\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 172, 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} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/172\hfil Uniqueness for differential systems] {Uniqueness for n-th order differential systems with strong singularities} \author[Y. Pan, M. Wang\hfil EJDE-2010/172\hfilneg] {Yifei Pan, Mei Wang} % in alphabetical order \address{Yifei Pan \newline Department of Mathematical Sciences \\ Indiana University - Purdue University Fort Wayne \\ Fort Wayne, IN 46805-1499, USA.\newline School of Mathematics and Informatics \\ Jiangxi Normal University, Nanchang, China} \email{pan@ipfw.edu} \address{Mei Wang \newline Department of Statistics\\ University of Chicago\\ Chicago, IL 60637, USA} \email{meiwang@galton.uchicago.edu} \thanks{Submitted July 26, 2010. Published December 6, 2010.} \subjclass[2000]{34A12, 65L05} \keywords{Unique continuation; uniqueness; Carath\'eodory theorem; \hfill\break\indent Gronwall inequality} \begin{abstract} Using a Lipschitz type condition, we obtain the uniqueness of solutions for a system of n-th order nonlinear ordinary differential equations where the coefficients are allowed to have singularities. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction and main results} Lipschitz condition was a key part in proving classical results on the existence and uniqueness of ordinary differential equations, as extensively surveyed and summarized in \cite{AL}. In this paper, we use a Lipschitz type condition to obtain the uniqueness of solutions of n-th order nonlinear ordinary differential systems where the coefficients are allowed to have singularities. Our results are in the spirit of the Carath\'eodory theorem on the existence of ordinary differential equations \cite[Chapter 2]{CL}, which gives a Lipschitz condition in first order differential equations. The conditions in this paper are in terms of absolute continuity, thus the uniqueness result is on solutions in weaker or more general sense. For first order differential equations, Nagumo's Theorem \cite{Nagumo} and its generalizations \cite{Athanassov90, Constantin10} give precise coefficients and sharp order for an isolated singularity in the Lipschitz condition. A natural generalization of the classical Carath\'eodory condition is to higher order linear and nonlinear differential systems (e.g. \cite{Bartusek97, Nth2004}). Our results are on higher order differential equations with coefficients of singularities under integrability conditions. The main results are stated below. The proofs are provided in the next section. In the last section, we provide two applicable forms of the main theorem as corollaries, and we give an example to illustrate the sharpness of the singularity order allowed in the main condition of the Lipschitz type in an $n$ order differential equation. Let $L^1(a,b)$ denote the set of real Lebesgue integrable functions on the interval $(a,b)$, $\|\cdot\|$ the Euclidean norm in $\mathbb{R}^d$, and $\mathcal{C}^{k}(a,b)$ the set of $d$-dimensional functions with $k$-th continuously differentiable components on $(a,b)$. The following theorem is on the uniqueness of solutions of differential systems. \begin{theorem} \label{f3-thm-AC} Consider the system of differential equations of $y: (a,b)\to \mathbb{R}^d$, % \label{f3-odesys} \begin{gathered} y^{(n)}= f(x,y,y',\dots,y^{(n-1)}), \quad x\in(a,b),\; a<00$, which contradicts the definition of$\varepsilon'$. Therefore we must have $u(t)-v(t)= \vec 0, \quad \forall t\in[0,b).$ To obtain the results on$(a,0]$, replacing$u(x)$,$v(x)$,$x\in(a,b)$by$u^*(x)=u(-x)$,$v^*(x)=v(-x)$,$x\in(-b,-a)$to obtain $u^*(t)-v^*(t)= \vec 0 \quad \forall t\in[0,-a).$ Combining the results we arrived at $u(t)-v(t)= \vec 0 \quad \forall t\in(a,b).$ This concludes the proof for Theorem \ref{f3-thm-AC}. \end{proof} The proof of Theorem \ref{f3-thm-global} is similar to the proof of Theorem \ref{f3-thm-AC}. \begin{proof}[Proof of Theorem \ref{f3-thm-global}] Notice that$y$satisfies the assumptions in Lemma \ref{f3-keylem}. Define $B(\varepsilon) = \int_0^\varepsilon \Big(\sum_{k=0}^{n-1} \lambda_k(t)\frac{\|y^{(k)}(t)\|}{t^{n-k-1}}\Big) dt$ for any$\varepsilon\in(0,b). Applying \eqref{f3-keyleminq} in Lemma \ref{f3-keylem} and then \eqref{f3-thm-globalinq}, \begin{align*} B(\varepsilon) & = \sum_{k=0}^{n-1} \int_0^\varepsilon \lambda_k(t)\frac{\|y^{(k)}(t)\|}{t^{n-k-1}} dt \le \sum_{k=0}^{n-1} \int_0^\varepsilon \lambda_k(t) dt \int_0^\varepsilon \|y^{(n)}(t)\| dt\\ &\le \sum_{k=0}^{n-1} \int_0^\varepsilon \lambda_k(t) dt \int_0^\varepsilon \Big(\sum_{k=0}^{n-1} \lambda_k(t) \frac{\|y^{(k)}(t)\|}{t^{n-k-1}}\Big) dt\\ &=\Big(\int_0^\varepsilon \sum_{k=0}^{n-1} \lambda_k(t) dt\Big) B(\varepsilon). \end{align*} %B(\varepsilon)\not=0$would imply $\int_0^{\varepsilon} \sum_{k=0}^{n-1} \lambda_k(t) dt \ge 1.$ On the other hand,$\lambda_k\in L^1(a,b)$implies $\int_0^{\varepsilon} \sum_{k=0}^{n-1} \lambda_k(t) dt \to 0 \quad \text{as } \varepsilon\to 0.$ Thus there must exist$\varepsilon\in(0,b)$such that$B(\tilde\varepsilon)=0, \forall \tilde\varepsilon \le \varepsilon$. Integrating \eqref{f3-thm-globalinq} on both sides, $\int_0^\varepsilon \|y^{(n)}(t)\| dt \le \int_0^\varepsilon \Big(\sum_{k=0}^{n-1} \lambda_k(t)\frac{\|y^{(k)}(t)\|}{t^{n-k-1}}\Big) dt =B(\varepsilon) =0$ which leads to $\|y^{(n)}(t)\| =0 \text{ a. e. } t\in(0,\varepsilon) \quad \Longrightarrow \quad y^{(n)}(t) =\vec 0\quad \text{a. e. } t\in(0,\varepsilon).$ Consequently, $y^{(n-1)}(\varepsilon) =\int_0^\varepsilon y^{(n)}(t)dt=\vec 0 \quad \Longrightarrow \quad y^{(n-1)}(t)= \vec 0 \quad\text{a. e. }t\in(0,\varepsilon).$ This argument leads to $y(t)= \vec 0 \quad \text{a. e. }t\in(0,\varepsilon).$ where a. e.'' can be removed by the continuity of$y$. The interval$(0,\varepsilon)$on which$y\equiv \vec 0$can be extended to$(0,b)$and$(a,0)$using arguments analogous to the ones used in the proof of Theorem \ref{f3-thm-AC}. This concludes the proof of Theorem \ref{f3-thm-global}. \end{proof} \section{Corollaries and an example} In Corollary \ref{f3-cor-jacobian} below, we give an explicit form of the$L^1$functions in the Lipschitz condition \eqref{f3-lip} in Theorem \ref{f3-thm-AC} in terms of the Jacobians, under stronger differentiability conditions on the function$f$in the differential system \eqref{f3-odesys} in Theorem \ref{f3-thm-AC}. Recall that $f(x,s_0,s_1,\dots,s_{n-1}): (a,b)\times \mathbb{R}^d \times \dots\times \mathbb{R}^d \quad \to \quad \mathbb{R}^d$ where$ f=(f_1,\dots,f_d)\in\mathbb{R}^d$,$s_k = \left(s_{k1},\dots,s_{kd}\right)\in \mathbb{R}^d$,$ k=0,\dots,n-1$. For each$x\in(a,b)$, denote the Jacobian $J_k = J_k(x) = \det\Big( \frac{\partial f(x,s_0,\dots,s_{n-1})}{\partial s_k}\Big) = \left| \begin{matrix} \frac{\partial f_1}{\partial s_{k1}} & \dots & \frac{\partial f_1}{\partial s_{kd}} \\ \vdots && \vdots \\ \frac{\partial f_d}{\partial s_{k1}} & \dots & \frac{\partial f_d}{\partial s_{kd}} \\ \end{matrix} \right|,$ for$k=0,1,\dots,n-1$. \begin{corollary}\label{f3-cor-jacobian} If$f(x, s_0,\dots,s_{n-1})$is differentiable on$(s_0,\dots,s_{n-1})\in\mathbb{R}^{nd}$, a. e.$x\in(a,b)$, and $J_k(x) ~x^{n-k-1}\in L^1(a,b), \quad k=0,\dots,n-1,$ then there are$\lambda_k(x) \in L^1(a,b)$such that the Lipschitz condition \eqref{f3-lip} holds in Theorem \ref{f3-thm-AC}. \end{corollary} \begin{proof} By the differentiability of$f$and the mean value theorem, $f(x,s_0,\dots,s_{n-1}) - f(x,r_0,\dots,r_{n-1}) = Df(x,s') (s_0-r_0,\dots,s_{n-1}-r_{n-1}),$ a. e.$x\in(a,b)$, where $Df(x,s') = \Big(\frac{\partial f(x, s_0,\dots,s_{n-1})}{\partial s_0}, \dots, \frac{\partial f(x, s_0,\dots,s_{n-1})}{\partial s_{n-1}} \Big) \Big|_{(x,s')}$ and$(x,s')=(x,s'_0,\dots,s'_{n-1})\in(a,b)\times \mathbb{R}^{nd}$is on the line connecting the two points$(x,s_0,\dots,s_{n-1})$and$(x,r_0,\dots,r_{n-1})$. By matrix multiplication, $f(x,s_0,\dots,s_{n-1}) - f(x,r_0,\dots,r_{n-1}) = \sum_{k=0}^{n-1} \frac{\partial f(x,s_0,\dots,s_{n-1})} {\partial s_k}(x,s') ~(s_k - r_k) ,$ a. e.$x\in(a,b)$. Taking the norm, we have $\|f(x,s_0,\dots,s_{n-1}) - f(x,r_0,\dots,r_{n-1})\| \le \sum_{k=0}^{n-1} | J_k(x,s'_k)| \,\|s_k - r_k\|,$ a. e.$x\in(a,b)$. Therefore, the functions $\lambda_k(x) = J_k(x,s') ~x^{n-k-1} \in L^1(a,b),\quad k=0,1,\dots,n-1$ satisfy the Lipschitz condition \eqref{f3-lip}. \end{proof} When$f$in Theorem \ref{f3-thm-AC} is linear, the results can be stated as the corollary below. \begin{corollary} Let$a<0