\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 43, pp. 1--7.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/43\hfil Oscillation theory]
{Oscillation theory for a pair of second order dynamic equations with
a singular interface}
\author[P. K. Baruah, D. K. K. Vamsi\hfil EJDE-2008/43\hfilneg]
{Pallav Kumar Baruah, Dasu Krishna Kiran Vamsi} % in alphabetical order
\address{Pallav Kumar Baruah \newline
Department of Mathematics and Computer Science \\
Sri Sathya Sai University, Prasanthinilayam, India}
\email{baruahpk@sssu.edu.in}
\address{Dasu Krishna Kiran Vamsi \newline
Department of Mathematics and Computer Science \\
Sri Sathya Sai University, Prasanthinilayam, India}
\email{dkkvamsi@gmail.com}
\thanks{Submitted July 18, 2007. Published March 20, 2008.}
\subjclass[2000]{45C05, 34C10}
\keywords{Eigenvalues; eigenfunctions; dynamic equations;
angle function; \hfill\break\indent zeros of a function}
\begin{abstract}
In this paper we consider a pair of second order dynamic equations
defined on the time scale $I = [a,c]\cup [\sigma(c),b]$. We impose
matching interface conditions at the singular interface $c$. We
prove a theorem regarding the relationship between the number of
eigenvalues and zeros of the corresponding eigenfunctions.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\section{Introduction}
The study of waves plays a important role in physical sciences.
Waves of simple nature oscillate with a fixed frequency and
wave-length. The study of these simple sinusoidal waves form the
basis for the study of almost all forms of linear wave motion. The
oscillation nature of waves can be modelled by differential
equations specifically by ordinary Sturm-Liouville operators. In
\cite{b1}, the oscillatory nature of the self-adjoint second
order dynamic equation
\[
Lx(t) = (px^{\Delta})^{\Delta}(t) + q(t)x^{\sigma}(t)
\]
is discussed.
Also, Sturm's comparison and separation theorems have been
proved for the self-adjoint matrix equations
\[
LX(t) = (PX^{\Delta})^{\Delta}(t) + Q(t)X^{\sigma}(t).
\]
In \cite{r1}, the oscillatory and nonoscillatory behaviour of
solutions of second-order linear difference equations is
discussed. In literature of time scales, substantial amount of
work has been done on oscillation behaviour of nonlinear dynamic
equations \cite{b2,e1,e2,l1,o1,s1}.
In the literature we find a new class of interface problems,
termed as mixed pair of equations, discussed in the papers
\cite{p1,p4,p5,p6,p7,p8,v1,v2,v3,v4,v5,v6,v7} where two different
differential equations are defined on adjacent intervals with a
common point of interface and the solutions satisfy a matching
condition at the point of interface. We observe that the above
problem for the regular case has been discussed in
\cite{v1,v2,v3,v4,v5,v7}. In \cite{p1} the authors discuss an
application of the classical Weyl limit criterion to define the
coefficients with well-known Wronskian boundary conditions to
tackle the singularity at the boundary for this class of problems.
Though this work is specifically for Sturm-Liouville problems, it
paves a way to study the problem of singularity at the end
boundary points. But the problem of having a singularity at the
point of interface remained unexplored. This problem of having
singularity at the point of interface is discussed in
\cite{p2,p3}. In \cite{p2}, the Green's matrix is obtained for a
boundary value problem involving a pair of dynamic equations with
a singular interface. In \cite{p3}, the existence of matching
solutions for an initial value problem involving a pair of dynamic
equations with a singular interface is discussed.
In this paper we study the oscillation theory for dynamic
equations and also deal with the problem of having singularity
at the point of interface. In this direction, we intend to study
the oscillation behaviour for a pair of dynamic equations having a
singularity at the point of interface. In this paper, we prove
a theorem that gives the relationship between the number of
eigenvalues and zeros of the corresponding eigenfunctions.
In Section 2, we give few mathematical definitions, which we use
through the rest of the paper and in Section 3, we define the
pair of second order dynamic equations with matching interface
conditions. We also define the angle functions $\Theta$ and
$\hat{\Theta}$. In Section 4, we prove a theorem involving the
angle functions $\Theta$ and $\hat{\Theta}$. Finally, in Section
5, we prove a theorem that gives the relationship between number
of eigenvalues and zeros of the corresponding eigenfunctions.
\section{Mathematical Preliminaries}
\begin{definition} \label{def2.1} \rm
Let $ \mathbb{T}$ be a time scale. For $t \in \mathbb{T} $
we define the forward jump operator $\sigma : \mathbb{T} \to
\mathbb{T}$ by
\[
\sigma(t) := \inf \{s \in \mathbb{T} : s > t \},
\]
while the backward jump operator $ \rho : \mathbb{T} \to \mathbb{T}$
is defined by
\[
\rho(t) := \sup \{s \in \mathbb{T} : s < t \}.
\]
If $\sigma(t)> t$, we say that $t$ is right-scattered, while
$\rho(t) < t$ we say that $t$ is left-scattered. Points that are
right-scattered and left-scattered at the same time are called
isolated. Also, if $t < \sup \mathbb{T} $ and
$\sigma(t) = t$, then $t$ is called right-dense, and if
$t > \inf \mathbb{T}$ and $\rho(t) = t$, then $t$ is called
left-dense. Points that are right-dense and left-dense at the
same time are called dense. Finally, the graininess function
$\mu : \mathbb{T} \to [0,\infty)$ is defined by
\[
\mu(t) := \sigma(t) - t.
\]
\end{definition}
\begin{definition} \label{def2.2} \rm
A function $f : \mathbb{T} \to \mathbb{R}$ is called
rd-continuous provided it is continuous at right-dense
points in $\mathbb{T}$ and its left-sided limits exist (finite)
at left-dense points in $\mathbb{T}$. The set of rd-continuous
functions $f : \mathbb{T} \to \mathbb{R} $ will be denoted by
\[
\mathcal{C}_{rd} = \mathcal{C}_{rd}(\mathbb{T}).
\]
The set of functions $f : \mathbb{T} \to \mathbb{R}$ that are
differentiable and whose derivative is rd-continuous is denoted by
\[
\mathcal{C}_{rd}^{1} = \mathcal{C}_{rd}^{1}(\mathbb{T})
= \mathcal{C}_{rd}^{1}(\mathbb{T},\mathbb{R}).
\]
\end{definition}
\begin{definition} \label{def2.3} \rm
The function $\Theta(x,y) = \tan^{-1}\big(\frac{x}{y}\big) $
that gives the angle that the point $(x,y)$ makes with the positive
$x$-axis is called the angle function.
\end{definition}
\section{Pair of Dynamic Equations and the Angle Functions}
Let $I_1 = [a,c]$, $I_2 = [\sigma(c),b]$ and
$I = I_1 \cup I_2$ for $-\infty < a,b,c < \infty$.
Let
\begin{equation}
L_1X_1 = \frac{1}{r_1}(-(p_1X_1^{\Delta})^{\Delta} + q_1X_1)
\label{eqn1}
\end{equation}
be defined on $I_1$, and let
\begin{equation}
L_2X_2 = \frac{1}{r_2}(-(p_2X_2^{\Delta})^{\Delta} + q_2X_2)
\label{eqn2}
\end{equation}
be defined on $I_2$,
where $p_i \in \mathcal{C}_{rd}^{1}(I_i), q_i$ and
$r_i \in \mathcal{C}_{rd}^{1}(I_i)$ are real valued functions,
$p_i(t) > 0$ and $r_i(t) > 0$ for all $t \in I_i, i = 1,2$.
We assume that the functions $X_1, X_2$ satisfy the matching conditions
\begin{gather}
X_1(c) = X_2(\sigma(c)), \label{eqn3} \\
p_1(c)X_1^{\Delta}(c) = p_2(\sigma(c))X_2^{\Delta}(\sigma(c)).
\label{eqn4}
\end{gather}
For a real number $\lambda$, let us consider the pair of dynamic
equations
\begin{gather}
L_1X_1 = \lambda X_1 \quad \text{on } I_1 , \label{eqn5} \\
L_2X_2 = \lambda X_2 \quad \text{on } I_2 , \label{eqn6}
\end{gather}
together with the matching interface conditions
(\ref{eqn3}), (\ref{eqn4}). It follows from
\cite[Corollary 5.90 and Theorem 5.119]{b1} that
problem (\ref{eqn5}), (\ref{eqn6}) along with conditions
(\ref{eqn3}), (\ref{eqn4}) has two linearly independent real
valued solutions. For a real valued nontrivial solution
$X = (X_1,X_2)$ of (\ref{eqn5}), (\ref{eqn6}) along with
conditions (\ref{eqn3}), (\ref{eqn4}), we define the new
dependent variables $\rho_1, \rho_2$ and $\Theta_1, \Theta_2$ by
\begin{gather}
X_i(t) = \rho_i(t)\sin\Theta_i(t), \label{eqn7} \\
p_i(t)X_i^{\Delta}(t) = \rho_i(t)\cos\Theta_i(t), \label{eqn8}
\end{gather}
where the angle function $\Theta_i(t)$ satisfies
\begin{equation}
\Theta_i(t) = tan^{-1} \frac{X_i(t)}{p_i(t)X_i^{\Delta}(t)}
\quad \text{for } t \in I_i, \;i = 1,2. \label{eqn9}
\end{equation}
From conditions $ X_1(c) = X_2(\sigma(c))$,
$p_1(c)X_1^{\Delta}(c) = p_2(\sigma(c))X_2^{\Delta}(\sigma(c))$,
and
\[
\Theta_i(t) = \tan^{-1} \frac{X_i(t)}{p_i(t)X_i^{\Delta}(t)} \quad
\text{for } t \in I_i,\; i = 1,2,
\]
we get that
\begin{equation}
\Theta_1(c) = \Theta_2(\sigma(c)). \label{eqn10}
\end{equation}
We define
\[
\Theta(t) = \begin{cases}
\Theta_1(t), & t \in I_1 \\
\Theta_2(t), & t \in I_2.
\end{cases}
\]
We notice that $\Theta$ is a continuous and almost everywhere
continuously differentiable real valued function defined on
$I = I_1 \cup I_2$. Also, from relations (\ref{eqn7}), (\ref{eqn8}),
for any nontrivial solutions of (\ref{eqn5}), (\ref{eqn6}) along
with conditions (\ref{eqn3}), (\ref{eqn4}) it follows that
$\rho_i(t) \neq 0$ for all $t \in I_i, i = 1,2$, as
$\rho_i(t) = 0$ implies that $X_i(t) = 0$ contradicting our
assumption of nontrivial solutions. Also, since
$X_1(c) = X_2(\sigma(c))$ and $\Theta_1(c) = \Theta_2(\sigma(c))$,
from (\ref{eqn7}) we have $\rho_1(c) = \rho_2(\sigma(c))$,
and hence, for $t \in I$. $X(t) = 0$ if and only if $\Theta(t) = n\pi$,
for some integer $n$. If we let
$g_i(t) = \lambda r_i(t) - q_i(t)$, $t \in I_i, i = 1,2$,
equations (\ref{eqn5}), (\ref{eqn6}) can be rewritten in the form
\begin{gather}
(p_1X_1^{\Delta})^{\Delta} + g_1X_1 = 0 \quad \text{on }
I_1 \label{eqn11} \\
(p_2X_2^{\Delta})^{\Delta} + g_2X_2 = 0 \quad \text{on } I_2\,.
\label{eqn12}
\end{gather}
Let $\hat{p_i}, \hat{q_i}, \hat{r_i}, \hat{g_i}, \hat{\rho_i},
\hat{\Theta_i}$, $i = 1,2, $ be another set of functions as
defined preceding discussions. We define
\begin{gather*}
p(t) = \begin{cases}
p_1(t), & t \in I_1 \\
p_2(t), & t \in I_2,
\end{cases} \quad
\hat{p}(t) =\begin{cases}
\hat{p}_1(t), & t \in I_1 \\
\hat{p}_2(t), & t \in I_2,
\end{cases}
\\
g(t) = \begin{cases}
g_1(t), & t \in I_1 \\
g_2(t), & t \in I_2,
\end{cases} \quad
\hat{g}(t) = \begin{cases}
\hat{g}_1(t), & t \in I_1 \\
\hat{g}_2(t), & t \in I_2,
\end{cases}
\\
\hat{\Theta}(t) = \begin{cases}
\hat{\Theta}_1(t), & t \in I_1 \\
\hat{\Theta}_2(t), & t \in I_2.
\end{cases}
\end{gather*}
Upon delta differentiation, with respect to $t$, we have
from (\ref{eqn9}) for $\Theta(t)$,
$\Theta^{\Delta}(t) = \Theta'(t) $
(as $I_1$ and $I_2$ are continuous intervals and (\ref{eqn4})),
which is equal to
\begin{align*}
\frac{1}{1 + {\big(\frac{X(t)}{p(t)X'(t)}\big)}^2}
\frac{d}{dx}\frac{X(t)}{p(t)X'(t)}
&= \frac{p(t)(X'(t))^{2} - X(t)(p(t)X'(t))'}{(p(t)X'(t))^{2}
+ X^{2}(t)} \\
&= \frac{p(t)(X'(t))^2 + g(t)X^{2}(t)}{(p(t)X'(t))^{2} + X^{2}(t)}.
\end{align*}
Equations (\ref{eqn7}), (\ref{eqn8}) imply
\[
\Theta^{\Delta}(t) = \frac{1}{\rho^{2}(t)}
\Big( \frac{1}{p(t)}\rho^2(t)\cos^2(\Theta(t))
+ g(t)\rho^2(t)\sin^{2}\Theta(t)\Big).
\]
Hence,
\begin{equation}
\Theta^{\Delta}(t) = \frac{1}{p(t)} \sin^{2}\Theta(t)
+ g(t)\sin^{2}\Theta(t), \quad \text{for } t \in I. \label{eqn13}
\end{equation}
Similarly, for $\hat{\Theta}(t)$, we have
\begin{equation}
{\hat{\Theta}}^{\Delta}(t) = \frac{1}{\hat{p}(t)} \sin^2\hat{\Theta}(t)
+ g(t)\sin^2 \hat {\Theta}(t), \quad \text{for } t \in I. \label{eqn14}
\end{equation}
\section{A Theorem Involving Angle Functions}
\begin{theorem} \label{thm4.1}
Let $\hat{p}(t) \leq p(t)$ and $g(t) \leq \hat{g}(t)$ for all
$t \in I_1 \cup I_2$. Let $X(t)$ be a solution of \eqref{eqn11},
\eqref{eqn12}, \eqref{eqn3}, \eqref{eqn4}, and let $\hat{X}$ be a solution of
\begin{gather}
(\hat{p}_1 {\hat{X}_1}^{\Delta})^{\Delta}
+ \hat{g}_1 \hat{X}_1 = 0 \quad \text{on } I_1, \label{eqn15}
\\
(\hat{p}_2 {\hat{X}_2}^{\Delta})^{\Delta}
+ \hat{g}_2 \hat{X}_2 = 0 \quad \text{on } I_2, \label{eqn16}
\end{gather}
satisfying the matching conditions
\begin{gather}
\hat{X}_1(c) = \hat{X}_2(\sigma(c)), \label{eqn17}\\
\hat{p}_1(c){\hat{X}_1}^{\Delta}(c)
= \hat{p}_2(\sigma(c)){\hat{X}_2}^{\Delta}(\sigma(c)) \label{eqn18}
\end{gather}
Then, if $\hat{\Theta}(d) \geq \Theta(d)$ for some
$d \in (a,c] \cup [\sigma(c),b)$, then
$\hat{\Theta}(t) \geq \Theta(t)$ for all
$t \in (d,c] \cup [\sigma(c),b)$.
\end{theorem}
\begin{proof}
\textbf{Case I}
Let us suppose that $d \in [\sigma(c),b)$.
Then $\hat{\Theta}(d) \geq \Theta(d)$ implies that
$\hat{\Theta_2}(d) \geq \Theta_2(d)$, and from hypothesis of
the theorem we have that $\hat{p_2}(t) \leq p_2(t)$ and
$\hat{g_2}(t) \geq g_2(t)$ for all $t \in [\sigma(c),b)$.
For $t \in I_2$, we have, from relations (\ref{eqn13})
and (\ref{eqn14})
\begin{gather*}
\Theta_2^{\Delta}(t) = \frac{1}{p_2(t)}\sin^{2}\Theta_2(t)
+ g_2(t)\sin^{2}\Theta_2(t),
\\
\hat{\Theta}_2^{\Delta}(t) = \frac{1}{\hat{p}_2(t)}\sin^{2}
\hat{\Theta}_2(t) + \hat{g}_2(t)\sin^{2}\hat{\Theta}_2(t).
\end{gather*}
Let $\delta_2 = \hat{\Theta}_2 - \Theta_2$. Then, we have
\begin{align*}
\delta_2^{\Delta} &= \frac{1}{\hat{p}_2(t)}\sin^{2}\hat{\Theta}_2(t)
+ \hat{g}_2(t)\sin^2\hat{\Theta}_2(t)
- \frac{1}{p_2(t)}\sin^{2}\Theta_2(t) -g_2(t)\sin^{2}\Theta_2(t)\\
&= \frac{1}{\hat{p}_2(t)}\sin^{2}\hat{\Theta}_2(t)
- \frac{1}{p_2(t)}\cos^{2}\Theta_2(t)
+ \hat{g}_2(t)\sin^2\hat{\Theta}_2(t)-g_2(t)\sin^{2}\Theta_2(t) \\
&= \Big( \frac{1}{\hat{p}_2} - \frac{1}{p_2} \Big)\cos^2\hat{\Theta}_2
+ (\hat{g}_2 - g_2)\sin^{2}\Theta_2
+ \frac{1}{p_2}(\cos^{2}\hat{\Theta}_2 - \cos^{2}\Theta_2)\\
&\quad - \hat{g}_2(\sin^{2}\Theta_2 - \sin^{2}\hat{\Theta}_2).
\end{align*}
As we know that $\cos^2{\Theta} = 1 - \sin^2{\Theta}$, we have
\begin{align*}
\delta_2^{\Delta}
&= \Big( \frac{1}{\hat{p}_2} - \frac{1}{p_2} \Big)
\cos^2\hat{\Theta}_2 + (\hat{g}_2 - g_2)\sin^{2}\Theta_2\\
&\quad + \Big(\hat{g}_2 - \frac{1}{p_2}\Big) (\sin \hat{\Theta}_2
+ \sin \Theta_2)(\sin \hat{\Theta}_2 - \sin \Theta_2)
\\
&\geq h_2(t) + f_2(t)\delta_2(t),
\end{align*}
where $0 \leq h_2(t) \leq \frac{1}{\hat{p}_2} - \frac{1}{p_2}
+ \hat{g}_2 - g_2$,
$|f_2(t)| \leq 2 \Big( |\hat{g}_2(t)| + \frac{1}{p_2(t)} \Big)$;
as we have $\sin\Theta_2, \sin\hat{\Theta}_2, \cos\Theta_2,
\cos\hat{\Theta}_2$ bounded above by one and for small values
of $\hat{\Theta}_2$ and $\Theta_2$, we have
$\sin\hat{\Theta}_2 \approx \hat{\Theta}_2$ and
$\sin\Theta_2 \approx \Theta_2$. Clearly, the functions $f_2$ and
$h_2$ both are locally integrable on $I_2$.
Let $k_2(t) = exp \Big( - \int_d^{t}f_2(t)dt \Big) > 0$. Then, we have,
$(k_2\delta_2)^{\Delta} = (k_2\delta_2' + k_2{'}\delta_2)$,
since $\Theta^{\Delta} = \Theta'$. So, we have
$ (k_2\delta_2)^{\Delta} = k_2\delta_2' - k_2f_2\delta_2 = k_2(\delta_2'
- f_2\delta_2) \geq k_2h_2 \geq 0$ since,
$\delta_2^{'} \geq h_2(t) + f_2(t)\delta_2(t))$, and therefore
$k_2\delta_2$ is an increasing function; i.e.,
$k_2(t_1)\delta_2(t_1) \leq k_2(t_2)\delta_2(t_2)$, for $t_1 \leq t_2$.
We have $\delta_2(d) = (\hat{\Theta}_2(d) - \Theta_2(d)) \geq 0$,
hence, it follows that $\delta_2(t) \geq 0$ for all $t \in (d,b)$,
as $k_2(d)\delta_2(d) \leq k_2(t)\delta_2(t)$, for $d \leq t$ and
$k_2(d)\delta_2(d) \geq 0, k_2(t) > 0$ for $t \in I_2$.
Hence, $\hat{\Theta}(t) \geq \Theta(t)$ for all $t \in (d,b)$.
\smallskip
\noindent\textbf{Case II} Let us suppose that $d \in (a,c]$.
Then, $\hat{\Theta}(d) \geq \Theta(d)$ implies
$\hat{\Theta}_1(d) \geq \Theta_1(d)$, from the hypothesis
of the theorem, we have from relations (\ref{eqn13}) and (\ref{eqn14})
\begin{gather*}
\Theta_1^{\Delta}(t) = \frac{1}{p(t)} \cos^2\Theta_1(t)
+ g_1(t)\sin^{2}\Theta_1(t), \\
{\hat{\Theta}_1}^{\Delta}(t) = \frac{1}{\hat{p}_1(t)}\cos^{2}
\hat{\Theta}_1(t) + g_1(t)\sin^{2}\hat{\Theta}_1(t).
\end{gather*}
Let $\delta_1 = \hat{\Theta}_1 - \Theta_1$. Then as proceeding in
Case I, we can show that $\hat{\Theta}(t) \geq \Theta(t)$, for all
$t \in (d,c]$. In fact, from the proof of Case I, it follows that
$\hat{\Theta}_1(c) \geq \Theta_1(c)$. Now, by continuity of
$\Theta$ function (see relation (\ref{eqn10})), we have that
$\hat{\Theta}_2(\sigma(c)) = \hat{\Theta}_1(c) \geq \Theta_1(c) =
\Theta_2(\sigma(c))$, and hence by Case I we get that
$\hat{\Theta}_2(t) \geq \Theta(t)$ for all $t \in [\sigma(c),b)$.
Therefore, $\hat{\Theta}(t) \geq \Theta(t)$ for all $t \in
(d,c]\cup[\sigma(c),b)$. Hence, the proof is complete.
\end{proof}
\section{Relationship Between Eigenvalues and Zeros of Eigenfunctions}
\begin{theorem} \label{thm5.1}
Let $u = (u_1,u_2)$ and $v = (v_1,v_2)$
be real valued solutions of \eqref{eqn5}, \eqref{eqn6} along with
conditions \eqref{eqn3}, \eqref{eqn4}. If $u$ and $v$ are linearly
independent, then between any two consecutive zeros of $u$, there
lies exactly one zero of $v$.
\end{theorem}
\begin{proof} Let $t_1, t_2$ be two consecutive zeros of
$u$. For $u$, let $\Theta(t) = \tan^{-1}
\frac{u(t)}{p(t)u^{\Delta}(t)}$ and for $v$, let
$\tilde{\Theta}(t) = \tan^{-1}
\frac{v(t)}{p(t)v^{\Delta}(t)}$, where
\begin{gather*}
u_i(t) = \rho_i(t)\sin\Theta_i(t), p_i(t)u_i^{\Delta}(t)
= \rho_i(t)\cos\Theta_i(t), \\
v_i(t) = \rho_i(t)\sin\tilde{\Theta}_i(t), p_i(t)v_i^{\Delta}(t)
= \rho_i(t)\cos\tilde{\Theta}_i(t).
\end{gather*}
Let $\Theta$ and $\tilde{\Theta}$, the angel functions defined above,
be such that $\Theta(t_1) = 0, 0 \leq \tilde{\Theta}(t_1) < \pi$.
This may be accomplished by taking $-u(-v)$ instead of $u(v)$,
if necessary.
\smallskip
\noindent\textbf{Case I}
Suppose that $t_1 \in I_1$.
Then, $\Theta_1(t_1) = 0$ and since $u_1$ and $v_1$ are linearly
independent $\tilde{\Theta}_1(t_1) \neq 0$. Since, $\rho_i(t) \neq 0$
and $\tilde{\Theta}_1(t_1) = 0$ violates the definition of $u_1$
and $v_1$ being linearly independent on $I_1$.
Hence, $\Theta_1(t_1) = 0 \leq \tilde{\Theta}_1(t_1) < \pi$;
i.e., $\Theta(t_1) = 0 \leq \tilde{\Theta}(t_1) < \pi$. Now,
from Theorem 4.1, it follows that
$\pi = \Theta(t_2) < \tilde{\Theta}(t_2)$.
Since, $t_1$ and $t_2$ are two consecutive zeros of
$u, \rho_i(t) \neq 0$ and $u_i(t) = \rho_i(t)\sin\Theta_i(t)$.
Hence (by continuity) $\tilde{\Theta}$ must take the value $\pi$
at some point $y_1 \in (t_1,t_2)$. Since
$v_i(t) = \rho_i(t)\sin\tilde{\Theta}_i(t), v$ has at
least one zero in $(t_1,t_2)$.
\smallskip
\noindent\textbf{Case II}
Suppose that $t_1 \in I_2$.
Then, $\Theta_2(t_1) = 0$, since $u_2$ and $v_2$ are linearly
independent, $\tilde{\Theta}_2 \neq 0$.
Hence, $\Theta_2(t_1) = 0 \leq \tilde{\Theta}_2(t_1) < \pi$;
i.e., $\Theta(t_1) = 0 \leq \tilde{\Theta}(t_1) < \pi$, and
therefore as in Case I, it follows that $v$ has at least one zero
in $(t_1,t_2)$. Now, let us suppose that $v$ has two(consecutive)
zeros $y_1, y_2 \in (t_1,t_2)$. Then as shown above,
there exists a point $t_3 \in (y_1,y_2)$ such that $u(t_3) = 0$,
a contradiction. Hence, $v$ has exactly one zero in $(t_1,t_2)$.
This completes the proof.
\end{proof}
\subsection*{Acknowledgments}
The authors dedicate this work to the Chancellor of
Sri Sathya Sai University, Bhagwan Sri Sathya Sai Baba.
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\end{document}