\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 73, 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 (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/73\hfil Existence of positive solutions] {Existence of positive solutions for nonlinear dynamic systems with a parameter on a measure chain} \author[S.-H. Ma, J.-P. Sun, D.-B. Wang\hfil EJDE-2007/73\hfilneg] {Shuang-Hong Ma, Jian-Ping Sun, Da-Bin Wang} % in alphabetical order \address{Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China} \email[S.-H. Ma]{mashuanghong@lut.cn} \email[J.-P. Sun]{jpsun@lut.cn} \email[D.-B. Wang (Corresponding author)]{wangdb@lut.cn} \thanks{Submitted January 9, 2007. Published May 15, 2007.} \subjclass[2000]{34B15, 39A10} \keywords{Dynamic system; positive solution; cone; fixed point; measure chain} \begin{abstract} In this paper, we consider the following dynamic system with parameter on a measure chain $\mathbb{T}$, \begin{gather*} u^{\Delta\Delta}_{i}(t)+\lambda h_{i}(t)f_{i}(u_{1}(\sigma(t)), u_{2}(\sigma(t)),\dots ,u_{n}(\sigma(t)))=0,\quad t\in[a,b], \\ \alpha u_{i}(a)-\beta u^{\Delta}_{i}(a)=0,\quad \gamma u_{i}(\sigma(b))+\delta u^{\Delta}_{i}(\sigma(b))=0, \end{gather*} where $i=1,2,\dots ,n$. Using fixed-point index theory, we find sufficient conditions the existence of positive solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \section{Introduction} The theory of dynamic equations on time scales has become a new important mathematical branch (see, for example, \cite{a1,a3,b1,b2}) since it was initiated by Hilger \cite{h2}. At the same time, boundary-value problems (BVPs) for scalar dynamic equations on time scales have received considerable attention \cite{a4,a5,a6,a7,c1,e1,h1,l1,l2}. However, to the best of our knowledge, only a few papers can be found in the literature for systems of BVPs for dynamic equations on time scales \cite{l2}. Sun, Zhao and Li \cite{s1} considered the following discrete system with parameter \begin{gather*} \Delta^{2}u_{i}(k)+\lambda h_{i}(k)f_{i}(u_{1}(k),u_{2}(k),\dots , u_{n}(k))=0,\quad k\in[0,T],\\ u_{i}(0)=u_{i}(T+2)=0, \end{gather*} where $i=1,2,\dots ,n$, $\lambda>0$ is a constant, $T$ and $n\geq2$ are two fixed positive integers. They established the existence of one positive solution by using the theory of fixed-point index \cite{g1}. Motivated by \cite{s1}, the purpose of this paper is to study the following more general dynamic system with parameter on a measure chain $\mathbb{T}$, \begin{gather} u^{\Delta\Delta}_{i}(t)+\lambda h_{i}(t)f_{i}(u_{1}(\sigma(t)),u_{2}(\sigma(t)), \dots ,u_{n}(\sigma(t)))=0,\quad t\in[a,b], \label{e1.3} \\ \alpha u_{i}(a)-\beta u^{\Delta}_{i}(a)=0,\quad \gamma u_{i}(\sigma(b))+\delta u^{\Delta}_{i}(\sigma(b))=0, \label{e1.4} \end{gather} where, $i=1,2,\dots ,n$, $\lambda>0$ is constant, $a, b\in \mathbb{T}$, $\alpha$, $\beta$, $\gamma$, $\delta\geq0$, $\gamma(\sigma(b)-\sigma^{2}(b))+\delta\geq0$, $r=\gamma\beta+\alpha\delta+\alpha\gamma(\sigma(b)-a)>0$, and the function $\sigma(t)$ and $[a,b]$ is defined as in Section 2 below. Let $\mathbb{R}$ be the set of real numbers, and $\mathbb{R}_{+}=[0, \infty)$. For $u= (u_{1},u_{2},\dots ,u_{n})\in \mathbb{R}^{n}_{+}$, let $\| u\|=\sum_{i=1}^{n}u_{i}$. We make the following assumptions for $i=1,2,\dots ,n$: \begin{itemize} \item[(H1)] $h_{i}: [a, b]\to (0, \infty)$ is continuous. \item[(H2)] $f_{i}: \mathbb{R}^{n}_{+} \to\mathbb{R}_{+}$ is continuous. \end{itemize} For convenience, we introduce the following notation \begin{gather*} f^{0}_{i}= \lim_{\| u\| \to 0}\frac{f_{i}(u)}{\| u\|},\quad f^{\infty}_{i}= \lim_{\| u\| \to\infty}\frac{f_{i}(u)}{\| u\|},\quad u\in \mathbb{R}^{n}_{+},\\ f^{0}= \sum_{i=1}^{n}f^{0}_{i}\quad \text{and}\quad f^{\infty}= \sum_{i=1}^{n}f^{\infty}_{i}. \end{gather*} \section{Preliminaries} In this section, we introduce several definitions on measure chains and some notation. Also we give some lemmas which are useful in proving our main result. \begin{definition} \label{def2.1} \rm Let $\mathbb{T}$ be a closed subset of $\mathbb{R}$ with the properties \begin{gather*} \sigma(t)=\inf\{\tau\in \mathbb{T}:\tau>t\}\in \mathbb{T} \\ \rho(t)=\sup\{\tau\in \mathbb{T}:\tau\inf \mathbb{T}$, respectively. We assume throughout that$\mathbb{T}$has the topology that it inherits from the standard topology on$\mathbb{R}$. We say$t$is right-scattered, left-scattered, right-dense and left-dense if$\sigma(t)>t$,$\rho(t)0$, there is an open neighborhood$U$of$t$, such that $$| x(\sigma(t))-x(s)-\theta[\sigma(t)-s]|\leq\varepsilon|\sigma(t)-s|, \quad s\in U.$$ In this case,$\theta$is called the$\Delta$-derivative of$x$at$t\in \mathbb{T}$and we denote it by$\theta=x^{\Delta }(t)$. It can be shown that if$x:\mathbb{T}\to \mathbb{R}$is continuous at$t\in \mathbb{T}$, then $x^{\Delta}(t)=\frac{x(\sigma(t))-x(t)}{\sigma(t)-t}$ if$t$is right-scattered, and $x^{\Delta}(t)=\lim_{s \to t}\frac{x(t)-x(s)}{t-s}$ if$t$is right-dense. \end{definition} In the rest of the paper, we assume that the set$[a,\sigma(b)]$is, such that $\xi=\min\{t\in \mathbb{T}: t\geq\frac{\sigma(b)+3a}{4}\},\quad \omega=\max\{t\in \mathbb{T}: t\leq\frac{3\sigma(b)+a}{4}\},$ exist and satisfy $$\frac{\sigma(b)+3a}{4}\leq\xi<\omega\leq\frac{3\sigma(b)+a}{4}.$$ We also assume that if$\sigma(\omega)=b$and$\delta=0$, then$\sigma(\omega)<\sigma(b)$. We denote by$G(t,s)$the Green function of the boundary-value problem \begin{gather*} -u^{\Delta\Delta}(t)=0,\quad t\in[a,b],\\ \alpha u(a)-\beta u^{\Delta}(a)=0,\quad \gamma u(\sigma(b))+\delta u^{\Delta}(\sigma(b))=0, \end{gather*} which is explicitly given in \cite{e1}, $$G(t,s)=\begin{cases} \frac{1}{r}\{\alpha(t-a)+\beta\}\{\gamma(\sigma(b)-\sigma(s))+\delta\}, &t\leq s,\\ \frac{1}{r}\{\alpha(\sigma(s)-a)+\beta\}\{\gamma(\sigma(b)-t)+\delta\}, &t\geq \sigma(s), \end{cases}$$ for$t\in[a,\sigma^{2}(b)]$and$s\in[a,b]$, where$r=\gamma\beta+\alpha\delta+\alpha\gamma(\sigma(b)-a)$. For this Green function, we have the following lemmas \cite{b1,b2,e1}. \begin{lemma} \label{lem2.1} Assume$\alpha, \beta, \gamma, \delta \geq 0$,$\gamma(\sigma(b)-\sigma^{2}(b))+\delta\geq0$, and $$r=\gamma\beta+\alpha\delta+\alpha\gamma(\sigma(b)-a)>0\,.$$ Then, for$(t,s)\in[a,\sigma^{2}(b)]\times[a,b]$,$0\leq G(t,s)\leq G(\sigma(s),s)$. \end{lemma} \begin{lemma} \label{lem2.2} (i) If$(t,s)\in[(\sigma(b)+3a)/4, (3\sigma(b)+a)/4]\times [a,b]$, then$G(t,s)\geq lG(\sigma(s),s)$, where $$l=\min\Big\{\frac{\alpha[\sigma(b)-a]+4\beta}{4\alpha[\sigma(b)-a]+4\beta},\, \frac{\gamma[\sigma(b)-a]+4\delta}{4\gamma[\sigma(b)-\sigma(a)] +4\delta}\Big\};$$ (ii) If$(t,s)\in[\xi,\sigma(\omega)]\times[a,b]$, then$G(t,s)\geq kG(\sigma(s),s)$, where $$k=\min\Big\{l,\min_{s\in[a,b]}\frac{G(\sigma(\omega),s)}{G(\sigma(s),s)}\Big\}.$$ \end{lemma} The following well-known result of the fixed-point index is crucial in our arguments. \begin{lemma}[\cite{g1}] \label{lem2.3} Let$E$be a Banach space and$K$a cone in$E$. For$r>0$, define$K_{r}=\{u\in K: \| u\|0$. If there exists$f_{i_{0}}$such that $$\label{e2.2} f_{i_{0}}(u_{1}(\sigma(t)),u_{2}(\sigma(t)),\dots ,u_{n}(\sigma(t)))\geq \eta\sum_{i=1}^{n}u_{i}(t),\quad t\in[\xi, \sigma(\omega)],$$ then$\| A(u)\| \geq \lambda k \eta\Gamma\| u\|$. \end{lemma} \begin{proof} From the definition of$Kand \eqref{e2.2}, we have \begin{align*} \| A(u)\| &=\sum_{i=1}^{n}| A_{i}(u)|_{0}\\ &\geq | A_{i_{0}}|_{0} =\lambda \max_{t\in[a, \sigma^{2}(b)]}\int^{\sigma(b)}_{a}G(t, s)h_{i_{0}}(s)f_{i_{0}}(u_{1}(\sigma(s)),\dots , u_{n}(\sigma(s)))\Delta s\\ &\geq \lambda\max_{t\in[a, \sigma^{2}(b)]} \int^{\sigma(\omega)}_{\xi}G(t, s)h_{i_{0}}(s)f_{i_{0}}(u_{1}(\sigma(s)),\dots , u_{n}(\sigma(s)))\Delta s\\ &\geq \lambda\eta \max_{t\in[a, \sigma^{2}(b)]}\int^{\sigma(\omega)}_{\xi}G(t, s)h_{i_{0}}(s)\sum_{i=1}^{n}u_{i}(s)\Delta s\\ &\geq k\lambda\eta\| u\| \gamma_{i_{0}}\\ &\geq k\lambda\eta \Gamma\| u\|. \end{align*} The proof is complete. \end{proof} For eachi=1,2,\dots , n$, we define a new function$\tilde{f_{i}}: \mathbb{R}_{+} \to \mathbb{R}_{+}$by $$\tilde{f_{i}}(t)=\max\{f_{i}(u): u\in \mathbb{R}^{n}_{+},\; \| u\|\leq t\}.$$ Denote $$\tilde{f^{0}_{i}}= \lim_{t\to 0}\frac{\tilde{f_{i}}(t)}{t},\quad \tilde{f^{\infty}_{i}}=\lim_{t\to\infty}\frac{\tilde{f_{i}}(t)}{t}.$$ As in \cite[Lemma 2.8]{w1}, we can obtain the following result. \begin{lemma} \label{lem2.6} Assume that (H2) holds. Then,$\tilde{f^{0}_{i}}= f^{0}_{i}$and$\tilde{f^{\infty}_{i}}= f^{\infty}_{i}$. \end{lemma} \begin{lemma} \label{lem2.7} Assume that (H1) and (H2) hold. Let$h>0$. If there exists$\varepsilon>0$, such that $$\label{e2.3} \tilde{f_{i}}(h) \leq \varepsilon h,\quad i=1,2,\dots , n,$$ then$\| A(u)\| \leq \lambda\varepsilon C\|u\|$, for$u\in \partial K_{h}$, where $$C= \sum^{n}_{i=1}[\max_{t\in[a, \sigma^{2}(b)]}\int^{\sigma(b)}_{a}G(t, s) h_{i}(s)\Delta s]\,.$$ \end{lemma} \begin{proof} Suppose$u\in\ \partial K_{h}$; i.e.,$u\in\ K$and$\| u\|=h, then it follows from \eqref{e2.3} that \begin{align*} A_{i}(u)(t)& =\lambda\int^{\sigma(b)}_{a}G(t,s)h_{i}(s)f_{i}(u_{1}(\sigma(s)), \dots , u_{n}(\sigma(s)))\Delta s\\ &\leq \lambda\int^{\sigma(b)}_{a}G(t,s)h_{i}(s)\tilde{f_{i}}(h)\Delta s\\ &\leq \lambda\varepsilon h\int^{\sigma(b)}_{a}G(t,s)h_{i}(s)\Delta s\\ &\leq \lambda\varepsilon h\max_{t\in[a, \sigma^{2}(b)]} \int^{\sigma(b)}_{a}G(t,s)h_{i}(s)\Delta s,\quad t\in[a, \sigma^{2}(b)],\; i=1,2,\dots , n. \end{align*} So, $$| A_{i}(u)|_{0}\leq\lambda\varepsilon h\max_{t\in[a, \sigma^{2}(b)]} \int^{\sigma(b)}_{a}G(t,s)h_{i}(s)\Delta s,\quad i=1,2,\dots , n.$$ Therefore, $$\| A(u)\|= \sum^{n}_{i=1}| A_{i}(u)|_{0} \leq\ \lambda\varepsilon h\sum^{n}_{i=1}[\max_{t\in[a, \sigma^{2}(b)]}\int^{\sigma(b)}_{a}G(t, s)h_{i}(s)\Delta s] = \lambda\varepsilon C\| u\|.$$ The proof is complete. \end{proof} \section{Main Result} Our main result is the following theorem. \begin{theorem} \label{thm3.1} Assume that (H1) and (H2) hold. Then, for all\lambda>0$, \eqref{e1.3} and \eqref{e1.4} has a positive solution if one of the following two conditions holds: \begin{itemize} \item[(a)]$f^{0}=0$and$f^{\infty}=\infty$; \item[(b)]$f^{0}=\infty$and$f^{\infty}= 0$. \end{itemize} \end{theorem} \begin{proof} First, we suppose that (a) holds. Since$f^{0}=0$implies that$f^{0}_{i}=0$,$i=1,2,\dots ,n$, it follows from Lemma \ref{lem2.6} that$\tilde{f^{0}_{i}}=0$,$i=1,2,\dots ,n$. Therefore, we can choose$r_{1}>\ 0$, such that $$\tilde{f_{i}}(r_{1}) \leq \varepsilon r_{1},\quad i=1,2,\dots , n,$$ where the constant$\varepsilon>0$satisfies$\lambda\varepsilon C < 1$, and$C$is defined in Lemma \ref{lem2.7}. By Lemma \ref{lem2.7}, we have $$\label{e3.1} \| A(u)\|\leq \lambda\varepsilon C\| u\|< \| u\|,\quad \text{for } u\in\ \partial K_{r_{1}}.$$ Now, since$f^{\infty}=\infty$, there exists$f_{i_{0}}$so that$f^{\infty}_{i_{0}}= \infty$. Therefore, there is$H>0$, such that $$f_{i_{0}}(u)\geq \eta\| u\|,\quad\text{for } u\in \mathbb{R}^{n}_{+}, \quad \text{and}\quad \| u\|\geq H,$$ where$\eta\ >0$is chosen so that$\lambda\eta k\Gamma\ >1$. Let$r_{2}=\max\{2r_{1}, \frac{1}{k}\ H\}$. If$u\in\ \partial K_{r_{2}}$, then $$\| u\|= \sum^{n}_{i=1}| u_{i}|_{0} \geq \sum^{n}_{i=1}u_{i}(t)\geq k\| u\| = kr_{2}\geq H,\quad t\in [\xi, \sigma(\omega)],$$ which implies that $$f_{i_{0}}(u_{1}(\sigma(t)),u_{2}(\sigma(t)),\dots ,u_{n}(\sigma(t))) \geq\eta\| u\|\geq \eta\sum_{i=1}^{n}u_{i}(t),\quad t\in[\xi, \sigma(\omega)].$$ It follows from Lemma \ref{lem2.5} that $$\label{e3.2} \| A(u)\|\geq \lambda\eta\Gamma k\| u\| > \| u\|,\quad \text{for } u\in \partial K_{r_{2}}.$$ By \eqref{e3.1}, \eqref{e3.2} and Lemma \ref{lem2.3}, $$i(A, K_{r_{1}}, K)=1\quad\text{and}\quad i(A, K_{r_{2}}, K)=0.$$ It follows from the additivity of the fixed-point index that $$i(A, K_{r_{2}}\backslash \bar{K}_{r_{1}}, K)=-1,$$ which implies that$A$has a fixed point$u\in K_{r_{2}}\backslash \bar{K}_{r_{1}}$. The fixed point$u\in K_{r_{2}}\backslash \bar{K}_{r_{1}}$is the desired positive solution of \eqref{e1.3} and \eqref{e1.4}. Next, we suppose that (b) holds. Since$f^{0}=\infty$, there exists$f_{i_{0}}$so that$f^{0}_{i_{0}}= \infty$. Therefore, there is$r_{1}> 0$, such that $$f_{i_{0}}(u)\geq \eta\| u\|,\quad \text{for } u\in \mathbb{R}^{n}_{+}, \quad \text{and}\quad \| u\|\leq r_{1},$$ where$\eta\ >0$is chosen so that$\lambda\eta k\Gamma\ >1$. If$u\in \partial K_{r_{1}}$, then $$f_{i_{0}}(u_{1}(\sigma(t)),u_{2}(\sigma(t)),\dots ,u_{n}(\sigma(t))) \geq\eta\| u\|\geq \eta\sum_{i=1}^{n}u_{i}(t),\quad t\in[\xi, \sigma(\omega)].$$ It follows from Lemma \ref{lem2.5} that $$\label{e3.3} \| A(u)\|\geq \lambda\eta\Gamma k\| u\| > \| u\|,\quad \text{for } u\in \partial K_{r_{1}}.$$ In view of$f^{\infty}= 0$implies that$f^{\infty}_{i}= 0$,$i=1,2,\dots ,n$, it follows from Lemma \ref{lem2.6} that$\tilde{f_{i}^{\infty}}=0$,$i=1,2,\dots ,n$. Therefore, we can choose$r_{2}> 2r_{1}$, such that $$\tilde{f_{i}}(r_{2})\leq \varepsilon r_{2},\quad i=1,2,\dots , n,$$ where the constant$\varepsilon>0$satisfies $$\lambda\varepsilon C< 1,$$ and$C$is defined in Lemma \ref{lem2.7}. We have by Lemma \ref{lem2.7} that $$\label{e3.4} \| A(u)\|\leq \lambda\varepsilon C\| u\|< \| u\|,\quad \text{for } u\in\ \partial K_{r_{2}}.$$ By \eqref{e3.3}, \eqref{e3.4} and Lemma \ref{lem2.3}, $$i(A, K_{r_{1}}, K)=0\quad \text{and}\quad i(A, K_{r_{2}}, K)=1.$$ It follows from the additivity of the fixed-point index that $$i(A, K_{r_{2}}\backslash \bar{K}_{r_{1}}, K)=1,$$ which implies that$A$has a fixed point$u\in K_{r_{2}}\backslash \bar{K}_{r_{1}}$, which is the desired positive solution of \eqref{e1.3} and \eqref{e1.4}. \end{proof} \begin{remark} \label{rmk3.1} \rm It is worth noting that these techniques can be extended to the following multi-point system based in \cite{a6}, \begin{gather*} (p_{i}y^{\Delta}_{i})^{\Delta}(t)-q_{i}(t)y_{i}(t) +\lambda h_{i}(t)f_{i}(y_{1}(\sigma(t)),y_{2}(\sigma(t)),\dots , y_{m}(\sigma(t)))=0, \quad t\in (t_{1},t_{n}), \\ \alpha y_{i}(t_{1})-\beta p_{i}(t_{1})y^{\Delta}_{i}(t_{1}) =\sum^{n-1}_{k=2}a_{ki}y_{i}(t_{k}), \quad \gamma y_{i}(t_{n})+\delta p_{i}(t_{n})y^{\Delta}_{i}(t_{n})=\sum^{n-1}_{k=2}b_{ki}y_{i}(t_{k}), \end{gather*} for$i=1,2,\dots , m$. \end{remark} \begin{example} \label{exa3.1}\rm Let$\mathbb{T}=\{1-(\frac{1}{2})^{\mathbb{N}_{0}}\}\cup[1,2]$. We consider the dynamic system \begin{gather} u^{\Delta\Delta}_{i}(t)+\lambda f_{i}(u_{1}(\sigma(t)),u_{2}(\sigma(t)), \dots ,u_{n}(\sigma(t)))=0,\quad t\in[0,1], \label{e3.5}\\ u_{i}(0)- u^{\Delta}_{i}(0)=0,\quad u_{i}(1)+ u^{\Delta}_{i}(1)=0, \label{e3.6} \end{gather}$i=1,2,\dots ,n$, where$f_{i}:\mathbb{R}^{n}_{+}\to\mathbb{R}_{+}$is define by $$f_{i}(u_{1},u_{2},\dots ,u_{n})=(u_{1}+u_{2}+\dots +u_{n})^{i+1},\quad i=1,2,\dots ,n.$$ It is easy to see that $$f^{0}= 0\quad \text{and}\quad f^{\infty}= \infty.$$ So, it follows from Theorem \ref{thm3.1} that for all$\lambda>0$, \eqref{e3.5}-\eqref{e3.6} has at least one positive solution. \end{example} \subsection*{Acknowledgment} The authors would like to thank the anonymous referees for their valuable suggestions which led to an improvement of this paper. \begin{thebibliography}{00} \bibitem{a1} B. Aulbach, S. Hilger; \emph{Linear dynamic processes with inhomogeneous time scale}, Nonlinear Dyn. Quantum Dyn. 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