\documentclass{amsart} 
\begin{document} 
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.\ 1998(1998), No.~35, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad  ftp ejde.math.unt.edu (login: ftp)}
\thanks{\copyright 1998 Southwest Texas State University  and 
University of North Texas.} 
\vspace{1.5cm}

\title[\hfilneg EJDE--1998/35\hfil Eigenvalue comparisons on a measure 
chain]
{Eigenvalue comparisons for differential equations on a measure chain} 

\author[C.~J.~Chyan, J.~M.~Davis, J.~Henderson, \& W.~K.~C.~Yin\hfil 
EJDE--1998/35\hfilneg]
{Chuan~Jen~Chyan, John~M.~Davis, Johnny~Henderson, \& William~K.~C.~Yin}

\address{Chuan~Jen~Chyan \hfill\break
          Department of Mathematics\\
          Tamkang University\\
          Taipei, Taiwan }
\email{chuanjen@mail.tku.edu.tw } 

\address{John~M.~Davis \hfill\break
        Department of Mathematics\\
        Auburn University\\
        Auburn, AL 36849 USA} 
\email{davis05@mail.auburn.edu}

\address{Johnny~Henderson \hfill\break
       Department of Mathematics\\
       Auburn University\\
       Auburn, AL 36849 USA} 
\email{hendej2@mail.auburn.edu}

\address{William~K.~C.~Yin \hfill\break
      Department of Mathematics\\
      LaGrange College\\
      LaGrange, GA 30240 USA} 
\email{wyin@lgc.edu}

\thanks{Submitted November 23, 1998. Published December 19, 1998.}
\subjclass{34B99, 39A99 }
\keywords{Measure chain, eigenvalue problem }


\begin{abstract}
The theory of $\mathbf{u_0}$-positive operators with respect to a cone in 
a Banach space is applied to eigenvalue problems associated with the 
second order $\Delta$-differential equation (often referred to as a 
differential equation on a measure chain) given by 
        $$
y^{\Delta\Delta}(t)+\lambda p(t)y(\sigma(t))=0, \qquad t\in[0,1],
        $$
satisfying the boundary conditions $y(0)=0=y(\sigma^2(1))$. The existence 
of 
a smallest positive eigenvalue is proven and then a theorem is 
established comparing the smallest positive eigenvalues for two problems 
of this type.  
\end{abstract}

\maketitle

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem*{remark}{Remark}
\newtheorem{lemma}{Lemma}

\theoremstyle{definition}
\newtheorem{definition}{Definition}
\numberwithin{equation}{section}
\numberwithin{theorem}{section}
\numberwithin{definition}{section}
\numberwithin{lemma}{section}
\numberwithin{corollary}{section}

\def\R{{\mathbb R}}													
\def\into{\rightarrow}
\def\B{{\mathcal B}}
\def\P{{\mathcal P}}
\def\e{\varepsilon}		
\newcommand{\norm}[1]{||#1||}

%%%

\section{Background} 
In this paper, we are concerned with comparing the smallest positive 
eigenvalues for second order $\Delta$-differential equations satisfying 
conjugate boundary conditions.  Much recent attention has been given to 
differential equations on measure chains, and we refer the reader to 
\cite{AuHi,ErHi,Hi} for some historical works as well as to the more 
recent papers \cite{AgBo,ErPe1,ErPe2} and the book \cite{KaLaSi} for 
excellent references on these types of equations.  Before introducing the 
problems of interest for this paper, we present some definitions and 
notation which are common to the recent literature. Our sources for this 
background material are the two papers by Erbe and Peterson 
\cite{ErPe1,ErPe2}.

\begin{definition}  
Let $T$ be a closed subset of $\R$, and let $T$ have the subspace 
topology inherited from the Euclidean topology on $\R.$  The set $T$ is 
referred to as a {\em measure chain} or, in some places in the 
literature, a {\em time scale}. For $t < \sup T$ and $r > \inf T$, define 
the {\em forward jump operator}, $\sigma$, and the {\em backward jump 
operator}, $\rho$, respectively, by 
	$$
	\begin{aligned}
	\sigma(t)&=\inf \{\tau \in T \ |\ \tau > t \} \in T,\\ 
	\rho(r)&=\sup \{\tau \in T \ |\ \tau < r \} \in T,
	\end{aligned}
	$$ 
for all $t, r \in T$. If $\sigma(t) > t$, $t$ is said to be {\em right 
scattered}, and if $\rho(r)<r$, $r$ is said to be {\em left scattered}. 
If $\sigma(t) = t$, $t$ is said to be {\em right dense}, and if $\rho(r) 
= 
r$, $r$ is said to be {\em left dense}. 
\end{definition}

\begin{definition} 
For $x:T \to \R$ and $t \in T$ (if $t = \sup T$, assume $t$ is not left 
scattered), define the {\em delta derivative of} $x(t)$, denoted by 
$x^\Delta 
(t)$, to be the number (when it exists), with the property that, for any 
$\epsilon > 0$, there is a neighborhood, $U$, of $t$ such that 
	$$
	\Big\vert[x(\sigma(t)) - x(s)] - x^\Delta (t) [\sigma(t) - s]
	\Big\vert \leq \epsilon \Big\vert\sigma(t) -s\Big\vert,
	$$
for all $s \in U$. The {\em second delta derivative} of $x(t)$ is defined 
by 		
	$$
	x^{\Delta\Delta}(t) = (x^\Delta)^\Delta(t).
	$$
If $F^\Delta (t) = h(t)$, then define the {\em integral} by 
	$$
	\int_a^t h(s) \Delta s = F(t) - F(a). 
	$$
\end{definition}

Throughout, we will assume that $T$ is a closed subset of $\R$ with 
$0,1\in T$.

\begin{definition}
Define the closed interval, $[0,1]\subset T$ by 
	$$
	[0, 1]:=\{t\in T \mid 0\leq t\leq 1\}.
	$$
Other closed, open, and half-open intervals in $T$ are similarly defined. 
\end{definition}

For convenience, we will use interval notation, $[0,1]$ and inequalities 
such as $0\leq t\leq 1$ interchangeably.

We are concerned with the comparison of the eigenvalues for the 
eigenvalue problems
	\begin{alignat}{2} 
	y^{\Delta\Delta}(t)+\lambda_1\, p(t)y(\sigma(t))&=0, \qquad &t\in[0,1], 
\label{e1}\\
	y^{\Delta\Delta}(t)+\lambda_2\, q(t)y(\sigma(t))&=0, \qquad &t\in[0,1], 
\label{e2}
	\end{alignat}
satisfying the two-point conjugate boundary conditions
	\begin{equation}\label{e3}
	y(0)=0=y(\sigma^2(1)),
	\end{equation}
where we assume $0<p(t)\leq q(t)$ for $t\in[0,1]$.

To be more precise, we will first establish the existence of smallest 
positive eigenvalues for \eqref{e1}, \eqref{e3} and \eqref{e2}, 
\eqref{e3}, respectively, and then we will compare these smallest 
positive eigenvalues. Our techniques involve applications from the theory 
of $\mathbf{u_0}$-positive operators with respect to a cone in a Banach 
space as it is developed in Krasnosel'skii's book \cite{Kr} or in the 
book by Krein and Rutman \cite{KrRu}. Also, we make use of the sign 
properties of an appropriate Green's function.

Results of this type are not without motivation. The cone theory 
techniques we apply here have been successfully applied by several 
authors in comparing eigenvalues for boundary value problems for ordinary 
differential equations including two-point, multipoint, focal, right 
focal, and Lidstone problems; for example, see 
\cite{AhLa,Am,ElHe1,ElHe2,GeTr,HaPe1,Ka,KeTr1,KeTr2,La, To1,To2,Tr}. In 
addition, a few smallest eigenvalue comparison results have been obtained 
for boundary value problems for finite difference equations. A 
representative set of references for these works would be 
\cite{DavElHe,HaPe2,HaPe3}.

In the development of this paper, we include in Section 2 preliminary 
definitions and fundamental results from the theory of 
$\mathbf{u_0}$-positive operators with respect to a cone in a Banach 
space. Then, in Section 3, we apply the results of Section 2 in comparing 
the smallest positive
eigenvalues of \eqref{e1}, \eqref{e3} and \eqref{e2}, \eqref{e3}. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%
%%%%  SECTION 2: Background

\section{Cones and $\mathbf{u_0}$-Positive Operators}

In this section, we provide definitions and auxillary results from cone 
theory which we will apply in the next section to the eigenvalue problems 
\eqref{e1}, \eqref{e3} and \eqref{e2}, \eqref{e3}. Most of the discussion 
of this section involving the theory of cones in a Banach space arises 
from results in Krasnosel'skii's book \cite{Kr}.

\begin{definition}
Let $\B$ be a Banach space over $\R$. A nonempty, closed set 
$\P\subset\B$ is said to be a {\it cone} provided
\begin{itemize}
	\item[(i)] $\alpha \mathbf{u}+\beta \mathbf{v} \in \P$ for all 
$\mathbf{u,v}\in\P$ and all $\alpha,\beta\geq 0$, and
	\item[(ii)] $\mathbf{u,-u}\in \P$ implies $\mathbf{u}=\mathbf{0}$.
\end{itemize}
A cone is said to be {\it reproducing\/} if $\B = \P - \P$. A cone is 
said to be {\it solid\/} if $\P^\circ\not=\emptyset$, where $\P^\circ$ 
denotes the interior of $\P$.
\end{definition}

\begin{remark} \label{r1} 
Krasnosel'skii {\cite{Kr}} proved that every solid cone is reproducing.
\end{remark}

\begin{definition}
A Banach space $\B$ is called a {\it partially ordered Banach space\/} if 
there exists a partial
ordering $\preceq$ on $\B$ satisfying 
	\begin{itemize}
	\item[(i)] $\mathbf{u}\preceq\mathbf{v}$, for $\mathbf{u,v}\in\B$ implies
	$t\mathbf{u}\preceq t\mathbf{v}$, for all $t\geq 0$, and 
	\item[(ii)] $\mathbf{u}_1\preceq\mathbf{v}_1$ and 
	$\mathbf{u}_2\preceq\mathbf{v}_2$, for
	$\mathbf{u}_1,\mathbf{u}_2,\mathbf{v}_1,\mathbf{v}_2\in\B$ imply 
	$\mathbf{u}_1 + \mathbf{u}_2 \preceq \mathbf{v}_1 + \mathbf{v}_2$.
	\end{itemize}
Let $\P\subset\B$ be a cone and define $\mathbf{u}\preceq\mathbf{v}$ if 
and only if $\mathbf{v}-\mathbf{u}\in\P$. Then $\preceq$ is a partial 
ordering on $\B$ and we will say that $\preceq$ is the partial ordering 
induced by $\P$. Moreover, $\B$ is a partially ordered Banach space with 
respect to $\preceq$.
\end{definition}

\begin{definition}
If $L_1,L_2:\B \rightarrow\B$ are bounded, linear operators, then we say 
that $L_1\preceq L_2$ {\it with respect to\/} $\P$ provided 
$L_1\mathbf{u}\preceq L_2\mathbf{u}$ for every $\mathbf{u}\in P$. A 
bounded, linear operator $L_1:\B\rightarrow \B$ is {\it 
$\mathbf{u_0}$-positive with respect to $\P$} if there exists 
$\mathbf{u}_0\in\P$, $\mathbf{u}_0\not = \mathbf{0}$, such that for each 
nonzero $\mathbf{u}\in\P$, there exist $k_1(\mathbf{u}), k_2(\mathbf{u})$ 
such that $k_1\mathbf{u}_0\preceq L_1\mathbf{u}\preceq k_2\mathbf{u}_0$.
\end{definition}

Of the next three results, the first two can be found in Krasnosel'skii's 
book \cite{Kr} and the third result is proved by Keener and Travis 
\cite{KeTr1} as an extension of results from \cite{Kr}.

\begin{theorem} \label{t1}
Let $\B$ be a Banach space over $\R$ and let $\P\subset\B$ be a solid 
cone. If $L_1:\B\rightarrow\B$ is a linear operator such that 
$L_1:\P\setminus\{\mathbf{0}\}\rightarrow\P^\circ$, then $L_1$ is 
$\mathbf{u_0}$-positive with respect to $\P$.
\end{theorem}

\begin{theorem} \label{t2}
Let $\B$ be a Banach space over $\R$ and let $\P\subset\B$ be a 
reproducing cone. Let $L_1:\B\rightarrow\B$ be a compact, linear operator 
which is $\mathbf{u_0}$-positive with respect to $\P$. Then $L_1$ has an 
essentially unique eigenvector in $\P$, and the corresponding eigenvalue 
is simple, positive, and larger than the absolute value of any other 
eigenvalue.
\end{theorem}

\begin{theorem} \label{t3}
Let $\B$ be a Banach space over $\R$ and let $\P\subset\B$ be a cone. Let 
$L_1,L_2:\B\rightarrow\B$ be bounded, linear operators, and assume that 
at least one of the operators is $\mathbf{u_0}$-positive with respect to 
$\P$. If $L_1\preceq L_2$ with respect to $\P$, and if there exist 
nonzero $\mathbf{u_1,u_2}\in\P$ and positive real numbers $\lambda_1$ and 
$\lambda_2$ such
that $\lambda_1\mathbf{u_1}\preceq L_1\mathbf{u_1}$ and 
$L_2\mathbf{u_2}\preceq\lambda_2\mathbf{u_2}$, then 
$\lambda_1\leq\lambda_2$. Moreover, 
if $\lambda_1=\lambda_2$, then $\mathbf{u_1}$ is a scalar multiple of 
$\mathbf{u_2}$.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%
%%%%% SECTION 3: Main

\section{Eigenvalue Comparisons for the Boundary Value Problems}

In order to apply the results of Section 2 concerning the theory of 
$\mathbf{u_0}$-positive operators, we now introduce a suitable Banach 
space, $\B$, and a cone, $\P$, in the Banach space.  Define $\B$ by
	$$
	\begin{aligned}
	\B:=\{x:[0,\sigma^2(1)]\rightarrow\R \mid\  &x^\Delta \text{ exists and 
is 
bounded 
	on } [0,\sigma(1)],\\	
	& \text{ and } x \text{ satisfies the boundary conditions } \eqref{e3}\}
	\end{aligned}
	$$
and let the norm $\norm{\cdot}$ on $\B$ be defined by
	$$ 
	\norm{x}:=\max\Big\{\sup_{t\in[0,\sigma^2(1)]} 
|x(t)|,\sup_{t\in[0,\sigma(1)]} 
	|x^\Delta (t)|\Big\}.
	$$
Notice that if $\norm{x}=0$, then $x(t)\equiv 0$. Define the cone 
$\P\subset\B$ by
	$$ 
	\P:=\{x\in\B \mid x(t)\geq 0 \text{ for } t\in[0,\sigma^2(1)]\}. 
	$$

\begin{lemma}\label{l1}
The cone $\P$ has nonempty interior and
	$$
	Q:=\{x\in\P\mid x(t)>0 \text{ on }(0,\sigma^2(1)),\ x^\Delta(0)>0,\ 
	x^\Delta(\sigma(1))<0\}\subset\P^\circ.
	$$
\end{lemma}

\begin{proof}
Choose $x(t)\in Q$. Our only concern is the positivity of $x(t)$ in a 
right deleted neighborhood of $t=0$ and in a left deleted neighborhood of 
$t=\sigma^2(1)$. If $t=0$ is right dense, then by the definition of $Q$ 
we 
have $x'(0)>0$. If $t=0$ is right scattered, then $x(\sigma(0))>0$. In 
either 
case, $x(t)>0$ on any right deleted neighborhood of $t=0$. Now consider 
the right endpoint. If $t=\sigma^2(1)$ is left dense, then 
$x^\Delta(\sigma(1))=x'(\sigma^2(1))<0$. If $t=\sigma^2(1)$ is left 
scattered, then 
$x(\sigma(1))>0$. Again, in either case, $x(t)>0$ on any left deleted 
neighborhood of $t=\sigma^2(1)$. 
\end{proof}

\begin{corollary}
The cone $\P$ is solid and hence reproducing.
\end{corollary}

Next we define the linear operators $L_1,L_2:\B\into\B$ by
	\begin{alignat}{1}
	L_1x(t)&=\int_0^{\sigma(1)}G(t,s)p(s)x(\sigma(s))\Delta s,\label{oper1}\\
	L_2x(t)&=\int_0^{\sigma(1)}G(t,s)q(s)x(\sigma(s))\Delta s,\label{oper2}
	\end{alignat}
respectively, where $G(t,s)$ is the Green's function for 
	$$
	-x^{\Delta\Delta}(t)=0
	$$
satisfying \eqref{e3}. That is, 
	$$
	G(t,s)=
	\begin{cases}
	\frac{t\big(\sigma^2(1)-\sigma(s)\big)}{\sigma^2(1)}, &\qquad 0\leq 
t\leq s\leq \sigma(1),\\
	\frac{\sigma(s)\big(\sigma^2(1)-t\big)}{\sigma^2(1)}, &\qquad 0\leq 
\sigma(s)\leq t\leq 
\sigma^2(1),
	\end{cases}
	$$
on $[0,\sigma^2(1)]\times [0,\sigma(1)]$; see Erbe and Peterson 
\cite{ErPe1,ErPe2}. Note that
	$$
	G(t,s)>0 \quad \text{ on } (0,\sigma^2(1))\times (0,\sigma(1)).
	$$
	
\begin{lemma}\label{l1.5}
Let $\lambda_1$ be an eigenvalue of \eqref{e1}, \eqref{e3} and $u(t)$ be 
the 
corresponding eigenvector. Then
	$$
	u(t)=\lambda_1\int_0^{\sigma(1)}G(t,s)p(s)u(\sigma(s))\Delta s.
	$$
That is, $\frac{1}{\lambda_1}u=L_1 u$. Hence, the eigenvalues of 
\eqref{e1}, 
\eqref{e3} are reciprocals of the eigenvalues of \eqref{oper1} and 
conversely.
\end{lemma}

\begin{lemma}\label{l2}
The linear operators $L_1$ and $L_2$ are $\mathbf{u_0}$-positive with 
respect to $\P$.
\end{lemma}

\begin{proof}
We prove the statement is true for the operator $L_1$. By 
Theorem~\ref{t1}, we only need to show that 
$L_1:\P\setminus\{\mathbf{0}\}\into\P^\circ$. To this end, choose 
$v\in\P\setminus\{\bf 0\}$. Then, for $t\in(0,\sigma^2(1))$,
	$$
	L_1v(t)=\int_0^{\sigma(1)}G(t,s)p(s)v(\sigma(s))\Delta s >0.
	$$
A direct computation yields
	\begin{alignat}{2}
	G^\Delta(0,s)&=\frac{\sigma^2(1)-\sigma(s)}{\sigma^2(1)}>0, 
	&\qquad &0\leq s< 1,\label{gf1}\\
	G^\Delta(\sigma(1),s)&=-\frac{\sigma(s)}{\sigma^2(1)}<0, 
	&\qquad &0<s\leq \sigma(1)\label{gf2}.
	\end{alignat}
By \eqref{gf1}, we obtain
	$$
	\begin{aligned}
	(L_1v)^\Delta(0)&=\int_0^{\sigma(1)}G^\Delta(0,s)p(s)v(\sigma(s))\Delta 
s\\
	&=\int_0^{\sigma(1)} 
\frac{\sigma^2(1)-\sigma(s)}{\sigma^2(1)}p(s)v(\sigma(s))\Delta s\\
	&>0.
	\end{aligned}
	$$
Similarly, $(L_1v)^\Delta(\sigma(1))<0$ by using \eqref{gf2}. Hence 
$L_1v\in 
Q\subset\P^\circ$.
\end{proof}

By the way the operators were defined, $L_1,L_2:\P\into\P$ and therefore 
$L_1$ and $L_2$ are bounded. It follows from standard arguments involving 
the Arzela-Ascoli Theorem that $L_1$ and $L_2$ are in fact compact 
operators. We may now apply Theorems~\ref{t2} and \ref{t3} to obtain the 
eigenvalue comparison we seek.

\begin{theorem}\label{t4}
Suppose $0<p(t)\leq q(t)$ for $0\leq t\leq 1$. Then the operator $L_1$ 
has an essentially unique eigenvector 
$u\in\P^\circ\setminus\{\mathbf{0}\}$, and the corresponding eigenvalue 
$\Lambda$ is simple, positive, and larger than the absolute value of any 
other 
eigenvalue.
\end{theorem}

\begin{proof}
The existence of such an eigenvalue $\Lambda$ with eigenvector $u\in\P$ 
follows from Theorem~\ref{t2}. Since $u\not\equiv\mathbf{0}$, the proof 
of Lemma~\ref{l2} shows $L_1u\in\P^\circ$. Since $L_1u=\Lambda u$, it 
follows 
that $u\in\P^\circ$.
\end{proof}

\begin{theorem}\label{t5}
Suppose $0<p(t)\leq q(t)$ for $0\leq t\leq 1$. Let $\Lambda_1$ and 
$\Lambda_2$ be 
the largest positive eigenvalues of $L_1$ and $L_2$, respectively. Then 
$\Lambda_1\leq\Lambda_2$. Furthermore, $\Lambda_1=\Lambda_2$ if and only 
if $p(t)\equiv q(t)$ 
for $0\leq t\leq 1$.
\end{theorem}

\begin{proof}
Let $\Lambda_1$ and $\Lambda_2$ be as in the statement of the theorem. 
Since by 
assumption $p(t)\leq q(t)$, we have, for $u\in\P$,
	$$
	\begin{aligned}
	L_1u(t)&=\int_0^{\sigma(1)}G(t,s)p(s)u(\sigma(s))\Delta s\\
	&\leq \int_0^{\sigma(1)}G(t,s)q(s)u(\sigma(s))\Delta s\\
	&=L_2u(t)
	\end{aligned}
	$$
and hence $L_1\preceq L_2$ with respect to $\P$. If $u_1,u_2\in\P^\circ$ 
are the essentially unique eigenvectors given by Theorem~\ref{t4} that 
correspond to $\Lambda_1$ and $\Lambda_2$, respectively. Theorem~\ref{t3} 
then 
yields $\Lambda_1\leq\Lambda_2$. 

For the final statement of the theorem, suppose that $p(t_0)<q(t_0)$ for 
some $t_0\in(0,1)$. The proof of Theorem~\ref{t4} shows $L_1u_1(t_0)>0$. 
It can be argued just as in Lemma~\ref{l2} that 
$(L_2-L_1)u_1\in\P^\circ$. But $u_1\in\P^\circ$ so for sufficiently small 
$\e>0$, it must be that $(L_2-L_1)u_1\geq \e u_1$. Therefore
	$$
	L_2u_1\geq L_1u_1+\e u_1=(\Lambda_1+\e)u_1.
	$$
Since $L_2u_2=\Lambda_2u_2$, if we apply Theorem~\ref{t2} to the operator 
$L_2$ we have $\Lambda_1+\e\leq \Lambda_2$ or equivalently 
$\Lambda_1<\Lambda_2$. Conversely, 
$\Lambda_1=\Lambda_2$ implies $p(t)=q(t)$ for all $t\in(0,1)$.
\end{proof}

In view that the eigenvalues of $L_1$ are reciprocals of the eigenvalues 
of \eqref{e1}, \eqref{e3}, and conversely, and in view of 
Theorems~\ref{t4} and \ref{t5}, we see that
	$$
	\lambda_1=\frac{1}{\Lambda_1}\geq \frac{1}{\Lambda_2}=\lambda_2.
	$$		
Moreover, if $p(t)\leq q(t)$ and $p(t)\not\equiv q(t)$, then
	$$
	\frac{1}{\Lambda_1}>\frac{1}{\Lambda_2}.
	$$	
We are now able to state the following comparison theorem for smallest 
positive eigenvalues, $\lambda_1$ and $\lambda_2$, of \eqref{e1}, 
\eqref{e3} and 
\eqref{e2}, \eqref{e3}.

\begin{theorem}
Assume the hypotheses of Theorem~\ref{t5}. Then there exist smallest 
positive eigenvalues $\lambda_1$ and $\lambda_2$ of \eqref{e1}, 
\eqref{e3} and 
\eqref{e2}, \eqref{e3}, respectively, each of which is simple and less 
than the absolute value of any other eigenvalue for the corresponding 
problem, and the eigenvectors corresponding to $\lambda_1$ and 
$\lambda_2$ may be 
chosen to belong to $\P^\circ$. Finally, $\lambda_1\geq \lambda_2$ with 
$\lambda_1=\lambda_2$ 
if and only if $p(t)\equiv q(t)$ on $0\leq t\leq 1$.
\end{theorem} 	

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}

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