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.
$$
A direct computation yields
\begin{alignat}{2}
G^\Delta(0,s)&=\frac{\sigma^2(1)-\sigma(s)}{\sigma^2(1)}>0,
&\qquad &0\leq s< 1,\label{gf1}\\
G^\Delta(\sigma(1),s)&=-\frac{\sigma(s)}{\sigma^2(1)}<0,
&\qquad &0~~0.
\end{aligned}
$$
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~~

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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% BIBLIOGRAPHY \begin{thebibliography}{99} \bibitem {AgBo} R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, preprint. \bibitem {AhLa} S. Ahmad and A. Lazer, Positive operators and Sturmian theory of nonselfadjoint second order systems, {\it Nonlinear Equations in Abstract Spaces}, Academic Press, New York, 1978, pp.25--42. \bibitem {Am} H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, {\it SIAM Rev.} {\bf 18} (1976), 620--709. \bibitem {AuHi} B. Aulback and S. Hilger, Linear dynamic processes with inhomogeneous time scale, Nonlinear Dynamics and Quantum Dynamical Systems, {\em Math. Res.} {\bf 59}, Akademie Verlag, Berlin, 1990. \bibitem {DavElHe} J.M. Davis, P.W. Eloe, and J. Henderson, Comparison of eigenvalues for discrete Lidstone boundary value problems, {\it Dynam. Systems Appl.}, in press. \bibitem {ElHe1} P.W. Eloe and J. Henderson, Comparison of eigenvalues for a class of two-point boundary value problems, {\it Appl. Anal.} {\bf 34} (1989), 25--34. \bibitem {ElHe2} P.W. Eloe and J. Henderson, Comparison of eigenvalues for a class of multipoint boundary value problems, {\it WSSIAA} {\bf 1} (1992), 179--188. \bibitem {ErHi} L.H. Erbe and S. Hilger, Sturmian theory on measure chains, {\em Differential Equations Dynam. Systems} {\bf 1}(1993), 223--246. \bibitem {ErPe1} L.H. Erbe and A. Peterson, Green's functions and comparison theorems for differential equations on measure chains, {\em Dynam. Contin. Discrete Impuls. Systems}, in press. \bibitem {ErPe2} L.H. Erbe and A. Peterson, Positive solutions for a nonlinear differential equation on a measure chain, preprint. \bibitem {GeTr} R.D. Gentry and C.C. Travis, Comparison of eigenvalues associated with linear differential equations of arbitrary order, {\it Trans. Amer. Math. Soc.} {\bf 223} (1976), 167--179. \bibitem{HaPe1} D. Hankerson and A. Peterson, Comparison theorems for eigenvalue problems for $n$th order differential equations, {\it Proc. Amer. Math. Soc.} {\bf 104} (1988), 1204--1211. \bibitem{HaPe2} D. Hankerson and A. Peterson, A positivity result applied to difference equations, {\it J. Approx. Theory} {\bf 59} (1989), 76--86. \bibitem{HaPe3} D. Hankerson and A. Peterson, Comparison of eigenvalues for focal point problems for $n$th order difference equations, {\it Differential Integral Equations} {\bf 3} (1990), 363--380. \bibitem {Hi} S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, {\em Results Math.} {\bf 18}(1990), 18--56. \bibitem{Ka} E.R. Kaufmann, Comparison of eigenvalues for eigenvalue problems for a right disfocal operator, {\it Panamer. Math. J.} {\bf 4} (1994), 103--124. \bibitem {KaLaSi} B. Kaymakcalan, V. Lakshmikantham and S. Sivasundaram, {\em Dynamical Systems on Measure Chains}, Kluwer Academic Publishers, Boston, 1996. \bibitem{KeTr1} M.S. Keener and C.C. Travis, Positive cones and focal points for a class of $n$th order differential equations, {\it Trans. Amer. Math. Soc.} {\bf 237} (1978), 331--351. \bibitem{KeTr2} M.S. Keener and C.C. Travis, Sturmian theory for a class of nonselfadjoint differential systems, {\it Ann. Mat. Pura Appl.} {\bf 123} (1980), 247--266. \bibitem{Kr} M.A. Krasnosel'skii, {\it Positive Solutions of Operator Equations}, P. Noordhoff Ltd., Groningen, The Netherlands, 1964. \bibitem{KrRu} M.G. Krein and M.A. Rutman, Linear operators leaving a cone invariant in a Banach space, in {\it American Mathematical Society Translations, Series 1}, Providence, 1962. \bibitem{La} T.M. Lamar, Analysis of a $2n$th Order Differential Equation With Lidstone Boundary Conditions, Ph.D. Dissertation, Auburn University, 1997. \bibitem{To1} E. Tomastik, Comparison theorems for conjugate points of $n$th order nonselfadjoint differential equations, {\it Proc. Amer. Math. Soc.} {\bf 96} (1986), 437--442. \bibitem{To2} E. Tomastik, Comparison theorems for focal points of systems of $n$th order nonselfadjoint differential equations, {\it Rocky Mountain J. Math.} {\bf 18} (1988), 1--12. \bibitem{Tr} C.C. Travis, Comparison of eigenvalues for linear differential equations of order $2n$, {\it Trans. Amer. Math. Soc.} {\bf 177} (1973), 363--374. \end{thebibliography} \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% ********************************************************************* * John M Davis Office: Math Annex 147 * * Dept of Mathematics Phone: (334) 844-3621 * * Auburn University Fax: (334) 844-6555 * * Auburn, AL 36849-5310 WWW: http://www.auburn.edu/~davis05/ * *********************************************************************