\documentclass[reqno]{amsart} \begin{document} {\noindent\small {\em Electronic Journal of Differential Equations}, Vol.~2000(2000), No.~54, pp.~1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)} \thanks{\copyright 2000 Southwest Texas State University and University of North Texas.} \vspace{1cm} \title[\hfilneg EJDE--2000/54\hfil Smoothness of solutions] {Smoothness of solutions of conjugate boundary-value problems on a measure chain } \author[Eric R. Kaufmann\hfil EJDE--2000/54\hfilneg] {Eric R. Kaufmann } \address{Eric R. Kaufmann \hfill\break\indent Department of Mathematics and Statistics\\ University of Arkansas at Little Rock\hfill\break\indent Little Rock, Arkansas 72204-1099 USA} \email{erkaufmann@@athena.ualr.edu} \date{} \thanks{Submitted June 27, 2000. Published July 14, 2000.} \subjclass{34B15, 34B99, 39A10, 34A99 } \keywords{ Measure chain, initial-value problem, boundary-value problem} \begin{abstract} In this paper we consider the $n^{th}$ order $\Delta$-differential equation (often refered to as a differential equation on a measure chain) $$u^{\Delta_n}(t) = f(t, u(\sigma(t)),\dots, u^{\Delta_{n-1}}(\sigma(t)))$$ satisfying $n$-point conjugate boundary conditions. We show that solutions depend continuously and smoothly on the boundary values. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Differential equations on a measure chain (also called differential equations on time scales) have received much attention since Hilger's \cite{sh} work unifying continuous and discrete calculus. Subsequent works by Agarwal and Bohner \cite{rpamb}, Aulback and Hilger \cite{bash}, Erbe and Hilger \cite{lhesh}, and Kaymakcalan, {\it et al.} \cite {kls} have furthered the development of calculus on measure chains. There are many recent papers that consider a variety of different problem for differential equations on a measure chain. See \cite{da, cdhy, lheap1, lheap2} for example. In this paper we are concerned with the continuous dependence and smoothness of solutions of differential equations on a measure chain with respect to boundary values. The results of this paper are patterned after those found in Henderson and Lee \cite{jhll} and Henderson \cite{jh}. In \cite{jhll}, the authors considered the continuous dependence and smoothness of solutions of conjugate boundary-value problems for difference equations with respect to boundary conditions. In \cite{jh}, the author considered the continuous dependence and smoothness of solutions of conjugate boundary-value problems for differential equations with respect to boundary conditions. Other works devoted to continuous dependence and smoothness of solutions with repsect to boundary values include \cite{adjh, je, jejh, dh, jhmhlh, jherk} and references therein. Let $T$ be a nonempty closed subset of ${\mathbb R},$ and let $T$ have the subspace topology inherited from the Euclidean topology on ${\mathbb R}.$ Then $T$ is called a {\em measure chain}, (in some of the literature $T$ is called a {\em time scale}). \begin{definition} For $t < \sup T$ and $r > \inf T$, we define the {\em forward jump operator}, $\sigma,$ and the {\em backward jump operator}, $\rho,$ respectively, by $$\gathered \sigma(t) = \inf \{\tau \in T \ |\ \tau > t \} \in T, \cr \rho(r) = \sup \{\tau \in T \ |\ \tau < r \} \in T, \endgathered$$ for all $t, r \in T.$ If $\sigma(t) > t, \ t$ is said to be {\em right scattered}, and if $\sigma(t) = t, \ t$ is said to be {\em right dense}. If $\rho(r) < r, \ r$ is said to be {\em left scattered}, and if $\rho(r) = r, \ r$ is said to be {\em left dense}. \end{definition} \begin{definition} For $x:T \to \mathbb{R}$ and $t \in T$ (assume $t$ is not left scattered if $t = \sup T$), we define the {\em delta derivative} of $x(t)$, $x^\Delta (t)$, to be the number (when it exists), with the property that, for each $\varepsilon > 0,$ there is a neighborhood, $U,$ of $t$ such that \begin{displaymath} \Big\vert[x(\sigma(t)) - x(s)] - x^\Delta (t) [\sigma(t) - s]\Big\vert \leq \varepsilon \Big\vert\sigma(t) - s\Big\vert, \end{displaymath} for all $s \in U$. Higher order delta derivatives are defined recursively, \begin{displaymath} x^{\Delta_n}(t) = (x^{\Delta_{n-1}})^\Delta(t). \end{displaymath} \end{definition} For convenience, we will use the notation $x^{\Delta_0}(t)$ to represent the function $x(t)$. That is, $x^{\Delta_0}(t) = x(t)$. {\bf Remarks:} If $x:T \to \mathbb{R}$ is continuous at $t \in T,$ $t < \sup T,$ and $t$ is right scattered, then \begin{displaymath} x^\Delta (t) = \frac {x(\sigma (t)) - x(t)}{\sigma (t) - t}. \end{displaymath} In particular, if $T = \mathbb{Z},$ the integers, then \begin{displaymath} x^\Delta (t) = \Delta x(t) = x(t+1) - x(t), \end{displaymath} whereas, if $t$ is right dense, then \begin{displaymath} x^\Delta (t) = x'(t). \end{displaymath} Let $a, b \in T$. We define the closed interval $[a, b]$ by $[a,b] = \{ t \in T| a \leq t \leq b\}$. Other closed, open, and half-open intervals in $T$ are similarly defined. We consider solutions of the $\Delta$-differential equation $$\label{de} u^{\Delta_n}(t) = f(t, u(\sigma(t)), u^{\Delta}(\sigma(t)), \dots , u^{\Delta_{n-1}}(\sigma(t)),) \, t \in T,$$ We will assume throughout that \begin{itemize} \item[(A)] $f(t, x_1, x_2, \dots , x_n)\!: T \times {\mathbb R}^n \to {\mathbb R}^n$ is continuous. \end{itemize} At times we will need to assume that \begin{itemize} \item[(B)] $\frac{\partial f}{\partial x_i}(t, x_1, x_2, \dots , x_n)\!: T \times {\mathbb R}^n \to {\mathbb R}^n$ is continuous, $1 \leq i \leq n$. \end{itemize} Given a solution, $u(t)$, of (\ref{de}), we will also have need of the {\em variational equation along $u(t)$}, $$\label{ve} z^{\Delta_n}(t) = \sum_{i=1}^{n} \frac{\partial f}{\partial x_i}(t, u(\sigma(t)), u^{\Delta}(\sigma(t)), \dots , u^{\Delta_{n-1}}(\sigma(t))) z^{\Delta_i}(\sigma(t)),$$ In Section 2 we state two results for solutions of initial-value problems of (\ref{de}). The first result is that solutions of initial-value problems depend continuously on initial data provided condition (A) holds. The second results states that if conditions (A) and (B) hold then solutions of initial-value problems can be differentiated with respect to initial values. In Section 3 we state our main results which are analogues of the Theorems in section 2 for $n$-point conjugate boundary-value problems. The proofs of these Theoresm depend on the uniqueness of solutions of conjugate boundary value problems. \section{Smoothness with Respect to Initial Values} In this section we present theorems on continuous dependence and smoothness of solutions of initial-value problems with respect to initial values. The $\Delta$-differential equation along with the conditions $$\label{ic} u^{\Delta_i}(t_0) = c_{i+1}, \, 0 \leq i \leq n-1,$$ where $t_0 \in T$ is called an initial-value problem. The authors in \cite{kls} have shown that under a weaker condition than (A) initial value problems of the form (\ref{de}), (\ref{ic}) have unique solutions. Furthermore they have shown that that the initial-value problem (\ref{de}), (\ref{ic}) depends continuously on the initial values under this weaker condition. Theorem \ref{cdic} is similar to the theorem on continuous dependence presented in \cite{kls}. \begin{theorem}[Continuous Dependence on Initial Values]\label{cdic} Suppose that condition (A) is satisfied. Let $u(t; t_0, c_1, c_2, \dots, c_n)$ be the solution of (\ref{de}), (\ref{ic}) where $t_0 \in T$ and $c_1, c_2, \dots c_n \in {\mathbb R}$. Then for each $\varepsilon > 0$ and $\tau$ such that $t_0 + \tau \in T$ there exists a $\delta(\varepsilon, t_0, \tau, c_1, \dots, c_n)$ such that if $|c_i - d_i| < \delta, \, 1 \leq i \leq n$ then \begin{displaymath} |u(t; t_0, c_1, c_2, \dots, c_n) - u(t; t_0, d_1, d_2, \dots, d_n)| < \varepsilon \end{displaymath} for all $t \in [t_0, t_0 + \tau]$. \end{theorem} \begin{theorem}\label{scic} Assume that conditions (A) and (B) are satisfied. Let $u(t) = u(t; t_0, c_1, c_2, \dots, c_n)$ denote the solution of the initial-value problem (\ref{de}), (\ref{ic}) where $t_0 \in T$ and $c_1, c_2, \dots c_n \in {\mathbb R}$. Then, given $\gamma_1, \dots , \gamma_n \in {\mathbb R}$, for each $1 \leq j \leq n$ \begin{displaymath} \beta_j(t) = \frac{\partial u}{\partial c_j}(t;t_0, \gamma_1, \dots, \gamma_n) \end{displaymath} exists and is the solution of the variational equation \begin{eqnarray*} \beta_j^{\Delta_n}(t) & = & \sum_{i=1}^{n} \frac{\partial f}{\partial x_i}(t, u(\sigma(t); t_0, \gamma_1, \dots, \gamma_n), u^{\Delta}(\sigma(t); t_0, \gamma_1, \dots, \gamma_n),\\ && \quad \quad \quad \quad \dots , u^{\Delta_{n-1}}(\sigma(t); t_0, \gamma_1, \dots, \gamma_n)) \beta_j^{\Delta_i}(\sigma(t)) \end{eqnarray*} and satisfies \begin{displaymath} \beta_j^{\Delta_i}(t_0) = \delta_{i,j}, \, 1 \leq i \leq n, \end{displaymath} where \begin{displaymath} \delta_{i,j} = \left \{ \begin{array}{cc} 1, & i = j,\\ 0, & i \neq j. \end{array} \right . \end{displaymath} \end{theorem} \section{Smoothness with Respect to Boundary Values} In this section we state and prove analogues to Theorems \ref{cdic} and \ref{scic} for $n$-point conjugate boundary-value problems. \begin{definition} Let $t_1 < t_2 < \cdots < t_n \in T$ and let $u_1, u_2, \dots, u_n \in {\mathbb R}$. A boundary-value problem satisfying $$\label{bc} u(t_i) = u_i, \, 1 \leq i \leq n,$$ is called an {\em $n$-point conjugate boundary-value problem}. \end{definition} We give some conditions characterizing disconjugacy for linear $\Delta$-differential equations in terms of {\em generalized zeros}. These conditions parallel those given by Hartman \cite{PH} for the disconjugacy for difference equations. \begin{definition} Let $u\!: T \to {\mathbb R}$. We say that $u$ has a {\em generalized zero} at $t_0$ if either $u(t_0) = 0$ or if there is a $k \in {\mathbb N}$ such that $(-1)^k u(\rho^k(t_0)) u(t_0) > 0$ and $u(\rho(t_0)) = u(\rho^2(t_0)) = \cdots = u(\rho^{k-1}(t_0)) = 0$. \end{definition} \vspace{0.1in} \begin{definition}\label{def1} The nonlinear $\Delta$-differential (\ref{de}) is said to be $n$-point {\em disconjugate on $T$} provided that whenever $u(t)$ and $v(t)$ are solutions of (\ref{de}) such that if $u(t) - v(t)$ has $n$ generalized zeros at $t_1 < t_2 < \cdots < t_n \in T$ then $u(t) - v(t) \equiv 0$ on $[t_1, +\infty)$. \end{definition} In the case when (\ref{de}) is linear, say $$\label{altde} v^{\Delta_n}(t) = \sum_{i=1}^n \alpha_i(t) v^{\Delta_{i-1}}(\sigma(t)),$$ where $\alpha_i: T \to {\mathbb R}, \, 1 \leq i \leq n$, we may reformulate Definition \ref{def1} as follows. \begin{definition} The linear equation (\ref{altde}) is said to be {\em $n$-point disconjugate on $T$} provided no nontrivial solution $u$ of (\ref{altde}) has $n$-generalized zeros on $T$. \end{definition} We adopt the following notation to distinguish initial-value problems from boundary value problems. Given $t_0 \in T$ and $c_1, \dots, c_n \in {\mathbb R}$, let $v(t) = v(t; t_0, c_1, \dots, c_n)$ denote the solution of the initial-value problem (\ref{de}), (\ref{ic}). Given $t_1, \dots, t_n \in T$ and $v_1, \dots, v_n \in {\mathbb R}$, let $v(t) = v(t; t_1, \dots, t_n, v_1, \dots, v_n)$ denote the solution of the boundary value problem (\ref{de}), (\ref{bc}). We will use the Brouwer Theorem on Invariance of Domain, Theorem \ref{bt} below, to prove that solutions of (\ref{de}) depend continuously on the boundary values when (\ref{de}) is $n$-point disconjugate. To show that (\ref{de}) depends smoothly on the boundary values we must further assume that the variational equation, (\ref{ve}), is $n$-point disconjugate. \begin{theorem}\label{bt} If $U$ is an open subset of ${\mathbb R}^n$, $n$ dimensional Euclidean space, and $\varphi: U \to {\mathbb R}^n$ is one-to-one and continuous on $U$ then $\varphi$ is a homeomorphism and $\varphi(U)$ is an open subset of ${\mathbb R}^n$. \end{theorem} \vspace{0.1in} \begin{theorem}[Continuous Dependence on Boundary Values]\label{cdbc} Suppose that condition (A) is satisfied and that (\ref{de}) is $n$-point disconjugate on $T$. Let $y(t)$ be a solution of (\ref{de}) on $[t_1, +\infty)$ and let $t_1 < t_2 < \cdots < t_n \in T$ be given. Then there exists an $\varepsilon > 0$ such that if $\gamma_i \in {\mathbb R}, \, 1 \leq i \leq n$ where $|\gamma_i| < \varepsilon, \, 1 \leq i \leq n$, then the boundary-value problem (\ref{de}) satisfying \begin{displaymath} u(t_i) = y(t_i) + \gamma_i, \quad 1 \leq i \leq n, \end{displaymath} has a unique solution $u(t; t_1, \dots, t_n, y(t_1) + \gamma_1, \dots, y(t_n) + \gamma_n)$. Furthermore we have $u(t; t_1, \dots, t_n$, $y(t_1) + \gamma_1, \dots, y(t_n) + \gamma_n)$ converging to $y(t)$ as $\varepsilon \to 0$. \end{theorem} \noindent {\bf Proof:} Let $t_1 < t_2 < \cdots < t_n \in T$ be given and define a mapping $\varphi: {\mathbb R}^n \to {\mathbb R}^n$ by $\varphi(c_1, c_2, \dots, c_n) = ( v(t_1), v(t_2), \dots , v(t_n) )$ where $v(t) = v(t; t_1, c_1, \dots c_n)$ is the solution of the (\ref{de}) satisfying the initial conditions \begin{displaymath} v^{\Delta_{i-1}}(t_1) = c_i, \quad 1 \leq i \leq n. \end{displaymath} We will show that $\varphi$ is one-to-one and continuous. It will then follow from Theorem \ref{bt}, that $\varphi$ is a homeomorphism. Suppose that $\varphi(c_1, c_2, \dots, c_n) = \varphi(c'_1, c'_2, \dots, c'_n)$. Then, \begin{eqnarray*} &&(v(t_1; t_1, c_1, \dots , c_n), v(t_2; t_1, c_1, \dots , c_n), \dots , v(t_n; t_1, c_1, \dots , c_n)) \\ && \quad \quad = (v(t_1; t_1, c'_1, \dots , c'_n), v(t_2; t_1, c'_1, \dots , c'_n), \dots , v(t_n; t_1, c'_1, \dots , c'_n)). \end{eqnarray*} Now, equation (\ref{de}) is $n$-point disconjugate on $T$ and hence solutions to (\ref{de}), (\ref{bc}) are unique. And so, for all $t \in [t_1, +\infty)$ we have \begin{displaymath} v(t;t_1, c_1, \dots, c_n) = v(t;t_1, c'_1, \dots, c'_n). \end{displaymath} In particular, \begin{displaymath} v^{\Delta_{i-1}}(t_1;t_1, c_1, \dots, c_n) = v^{\Delta_{i-1}}(t_1;t_1, c'_1, \dots, c'_n), \, 1 \leq i \leq n. \end{displaymath} Recalling our notation, we see that $(c_1, c_2, \dots, c_n) = (c'_1, c'_2, \dots, c'_n)$. Hence $\varphi$ is one-to-one. To show that $\varphi$ is continuous we consider a sequence $\{\!(c^{\ell}_1, c^{\ell}_2, \dots, c^{\ell}_n)\!\}_{\ell = 1}^{\infty}$ which converges to $(c_1, c_2, \dots, c_n)$ as $\ell \to \infty$. By the continuous dependence on initial values, Theorem \ref{cdic}, $v(t;t_1, c^{\ell}_1, \dots, c^{\ell}_n) \to v(t;t_1, c_1, \dots, c_n)$ for all $t \in [t_1, +\infty)$ as $\ell \to \infty$. That is, \begin{displaymath} \lim_{\ell \to \infty} v(t;t_1, c^{\ell}_1, \dots, c^{\ell}_n) = v(t;t_1, c_1, \dots, c_n). \end{displaymath} Thus, $\{\!\varphi(c^{\ell}_1, c^{\ell}_2, \dots, c^{\ell}_n)\!\}$ converges to $\varphi(c_1, c_2, \dots, c_n)$ as $\ell \to \infty$ and so $\varphi$ is continuous. By the Brouwer Theorem on Invariance of Domain, $\varphi$ is a homeomorphism onto its range, $\varphi({\mathbb R}^n)$, and $\varphi({\mathbb R}^n)$ is open in ${\mathbb R}^n$. Let $y(t)$ be a solution of (\ref{de}). Then $(y(t_1), \dots, y(t_n)) \in \varphi({\mathbb R}^n)$. Since $\varphi({\mathbb R}^n)$ is open, there exists an $\varepsilon > 0$ such that if $|\gamma_i| < \varepsilon, \, 1 \leq i \leq n,$ then $(y(t_1) + \gamma_1, \dots, y(t_n) + \gamma_n) \in \varphi({\mathbb R}^n)$. Since $\varphi$ is one-to-one there exists a unique $r = (r_1, \dots , r_n) \in {\mathbb R}^n$ such that $\varphi(r_1, \dots , r_n) = (y(t_1) + \gamma_1, \dots, y(t_n) + \gamma_n)$. By our definition, \begin{displaymath} \varphi(r_1, \dots , r_n) = (v(t_1;t_1, r_1, \dots, r_n), \dots, v(t_n;t_1, r_1, \dots, r_n)) \end{displaymath} where $v(t;t_1, r_1, \dots, r_n)$ is the solution of (\ref{de}) satisfying the initial conditions \begin{displaymath} v^{\Delta_{i-1}}(t_1) = r_{i}, \quad 1 \leq i \leq n. \end{displaymath} Thus, \begin{displaymath} (y(t_1) + \gamma_1, \dots, y(t_n) + \gamma_n) = (v(t_1;t_1, r_1, \dots, r_n), \dots, v(t_n;t_1, r_1, \dots, r_n)). \end{displaymath} That is, $v(t;t_1, r_1, \dots, r_n)$ is the solution of (\ref{de}) satisfying the boundary conditions, \begin{displaymath} v(t_i;t_1, r_1, \dots, r_n) = y(t_i) + \gamma_i, \quad 1 \leq i \leq n. \end{displaymath} Now consider a sequence $\{(y(t_1) + \gamma^{\ell}_1, \dots, y(t_n) + \gamma^{\ell}_n)\}_{\ell = 1}^{\infty} \subset \varphi( {\mathbb R}^n)$ where $|\gamma^{\ell}_i| < \varepsilon$, $1 \leq i \leq n$ and \begin{displaymath} \lim_{\ell \to \infty} (y(t_1) + \gamma^{\ell}_1, \dots, y(t_n) + \gamma^{\ell}_n) = (y(t_1), \dots, y(t_n)). \end{displaymath} Let \begin{displaymath} u_{\ell}(t) = u(t; t_1, \dots, t_n, y(t_1) + \gamma^{\ell}_1, \dots, y(t_n) + \gamma^{\ell}_n). \end{displaymath} Since $\varphi$ is a homeomorphism then $\varphi^{-1}$ is continuous and so, \begin{eqnarray*} \lim_{\ell \to \infty} \varphi^{-1}( u_{\ell}(t_1), \dots , u_{\ell}(t_n)) & = & \lim_{\ell \to \infty} \varphi^{-1}(y(t_1) + \gamma^{\ell}_1, \dots, y(t_n) + \gamma^{\ell}_n)\\ & = & \varphi^{-1}(\lim_{\ell \to \infty} y(t_1) + \gamma^{\ell}_1, \dots, \lim_{\ell \to \infty} y(t_n) + \gamma^{\ell}_n)\\ & = & \varphi^{-1}(y(t_1), \dots, y(t_n)). \end{eqnarray*} That is, the initial values of $u_{\ell}(t)$ converge to the initial values of $y(t)$. By Theorem \ref{cdic} $u_{\ell}(t)$ converges uniformly to $y(t)$ on each compact subset of $[t_1, +\infty)$. Thus, $u(t; t_1, \dots, t_n, y(t_1) + \gamma^{\ell}_1, \dots, y(t_n) + \gamma^{\ell}_n)$ converges to $y(t)$ as $\varepsilon \to 0$ and the proof is complete. \begin{theorem} Assume that $f$ satisfies (A) and (B), that (\ref{de}) is $n$-point disconjugate on $T$, and that the variational equation (\ref{ve}) is $n$-point disconjugate along all solutions of (\ref{de}). Let $u(t) = u(t; t_1, \dots t_n, u_1, \dots, u_n)$ be the solution of (\ref{de}), (\ref{bc}) on $[t_1, +\infty)$. Then for $1 \leq j \leq n, \frac{\partial u}{\partial u_j}$ exists on $[t_1, +\infty)$ and $z_j(t) = \frac{\partial u}{\partial u_j}$ is the solution of the variational equation (\ref{ve}) along $u(t)$ and satisfies \begin{displaymath} z_j(t_i) = \delta_{ij}, \quad 1 \leq i \leq n. \end{displaymath} \end{theorem} \noindent {\bf Proof:} Fix $j, \, 1 \leq j \leq n$. Let $\varepsilon > 0$ be as Theorem \ref{cdbc} and let $h$ be such that $0 < |h| < \varepsilon$. Define the difference quotient \begin{displaymath} z_{jh}(t) = \frac{1}{h}[ u(t; t_1, \dots, t_n, u_1, \dots, u_j + h, \dots, u_n) - u(t; t_1, \dots, t_n, u_1, \dots, u_n)] \end{displaymath} It suffices to show that $\lim_{h \to \infty} z_{jh}(t)$ exists on $[t_1, +\infty)$. Note that for all $h \neq 0$, \begin{displaymath} z_{jh}(t_i) = \delta_{ij}, \quad 1 \leq i \leq n. \end{displaymath} For each $2 \leq i \leq n,$ define $\alpha_i = u^{\Delta_{i-1}}(t_j; t_1, \dots , t_n, u_1, \dots u_n)$ and $\varepsilon_i = \varepsilon_i(h) = u(t_i; t_1, \dots, t_n, u_1, \dots, u_i + h, \dots, u_n) - \alpha_i$. Recalling our notation we see that $u(t_i; t_1, \dots, t_n,$ $u_1, \dots, u_i + h, \dots, u_n) = u_i + h$ and $u(t_i; t_1, \dots, t_n, u_1, \dots, u_n) = u_i$. As a consequence of Theorem \ref{cdbc} $\varepsilon_i \to 0$ as $h \to 0$ for $2 \leq i \leq n$. Recall that $v(t; t_j, v_1, v_2, \dots v_n)$ is the solution of (\ref{de}) satisfying the initial conditions \begin{eqnarray*} &&v^{\Delta_{i-1}}(t_j) = v_i, \, 1 \leq i \leq n. \end{eqnarray*} In particular, $v(t; t_j, u_j, \alpha_2, \dots, \alpha_n)$ is the solution of (\ref{de}) satisfying $v(t_j) = u_j$ and for $2 \leq i \leq n, \, v^{\Delta_{i-1}}(t_j) = \alpha_i$. Likewise $v(t; t_j, u_j + h, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n)$ is the solution of (\ref{de}) satisfying $v(t_j) = u_j + h$ and for $2 \leq i \leq n, \, v^{\Delta_{i-1}}(t_j) = \alpha_i + \varepsilon_i$. Since solutions to initial-value problems are unique then $v(t; t_j, u_j, \alpha_2, \dots, \alpha_n) = u(t; t_1, \dots, t_n, u_1, \dots, u_n)$. Similarly, we have $v(t; t_j, u_j + h, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n) = u(t, t_1, \dots, t_n,$ $u_1, \dots, u_j + h, \dots, u_n)$. Using a telescoping sum, we have \begin{eqnarray*} \lefteqn{z_{jh}(t)}\\ & = & \frac{1}{h}[u(t; t_1, \dots, t_n, u_1, \dots, u_j + h, \dots, u_n) - u(t; t_1, \dots, t_n, u_1, \dots, u_n)]\\ & = & \frac{1}{h}[v(t; t_j, u_j + h, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n) - v(t; t_j, u_j, \alpha_2, \dots, \alpha_n)]\\ & = & \frac{1}{h}[[v(t; t_j, u_j + h, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n) - v(t; t_j, u_j, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n)]\\ & & + [v(t; t_j, u_j, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n) - v(t; t_j, u_j, \alpha_2 , \dots, \alpha_n + \varepsilon_n)]\\ & & + \dots + [v(t; t_j, u_j, \alpha_2, \dots, \alpha_{n-1}, \alpha_n + \varepsilon_n) - v(t; t_j, u_j, \alpha_2 , \dots, \alpha_{n-1}, \alpha_n)]]\\ \end{eqnarray*} By Theorem \ref{scic}, solutions of (\ref{de}) can be differentiated with respect to initial values. That is $\beta_1 = \frac{\partial v}{\partial v_1}, \beta_2 = \frac{\partial v}{\partial v_2}, \dots, \beta_n = \frac{\partial v}{\partial v_n}$ exist. By Theorem \ref{scic} and the Mean Value Theorem, we see that \begin{eqnarray} z_{jh}(t) & = & \frac{1}{h}[\beta_1(t, v(t; t_1, u_j + \bar{h}, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n))h \label{eq1}\\ & & + \, \, \beta_2(t, v(t; t_1, u_j, \alpha_2 + \bar{\varepsilon}_2, \dots, \alpha_n + \varepsilon_n))\varepsilon_2\nonumber\\ & & + \, \dots \, + \, \, \beta_n(t; v(t, t_1, u_j, \alpha_2, \dots, \alpha_n + + \bar{\varepsilon}_n))\varepsilon_n]\nonumber \end{eqnarray} where \begin{eqnarray*} &&\beta_1(t; v(t; t_1, u_j + \bar{h}, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n))\\ && \hspace{1.6in} = \frac{\partial v}{\partial v_1} (t; t_1, u_j + \bar{h}, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n)\\ &&\hspace{0.4in} \vdots \\ &&\beta_n(t; v(t; t_1, u_j, \alpha_2, \dots, \alpha_n + \bar{\varepsilon}_n)) = \frac{\partial v}{\partial v_n} (t; t_1, u_j, \alpha_2, \dots, \alpha_n + + \bar{\varepsilon}_n) \end{eqnarray*} and $\bar{h}$ is between $0$ and $h$ and $\bar{\varepsilon}_{\ell}$ is between $0$ and $\varepsilon_{\ell}$, $2 \leq \ell \leq n$. That is, $\beta_1(t; v(t, t_1, u_j + \bar{h}, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n))$ is the solution of the variational equation (\ref{ve}) along $v(t; t_1, u_j + \bar{h}, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n)$ satisfying $\beta_1^{\Delta_{i-1}}(t_j) = \delta_{i1}, 1 \leq i \leq n$. For $2 \leq \ell \leq n$, $\beta_\ell(t; v(t; t_1, u_j, \alpha_2 + \varepsilon_2, \dots, \alpha_\ell + \bar{\varepsilon}_\ell, \dots, \alpha_n + \varepsilon_n))$ is the solution of the variational equation (\ref{ve}) along $v(t; t_1, u_j, \alpha_2 + \varepsilon_2, \dots, \alpha_\ell + \bar{\varepsilon}_\ell, \dots, \alpha_n + \varepsilon_n)$ satisfying $\beta_\ell^{\Delta_{i-1}}(t_j) = \delta_{i\ell}, 1 \leq i \leq n$. In particular note that \begin{displaymath} \beta_2(t_j) = \cdots = \beta_n(t_j) = 0. \end{displaymath} Distribute the factor $\frac{1}{h}$ in equation (\ref{eq1}). \begin{eqnarray} z_{jh}(t) & = & \beta_1(t, v(t; t_1, u_j + \bar{h}, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n))\label{eq2}\\ & & + \, \beta_2(t, v(t; t_1, u_j, \alpha_2 + \bar{\varepsilon}_2, \dots, \alpha_n + \varepsilon_n))\frac{\varepsilon_2}{h}\nonumber\\ & & + \dots + \beta_n(t; v(t, t_1, u_j, \alpha_2, \dots, \alpha_n + + \bar{\varepsilon}_n))\frac{\varepsilon_n}{h}.\nonumber \end{eqnarray} To show that $\lim_{h \to 0} z_{jh}(t)$ exists, it suffices to show that $\lim_{h \to 0} \frac{\varepsilon_\ell}{h}$ exists for $2 \leq \ell \leq n$. Recall that $z_{jh}(t_1) = \cdots = z_{jh}(t_{j-1}) = z_{jh}(t_{j+1}) = \cdots = z_{jh}(t_n) = 0$. Evaluate (\ref{eq2}) at $t_\ell, 1 \leq \ell \leq n, \ell \neq j$ to obtain the system of equations \begin{eqnarray*} &&-\beta(t_\ell; v(t_j; u_j + \bar{h}, \alpha_2 + \varepsilon_2, \dots, \alpha_n + \varepsilon_n)\\ &&\quad \quad \quad = \beta_2 (t_\ell; v(t_j; u_j, \alpha_2 + \bar{\varepsilon}_2, \dots, \alpha_n + \varepsilon_n)\frac{\varepsilon_2}{h}\\ &&\quad \quad \quad + \cdots + \beta_n (t_\ell; v(t_j; u_j, \alpha_2, \dots, \alpha_n + \bar{\varepsilon}_n)\frac{\varepsilon_n}{h}, \, 1 \leq \ell \leq n, \, \ell \neq j. \end{eqnarray*} This is a system of $n-1$ equations in the $n-1$ unknowns $\frac{\varepsilon_2}{h}, \frac{\varepsilon_3}{h}, \dots , \frac{\varepsilon_n}{h}$. By Cramer's rule we have, (after surpressing the variable dependency in $v( \cdot )$), \begin{eqnarray*} \frac{\varepsilon_2}{h} & = & \frac{\left | \begin{array}{cccc} -\beta_1(t_1; v(\cdot)) & \beta_3(t_1; v(\cdot)) & \cdots & \beta_n(t_1; v(\cdot))\\ \vdots & \vdots & & \vdots\\ -\beta_1(t_{j-1}; v(\cdot)) & \beta_3(t_{j-1}; v(\cdot)) & \cdots & \beta_n(t_{j-1}; v(\cdot))\\ -\beta_1(t_{j+1}; v(\cdot)) & \beta_3(t_{j+1}; v(\cdot)) & \cdots & \beta_n(t_{j+1}; v(\cdot))\\ \vdots & \vdots & & \vdots\\ -\beta_1(t_n; v(\cdot)) & \beta_3(t_n; v(\cdot)) & \cdots & \beta_n(t_n; v(\cdot))\\ \end{array} \right |}{D(h)},\\ \\ & \vdots &\\ \\ \frac{\varepsilon_n}{h} & = & \frac{\left | \begin{array}{cccc} \beta_2(t_1; v(\cdot)) & \beta_3(t_1; v(\cdot)) & \cdots & -\beta_1(t_1; v(\cdot))\\ \vdots & \vdots & & \vdots\\ \beta_2(t_{j-1}; v(\cdot)) & \beta_3(t_{j-1}; v(\cdot)) & \cdots & -\beta_1(t_{j-1}; v(\cdot))\\ \beta_2(t_{j+1}; v(\cdot)) & \beta_3(t_{j+1}; v(\cdot)) & \cdots & -\beta_1(t_{j+1}; v(\cdot))\\ \vdots & \vdots & & \vdots\\ \beta_1(t_n; v(\cdot)) & \beta_3(t_n; v(\cdot)) & \cdots & -\beta_1(t_n; v(\cdot))\\ \end{array} \right |}{D(h)}, \end{eqnarray*} provided that \begin{displaymath} D(h) \equiv \left | \begin{array}{cccc} \beta_2(t_1; v(\cdot)) & \beta_3(t_1; v(\cdot)) & \cdots & \beta_n(t_1; v(\cdot))\\ \vdots & \vdots & & \vdots\\ \beta_2(t_{j-1}; v(\cdot)) & \beta_3(t_{j-1}; v(\cdot)) & \cdots & \beta_n(t_{j-1}; v(\cdot))\\ \beta_2(t_{j+1}; v(\cdot)) & \beta_3(t_{j+1}; v(\cdot)) & \cdots & \beta_n(t_{j+1}; v(\cdot))\\ \vdots & \vdots & & \vdots\\ \beta_1(t_n; v(\cdot)) & \beta_3(t_n; v(\cdot)) & \cdots & \beta_n(t_n; v(\cdot))\\ \end{array} \right | \, \neq \, 0. \end{displaymath} To see that $D(h) \neq 0$ for small values of $h$, consider the determinant \begin{displaymath} D = \left | \begin{array}{ccc} \beta_2(t_1; v(t;t_j, u_j, \alpha_2, \dots, \alpha_n)) & \cdots & \beta_n(t_1; v(t;t_j, u_j, \alpha_2, \dots, \alpha_n))\\ \vdots & & \vdots\\ \beta_2(t_{j-1}; v(t;t_j, u_j, \alpha_2, \dots, \alpha_n)) & \cdots & \beta_n(t_{j-1}; v(t;t_j, u_j, \alpha_2, \dots, \alpha_n))\\ \beta_2(t_{j+1}; v(t;t_j, u_j, \alpha_2, \dots, \alpha_n)) & \cdots & \beta_n(t_{j+1}; v(t;t_j, u_j, \alpha_2, \dots, \alpha_n))\\ \vdots & & \vdots\\ \beta_1(t_n; v(t;t_j, u_j, \alpha_2, \dots, \alpha_n)) & \cdots & \beta_n(t_n; v(t;t_j, u_j, \alpha_2, \dots, \alpha_n))\\ \end{array} \right | \end{displaymath} If $D = 0$ then there exists a set of numbers $r_2, \dots, r_n$, at least one of which is nonzero, such that \begin{displaymath} \gamma(t) = \sum_{\ell = 2}^n r_\ell \beta_\ell(t; v(t;t_j, u_j, \alpha_2, \dots, \alpha_n)) \end{displaymath} is a nontrivial solution of (\ref{ve}) along $v(t;t_j, u_j, \alpha_2, \dots, \alpha_n)$ that vanishes at $t = t_1, \dots t_{j-1}$, $t_{j+1}, \dots, t_n$. Since $\beta_\ell(t_j) = 0$ for $2 \leq \ell \leq n$ then $\gamma(t_j) = 0$. That is $\gamma(t)$ is a nontrivial solution of (\ref{ve}) that has $n$ zeros in $T$ contradicting the $n$-point disconjugacy of the variational equation. Consequently $D \neq 0$. By continuity, $D(h) \neq 0$ for $h$ sufficiently small. Thus $\lim_{h \to 0} \frac{\varepsilon_\ell}{h}$ exists for each $2 \leq \ell \leq n$. Let \begin{displaymath} \lim_{h \to 0} \frac{\varepsilon_\ell}{h} = k_\ell, \, 2 \leq \ell \leq n. \end{displaymath} Then, \begin{eqnarray*} z_j(t) & = & \lim_{h \to 0} z_{jh}(t)\\ & = & \beta_1(t; v(t_j; u_j, \alpha_2, \dots , \alpha_n)) + \sum_{\ell = 2}^n k_\ell \beta_\ell(t; v(t_j; u_j, \alpha_2, \dots , \alpha_n)) \end{eqnarray*} exists. That is, $\frac{\partial u}{\partial u_j}(t; t_1, \dots, t_n, u_1, \dots, u_n)$ exists and $z_j(t) = \frac{\partial u}{\partial u_j}$. Furthermore, since each $\beta_\ell(t; v(t; t_j, u_j, \alpha_2, \dots , \alpha_n)), \, 1 \leq \ell \leq n$ is a solution of the variational equation (\ref{ve}) along $v(t; t_j, u_j, \alpha_2, \dots , \alpha_n) = u(t; t_1, \dots, t_n, u_1, \dots, u_n)$ then $z_j(t) = \frac{\partial u}{\partial u_j}$ is also a solution of (\ref{ve}) along $u(t;t_1, \dots, t_n, u_1, \dots, u_n)$. Finally we note that \begin{displaymath} z_j(t_i) = \lim_{h \to 0} z_{jh}(t_i) = \delta_{ij}, \, 1 \leq i \leq n \end{displaymath} and the proof is complete. \begin{thebibliography}{99} \bibitem {rpamb} R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, {\it Results Math.} {\bf 35} (1999), 3 - 22. \bibitem {da} D. Anderson, Positivity of Green's functions for $n$-point right focal boundary-value problem on a measure chain, preprint. \bibitem {bash} B. Aulback and S. 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