\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Sixth Mississippi State Conference on Differential Equations and Computational Simulations, {\em Electronic Journal of Differential Equations}, Conference 15 (2007), pp. 159--162.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{159} \title[\hfilneg EJDE-2006/Conf/15\hfil Error estimates for asymptotic solutions] {Error estimates for asymptotic solutions of dynamic equations on time scales} \author[G. Hovhannisyan\hfil EJDE/Conf/15 \hfilneg] {Gro Hovhannisyan} % in alphabetical order \address{Gro R. Hovhannisyan \newline Kent State University, Stark Campus \\ 6000 Frank Ave. NW, Canton, OH 44720-7599, USA} \email{ghovhannisyan@stark.kent.edu} \thanks{Published February 28, 2007.} \subjclass[2000]{39A10} \keywords{Time scale; asymptotic representation; error estimates; \hfill\break\indent second order differential equation; first order system } \begin{abstract} We establish error estimates for first-order linear systems of equations and linear second-order dynamic equations on time scales by using calculus on a time scales \cite{a1,b3,b4} and Birkhoff-Levinson's method of asymptotic solutions \cite{b2,l1,h2,h3}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \section{Results} Asymptotic behavior of solutions of dynamic equations and systems on time scales was investigated in \cite{b4}. In this paper we establish error estimates of such asymptotic representations, which may be applied to the investigation of stability of dynamic equations (see f.e. \cite{h3}). Consider the system of ordinary differential equations on time scales \begin{equation}\label{e1} a^{\Delta}(t)=A(t)a(t),\quad t>T, \end{equation} where $a^{\Delta}$ is delta (Hilger) derivative, $a(t)$ is a n-vector function, and $A(t)$ is a $n\times n$ matrix function from $ C_{rd}(T,\infty)$ (definition of rd-continuous functions see in \cite{b3}). A time scale is an arbitrary nonempty closed subset of the real numbers. 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 $$ \sigma(t)=\inf\{s\in \mathbb{T}:s>t\}. $$ The graininess function $\mu:\mathbb{T}\to[0,\infty]$ is defined by $$ \mu(t)=\sigma(t)-t. $$ We assume that $\sup\mathbb{T}=\infty$. Suppose we can find the exact solutions of the auxiliary system \begin{equation}\label{e2} \psi^{\Delta}(t)=A_1(t)\psi(t),\quad t>T, \end{equation} with the matrix function $A_1(t)\in C_{rd}(T,\infty)$ close to the matrix function A(t), which means that condition \eqref{e5} below is satisfied. Let $\Psi(t)$ be the fundamental matrix of the system \eqref{e2}. Then the solutions of \eqref{e1} can be represented in the form \begin{equation}\label{e3} a(t)=\Psi(t)(C+\varepsilon(t)), \end{equation} where $a(t),\varepsilon(t), C $ are the vector columns. We can consider \eqref{e3} as a definition of the error vector function $\varepsilon(t)$. Denote \begin{equation}\label{e4} H(t)\equiv\left(1+\mu(t)\Psi^{-1}(t)\Psi^{\Delta}(t)\right)^{-1}\Psi^{-1}(t) \left(A(t)\Psi(t)-\Psi^{\Delta}(t)\right). \end{equation} \begin{theorem} \label{thm1} Assume there exist an invertible and differentiable matrix function $\Psi(t)\in C_{rd}(T,\infty)$ such that $1+\mu(t)\Psi^{-1}(t)\Psi^{\Delta}(t)$ is invertible and \begin{equation}\label{e5} \int_t^{\infty}\Big(\lim_{m\searrow\mu(s)}\frac{\log(1+m\|H(s)\|)}{m} \Delta s\Big)<\infty. \end{equation} Then every solution of \eqref{e1} can be represented in form \eqref{e3} and the error function $\varepsilon(t)$ can be estimated as \begin{equation}\label{e6} \|\varepsilon(t)\|\le \|C\|\left(-1+e_{\|H\|}(\infty,t)\right), \end{equation} where $\|.\|$ is the Euclidean vector (or matrix) norm: $\|C\|=\sqrt{C_1^2+\dots +C_n^2}$, and expression in \eqref{e5} usually is used to define the exponential function on time scales (see \cite{a1,b3}): \begin{equation}\label{e7} e_{\|H\|}(\infty,t)=\exp{\left(\int_t^{\infty}\lim_{m\searrow\mu(s)} \frac{\log(1+m\|H(s)\|)}{m}\Delta s\right)}. \end{equation} \end{theorem} \begin{remark} \label{rmk1} \rm Comparing with the similar result from \cite{b4} advantage of Theorem \ref{thm1} is that it not only proves that error vector function approaches to zero as $t$ approaches to infinity, but inequality \eqref{e6} also estimates the speed of that approach to zero. >From the estimate \eqref{e6} it follows also that the error vector function $\varepsilon(t)$ is small when $\int_t^{\infty}\lim_{m\searrow\mu(s)}\frac{\log(1+m\|H(s)\|)}{m}\Delta s$ is small. \end{remark} \begin{proof}[Proof of Theorem \ref{thm1}] Let $a(t)$ be a solution of \eqref{e1}. The substitution $a(t)=\Psi(t)u(t)$ transforms \eqref{e1} into \[ u^{\Delta}=H(t)u(t),\quad t>T, \] where $H$ is defined by \eqref{e4}. By integration we get \begin{equation}\label{e8} u(t)=C-\int_t^bH(s)u(s)\Delta s,\quad tt_0>0,\; t\in\mathbb{T}. \end{equation} >From the functions $\varphi_{1,2}(t)\in C_{rd}^2(T,\infty)$ let us construct auxiliary matrix-functions \begin{equation}\label{e11} \begin{gathered} \Phi(t)=\begin{pmatrix} \varphi_1(t) & \varphi_2(t)\\ \varphi_1^{\Delta}(t) & \varphi_2^{\Delta}(t)\end{pmatrix},\quad H(t)=(1+\mu(t)\Phi^{-1}(t)\Phi^{\Delta}(t))^{-1}B(t), \\ B(t)=\begin{pmatrix} B_{21}(t) & B_{22}(t)\\ -B_{11}(t) & -B_{12}(t)\end{pmatrix}, \quad B_{kj}(t)\equiv\frac{\varphi_{k}(t)L[\varphi_{j}(t)]}{ W(\varphi_1,\varphi_2)}, \quad j=1,2. \end{gathered} \end{equation} \begin{theorem}\label{thm2} Let $\varphi_{1,2}(t)\in C_{rd}^2(T,\infty)$ be complex-valued functions such that \begin{equation}\label{e12} \int_T^{\infty}\Big(\lim_{m\searrow\mu(s)}\frac{ \log\left(1+m\|\left(1+m\Phi^{-1}(t)\Phi^{\Delta}(t)\right)^{-1} B(t)\|\right)}{m}\Big)\Delta t <\infty, \quad k,j=1,2, \end{equation} where $\|.\|$ is Euclidean matrix norm. Then for arbitrary constants $C_1$,$C_2$ there exist solution of \eqref{e1} that can be written in the form \begin{gather} x(t)=\left[C_1+\varepsilon_1(t)\right]\varphi_1(t)+\left[C_2+\varepsilon_2(t)\right] \varphi_2(t), \label{e13} \\ x^{\Delta}(t)=\left[C_1+\varepsilon_1(t)\right]\varphi_1^{\Delta}(t)+\left[C_2 +\varepsilon_2(t)\right]\varphi_2^{\Delta}(t). \label{e14} \end{gather} The error vector-function $\varepsilon(t)=(\varepsilon_1(t),\varepsilon_2(t))$ is estimated as \begin{equation}\label{e15} \|\varepsilon(t)\|\le \|C\|\big(-1 +e_{\|H(t)\|}(\infty,t)\big), \end{equation} where $C=(C_1,C_2)$ is an arbitrary constant vector, and the matrix function $H(t)$ is defined in \eqref{e11}. \end{theorem} \begin{proof} Rewrite equation \eqref{e10} in form \eqref{e1}: \begin{equation}\label{e16} a^{\Delta}(t)=A(t)a(t), \end{equation} where \[ a(t)=\begin{pmatrix} x(t)\\ x^{\Delta}(t)\end{pmatrix},\quad A(t)=\begin{pmatrix}0 & 1\\ -q(t)& -p(t)\end{pmatrix}. \] By substitution \begin{equation}\label{e17} a(t)=\Phi(t)w(t), \end{equation} in \eqref{e16} we get \begin{equation}\label{e18} w^{\Delta}=H(t)w(t), \end{equation} where $H(t)$ defined by \eqref{e11}. To apply Theorem \ref{thm1} to system \eqref{e16} we choose $A(t)=H(t)$ and $A_1\equiv 0$. Then the identity matrix is fundamental solution of \eqref{e2}, so conditions \eqref{e5} turns to \eqref{e12}. >From Theorem \ref{thm1} we have \[ w(t)=C+\varepsilon(t),\quad \hbox{or} \quad a(t)=\Phi(t)w(t)=\Phi(t)(C+\varepsilon(t)). \] Representations \eqref{e13},\eqref{e14} and estimates \eqref{e15} follow from Theorem \ref{thm1}. \end{proof} \begin{example} \rm For solutions of the equation $$ x^{\Delta\Delta}(t)+\big(\gamma^2+\frac{1}{t^2}\big)x(t)=0,\quad t>t_0, $$ we get representations \eqref{e13}, \eqref{e14}, where $$ \varphi_1= \cos_{\gamma}(t,t_0),\quad \varphi_2= \sin_{\gamma}(t,t_0) $$ are trigonometric functions on time scales \cite{b3}. By direct calculations $H=O(t^{-2})$ as $t\to\infty$, and $$ |\varepsilon_j(t)|\le \|C\|\Big[-1 +\exp\Big(\int_t^{\infty}\lim_{m\searrow\mu(s)} \frac{\log\left(1+C_1 m s^{-2}\right)}{m}\Delta s\Big)\Big], \quad j=1,2. $$ \end{example} \subsection*{Acknowledgments} The author wants to thank the anonymous referee for his/her comments that helped improving the original paper. \begin{thebibliography}{00} \bibitem{a1} R. Aulbach and S. Hilger. \emph{Linear dynamic processes with inhomogeneous time scales.} In Non-linear Dynamics and Quantum Dynamical systems. Academie Verlag, Berlin, 1990. \bibitem{b1} Z. Benzaid and D. A. Lutz. \emph{Asymptotic representation of solutions of perturbed systems of linear difference equations.} Studies in Applied Mathematics, 1987, 77, 195-221. \bibitem{b2} J. D. Birkhoff \emph{Quantum mechanics and asymptotic series.} Bull. Amer. Math., Soc. 32 1933, 681-700 \bibitem{b3} M. Bohner, A. Peterson. \emph{Dynamic equations on time scales: An Introduction with Applications.} Birkhauser, Boston, 2001. \bibitem{b4} M. Bohner, D. A. Lutz. \emph{Asymptotic behavior of Dynamic Equations on Time Scales.} J. Differ. Equations Appl, 2001, 7 (1): 21-50. \bibitem{l1} N. Levinson \emph{The asymptotic nature of solutions of linear systems of differential equations.} Duke Math. J., 1948, 15, 111-126. \bibitem{h1} W.A. Harris and D.A. Lutz. \emph{A unified theory of asymptotic integration.} J. Math. Anal. Appl., 1977, 57 (3), 571-586. \bibitem{h2} G. Hovhannisyan. \emph{Estimates for error functions of asymptotic solutions of ordinary linear differential equations}. Izv. NAN Armenii, Matematika, [English translation: Journal of Contemporary Math. Analysis - (Armenian Academy of Sciences) 1996, v. 31, no.1, 9-28. \bibitem{h3}G. Hovhannisyan. \emph{Asymptotic stability for second-order differential equations with complex coefficients} Electron. J. Diff. Eqns., Vol. 2004 (2004), No. 85, 1-20. \end{thebibliography} \end{document}