\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 52, pp. 1--10.\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{8mm}} \begin{document} \title[\hfilneg EJDE-2010/52\hfil Multiplicity of positive solutions] {Multiplicity of positive solutions for four-point boundary value problems of impulsive differential equations with $p$-Laplacian} \author[L. Shen, X. Liu, Z. Lu \hfil EJDE-2010/52\hfilneg] {Li Shen, Xiping Liu, Zhenhua Lu} \address{Li Shen \newline College of Science, University of Shanghai for Science and Technology\\ Shanghai 200093, China} \email{eric0shen@gmail.com} \address{Xiping Liu \newline College of Science, University of Shanghai for Science and Technology\\ Shanghai 200093, China} \email{xipingliu@163.com} \address{Zhenhua Lu\newline College of Science, University of Shanghai for Science and Technology\\ Shanghai 200093, China} \email{ehuanglu@163.com} \thanks{Submitted November 16, 2009. Published April 14, 2010.} \thanks{Supported by grant 10ZZ93 from Innovation Program of Shanghai Municipal \hfill\break\indent Education Commission} \subjclass[2000]{34A37, 34B37} \keywords{$p$-Laplacian; impulsive; positive solutions; boundary value problem} \begin{abstract} Using a fixed-point theorem in cones, we obtain sufficient conditions for the multiplicity of positive solutions for four-point boundary value problems of third-order impulsive differential equations with $p$-Laplacian. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Recently, there has been much attention focused on the theory of impulsive differential equation as it is widely used in various areas such as mechanics, electromagnetism, chemistry. A lot of theories have been established to solve these problems, see \cite{Lak}, \cite{Fu} and the references therein. Guo \cite{G1} obtained the existence of solutions, via cone theory, for second-order impulsive differential equation \begin{gather*} x''=f(t,x,Tx), \quad t\geq 0, \; t\neq t_{k}\; k=1,2,3,\dots, \\ \Delta x|_{t=t_{k}} =I_{k}(x(t_{k})), \quad k =1,2,3,\dots,\\ \Delta x'|_{t=t_{k}} =\overline{I}_{k}(x(t_{k})), \quad k = 1,2,3,\dots,\\ x(0)=x_{0}, \quad x'(0)=x_{0}^{*}. \end{gather*} In \cite{AR}, using Leggett-Williams fixed point theorem, authors studied the multiplicity result for second order impulsive differential equations \begin{gather*} y''+\phi(t)f(y(t))=0, \quad t\in (0,1)\setminus \{t_1,t_2,\dots,t_m\}, \\ \Delta y(t_{k}) =I_{k}(y(t_{k}^{-})), \;k =1,2,3,\dots,m,\\ \Delta y'(t_{k}) =J_{k}(y(t_{k}^{-})), \;k =1,2,3,\dots,m,\\ y(0)=y(1)=0. \end{gather*} Kaufmann \cite{Kaufmanna} studied a second-order nonlinear differential equation on an unbounded domain with solutions subject to impulsive conditions and the Sturm-Liouville type boundary conditions. In \cite{J1}-\cite{JL}, the authors studied positive solutions of multiple points boundary value problems for second order impulsive differential equations. All the works above concern boundary value problems with second-order impulsive equations, and there are just a few works that consider multiplicity of positive solutions for third-order impulsive equations with $p$-Laplacian. Motivated by all the works above, we concentrate on getting multiple positive solutions for four-point boundary value problems of third-order impulsive differential equations with $p$-Laplacian $$\label{eq1.1} \begin{gathered} (\phi_p(u''(t)))'=f(t,u(t),u'(t)), \quad t \in (0,1) \backslash \{t_1,t_2,\dots,t_m\}, \\ \Delta u''(t)|_{t=t_{k}}=0, \quad k =1,2,\dots,m,\\ \Delta u'(t)|_{t=t_{k}} =I_{k}(u(t_{k})), \quad k =1,2,\dots,m,\\ \Delta u(t)|_{t=t_{k}} =J_{k}(u(t_{k})), \quad k = 1,2,\dots,m,\\ u''(0)=0,\quad u'(0)=\alpha u'(\xi)+\beta u'(\eta),\quad u(1)=\delta u(0), \end{gathered}$$ where $\phi_p$ is $p$-Laplacian operator $$\phi_p(s)=|s|^{p-2}s, p>1,\quad (\phi_p)^{-1}=\phi_q,\quad \frac{1}{p}+\frac{1}{q}=1,$$ $t_{k}$, $k = 0,1,2,\dots,m,m+1$, are constants which satisfy $$0=t_00, \alpha+\beta<1; 0<\xi, \eta<1; \xi,\eta\neq t_{k} (k = 1,2,\dots,m); \delta>1; f\in C([0,1]\times[0,+\infty)\times\mathbb{R},[0,+\infty)), I_k, J_k\in C([0,+\infty),[0,+\infty)). \section{Preliminaries} Let J=[0,1]\backslash\{t_1,t_2,\dots,t_m\},  PC[0,1]=\{ u: [0,1]\to R,\; u is continuous at t\neq t_k, u(t_k^+),\; u(t_k^-) exist, and u(t_k^-)=u(t_k), k=1,2,\dots,m\}, PC^1[0,1]=\{ u\in PC[0,1]\;|\ u'  is continuous at t\neq t_k, u'(t_k^+),\; u'(t_k^-) exist, k=1,2,\dots,m\}, with the norm$$ \|u\|_{PC}=\sup_{t\in J}|u(t)|,\quad \|u\|_{PC^1}=\mathrel{\max _{{t\in J}}} \{\|u\|_{PC}, \|u'\|_{PC}\}. Obviously PC[0,1] and PC^1[0,1] are Banach spaces. \begin{lemma} \label{lem2.1} u\in PC^1[0,1]\bigcap C^3[J]  is a solution of \eqref{eq1.1} if and only if \label{eq2.1} \begin{aligned} u(t)&= u(0)+u'(0)t+\int_0^t(t-s)\phi_q\Big(\int_0^s f(r,u(r),u'(r))dr\Big)ds \\ &\quad +\sum_{t_k0,L>0. Let r>a>0, L>0 be constants, \theta, \psi: P \to[0,+\infty) be two nonnegative continuous convex functions which satisfy \eqref{eq2.5} and \eqref{eq2.6}, and \gamma be a nonnegative concave function on P. We define convex sets as follows \begin{gather*} P(\theta,r;\psi,L)=\{x\in P: \theta(x)a\}, \\ \overline{P}(\theta,r;\psi,L;\gamma,a)=\{x\in P: \theta(x)\leq r, \psi(x)\leq L,\gamma(x)\geq a\}. \end{gather*} \begin{lemma}[\cite{BG}] \label{lem2.2} Let E be Banach space, P\subset E be a cone and r_2\geq d>b>r_1>0, L_2\geq L_1>0 be constants. Assume \theta, \psi: P \to[0,+\infty) are nonnegative continuous convex functions which satisfy \eqref{eq2.5} and \eqref{eq2.6}. \gamma is a nonnegative concave function on P such that for all x in \overline{P}(\theta,r_2;\psi,L_2) satisfies \gamma(x)\leq\theta(x). T:\overline{P}(\theta,r_2;\psi,L_2)\to\overline{P}(\theta,r_2;\psi,L_2) is a completely continuous operator. Suppose \begin{itemize} \item[(C1)] \{x\in \overline{P}(\theta,d;\psi,L_2;\gamma,b): \gamma(x)>b\}\neq\phi, and \gamma(Tx)>b, for \\ x\in \overline{P}(\theta,d;\psi,L_2;\gamma,b); \item[(C2)] \theta(Tx)b, for x\in \overline{P}(\theta,r_2;\psi,L_2;\gamma,b) with \theta (Tx)>d. \end{itemize} Then T has at least three fixed points x_1,x_2,x_3 in \overline{P}(\theta,r_2;\psi,L_2). Further, \begin{gather*} x_1 \in \overline{P}(\theta,r_1;\psi,L_1),\; \ x_2\in\{\overline{P}(\theta,r_2;\psi,L_2;\gamma,b) : \gamma(x)>b\},\\ x_3\in \overline{P}(\theta,r_2;\psi,L_2)\backslash \big(\overline{P}(\theta,r_1;\psi,L_1) \cup\overline{P}(\theta,r_2;\psi,L_2;\gamma,b)\big). \end{gather*} \end{lemma} \section{Main results} Let closed cone P be defined by P=\{ u\in PC^1[0,1]: u(t)\geq 0\}. Define operator T:P\to PC^1[0,1] by \begin{align*} Tu(t) &= u(0)+u'(0)t+\int _0^t(t-s)\phi_q\Big(\int_0^s f(r,u(r),u'(r))dr\Big)ds\\ &\quad +\sum_{t_kb>r_1>0, L_2\geq L_1>0 such that r_2\geq\frac{bl\delta(m+1/q)}{(\delta-1)(1-\alpha-\beta)}, \quad L_2\geq \frac{bl\delta (m+1/q)}{1-\alpha-\beta}, and the following conditions hold \begin{itemize} \item[(H1)] f(t,u,v)<\phi_p(\min\{\frac{\delta-1}{\delta}M_1r_1,M_1L_1\}), (t,u,v)\in [0,1]\times [0,r_1]\times [-L_1,L_1]; \item[(H2)] \phi_p(bl)J_k(u) for u\in[0,r_2], I_u^{r_1}<\min\{\frac{\delta-1}{\delta}M_1r_1,M_1L_1\},\\ I_u^{r_2}<\min\{\frac{\delta-1}{\delta}M_1r_2,M_1L_2\}. \end{itemize} Then boundary-value problem \eqref{eq1.1} has at least three positive solutions u_1,u_2,u_3\in \overline{P}(\theta,r_2;\psi,L_2) which satisfy \begin{gather*} \sup_{0\leq t\leq 1}u_1(t)\leq r_1,\quad \sup_{0\leq t\leq 1}|u_1'(t)| \leq L_1; \\ b<\min_{t\in[a_m,b_m]}u_2(t)\leq\sup_{0\leq t\leq 1}u_2(t)\leq r_2, \quad \sup_{0\leq t\leq 1}|u_2'(t)| \leq L_2;\\ \sup_{0\leq t\leq 1}u_3(t)\leq\delta d,\quad \sup_{0\leq t\leq 1}|u_3'(t)|\leq L_2. \end{gather*} \end{theorem} \begin{proof} We need to prove \frac{\delta-1}{\delta}M_1r_1\geq bl in order to make sure that the theorem makes sense, since we have r_2\geq bl\delta \frac{ m+1/q}{(\delta-1)(1-\alpha-\beta)}, and \begin{align*} \frac{\delta-1}{\delta}M_1r_1 & =\frac{(\delta-1)(1-\alpha-\beta)}{ \delta(1/q +m(1-\alpha-\beta)+x\alpha+y\beta)}r_1 \\ & \geq\frac{\delta-1}{\delta}\times \frac{1-\alpha-\beta}{1/q+m}\times\frac{m+1/q}{ (\delta-1)(1-\alpha-\beta)}bl\delta\\ & =bl. \end{align*} Similarly, we have M_1L_2\geq M_1r_2(\delta -1)\geq bl\delta>bl, so there has no contradiction among conditions. It is easy to see that \eqref{eq1.1} has a solution if and only if \begin{align*} Tu(t)&= u(0)+u'(0)t+\int _0^t(t-s)\phi_q\left(\int_0^s f(r,u(r),u'(r))dr\right)ds\\ &\quad +\sum_{t_kJ_k(u) and I_u^{r_2}<\min\{\frac{\delta-1}{\delta}M_1r_2,M_1L_2\}, we get \sum_{k=1}^{m}\Big((1-t_k)I_k(u(t_k))+J_k(u(t_k))\Big)\leq \sum_{k=1}^{m}I_k(u(t_k))\leq m\frac{\delta-1}{\delta}M_1r_2. $$By condition (H3), f(t,u,v)<\phi_p(\frac{\delta-1}{\delta}M_1r_2), we obtain$$ \phi_q\Big(\int_0^s f(t,u(r),u'(r))dr\Big)\leq \frac{\delta-1}{\delta}M_1r_2\phi_q(s). $$Hence,$$ u(0)\leq\frac{ \alpha/q+\beta/q+x\alpha+y\beta}{\delta(1-\alpha-\beta)}M_1r_2 +\frac{1}{q\delta}M_1r_2+\frac{m}{\delta}M_1r_2. $$Similarly,$$ u'(0)\leq\frac{(\delta-1)(\alpha+\beta+q(x\alpha +y\beta))} {q\delta(1-\alpha-\beta)}M_1r_2. Therefore, we can show that \begin{align*} \theta(Tu) &= \sup_{0\leq t\leq1}(u(0)+u'(0)t+\int _0^t(t-s)\phi_q\Big(\int_0^s f(r,u(r),u'(r))dr\Big)ds\\ &\quad +\sum_{t_kb, so \{x\in \overline{P}(\theta,d;\psi,L_2;\gamma,b): \gamma(x)>b\}\neq\phi. For u\in \overline{P}(\theta,d;\psi,L_2;\gamma,b), we have b\leq u(t)\leq d, |u'(t)|\leq L_2 for all t \in [a_m,b_m]. Since Tu is a monotone increasing function, and (Tu)(t)\geq 0, t\in [0,1], we have \begin{align*} \gamma (Tu) &= \min_{t\in [a_m,b_m]}\big(u(0)+u'(0)t+\int _0^t(t-s)\phi_q(\int_0^s f(r,u(r),u'(r))dr)ds\\ &\quad +\sum_{t_k \frac{1}{\delta-1}\int _{a_m}^{b_m}ds\int_{a_m}^s\phi_q(\int_{a_m}^r \phi_p (bl)dw)dr\\ & = \frac{bl}{ \delta-1}\int _{a_m}^{b_m}ds\int_{a_m}^s\phi_q(r-a_m)dr\\ & = \frac{bl}{\delta-1}\int_{a_m}^{b_m}(b_m-r)\phi_q(r-a_m)dr = b. \end{align*} Thus \gamma(Tu)>b  and the condition (C1) of Lemma \ref{lem2.2} also holds. Finally to prove (C3) of Lemma \ref{lem2.2}, we check \gamma(Tu)>b to be satisfied for all u\in\overline{P}(\theta,r_2;\psi,L_2;\gamma,b) with \theta (Tu)>d. Since Tu is a nonnegative monotone increasing function, we can get \begin{gather*} \theta (Tu)=\sup_{0\leq t\leq 1}Tu(t)=Tu(1),\\ \gamma(Tu)=\min_{t\in [a_m,b_m]}Tu(t)=Tu(a_m), \\ Tu(a_m)\geq Tu(0)=\frac {1}{\delta}Tu(1)>\frac{d}{\delta} \geq b; \end{gather*} that is, \gamma (Tu)>b. We have checked Lemma \ref{lem2.2} to make sure all the conditions are satisfied with the work we have done in the section above. Then T has at least three fixed points u_1,u_2,u_3 in \overline{P}(\theta,r_2;\psi,L_2). Further, \begin{gather*} u_1 \in \overline{P}(\theta,r_1;\psi,L_1),\quad u_2\in\{\overline{P}(\theta,r_2;\psi,L_2;\gamma,b): \gamma(x)>b\},\\ u_3\in\overline{P}(\theta,r_2;\psi,L_2)\backslash\{\overline{P} (\theta,r_1;\psi,L_1)\cup\overline{P}(\theta,r_2;\psi,L_2;\gamma,b)\}. \end{gather*} Therefore \eqref{eq1.1} has at least three positive solutions u_1,u_2,u_3. From the boundary conditions we have u_3(1)=\delta u_3(0) and u_3 is a monotone increasing function, so we have b>\gamma(u_3)=\min_{a_m\leq t\leq b_m} u_3(t)=u_3(a_m)\geq u_3(0) =\frac{1}{\delta}u_3(1)=\frac{1}{\delta}\theta(u_3),  so $\theta(u_3)\leq \delta b$, that means $\sup_{0\leq t\leq 1} u_3(t)\leq \delta d$, and $u_1,u_2,u_3$ satisfy \begin{gather*} \sup_{0\leq t\leq 1}u_1(t)\leq r_1, \quad \sup_{0\leq t\leq 1}|u_1'(t)| \leq L_1; \\ b<\min_{t\in[a_m,b_m]}u_2(t)\leq \sup_{0\leq t\leq 1}u_2(t)\leq r_2, \quad \sup_{0\leq t\leq 1}|u_2'(t)| \leq L_2; \\ \sup_{0\leq t\leq 1}u_3(t)\leq\delta d,\quad \sup_{0\leq t\leq1}|u_3'(t)|\leq L_2. \end{gather*} \end{proof} \begin{thebibliography}{0} \bibitem{AR} Ravi P. Agarwal, Donal O'Regan; \emph{A multiplicity result for second order impulsive differential equations via the Leggett-Williams fixed point therom}, Appl. Math. Comput. 161 (2005): 433-439. \bibitem{BG} Z. Bai, W. 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