}
\dfrac{\lambda ^{-\kappa }e^{-\lambda
W\sum_{m=1}^{n}B_{ms}(\lambda )\Delta _{ml}(\lambda
)}}{\lambda ^{-\kappa }e^{-\lambda W}\Delta (\lambda )}
& \text{if }  l\leq \tau _{j} \\[8pt]
\dfrac{\lambda ^{-\kappa }e^{-\lambda W+\lambda \pi
_{l}x_{0l}\sum_{m=1}^{n}B_{ms}(\lambda )\Delta _{ml}(\lambda )}}{
\lambda ^{-\kappa }e^{-\lambda W}\Delta (\lambda )} & \text{if }
 l\geq \tau _{j}+1,
\end{cases}   \label{eq16}
\end{equation}
where
\begin{equation}
g_{pq}^{0}(x,\xi ,\lambda )=\begin{cases}
\sum_{s=1}^{\tau
_{j}}\widehat{M}_{ps}(x)\widehat{V}_{sq}(\xi )e^{\lambda \pi
_{s}(x_{s}-\xi _{s})} & \text{if } a\leq \xi \leq x\leq b
\\[3pt]
-\sum_{s=\tau
_{j}+1}^{n}\widehat{M}_{ps}(x)\widehat{V}_{sq}(\xi
)e^{\lambda \pi _{s}(x_{s}-\xi _{s})} & \text{if }  a\leq x\leq \xi \leq b,
\end{cases}  \label{eq17}
\end{equation}
$W=\sum_{s=\kappa _{j}+\nu _{j}+1}^{n}\omega _{s}^{(2)}$,
the $V_{sq}(\xi )$ is the element of the matrix $V(x)$ which verifies
$M(x)V(x)=I$, $\Delta _{ms}(\lambda )$ is the complement algebraic of
the element $(m,s)$ in $\Delta (\lambda )$.

\begin{theorem} \label{th2}
Suppose that the boundary-value problem
\eqref{eq1}--\eqref{eq2} is regular, and the conditions
\ref{cond1}, \ref{cond2}, \ref{cond3}, of section \ref{sec2} are satisfied.
Then, in each sector $R_{j}(\delta)$ the elements
$G_{pq}(x,\xi ,\lambda )$ of
the Green matrix admits the  estimate
\begin{equation}
G_{pq}(x,\xi ,\lambda )=0(1)\,.   \label{eq18}
\end{equation}
\end{theorem}

\begin{proof}
 Numerators in (\ref{eq15}), (\ref{eq16}) are bounded in
$R_{j}(\delta)$ for large $\lambda $. It follows from Theorem
\ref{th1} that the denominators are bounded below by a positive number in
$R_{j}(\delta)$. In other words, the functions $P_{ls}(\lambda )$ and
$Q_{ls}(\lambda )$ are uniformly bounded outside $\delta$-neighborhoods of
the zeros. Then (\ref{eq18}) follows directly from
\eqref{eq14}--\eqref{eq17}.
\end{proof}

\subsection*{An Expansion Formula}

\begin{theorem}  \label{th3}
If the boundary-value problem \eqref{eq1}--\eqref{eq2} is regular,
the Holder power satisfies  $\frac{1}{2}<\alpha \leq 1$, and the
conditions \ref{cond1}, \ref{cond2}, \ref{cond3}, of
section \ref{sec2}, are satisfied, then for all $f(x)\in L_{2}[a,b]$, the
following expansion formula holds in the sense of $L_{2}[a,b]$:
\begin{equation}
\frac{-1}{2\pi \sqrt{-1}}\sum_{\nu }\int_{\Gamma
_{\nu }}y(x,\lambda ,f)d\lambda =\sum_{\nu
}\mathop{\rm Res} y(x,\lambda ,f)=D^{-1}(x)f(x)\,,  \label{eq19}
\end{equation}
where $\Gamma _{\nu }$ is a simple closed contour containing only one pole
$\lambda _{\nu }$ of the integrand, and the sum over $\nu $ is extended to
all poles of this function. Here, $\mathop{\rm Res}_{z_{\nu }}F(z)$ denotes
the residual of $F(z)$ at $z_{\nu }$.
\end{theorem}

\begin{proof}
Theorem \ref{th1} implies that the distance between the zeros of
$\Delta(\lambda )$ is larger than some sufficiently small positive
 number $2\delta$. Then, we can choose a sequence of closed expanding
contours $\Gamma_{\nu }$, which does not intersect circles of radius
$\delta $ centered at the zeros of $\Delta (\lambda )$.
Since each $\Gamma _{\nu }$ is the union
of its parts in the sectors $R_{j}$, we can conclude from (\ref{eq14}),
 that
\begin{equation}
\begin{aligned}
&\int_{\Gamma _{\nu }}d\lambda
\sum_{q=1}^{n}\int_{a}^{b}G_{pq}(x,\xi ,\lambda
)f_{q}(\xi )d\xi \\
&=\sum_{j}\int_{\Gamma _{\nu }\cap R_{j}}d\lambda
\Big(\sum_{q=1}^{n}\int_{a}^{b}g_{pq}^{0}(x,\xi ,\lambda
)f_{q}(\xi )d\xi  \\
&\quad +\sum_{q=1}^{n}\int_{a}^{b}\Big(\sum_{l=1}^{\tau
_{j}}\sum_{s=\tau _{j}+1}^{n}P_{ls}(\lambda )\widehat{M}_{pl}(x)
\widehat{V}_{sq}(\xi )e^{\lambda \pi _{l}x_{l}-\lambda \pi _{s}\xi
_{s})} \\
&\quad +\sum_{l=\tau _{j}+1}^{n}\sum_{s=\tau
_{j}+1}^{n}P_{ls}(\lambda )\widehat{M}_{pl}(x)\widehat{V}_{sq}(\xi
)e^{\lambda \pi _{l}(x_{l}-x_{0l})-\lambda \pi _{s}\xi _{s})} \\
&\quad +\sum_{l=1}^{\tau _{j}}\sum_{s=1}^{\tau _{j}}Q_{ls}(\lambda )
\widehat{M}_{pl}(x)\widehat{V}_{sq}(\xi )e^{\lambda \pi _{l}x_{l}-\lambda
\pi _{s}(\xi _{s}-x_{0s})} \\
&\quad  +\sum_{l=\tau_{j}+1}^{n}\sum_{s=1}^{\tau _{j}}Q_{ls}(\lambda
)\widehat{M}_{pl}(x)\widehat{V}_{sq}(\xi )e^{\lambda \pi
_{l}(x_{l}-x_{0l})-\lambda \pi _{s}(\xi _{s}-x_{0s})}\Big) \Big) ,
\end{aligned}
\label{eq20}
\end{equation}
 here, $\sum_{j}$ denotes the sum over all $R_{j}$. From
(\ref{eq15})-(\ref{eq16}), the regularity of problem
\eqref{eq1}--\eqref{eq2} and the choice of $\Gamma _{\nu }$, it
follows that the $P_{ls}(\lambda )$, $Q_{ls}(\lambda )$ are
uniformly bounded on all $\Gamma _{\nu }$. Inequalities
(\ref{eq13}) imply that the real parts of all exponents in the
right side of (\ref{eq20}) are non-positive.
Using \cite[Lemma 1]{ras1}, \cite[Lemma 3]{ras1}
and Lemma \ref{le1}, it follows that
\begin{equation}
\begin{aligned}
&\lim_{\nu \to +\infty }\int_{\Gamma _{\nu
}}d\lambda \sum_{q=1}^{n}\int_{a}^{b}G_{pq}(x,\xi
,\lambda )f_{q}(\xi )d\xi\\
&=\lim_{\nu \to +\infty }\sum_{j}\int_{\Gamma
_{\nu }\cap R_{j}}d\lambda
\sum_{q=1}^{n}\int_{a}^{b}g_{pq}^{0}(x,\xi ,\lambda
)f_{q}(\xi )d\xi .
\end{aligned} \label{eqq21}
\end{equation}
By substituting the expression (\ref{eq17}) into (\ref{eqq21}), and using
Lemma \ref{le1}, appropriate transformations yield
\begin{equation*}
\sum_{\nu }\int_{\Gamma _{\nu }}y(x,\lambda
,f)d\lambda =\sum_{\nu }\mathop{\rm Res}\int_{a}^{b}G(x,\xi
,\lambda )f(\xi )d\xi =-2\pi \sqrt{-1}D^{-1}(x)f(x).
\end{equation*}
\end{proof}

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\end{document}