\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 151, pp. 1--8.\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-2010/151\hfil Existence of solutions?] {Existence of solutions to n-th order neutral dynamic equations on time scales} \author[Q. Li, Z. Zhang\hfil EJDE-2010/151\hfilneg] {Qiaoluan Li, Zhenguo Zhang} % in alphabetical order \address{Qiaoluan Li \newline College of Mathematics and Information Science, Hebei Normal University, \newline Shijiazhuang, 050016, China} \email{qll71125@163.com} \address{Zhenguo Zhang \newline College of Mathematics and Information Science, Hebei Normal University, \newline Shijiazhuang, 050016, China. \newline Information College, Zhejiang Ocean University, Zhoushan, 316000, China} \email{zhangzhg@mail.hebtu.edu.cn} \thanks{Submitted June 6, 2010. Published October 21, 2010.} \subjclass[2000]{34K40, 34N99, 39A10} \keywords{Time scales; dynamic equations; non-oscillatory solution.} \begin{abstract} In this article, we study n-th order neutral nonlinear dynamic equation on time scales. We obtain sufficient conditions for the existence of non-oscillatory solutions by using fixed point theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} This article concerns the n-th order neutral dynamic equation $$(x(t)+p(t)x(\tau(t)))^{\Delta^n}+f_1(t,x(\tau_1(t))) -f_2(t,x(\tau_2(t)))=0,\label{e1}$$ for $t\geq t_0$, where $t\in \mathbb{T}$, $n\in \mathbb{N}$. We assume $p\in C_{rd}(\mathbb{T}$, $\mathbb{R}),\tau,\tau_i\in C_{rd}(\mathbb{T},\mathbb{T})$, $\tau$ is strictly increasing, $\tau(t)0$, and $f_i$ is non-decreasing in $u$. In the sequel, without loss of generality, we assume that $f_i(t, u)>0$, $i=1,2$. In 1988, Stephan Hilger \cite{h1} introduced the theory of time scales as a means of unifying discrete and continuous calculus. Several authors have expounded on various aspects of this new theory, see \cite{b1,e3,z2} and references therein. Recently, much attention is concerned with questions of existence of non-oscillatory solutions for dynamic equations on time scales. For significant works along this line, see \cite{e2,h2,l1,s1}. Many results have been obtained for first and second order dynamic equations, however, few results are available for higher order dynamic equations. Motivated by these works, we investigate the existence of non-oscillatory solutions of \eqref{e1}. In Section 2, we present some preliminary material that we will need to show the existence of solutions of \eqref{e1}. We present our main results in Section 3. \section{Preliminaries} We assume the reader is familiar with the notation and basic results for dynamic equations on time scales. For a review of this topic we direct the reader to the monographs \cite{b2,b3}. We recall $x$ is a solution of \eqref{e1} provided that $x(t)+p(t)x(\tau(t))$ is n times differentiable, and $x$ satisfies \eqref{e1}. A solution $x$ of \eqref{e1} is called non-oscillatory if $x$ is of one sign when $t\geq T$. We define a sequence of functions $g_k(s,t)$, $k=1,2,\dots$ as follows. $$\begin{gathered} g_0(s,t)\equiv 1,\quad s,t\in \mathbb{T}^{\kappa},\\ g_{k+1}(s,t)=\int_{t}^{s}g_k(\sigma(u),t)\Delta u,\quad s,t\in \mathbb{T}^{\kappa}. \end{gathered}\label{e2}$$ For $g_{k}(s,t)$, we have the following Lemma. \begin{lemma}[\cite{z1}] \label{lem1} Assume $s$ is fixed, and let $g_k^{\Delta}(s, t)$ be the derivative of $g_{k}(s,t)$ with respect to $t$. Then $$g_k^{\Delta}(s, t)=-g_{k-1}(s, t), \quad k\in \mathbb{N},\; t\in \mathbb{T}^{\kappa}.\label{e3}$$ \end{lemma} \begin{lemma}[\cite{e1}] \label{lem2} Let $X$ be a Banach space, $\Omega$ be a bounded closed convex subset of $X$ and let $A, B$ be maps from $\Omega$ to $X$ such that $Ax+By\in \Omega$ for every pair $x, y\in \Omega$. If $A$ is a contraction and $B$ is completely continuous, then the equation $Ax+Bx=x$ has a solution in $\Omega$. \end{lemma} \begin{lemma}[\cite{e1}] \label{lem3} Let $X$ be a locally convex linear space, $S$ be a compact convex subset of $X$, and $T: S\to S$ be a continuous mapping with $T(S)$ compact. Then $T$ has a fixed point in $S$. \end{lemma} \section{Main Results} \begin{theorem} \label{thm1} Assume that $0< p(t)\leq p<1$, and there exists $b>0$ such that $$\int_{t_0}^{\infty}g_{n-1}(\sigma(s), 0)f_i(s, b)\Delta s<\infty,\quad i=1, 2.\label{e4}$$ Then \eqref{e1} has a bounded non-oscillatory solution which is bounded away from zero. \end{theorem} \begin{proof} Let $BC$ be the set of bounded functions on $[t_0,\infty)$ with sup norm $\|x\|=\sup_{t\geq t_0}|x(t)|$, $t\in \mathbb{T}$. Let $\Omega\subset BC$, $\Omega=\{x\in BC, 0t_0$, such that $\tau(t)\geq t_0$, $\tau_i(t)\geq t_0$, $i=1,2$ $t\geq t_1$ and $\int_{t_1}^{\infty}g_{n-1}(\sigma(s), 0)f_i(s, b)\Delta s\leq c$, $i=1,2$. Define a mapping $\Gamma$ on $\Omega$ as follows. $$(\Gamma x)(t)=(\Gamma_1 x)(t)+(\Gamma_2 x)(t),$$ where \begin{gather*} (\Gamma_1 x)(t)=\begin{cases} \alpha-p(t)x(\tau(t)),& t\geq t_1,\; t\in\mathbb{T},\\ (\Gamma_1 x)(t_1), & t_{0}\leq t\leq t_1,\; t\in\mathbb{T}. \end{cases} \\ (\Gamma_2 x)(t)=\begin{cases} (-1)^{n-1}\int_{t}^{\infty}g_{n-1}(\sigma(s), t)\big[f_1(s, x(\tau_1(s)))\\ -f_2(s, x(\tau_2(s)))\big]\Delta s, & t\geq t_1,\\[4pt] (\Gamma_2 x)(t_1), & t_{0}\leq t\leq t_1. \end{cases} \end{gather*} For any $x, y \in \Omega$, $t\ge t_0$, $t\in \mathbb{T}$, we have \begin{gather*} (\Gamma_1 x)(t)+(\Gamma_2 y)(t)\leq \alpha+c\leq M_2,\\ (\Gamma_1 x)(t)+(\Gamma_2 y)(t)\geq \alpha-pM_2-c\geq M_1. \end{gather*} Hence for $t\geq t_0$, $t\in \mathbb{T}$, $\Gamma_1x+\Gamma_2y\in \Omega$. Clearly, $\Gamma_1$ is a contraction mapping on $\Omega$ and $\Gamma_2$ is continuous. We shall show that $\Gamma_2$ is completely continuous. In fact, for any $x\in \Omega$, for $t_0\leq t\leq t_1$, $(\Gamma_2x)(t)=(\Gamma_2x)(t_1)$, and for $t\geq t_1$, we have \begin{align*} |(\Gamma_2x)(t)|&\leq \int_{t}^{\infty}g_{n-1}(\sigma(s), t)|f_1(s, x(\tau_1(s)))-f_2(s, x(\tau_2(s)))|\Delta s\\ &\leq \int_{t}^{\infty}g_{n-1}(\sigma(s), t)f_1(s, x(\tau_1(s)))\Delta s\\ &\leq \int_{t}^{\infty}g_{n-1}(\sigma(s), 0)f_1(s, b)\Delta s\leq c. \end{align*} Hence $\Gamma_2\Omega$ is uniformly bounded. For $\varepsilon>0$, there exists a $T$, such that for $t\geq T$, $$\int_{t}^{\infty}g_{n-1}(\sigma(s), 0)f_i(s, b)\Delta s<\frac{\varepsilon}{2}.$$ For $t, t'>T$, we have $$|(\Gamma_2x)(t)-(\Gamma_2x)(t')|\leq 2\int_{T}^{\infty} g_{n-1}(\sigma(s), 0)f_i(s, b)\Delta s< \varepsilon.$$ For $t, t'\in [t_1, T]$, we have \begin{align*} &|(\Gamma_2x)(t)-(\Gamma_2x)(t')|\\ &= |\int_{t}^{\infty}g_{n-1}(\sigma(s), t)[f_1(s, x(\tau_1(s)))- f_2(s, x(\tau_2(s)))]\Delta s\\ &\quad -\int_{t'}^{\infty}g_{n-1}(\sigma(s), t')[f_1(s, x(\tau_1(s)))- f_2(s, x(\tau_2(s)))]\Delta s|\\ &\leq |\int_{t}^{t'}g_{n-1}(\sigma(s), t)[f_1(s, x(\tau_1(s)))- f_2(s, x(\tau_2(s)))]\Delta s|\\ &\quad + \int_{t'}^{T}|g_{n-1}(\sigma(s), t)-g_{n-1}(\sigma(s), t')\|f_1(s, x(\tau_1(s)))- f_2(s, x(\tau_2(s)))|\Delta s\\ &\quad + \int_{T}^{\infty}|g_{n-1}(\sigma(s), t)-g_{n-1}(\sigma(s), t')\|f_1(s, x(\tau_1(s)))- f_2(s, x(\tau_2(s)))|\Delta s. \end{align*} There exists a $\delta$, so that when $|t-t'|<\delta$, $|(\Gamma_2x)(t)-(\Gamma_2x)(t')|<\varepsilon$, which shows that the family $\Gamma_2\Omega$ is equicontinuous, $\Gamma_2$ is completely continuous. By Lemma 2, there exists a fixed point $x\in \Omega$, such that $\Gamma x=x$. It is easily to see that $x$ is a bounded non-oscillatory solution which is bounded away from zero. \end{proof} \begin{theorem} \label{thm2} Assume that $1t_0$ such that $$T_0=\min\{\tau(t_1),\inf_{t\geq t_1}(\tau_1(t)), \inf_{t\geq t_1}(\tau_2(t))\}\geq t_0.$$ Let $BC$ be the set of bounded functions on $[t_0,\infty)$ with supremum norm $\|x\|=\sup_{t\geq t_0}|x(t)|$, $t\in \mathbb{T}$. Define a set $\Omega \subset BC$ as follows: \begin{align*} \Omega=\Big\{&x\in BC, x^{\Delta}(t)\leq 0, 00$, there exists a$T>t_1$such that $$\int_{\tau^{-1}(T)}^{\infty}g_{n-1}(\sigma(s), 0)f_i(s, b)\Delta s<\frac{-p_2\varepsilon}{2}.$$ For all$x\in \Omega$,$t,t'\geq T$, we have $$|(\Gamma_2x)(t)-(\Gamma_2x)(t')|\leq -\frac{2}{p_2} \int_{\tau^{-1}(T)}^{\infty}g_{n-1}(\sigma(s), 0)f_i(s, b)\Delta s<\varepsilon.$$ Since$\tau^{-1}(t),\frac{1}{p(\tau^{-1}(t))}$are continuous on$[t_1,T]$, they are uniformly continuous on$[t_1, T]$. Let$|g_{n-1}(\sigma(t),0)f_i(t, b)|\leq M$, when$t\in [t_1, T]$. Hence for each$\varepsilon>0$, there exists a$\delta>0$such that for$ t, t'\in [t_1, T]$,$|t-t'|<\delta$, we have \begin{gather*} |\frac{1}{p(\tau^{-1}(t))}-\frac{1}{p(\tau^{-1}(t'))}| <\frac{\varepsilon}{3c},\quad |\tau^{-1}(t)-\tau^{-1}(t')|<\frac{-p_2 \varepsilon}{3M},\\ \int_{\tau^{-1}(t_1)}^{\infty}|g_{n-1}(\sigma(s), t)-g_{n-1}(\sigma(s), t')|f_i(s, b)\Delta s<\frac{|p_2|\varepsilon}{3}. \end{gather*} For all$x\in\Omega$, when$t,t'\in [t_1,T]$and$|t-t'|<\delta, we have \begin{align*} &|(\Gamma_2x)(t)-(\Gamma_2x)(t')|\\ &= |\frac{1}{p(\tau^{-1}(t))}\int_{\tau^{-1}(t)}^{\infty}g_{n-1} (\sigma(s), t)[f_1(s, x(\tau_1(s)))-f_2(s, x(\tau_2(s)))]\Delta s\\ &\quad -\frac{1}{p(\tau^{-1}(t'))}\int_{\tau^{-1}(t')}^{\infty}g_{n-1} (\sigma(s), t')[f_1(s, x(\tau_1(s)))-f_2(s, x(\tau_2(s)))]\Delta s|\\ &\leq |\frac{1}{p(\tau^{-1}(t))}-\frac{1}{p(\tau^{-1}(t'))}| \int_{\tau^{-1}(t)}^{\infty}g_{n-1}(\sigma(s), t)|f_1(s, x(\tau_1(s)))-f_2(s, x(\tau_2(s)))|\Delta s\\ &\quad + |\frac{1}{p(\tau^{-1}(t'))}\|\int_{\tau^{-1}(t)}^{\tau^{-1}(t')} g_{n-1}(\sigma(s), t)[f_1(s, x(\tau_1(s)))-f_2(s, x(\tau_2(s)))]\Delta s|\\ &\quad +|\frac{1}{p(\tau^{-1}(t'))}|\int_{\tau^{-1}(t')}^{\infty} |g_{n-1}(\sigma(s), t)-g_{n-1}(\sigma(s),t')|\\ &\quad\times |\sum_{i=1}^{2}(-1)^{i+1}f_i(s, x(\tau_i(s)))|\Delta s\\ &< \frac{\varepsilon c}{3c}+\frac{M}{|p_2|}\cdot\frac{|p_2|\varepsilon}{3M} +\frac{|p_2|\varepsilon}{|p_2|3}=\varepsilon, \end{align*} which shows that the family\Gamma_2\Omega$is equicontinuous, so$\Gamma_2$is completely continuous. By Lemma 2, there exists a fixed point$x\in \Omega$such that$\Gamma x=x$. It is easily to see that$x$is a bounded non-oscillatory solution which is bounded away from zero. \end{proof} \subsection*{Example} On the time scale$\mathbb{T} =\{q^{n}: n\in \mathbb{N}_0,\,q>1\}, consider the dynamic equation \begin{aligned} &(x(t)-\frac{1}{\sqrt{q}}x(\rho(t)))^{\Delta^4} +2\frac{(\sqrt{q}-1)(q+1)^2(q^2+1)(q^2+q+1)} {q^{10}t^3(t+q^2)^2}x^2(\rho^2(t))\\ &- \frac{(\sqrt{q}-1)(q+1)^2(q^2+1)(q^2+q+1)} {q^{10}t^3(t+q^2)^3}x^2(\rho^3(t))=0, \end{aligned}\label{e5} where\rho$is the backward operator,$\rho^2(t)=\rho(\rho(t)), \rho^3(t)=\rho(\rho^2(t))$. In this equation,$n=4$,$p(t)=-\frac{1}{\sqrt{q}}$,$\tau(t)=\rho(t)=\frac{t}{q}$,$\tau_1(t)=\rho^2(t)$,$\tau_2(t)=\rho^3(t)$, \begin{gather*} f_1(t,b)=2\frac{(\sqrt{q}-1)(q+1)^2(q^2+1)(q^2+q+1)} {q^{10}t^3(t+q^2)^2}b^2, \\ f_2(b)=\frac{(\sqrt{q}-1)(q+1)^2(q^2+1)(q^2+q+1)} {q^{10}t^3(t+q^2)^3}b^2. \end{gather*} By the definition of$g_k(s, t), \begin{align*} g_{4-1}(\sigma(s), 0)\cdot f_1(s, b) & \leq s^3 2\frac{(\sqrt{q}-1)(q+1)^2(q^2+1)(q^2+q+1)} {q^{10}s^3(s+q^2)^2}b^2\\ &\leq 2\frac{(\sqrt{q}-1)(q+1)^2(q^2+1)(q^2+q+1)} {q^{10}s^2}b^2, \end{align*} \begin{align*} g_{4-1}(\sigma(s), 0)\cdot f_2(s, b) & \leq s^3 \frac{(\sqrt{q}-1)(q+1)^2(q^2+1)(q^2+q+1)} {q^{10}s^3(s+q^2)^3}b^2\\ &\leq \frac{(\sqrt{q}-1)(q+1)^2(q^2+1)(q^2+q+1)} {q^{10}s^3}b^2, \end{align*} and \begin{gather*} \int_{t_0}^{\infty}\frac{(\sqrt{q}-1)(q+1)^2(q^2+1)(q^2+q+1)} {q^{10}s^2}b^2\Delta s<\infty,\\ \int_{t_0}^{\infty} \frac{(\sqrt{q}-1)(q+1)^2(q^2+1)(q^2+q+1)} {q^{10}s^3}b^2\Delta s<\infty. \end{gather*} It is obviously that \eqref{e5} satisfies all conditions of Theorem 3. Hence \eqref{e5} has a bounded non-oscillatory solution which is bounded away from zero. In factx(t)=1+\frac{1}{t} \$ is a solution of \eqref{e5}. \subsection*{Acknowledgements} This research was supported by grants L2009Z02 from the Main Foundation of Hebei Normal University, and L2006B01 from the Doctoral Foundation of Hebei Normal University. The authors would like to thank the anonymous referee for his or her careful reading and the comments on improving the presentation of this article. \begin{thebibliography}{00} \bibitem{b1} M. Bohner, G. Guseinov; {\it Line integrals and Green's formula on time scales,} J. Math. Anal. Appl., 326 (2007), 1124-1141. \bibitem{b2} M. Bohner, A. Peterson; {\it Dynamic equations on time scales: An introduction with applications,} Birkh\"{a}user, Boston, Massachusetts, 2001. \bibitem{b3} M. Bohner, A. Peterson; {\it Advances in dynamic equations on time scales,} Birkh\"{a}user, Boston, Massachusetts, 2003. \bibitem{e1} L. Erbe, Q. Kong and B. 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