\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 $$\begin{gathered} x^{\Delta }(t)+f(t,x(\sigma (t)))=0,\quad t\in J:=[ 0,T] _{\mathbb{T}},\; t\neq t_k,\; k=1,2,\dots ,m, \\ x(t_k^{+})-x(t_k^{-})=I_k(x(t_k^{-})),\quad k=1,2,\dots ,m, \\ x(0)=x(\sigma (T)), \end{gathered} \label{e1.1}$$ where $\mathbb{T}$ is a time scale, $T>0$ is fixed, $0,T\in \mathbb{T}$, $f\in C( J\times [ 0,\infty ),( -\infty,\infty ) )$, $I_k\in C( [ 0,\infty ) ,(-\infty ,\infty ) )$, $t_k\in ( 0,T) _{\mathbb{T}}$, $00$. \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 t0 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 $$x\neq \Phi x+\lambda e\quad \text{for }x\in K\cap \partial \Omega _1\text{ and } \lambda >0. \label{e3.1}$$ 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 $$\| \Phi x\| \leq \| x\| \text{ for }x\in K\cap \partial \Omega _2. \label{e3.2}$$ 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 $$\| \Phi x\| <\| x\| \quad \text{ for }x\in K\cap \partial \Omega _3. \label{e3.5}$$ 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 $\mathbb{T}$: $$\begin{gathered} x^{\Delta }(t)+f(t,x(\sigma (t)))=0,\quad t\in [ 0,3] _{\mathbb{T}},\; t\neq \frac 12, \\ x( \frac 12^{+}) -x( \frac 12^{-}) =I(x(\frac 12)), \\ x(0)=x(3), \end{gathered} \label{e4.1}$$ 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} \begin{thebibliography}{99} \bibitem{b1} M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab; On first order impulsive dynamic equations on time scales, \emph{J. Difference Equ. 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