\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<t\}\in \mathbb{T}
\end{gather*}
for all $t\in \mathbb{T}$ with $t<\sup \mathbb{T}$ and
$t>\inf \mathbb{T}$, 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 $\sigma(t)>t$, $\rho(t)<t$,
$\sigma(t)=t$, $\rho(t)=t$, respectively.
\end{definition}

Throughout this paper we  assume that $a\leq b$ are points in
$\mathbb{T}$.

\begin{definition} \label{def2.2}\rm
If $r,s\in \mathbb{T}\cup \{-\infty,+\infty\}$, $r<s$, then an open
interval $(r,s)$ in $\mathbb{T}$ is defined by
$$
(r,s)=\{t\in \mathbb{T} : r<t<s\}.
$$
Other types of intervals are defined similarly.
\end{definition}

\begin{definition} \label{def2.3} Assume that
$x:\mathbb{T}\to \mathbb{R}$ and fix $t\in \mathbb{T}$.
Then, $x$ is called differentiable at $t\in \mathbb{T}$ if there
exists a $\theta\in \mathbb{R}$, such that, for any given
$\varepsilon>0$, 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
$K_{r}=\{u\in K: \| u\|<r\}$. Assume that $A: \bar{K}_{r}\to K$
is completely continuous, such that $Ax\neq x$ for $x\in \partial
K_{r}=\{u\in K: \| u\|=r\}$.
\begin{itemize}
\item[(i)] If $\| Ax\|\geq\| x\|$, for
$x\in \partial K_{r}$, then
$i(A, K_{r}, K)=0$.

\item[(ii)] If $\| Ax\|\leq\| x\|$, for $x\in \partial K_{r}$, then
$i(A, K_{r}, K)=1$.
\end{itemize}
\end{lemma}

 To apply Lemma \ref{lem2.3} to \eqref{e1.3} and \eqref{e1.4}, we define the Banach
space $B=\{x|x:[a,\sigma^{2}(b)]\to \mathbb{R} \text{ is continuous }\}$,
for $x\in B$, let $|x|_{0}=\max_{t\in[a,\sigma^{2}(b)]}| x(t)|$ and
$E=B^{n}$,  for  $u=(u_{1},u_{2},\dots ,u_{n})\in E$,
$\| u\|=\sum_{i=1}^{n}| u_{i}|_{0}$.

For $u\in E$ or $\mathbb{R}^{n}_{+}$, $\| u\|$ denotes
the norm of $u$ in $E$ and $\mathbb{R}^{n}_{+}$, respectively.

Define $K$ to be a cone in  $E$ by
\begin{align*}
K =\big\{&u=(u_{1},u_{2},\dots ,u_{n})\in E: u_{i}(t)\geq0,
t\in[a, \sigma^{2}(b)], i=1,2,\dots ,n, \\
&\text{and }  \min_{t\in[\xi, \sigma(\omega)]}\sum_{i=1}^{n}u_{i}(t)\geq k\|
u\| \big\}.
\end{align*}
For $u=(u_{1},u_{2},\dots ,u_{n})\in K$, let
$$
A(u)=(A_{1}(u), A_{2}(u),\dots , A_{n}(u)),
$$
where
$$
A_{i}(u)=\lambda\int^{\sigma(b)}_{a}G(t,
s)h_{i}(s)f_{i}(u_{1}(\sigma(s)),\dots ,
u_{n}(\sigma(s)))\Delta s,\quad  t\in[a,\sigma^{2}(b)].
$$

\begin{lemma} \label{lem2.4}
Assume that (H1) and (H2)
hold, then $A: K\to K$ is completely continuous.
\end{lemma}

\begin{proof}
For $u=(u_{1},u_{2},\dots , u_{n})\in K$, and
$i=1,2,\dots , n$, it follows from Lemma \ref{lem2.1} that
\begin{align*}
0&\leq 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(\sigma(s),s)h_{i}(s)
f_{i}(u_{1}(\sigma(s)),\dots , u_{n}(\sigma(s)))\Delta s, \quad
 t\in[a, \sigma^{2}(b)].
\end{align*}
So, for $i=1,2,\dots ,n$,
\[
| A_{i}(u)|_{0}\leq
 \lambda\int^{\sigma(b)}_{a}G(\sigma(s),s)h_{i}(s)f_{i}(u_{1}(\sigma(s)),
\dots ,u_{n}(\sigma(s)))\Delta s,
\]
For $t\in [\xi, \sigma(\omega)]$, from Lemma \ref{lem2.2} and the above inequality,
we have
\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\\
&\geq k\lambda\int^{\sigma(b)}_{a}G(\sigma(s),s)h_{i}(s)f_{i}
(u_{1}(\sigma(s)),\dots , u_{n}(\sigma(s)))\Delta s\\
&\geq k| A_{i}(u)|_{0},\quad  i=1,2,\dots , n.
\end{align*}
So, for $t\in [\xi, \sigma(\omega)]$,
$$
\sum_{i=1}^{n}A_{i}(u)(t)\geq k\sum_{i=1}^{n}| A_{i}(u)|_{0}= k \| Au\|.
$$
Hence,
$$
\min_{t\in[\xi, \sigma(\omega)]}\sum_{i=1}^{n}A_{i}(u)(t)\geq k\| Au\|;
$$
i.e., $A(u)\in K$. Further, it is easy to see that $A: K\to K$ is
completely continuous. The proof is complete.
\end{proof}

Now, it is not difficult to show that the problem \eqref{e1.3} and\eqref{e1.4}
is equivalent to the fixed-point equation
$A(u)= u$ in $K$.
Let
$$
\gamma_{i}= \max_{t\in[a,
\sigma^{2}(b)]}\int^{\sigma(\omega)}_{\xi}G(t, s)h_{i}(s) \Delta
s,\quad \text{and}\quad \Gamma= \min_{1\leq i\leq n}\{\gamma_{i}\}.
$$

\begin{lemma} \label{lem2.5}
Assume that (H1) and (H2) hold. Let $u= (u_{1},u_{2},\dots ,u_{n})\in K$
and $\eta>0$. If there exists $f_{i_{0}}$ such that
\begin{equation} \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)],
\end{equation}
then
$\| A(u)\| \geq \lambda k \eta\Gamma\| u\|$.
\end{lemma}

\begin{proof} From the definition of $K$ and \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 each $i=1,2,\dots , n$, 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
\begin{equation} \label{e2.3}
\tilde{f_{i}}(h) \leq \varepsilon h,\quad i=1,2,\dots , n,
\end{equation}
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
\begin{equation} \label{e3.1}
\| A(u)\|\leq \lambda\varepsilon C\| u\|< \| u\|,\quad
 \text{for } u\in\ \partial K_{r_{1}}.
\end{equation}
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
\begin{equation} \label{e3.2}
\| A(u)\|\geq \lambda\eta\Gamma k\| u\| > \| u\|,\quad
 \text{for } u\in \partial K_{r_{2}}.
\end{equation}
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
\begin{equation} \label{e3.3}
\| A(u)\|\geq \lambda\eta\Gamma k\| u\| > \| u\|,\quad
 \text{for } u\in \partial K_{r_{1}}.
\end{equation}
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
\begin{equation} \label{e3.4}
\| A(u)\|\leq \lambda\varepsilon C\| u\|< \| u\|,\quad
 \text{for } u\in\ \partial K_{r_{2}}.
\end{equation}
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. Sys., (Gaussig, 1990) volume 59 of
Math. Res., 9-20. Akademie Verlag, Berlin, 1990.

\bibitem{a2} R. P. Agarwal, M. Bohner;
\emph{Basic calculus on time scales and some of its applications}.
Results Math., 35 (1999), 3-22.

\bibitem{a3}  R. P. Agarwal, M. Bohner, P. Wong;
\emph{Sturm-Liouville eigenvalue problems on time scales}.
Apply. Math. Comput., 99 (1999), 153-166.

\bibitem{a4} R. P. Agarwal, D. O'Regan;
\emph{Triple solutions to boundary value problems on time scales}.
Appl. Math. Lett., 13(4) (2000), 7-11.

\bibitem{a5} R. P. Agarwal, D. O'Regan;
\emph{Nonlinear boundary value problems on time scales},
Nonlinear Anal., 44 (2001), 527-535.

\bibitem{a6}  D. R. Anderson;
\emph{Second-order $n$-point problems on time scales with changing-sign
nonlinearity},
Advances in Dynamical Systems and Applications, 1:1 (2006), 17-27.

\bibitem{a7} R. I. Avery, D. R. Anderson; \emph{Existence of three
positive solutions to a second-order boundary value problem on a
measure chain}. J. Comput. Appl. Math., 141 (2002), 65-73.

\bibitem{b1} M. Bohner,  A. Peterson,
\emph{Dynamic Equations on Time scales, An Introduction with
Applications}, Birkh\"{a}user, Boston, 2001.

\bibitem{b2} M. Bohner, A. Peterson, editors,
\emph{Advances in Dynamic Equations on
Time Scales}, Birkh\"{a}user, Boston, 2003.

\bibitem{c1} C. J. Chyan, J. Henderson;
\emph{Twin solutions of boundary value problems for
differential equations on measure chains}. J. Comput. Appl.
Math., 141 (2002), 123-131.

\bibitem{e1}  L. Erbe,  A. Peterson;
 \emph{Positive Solutions for a nonlinear differential equation on a
measure chain}. Math. Comput. Modelling, 32 (5-6) (2000), 571-585.

\bibitem{g1}  D. Guo,  V. Lakshmikantham, \emph{Nonlinear
Problems in Abstract Cones}, Academic Press, San Diego, CA, 1988.

\bibitem{h1} J. Henderson; \emph{Multiple solutions for
$2m^{th}$-order Sturm-Liouville boundary value problems on a measure
chain}. J. Difference Equations and Appl., 6 (2-3) (2000), 417-429.

\bibitem{h2} S. Hilger; \emph{Analysis on measure chains-A unified
approach to continuous and discrete calculus}. Results Math., 18 (1990),
18-56.

\bibitem{l1} W. C. Lian,  C. C. Chou,  C. T. Liu, F. H. Wong;
\emph{Existence of solutions for nonlinear BVPs of second-order
differential equations on  measure chains}. Math. Comput. Modelling,
34 (7/8) (2001), 821-837.

\bibitem{l2} W. T. Li, H. R. Sun;
\emph{Multiple positive solutions for nonlinear dynamical system on a
measure chain}. J. Comput. Appl. Math., 162 (2004), 421-430.

\bibitem{s1} J. P. Sun,  Y. H. Zhao,  W. T. Li;
\emph{Existence of positive solution for second-order nonlinear
discrete system with parameter}. Math. Comput. Modelling, 41
(2005), 493-499.

\bibitem{w1} H. Wang; \emph{On the number of positive solutions of
nonlinear systems}. J. Math. Anal. Appl., 281 (2003), 287-306.

\end{thebibliography}

\end{document}