\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 \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 $$\Delta (\lambda )=\det U(\lambda ) \label{eq7}$$ 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 $$y^{0}(x,\lambda )=\Big( M(x)+0\big( \frac{1}{|\lambda | ^{\alpha }}\big) \Big) \exp \Big( \lambda \int_{a}^{x}D(\xi )d\xi \Big) , \label{eq8}$$ 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 $$U_{pq}(\lambda )=A_{pq}(\lambda )+B_{pq}(\lambda )e^{\lambda \omega _{q}}\,,\quad p,q=\overline{1,n}, \label{eq9}$$ where $$A_{pq}(\lambda )=\sum_{s=1}^{n}\sum_{k=0}^{P}\lambda ^{k}\alpha _{ps}^{(k)}\widehat{M}_{sq}(a)\,, \label{eq10}$$ and $$B_{pq}(\lambda )=\sum_{s=1}^{n}\sum_{k=0}^{P}\lambda ^{k}\beta _{ps}^{(k)}\widehat{M}_{sq}(b). \label{eq10'}$$ 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 $$\Delta (\lambda )=\det\begin{pmatrix} A^{(1)}+B^{(1)}e^{\lambda \omega _{1}} & \dots & A^{(n)}+B^{(n)}e^{\lambda \omega _{n}} \end{pmatrix}. \label{eq11}$$ Using (\ref{eq7}), (\ref{eq9}), (\ref{eq10}), and (\ref{eq10'}) we conclude from (\ref{eq11}) that the following asymptotic relations hold: $$\Delta (\lambda )e^{-\lambda \sum_{s=\kappa _{j}+\nu _{j}+1}^{n}\omega _{s}^{(2)}}=\widehat{M}_{1j}(\lambda )e^{m_{1j}Z}+\dots + \widehat{M}_{\sigma _{j}j}(\lambda )e^{m_{\sigma _{j}j}Z}\,, \label{eq12}$$ where m_{1j}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 functionf(\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 $$\mathop{\rm Re}\lambda \pi _{1}\leq \mathop{\rm Re}\lambda \pi _{2}\leq \dots \leq \mathop{\rm Re}\lambda \pi _{\tau_{j}}\leq 0\leq \mathop{\rm Re}\lambda \pi _{\tau _{j+1}} \leq \dots \leq \mathop{\rm Re}\lambda \pi_{n}\,. \label{eq13}$$ 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{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} where $$P_{ls}(\lambda )=\begin{cases} \dfrac{\lambda ^{-\kappa }e^{-\lambda W\sum_{m=1}^{n}A_{ms}(\lambda )\Delta _{ml}(\lambda )}}{\lambda ^{-\kappa }e^{-\lambda W}\Delta (\lambda )} & \text{if } l\leq \tau _{j} \\[8pt] \dfrac{\lambda ^{-\kappa }e^{-\lambda W+\lambda \pi _{l}x_{0l}\sum_{m=1}^{n}A_{ms}(\lambda )\Delta _{ml}(\lambda )}}{ \lambda ^{-\kappa }e^{-\lambda W}\Delta (\lambda )} &\text{if } l\geq \tau _{j}+1 \end{cases} \label{eq15}$$ $$Q_{ls}(\lambda )=\begin{cases} \dfrac{\lambda ^{-\kappa }e^{-\lambda W\sum_{m=1}^{n}B_{ms}(\lambda )\Delta _{ml}(\lambda )}}{\lambda ^{-\kappa }e^{-\lambda W}\Delta (\lambda )} & \text{if } l\leq \tau _{j} \\[8pt] \dfrac{\lambda ^{-\kappa }e^{-\lambda W+\lambda \pi _{l}x_{0l}\sum_{m=1}^{n}B_{ms}(\lambda )\Delta _{ml}(\lambda )}}{ \lambda ^{-\kappa }e^{-\lambda W}\Delta (\lambda )} & \text{if } l\geq \tau _{j}+1, \end{cases} \label{eq16}$$ where $$g_{pq}^{0}(x,\xi ,\lambda )=\begin{cases} \sum_{s=1}^{\tau _{j}}\widehat{M}_{ps}(x)\widehat{V}_{sq}(\xi )e^{\lambda \pi _{s}(x_{s}-\xi _{s})} & \text{if } a\leq \xi \leq x\leq b \\[3pt] -\sum_{s=\tau _{j}+1}^{n}\widehat{M}_{ps}(x)\widehat{V}_{sq}(\xi )e^{\lambda \pi _{s}(x_{s}-\xi _{s})} & \text{if } a\leq x\leq \xi \leq b, \end{cases} \label{eq17}$$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 $$G_{pq}(x,\xi ,\lambda )=0(1)\,. \label{eq18}$$ \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]$: $$\frac{-1}{2\pi \sqrt{-1}}\sum_{\nu }\int_{\Gamma _{\nu }}y(x,\lambda ,f)d\lambda =\sum_{\nu }\mathop{\rm Res} y(x,\lambda ,f)=D^{-1}(x)f(x)\,, \label{eq19}$$ 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{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} 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{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} 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} \begin{thebibliography}{99} \bibitem{benz} H. E. Benzinger,\emph{Green's function for ordinary differential operators,} J. Diff. Equations, \textbf{7 } (1970), 3, 478-496 . \bibitem{birkh1} \ G. D. 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