\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 198, 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 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/198 Impulsive dynamic equations \hfil ]
{Nonlinear first-order periodic boundary-value problems of impulsive
dynamic equations\\ on time scales}

\author[W. Guan, D.-G. Li, S.-H. Ma \hfil EJDE-2012/198\hfilneg]
{Wen Guan, Dun-Gang Li, Shuang-Hong Ma}  % in alphabetical order

\address{Wen Guan \newline
Department of Applied Mathematics,
Lanzhou University of Technology,
Lanzhou, Gansu, 730050, China}
\email{mathgw@sohu.com}

\address{Dun-Gang Li \newline
Department of Applied Mathematics,
Lanzhou University of Technology,
Lanzhou, Gansu, 730050, China}
\email{dungangli@gmail.com}

\address{Shuang-Hong Ma \newline
Department of Applied Mathematics,
Lanzhou University of Technology,
Lanzhou, Gansu, 730050, China}
\email{mashuanghong@lut.cn}

\thanks{Submitted August 20, 2012. Published November 10, 2012.}
\subjclass[2000]{39A10, 34B15}
\keywords{Periodic boundary value problem; positive
solution; fixed point; \hfill\break\indent
time scale; impulsive dynamic equation}

\begin{abstract}
 By using the fixed point theorem in cones, in this paper, existence criteria
 for single and multiple positive solutions to a class of nonlinear
 first-order periodic boundary value problems of impulsive dynamic equations
 on time scales are obtained. An example is given to illustrate the main
 results in this article.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Let $\mathbb{T}$ be a time scale; i.e.,  is a nonempty closed
subset of $\mathbb{R}$. Let $0$, $T$ be points in $\mathbb{T}$, an interval
$(0,T) _{\mathbb{T}}$ denoting time scales interval, that is,
$(0,T) _{\mathbb{T}}:=( 0,T) \cap \mathbb{T}$. Other types of
intervals are defined similarly.

The theory of impulsive differential equations is emerging as an
important area of investigation, since it is a lot richer than the
corresponding theory of differential equations without impulse
effects. Moreover, such equations may exhibit several real world
phenomena in physics, biology, engineering, etc. (see \cite{b3,s1}). At
the same time, the boundary value problems for impulsive
differential equations and impulsive difference equations have
received much attention \cite{f1,l1,l2,s2,t1,z1}. On the other hand,
recently, the theory of dynamic equations on time scales has become
a new important branch (See, for example, \cite{b4,b5,h2}). Naturally,
some authors have focused their attention on the boundary value
problems of impulsive dynamic equations on time scales
\cite{b1,b2,g1,h1,l3,l4,l5,w1,w2}.
 However, to the best of our knowledge, few papers
concerning PBVPs of impulsive dynamic equations on time scales with
semi-position condition \cite{w1,w2}.

In this paper, we are concerned with the existence of positive solutions for
the following PBVPs of impulsive dynamic equations on time scales with
semi-position condition
\begin{equation}
\begin{gathered}
x^{\Delta }(t)+f(t,x(\sigma (t)))=0,\quad t\in J:=[
0,T] _{\mathbb{T}},\; t\neq t_k,\; k=1,2,\dots ,m, \\
x(t_k^{+})-x(t_k^{-})=I_k(x(t_k^{-})),\quad k=1,2,\dots ,m, \\
x(0)=x(\sigma (T)),
\end{gathered}  \label{e1.1}
\end{equation}
where $\mathbb{T}$ is a time scale, $T>0$ is fixed,
$0,T\in \mathbb{T}$, $f\in C( J\times [ 0,\infty ),( -\infty,\infty ) ) $,
$I_k\in C( [ 0,\infty ) ,(-\infty ,\infty ) ) $, $t_k\in ( 0,T) _{\mathbb{T}}$,
$0<t_1<\dots <t_m<T$, and for each $k=1,2,\dots ,m$,
$x(t_k^{+})=\lim_{h\to 0^{+}}x(t_k+h)$ and
$x(t_k^{-})=\lim_{h\to 0^{-}}x(t_k+h) $ represent the right and left limits of $x(t)$ at $t=t_k$.

Using fixed point theorems, Wang  \cite{w1,w2} considered the
existence of one or two positive solution to  \eqref{e1.1} when
the following hypothesis holds (semi-position condition):
\begin{itemize}
\item[(A)] There exists a positive number $M$ such that
\[
Mx-f(t,x)\geq 0\text{ for }x\in [ 0,\infty ) ,\quad t\in [0,T] _{\mathbb{T}}.
\]
\end{itemize}
Motivated by the results mentioned above, in this paper, we shall
obtain existence criteria for single and multiple positive solutions
to  \eqref{e1.1} by means of a fixed point theorem in cones. It
is worth noticing that:
 (i) Our hypotheses on nonlinearity $f$ in
this paper are weaker than condition (A) of \cite{w1,w2};
(ii) For the case $\mathbb{T}=\mathbb{R}$ and $I_k(x)\equiv 0,k=1,2,\dots ,m$,
problem \eqref{e1.1} reduces to the problem studied in \cite{p1} and for the
case $I_k(x)\equiv 0,k=1,2,\dots ,m$,  problem \eqref{e1.1} reduces to
the problem (in the one-dimension case) studied by \cite{s3}. The
ideas in this article come from \cite{t2}.

\begin{theorem}[\cite{g2}] \label{thm1.1}
 Let $X$ be a Banach space and
$K$ is a cone in $X$. Assume $\Omega _1,\Omega _2$ are open subsets
of $X$ with $0\in \Omega _1$, $\overline{\Omega }_1\subset \Omega_2$. Let
\[
\Phi :K\cap (\overline{\Omega }_2\setminus \Omega _1)\to K
\]
be a continuous and completely continuous operator such that
\begin{itemize}
\item[(i)] $\| \Phi x\| \leq \| x\| $ for $x\in K\cap
\partial \Omega _1$;

\item[(ii)] there exists $e\in K\backslash \{0\}$ such that
 $x\neq \Phi x+\lambda e$ for $x\in K\cap \partial \Omega _2$ and $\lambda >0$.
\end{itemize}
Then $\Phi $ has a fixed point in $K\cap (\overline{\Omega }_2\setminus
\Omega _1)$.
\end{theorem}

\begin{remark} \label{rmk1.1}\rm
In Theorem \ref{thm1.1}, if (i) and (ii) are replaced by
\begin{itemize}
\item[(i)] $\| \Phi x\| \leq \| x\| $ for $x\in K\cap
\partial \Omega _2$;

\item[(ii)] there exists $e\in K\backslash \{0\}$ such that
$x\neq \Phi x+\lambda e$ for $x\in K\cap \partial \Omega _1$ and $\lambda >0$,
 then $\Phi $ has also a fixed point in
$K\cap (\overline{\Omega }_2\setminus \Omega _1)$.
\end{itemize}
\end{remark}

\section{Preliminaries}

Throughout the rest of this paper, we  assume that the points of
impulse $t_k$ are right-dense for each $k=1,2,\dots ,m$.
We define
\begin{align*}
PC=\Big\{&x\in [0,\sigma (T)]_{\mathbb{T}}\to \mathbb{R}:x_k\in C(J_k,R),\;
 k=0,1,2,\dots ,m \text{ and}\\
&\text{there exist $x(t_k^{+})$  and $x(t_k^{-})$ with
$x(t_k^{-})=x(t_k)$,\; $k=1,2,\dots ,m$}\Big\},
\end{align*}
 where $x_k$ is the restriction of $x$ to
 $J_k=(t_k,t_{k+1}] _{\mathbb{T}}\subset (0,\sigma (T)]_{\mathbb{T}}$,
$k=1,2,\dots ,m$ and $J_0=[0,t_1]_{\mathbb{T}}$, $t_{m+1}=\sigma (T)$.
Let
\[
X=\{ x:x\in PC,\quad x(0)=x(\sigma (T))\}
\]
with the norm $\| x\| =\sup_{t\in [0,\sigma (T)]_{\mathbb{T}}}| x(t)|$, 
then $X$ is a Banach space.

\begin{lemma}[\cite{w1,w2}] \label{lem2.1}
 Suppose $M>0$ and $h:[0,T]_{\mathbb{T}}\to \mathbb{R}$ is rd-continuous,
then $x$ is a solution of
\[
x(t)=\int_0^{\sigma (T)}G(t,s)h(s)\triangle
s+\sum_{k=1}^mG(t,t_k)I_k(x(t_k)),\quad t\in [0,\sigma (T)]_{\mathbb{T}},
\]
where
\[
G(t,s)=\begin{cases}
\frac{e_M(s,t)e_M(\sigma (T),0)}{e_M(\sigma (T),0)-1}, & 0\leq s\leq t\leq
\sigma (T), \\
\frac{e_M(s,t)}{e_M(\sigma (T),0)-1}, & 0\leq t<s\leq \sigma (T),
\end{cases}
\]
 if and only if $x$ is a solution of the boundary-value problem
\begin{gather*}
x^{\Delta }(t)+Mx(\sigma (t))=h(t),\quad t\in J:=[ 0,T]
_{\mathbb{T}},\quad t\neq t_k,\quad k=1,2,\dots ,m, \\
x(t_k^{+})-x(t_k^{-})=I_k(x(t_k^{-})),\quad k=1,2,\dots ,m, \\
x(0)=x(\sigma(T)).
\end{gather*}
\end{lemma}

\begin{lemma} \label{lem2.2}
Let  $G(t,s)$ be defined as in Lemma \ref{lem2.1}. Then
\[
\frac 1{e_M(\sigma (T),0)-1}\leq G(t,s)
\leq \frac{e_M(\sigma (T),0)}{e_M(\sigma (T),0)-1}
\]
for all $t,s\in [0,\sigma (T)]_{\mathbb{T}}$.
\end{lemma}

\begin{remark} \label{rmk2.1}\rm
 Let $G(t,s)$ be defined as in Lemma \ref{lem2.1},
then $\int_0^{\sigma (T)}G(t,s)\triangle s=1/ M$.
Let
\[
K=\{ x\in X:x(t)\geq \delta \| x\| ,\; t\in [0,\sigma (T)]_{\mathbb{T}}\} ,
\]
where $\delta =\frac 1{e_M(\sigma (T),\quad 0)}\in (0,1)$. It is not
difficult to verify that $K$ is a cone in $X$.
\end{remark}

For $u\in K$, we consider the problem
\begin{equation}
\begin{gathered}
x^{\Delta }(t)+Mx(\sigma (t))=Mu(\sigma (t))-f(t,u(\sigma (t))),\\
 t\in [ 0,T] _{\mathbb{T}},\; t\neq t_k,\; k=1,2,\dots ,m, \\
x(t_k^{+})-x(t_k^{-})=I_k(x(t_k^{-})),\quad k=1,2,\dots ,m, \\
x(0)=x(\sigma (T)).
\end{gathered}  \label{e2.1}
\end{equation}
It follows from Lemma \ref{lem2.1} that  \eqref{e2.1} has a unique solution,
\[
x(t)=\int_0^{\sigma (T)}G(t,s)h_u(s)\triangle
s+\sum_{k=1}^mG(t,t_k)I_k(x(t_k)),\quad t\in [0,\sigma (T)]_{\mathbb{T}},
\]
where $h_u(s)=Mu(\sigma (s))-f(s,u(\sigma (s)))$,
$s\in [0,T]_{\mathbb{T}}$.

We define an operator $\Phi :K\to X$ by
\[
\Phi_x(t)=\int_0^{\sigma (T)}G(t,s)h_x(s)\triangle
s+\sum_{k=1}^mG(t,t_k)I_k(x(t_k)),\quad t\in [0,\sigma (T)]_{\mathbb{T}}.
\]
It is obvious that fixed points of $\Phi $ are solutions of \eqref{e1.1}.

\begin{lemma} \label{lem2.3}
The operator $\Phi :K\to X$ is completely continuous.
\end{lemma}
The proof similar to that in \cite{w1,w2}, so  we omit it here.

\section{Main results}

In this section, by defining an appropriate cones, we impose the conditions
on $f$ which allow us to apply the fixed point theorem in cones to establish
the existence criteria for single and multiple positive solutions of the
problem \eqref{e1.1}.

\begin{theorem} \label{thm3.1}
Suppose that there exist a positive number $M>0$ and $0<\alpha <\beta $ such that
\[
Mx-f(t,\quad x)\geq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in
[\delta \alpha ,\beta ].
\]
Then  \eqref{e1.1} has at least one positive solution if one of the
following two conditions holds:
(i)
\begin{gather*}
f(t,x) \leq 0 \quad\text{ for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \alpha ,\alpha ];\; \forall k,I_k(x)\geq 0,\; x\in [ \delta
\alpha ,\alpha ] , \\
f(t,x) \geq 0\text{ for }t\in [0,T]_{\mathbb{T}},\quad x\in [
\delta \beta ,\beta ];\; \forall k,I_k(x)\leq 0,\; x\in [ \delta \beta
,\beta ] ,
\end{gather*}
(ii)
\begin{gather*}
f(t,x) \geq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \alpha ,\alpha ];\; \forall k,\; I_k(x)\leq 0,\; x\in [ \delta
\alpha ,\alpha ] , \\
f(t,x) \leq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \beta ,\beta ];\; \forall k,\; I_k(x)\geq0,\; x\in [ \delta \beta
,\beta ] .
\end{gather*}
\end{theorem}

\begin{proof} Define the open sets
\[
\Omega _1 =\{x\in X:\| x\| <\alpha \}, \quad
\Omega _2 =\{x\in X:\| x\| <\beta \}.
\]
Firstly, we claim that $\Phi :K\cap (\overline{\Omega }_2\setminus \Omega
_1)\to K$.
In fact, for any $x\in K\cap (\overline{\Omega }_2\setminus \Omega _1)$, we
have $\delta \alpha \leq x\leq \beta $, by Lemma \ref{lem2.2}
\[
\| \Phi x\| \leq \frac{e_M(\sigma (T),0)}{e_M(\sigma (T),0)-1}
\Big[ \int_0^{\sigma (T)}(Mx(\sigma (s))-f(s,x(\sigma (s))))\triangle
s+\sum_{k=1}^mI_k(x(t_k))\Big]
\]
and
\begin{align*}
( \Phi x) (t) &= \int_0^{\sigma (T)}G(t,s)h_x(s)\triangle
s+\sum_{k=1}^mG(t,t_k)I_k(x(t_k)) \\
&\geq  \frac 1{e_M(\sigma (T),0)-1}[ \int_0^{\sigma (T)}(Mx(\sigma
(s))-f(s,x(\sigma (s))))\triangle s+\sum_{k=1}^mI_k(x(t_k))] .
\end{align*}
So
\[
( \Phi x) (t)\geq \frac 1{e_M(\sigma (T),0)}\| \Phi
x\| =\delta \| \Phi x\|;\quad \text{i.e., }\Phi x\in K.
\]
Therefore, $\Phi :K\cap (\overline{\Omega }_2\setminus \Omega _1)\to K$.

Secondly, we prove the result provided conditions (i) holds.
By the first inequality of (i), we have
\[
Mx-f(t,\quad x)\geq Mx,\quad t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \alpha ,\alpha ] .
\]
Let $e\equiv 1$, then $e\in K$. We assert that
\begin{equation}
x\neq \Phi x+\lambda e\quad \text{for }x\in K\cap \partial \Omega _1\text{ and }
\lambda >0.  \label{e3.1}
\end{equation}
If not, there would exist $x_0\in K\cap \partial \Omega _1$ and
$\lambda _0>0 $ such that $x_0=\Phi x_0+\lambda _0e$.

Since $x_0\in K\cap \partial \Omega _1$, it follows that
 $\delta \alpha =\delta \|x_0\| \leq x_0(t)\leq \alpha $.
Let $\mu =\min_{t\in [0,\sigma (T)]_{\mathbb{T}}}x_0(t)$, then for any
$t\in [0,\sigma (T)]_{\mathbb{T}}$, we have
\begin{align*}
x_0(t) &=( \Phi x_0) (t)+\lambda _0 \\
&=\int_0^{\sigma (T)}G(t,s)[Mx_0(\sigma (s))-f(s, x_0(\sigma
(s)))]\triangle s+\sum_{k=1}^mG(t,t_k)I_k(x_0(t_k))+\lambda _0 \\
&\geq \int_0^{\sigma (T)}G(t,s)Mx_0(\sigma (s))\triangle s+\lambda _0 \\
&\geq \mu \int_0^{\sigma (T)}G(t,s)M\triangle s+\lambda _0
= \mu +\lambda _0.
\end{align*}
This implies that $\mu \geq \mu +\lambda _0$, and this is a contradiction.
Therefore \eqref{e3.1} holds.

On the other hand, by using the second inequality of (i), we have
\[
Mx-f(t,\quad x)\leq Mx,\quad t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \beta ,\beta ] .
\]
We assert that
\begin{equation}
\| \Phi x\| \leq \| x\| \text{ for }x\in K\cap \partial \Omega _2.  \label{e3.2}
\end{equation}
In fact, if $x\in K\cap \partial \Omega _2$, then $\delta \beta =\delta
\| x\| \leq x(t)\leq \beta $; we have
\begin{align*}
( \Phi x) (t) &=\int_0^{\sigma (T)}G(t,s)[Mx(\sigma (s))-f(s,
\quad x(\sigma (s)))]\triangle s+\sum_{k=1}^mG(t,t_k)I_k(x(t_k)) \\
&\leq \int_0^{\sigma (T)}G(t,s)Mx(\sigma (s))\triangle s \\
&\leq \int_0^{\sigma (T)}G(t,s)M\triangle s\| x\|
=\| x\| .
\end{align*}
Therefore, $\| \Phi x\| \leq \| x\| $.

It follows from Remark \ref{rmk1.1}, \eqref{e3.1} and \eqref{e3.2} that $\Phi $
has a fixed point $x\in K\cap (\overline{\Omega }_2\setminus \Omega _1)$.
In a similar way, we can prove the result by Theorem \ref{thm1.1}
 if condition (ii) holds.
\end{proof}


\begin{theorem} \label{thm3.2}
Suppose that there exist a positive number $M>0$ and
 $0<\alpha <\rho <\beta $ such that
\[
Mx-f(t,\quad x)\geq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in
[\delta \alpha ,\beta ].
\]
Then \eqref{e1.1} has at least two positive solutions if one of the
following two conditions holds
(i)
\begin{gather*}
f(t,x) \leq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \alpha ,\alpha ];\; \forall k,\; I_k(x)\geq 0,\; x\in [ \delta
\alpha ,\alpha ] , \\
f(t,x) >0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [ \delta
\rho ,\rho ];\; \forall k,\; I_k(x)<0,\; x\in [ \delta \rho ,\rho ]
, \\
f(t,x) \leq 0 \quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \beta ,\beta ];\; \forall k,\; I_k(x)\geq 0,\; x\in [ \delta \beta
,\beta ] ,
\end{gather*}
(ii)
\begin{gather*}
f(t,x) \geq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \alpha ,\alpha ];\; \forall k,\; I_k(x)\leq 0,\; x\in [ \delta
\alpha ,\alpha ] , \\
f(t,x) < 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [ \delta
\rho ,\rho ];\; \forall k,\; I_k(x)>0,\; x\in [ \delta \rho ,\rho ]
, \\
f(t,x) \geq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \beta ,\beta ];\; \forall k,\; I_k(x)\leq 0,\; x\in [ \delta \beta
,\beta ] ,
\end{gather*}
\end{theorem}

\begin{proof}
We prove only  the result when condition (i) holds. In a
similar way we can obtain the result if condition (ii) holds.
Define $\Omega _1$, $\Omega _2$ as in Theorem \ref{thm3.1} and define
\[
\Omega _3=\{x\in X:\| x\| <\rho \}.
\]
Similar to the proof of Theorem \ref{thm3.1}, we can prove that
\begin{gather}
x\neq \Phi x+\lambda e\text{ for }x\in K\cap \partial \Omega _1\text{ and }
\lambda >0,  \label{e3.3} \\
x\neq \Phi x+\lambda e\text{ for }x\in K\cap \partial \Omega _2\text{ and }
\lambda >0,  \label{e3.4}
\end{gather}
where $e\equiv 1\in K$, and
\begin{equation}
\| \Phi x\| <\| x\| \quad \text{ for }x\in K\cap \partial
\Omega _3.  \label{e3.5}
\end{equation}
Thus we can obtain the existence of two positive solutions $x_1$ and $x_2$
by using Theorem \ref{thm1.1} and Remark \ref{rmk1.1}, respectively. It is easy to see that
$\alpha \leq \| x_1\| <\rho <\| x_2\| \leq \beta $.
\end{proof}

\begin{theorem} \label{thm3.3}
Suppose that there exist a positive number $M>0$ and
$0<\alpha _1<\beta _1<\alpha _2<\beta _2<\dots <\alpha _n<\beta_n$ such that
\[
Mx-f(t,\quad x)\geq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in
[\delta \alpha _1,\beta _n].
\]
Then \eqref{e1.1} has at least $n$ multiple positive solutions $x_i$
($1\leq i\leq n$) satisfying
$\alpha _i\leq \| x_i\| \leq \beta_i$, $1\leq i\leq n$, if one of the
following two conditions holds
(i)
\begin{gather*}
f(t,x) \leq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \alpha _i,\alpha _i] ;\; \forall k,\; I_k(x)\geq 0,\; x\in
[ \delta \alpha _i,\alpha _i] ,\; 1\leq i\leq n, \\
f(t,x) \geq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \beta _i,\beta _i] ;\; \forall k,I_k(x)\leq 0,\; x\in
[ \delta \beta _i,\beta _i] ,\; 1\leq i\leq n,
\end{gather*}
(ii)
\begin{gather*}
f(t,x) \geq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \alpha _i,\alpha _i] ;\; \forall k,I_k(x)\leq 0,\; x\in
[ \delta \alpha _i,\alpha _i] ,\; 1\leq i\leq n, \\
f(t,x) \leq 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [
\delta \beta _i,\beta _i] ;\; \forall k,I_k(x)\geq 0,\; x\in [
\delta \beta _i,\beta _i] ,\; 1\leq i\leq n.
\end{gather*}
\end{theorem}

\begin{remark} \label{rmk3.1}\rm
 In theorem \ref{thm3.3}, if (i) and (ii) are replaced by
(iii)
\begin{gather*}
f(t,x) <0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [ \delta
\alpha _i,\alpha _i] ;\; \forall k,I_k(x)>0,\; x\in [ \delta
\alpha _i,\alpha _i] ,\; 1\leq i\leq n, \\
f(t,x) > 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [ \delta
\beta _i,\beta _i] ;\; \forall k,I_k(x)<0,\; x\in [
\delta \beta _i,\beta _i] ,\; 1\leq i\leq n;
\end{gather*}
(iv)
\begin{gather*}
f(t,x) > 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [ \delta
\alpha _i,\alpha _i] ;\; \forall k,I_k(x)<0,\; x\in [ \delta
\alpha _i,\alpha _i] ,\; 1\leq i\leq n, \\
f(t,x) < 0\quad \text{for }t\in [0,T]_{\mathbb{T}},\; x\in [ \delta
\beta _i,\beta _i] ;\; \forall k,I_k(x)>0,\; x\in [
\delta \beta _i,\beta _i] ,\; 1\leq i\leq n.
\end{gather*}
Then  \eqref{e1.1} has at least $2n-1$ multiple positive solutions.
\end{remark}

\section{Examples}

\begin{example} \label{examp4.1}\rm
 Let $\mathbb{T}=[0,1]\cup [2,3]$. We consider
the following problem on $\mathbb{T}$:
\begin{equation}
\begin{gathered}
x^{\Delta }(t)+f(t,x(\sigma (t)))=0,\quad t\in [ 0,3] _{\mathbb{T}},\;
 t\neq \frac 12, \\
x( \frac 12^{+}) -x( \frac 12^{-}) =I(x(\frac 12)), \\
x(0)=x(3),
\end{gathered} \label{e4.1}
\end{equation}
where $T=3$, $f(t,x)=x-x^{1/2}+\frac 7{64}$, and $I(x)=x^{1/2}-x$.

Let $M=1$, $\alpha =e^2/32$, $\beta =4e^2$. Then
$e_M(\sigma (T),0)=2e^2$, $\delta =1/(2e^2)$, it is easy to see that
\[
Mx-f(t,x)=x^{1/2}-\frac 7{64}\geq \frac 18-\frac 7{64}=\frac 1{64}>0,
\quad \text{for }x\in [\frac 1{64},4e^2]=[\delta \alpha ,\beta ],
\]
and
\begin{gather*}
f(t,x) =x-x^{1/2}+\frac 7{64}\leq \frac 1{64}-\frac 18+\frac 7{64}=0,
\quad \text{for }x\in [\frac 1{64},\frac{e^2}{32}]=[\delta \alpha ,\alpha ]; \\
f(t,x) =x-x^{1/2}+\frac 7{64}>0,\quad \text{for }x\in [2,4e^2]=[\delta
\beta ,\beta ]; \\
I(x) =x^{1/2}-x\geq \frac 18-\frac 1{64}>0,\quad \text{for }x\in [\frac
1{64},\frac{e^2}{32}]=[\delta \alpha ,\alpha ]; \\
I(x) = x^{1/2}-x\leq 2^{1/2}-2<0,\quad \text{for }x\in [2,4e^2]=[\delta
\beta ,\beta ].
\end{gather*}
Therefore, by Theorem \ref{thm3.1}, it follows that \eqref{e4.1} has at least
one positive solution.
\end{example}

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\end{document}