\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 24, 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 (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/24\hfil Positive solutions] {Positive solutions for second-order m-point boundary-value problems with nonlinearity depending on the first derivative} \author[L. Yang, X. Liu, C. Shen\hfil EJDE-2006/24\hfilneg] {Liu Yang, Xiping Liu, Chunfang Shen} \address{College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China} \email[Liu Yang]{xjiangfeng@163.com} \email[Chunfang Shen]{minirose1982@163.com} \email[Xiping Liu]{xipingliu@163.com} \date{} \thanks{Submitted June 6, 2005. Published February 23, 2006.} \thanks{Supported by the Foundation of Educational Commission of Shanghai} \subjclass[2000]{34B10, 34B15} \keywords{Boundary value problem; positive solution; cone; fixed point theorem} \begin{abstract} We consider multiplicity of positive solutions for second-order $m$-point boundary-value problems, with the first order derivative involved in the nonlinear term. Using a fixed point theorem, we show the existence of at least three positive solutions. By giving an example we illustrate the main result of the article. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Multi-point boundary-value problems for ordinary differential equations arise in different areas of applied mathematics and physics. For example, the vibrations of a guy wire of uniform cross-section and composed of N parts of different densities can be set up as a multi-point boundary-value problem,many problems in the theory of elastic stability can be handled as multi-point boundary-value problems too.Recently, the existence and multiplicity of positive solutions for nonlinear ordinary differential equations and difference equations have received a great deal of attentions.To identify a few,we refer the reader to \cite{a1,c1,m1,m2,m3} and references therein. Ma and Wang \cite{m4} obtained the existence of one positive solution for more general three-point boundary-value problem \begin{gather} u''(t)+a(t)u'(t)+b(t)u(t)+h(t)f(u)=0,\quad t\in (0,1),\label{e1.1}\\ u(0)=0,\quad u(1)=\alpha u(\eta),\quad 0<\eta <1,\label{e1.2} \end{gather} under the assumption that $f$ is either suplinear or sublinear, and that the following conditions are satisfied: \begin{itemize} \item[(H1)] $f\in C([0,+\infty),[0,+\infty))$ \item[(H2)] $h\in C([0,1],[0,+\infty))$ and there exists $x_{0}\in (0,1)$ such that $h(x_{0})>0$ \item[(H3)] $a\in C[0,1]$, $b\in C([0,1],(-\infty,0])$ \item[(H4)] $0<\alpha \phi_{1}(\eta)<1$, where $\phi_{1}$ is the unique solution of the linear problem \begin{gather} \phi_{1}''(t)+a(t)\phi_{1}'(t)+b(t)\phi_{1}(t)=0,\quad t\in (0,1),\label{e1.3}\\ \phi_{1}(0)=0,\quad \phi_{1}(1)=1.\label{e1.4} \end{gather} \end{itemize} In \cite{m4}, the authors used a fixed point theorem for a mapping defined on Banach spaces with cones, by Guo and Krasnosel'skii \cite{g1}. However all the above works about positive solutions were done under the assumption that the first order derivative $x'$ is not involved in the nonlinear term. On the other hand, to the best of our knowledge, there are very few work considering the multiplicity of positive solutions with dependence on derivatives. In this paper, we consider the existence of at least three positive solutions for the equation $$x''(t)+a(t)x'(t)+b(t)x(t)+f(t,x(t),x'(t))=0,\quad t\in (0,1),\label{e1.5}$$ subject to the boundary conditions $$x(0)=0,\quad x(1)=\sum^{m-2}_{i=1} \alpha_{i} x(\xi_{i}),\label{e1.6}$$ or to the boundary conditions $$x'(0)=0, \quad x(1)=\sum^{m-2}_{i=1} \alpha_{i} x(\xi_{i}),\label{e1.7}$$ where $\xi_{i}\in(0,1)$, $\alpha_{i}>0$, $i=1,2,\dots,m-2$ are given constants. The interest in triple solutions evolved from the Leggett-Williams fixed point theorem \cite{l1}. When $x'$ does not appear in nonlinear term there are results about several nonlinear ordinary differential equations, obtained by the Leggett-Williams fixed point theorem; see \cite{g2,h1}. Recently Avery and Peterson \cite{a2}, Bai and Ge \cite{b1} generalized the fixed point theorem of Leggett-Williams by using theorem of fixed point index and Dugundji extension theorem. As applications of the results in \cite{b1,b2}, it has been obtained the existence of triple positive solutions of the boundary-value problem \begin{gather} x''(t)+a(t)f(t,x(t),x'(t))=0,\quad 00,\; L>0.\label{e2.2} Let $r>a>0,L>0$ be given, $\gamma,\theta:P\to [0,+\infty)$ be nonnegative continuous convex functionals satisfying \eqref{e2.1} and \eqref{e2.2}, $\alpha$ be a nonnegative continuous concave functional on $P$. Define the following convex sets: \begin{gather*} P(\gamma,L;\theta,r)=\{x\in P|\gamma(x)a\}, \\ \overline{P}(\gamma,L;\theta,r;\alpha,a)=\{x\in P|\gamma(x)\leq L,\theta(x)\leq r,\alpha(x)\geq a \}. \end{gather*} \begin{lemma} \label{lem2.1} Let E be a Banach space,$P\subset E$ be a cone and $r_{2}\geq c>b>r_{1}>0$, $L_{2}\geq L_{1}>0$ be given. Assume that $\gamma,\theta$ are nonnegative continuous convex functionals on P such that \eqref{e2.1}, \eqref{e2.2} are satisfied. $\alpha$ is a nonnegative continuous concave functional on $P$ such that $\alpha(x)\leq \theta(x)$ for all $x\in \overline{P}(\gamma,L_{2};\theta,r_{2})$ and let $T:\overline{P}(\gamma,L_{2};\theta,r_{2})\to \overline{P}(\gamma,L_{2}; \theta,r_{2})$ be a completely continuous operator. Suppose that \begin{itemize} \item[(S1)] The set $\{x\in \overline{P}(\gamma,L_{2};\theta,c;\alpha,b): \alpha(x)>b\}$ is not empty, and $\alpha(Tx)>b$ for $x$ in $\overline{P}(\gamma,L_{2};\theta,c;\alpha,b)$; \item[(S2)] $\gamma(Tx)b$, for all $x\in \overline{P}(\gamma,L_{2};\theta,r_{2};\alpha,b)$ with $\theta(Tx)>c$. \end{itemize} Then $T$ has at least three fixed points $x_{1},x_{2},x_{3}$ in $\overline{P}(\gamma,L_{2};\theta,r_{2})$. Further, \begin{gather*} x_{1}\in P(\gamma,L_{1};\theta,r_{1});\quad x_{2}\in \{\overline{P}(\gamma,L_{2};\theta,r_{2};\alpha,b):\alpha(x)>b\}, \\ x_{3}\in \overline{P}(\gamma,L_{2};\theta,r_{2})\setminus(\overline{P} (\gamma,L_{2};\theta,r_{2};\alpha,b)\cup \overline{P} (\gamma,L_{1};\theta,r_{1})). \end{gather*} \end{lemma} \section{Positive solutions of \eqref{e1.5}, \eqref{e1.6}} To state the main results of this section,we need the following lemma, which was established by Ma and Wang \cite{m4}. \begin{lemma} \label{lem3.1} Assume that (H3) holds. Let $\phi_{1},\phi_{2}$ be solutions of \eqref{e1.3}, \eqref{e1.4}, and \begin{gather} \phi_{2}''(t)+a(t)\phi_{2}'(t)+b(t)\phi_{2}(t)=0,\quad t\in (0,1),\label{e3.1}\\ \phi_{2}(0)=1,\quad \phi_{2}(1)=0.\label{e3.2} \end{gather} Then $\phi_{1}$ is strictly increasing and $\phi_{2}$ is strictly decreasing on [0,1]. \end{lemma} Inspired by \cite{m4}, we state following lemma which can be regard as a natural extension. \begin{lemma} \label{lem3.2} Suppose (H3) and $$\label{H5} 0<\sum_{i=1}^{m-2}\alpha_{i} \phi_{1}(\xi_{i})<1.$$ Then the problem \begin{gather} x''(t)+a(t)x'(t)+b(t)x(t)+y(t)=0,\quad t\in(0,1)\label{e3.3}\\ x(0)=0,\quad x(1)=\sum^{m-2}_{i=1} \alpha_{i}x(\xi_{i}),\label{e3.4} \end{gather} is equivalent to integral equation $$x(t)=\int^{1}_{0}G(t,s)p(s)y(s)ds+A \phi_{1}(t),\label{e3.5}$$ where \begin{gather} A=\frac{\sum^{m-2}_{i=1}\alpha_{i}}{1-\sum^{m-2}_{i=1}\alpha_{i}\xi_{i}} \int^{1}_{0}G(\xi_{i},s)p(s)y(s)ds,\label{e3.6} \\ p(t)=\exp\Big(\int^{t}_{0}a(s)ds\Big), \quad \rho=\phi'_{1}(0), \label{e3.7} \\ G(t,s)=\frac{1}{\rho}\begin{cases} \phi_{1}(t)\phi_{2}(s) &{s\geq t}\\ \phi_{1}(s)\phi_{2}(t) &{s\leq t}, \end{cases} \nonumber\\ u(t)\geq 0\quad\mbox{if } y(t)\geq 0. \nonumber \end{gather} \end{lemma} The proof of this lemma is very similar to a proof in \cite{m4}, so we omit it here. Let $$q(t)=\min\{\frac{\phi_{1}(t)}{|\phi_{1}|_{0}}, \frac{\phi_{2}(t)}{|\phi_{2}|_{0}}\},\quad t\in [0,1]$$ where $|y(t)|_{0}=\max|y(t)|,t\in [0,1]$. The following Lemma was also established by Ma and Wang. \begin{lemma} \label{lem3.3} Suppose (H3) and \eqref{H5} are satisfied, $y\in C[0,1],y\geq 0$, then the solution of \eqref{e3.3}-\eqref{e3.4} satisfies $$u(t)\geq |u|_{0}q(t),t\in [0,1].\label{e3.8}$$ \end{lemma} Thus, for any $\delta \in [0,1/2]$, there exists $\lambda$ such that $$u(t)\geq \lambda|u|_{0},t\in [\delta,1-\delta],\label{e3.9}$$ where $\lambda=\min\{q(t):t\in [\delta,1-\delta]\}$. Let \begin{gather*} M=\max_{0\leq t\leq 1} \int^{1}_{0}G(t,s)p(s)ds +\frac{\sum^{m-2}_{i=1}\alpha_{i}}{1-\sum^{m-2}_{i=1} \alpha_{i}\phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)ds; \\ N=\max_{0\leq t\leq 1}|\int^{1}_{0}\frac{\partial G(t,s)}{\partial t}p(s)ds +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}'(t)}{1-\sum^{m-2}_{i=1}\alpha_{i} \phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)ds|; \\ m=\min_{\delta\leq t\leq 1-\delta}\int^{1-\delta}_{\delta}G(t,s)p(s)ds +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}(\delta)}{1-\sum^{m-2}_{i=1} \alpha_{i}\phi_{1}(\xi_{i})}\int^{1-\delta}_{\delta}G(\xi_{i},s)p(s)ds. \end{gather*} To present our main results, we assume there exist constants $r_{2}\geq \frac{b}{\lambda}>b>r_{1}>0$, $L_{2}\geq L_{1}>0$ such that $\frac{b}{m}<\min\{\frac{r_{2}}{M},\frac{L_{2}}{N}\}$ and the following assumptions hold: \begin{itemize} \item[(A1)] $f(t,u,v)\in C([0,1]\times[0,+\infty)\times R,[0,+\infty))$; \item[(A2)] $f(t,u,v)< \min\{r_{1}/M,L_{1}/N\},(t,u,v)\in [0,1]\times [0,r_{1}]\times [-L_{1},L_{1}]$; \item[(A3)] $f(t,u,v)> b/m,(t,u,v)\in [\delta,1-\delta]\times [b,b/\lambda]\times [-L_{2},L_{2}]$; \item[(A4)] $f(t,u,v)\leq \min\{r_{2}/M,L_{2}/N\},(t,u,v) \in [0,1]\times [0,r_{2}]\times [-L_{2},L_{2}]$. \end{itemize} \begin{theorem} \label{thm3.1} Under assumption {\rm (A1)-(A4), (H3)}, \eqref{H5}, Problem \eqref{e1.5}-\eqref{e1.6} has at least three positive solutions $x_{1},x_{2},x_{3}$ satisfying $$\begin{gathered} \max_{0\leq t\leq1}x_{1}(t)\leq r_{1},\max_{0\leq t\leq1}|x'_{1}(t)| \leq L_{1};\\ b<\min_{\delta\leq t\leq 1-\delta}x_{2}(t)\leq \max_{0\leq t \leq 1}x_{2}(t)\leq r_{2},\max_{0\leq t\leq 1}|x'_{2}(t)|\leq L_{2}; \\ \max_{0\leq t\leq 1} x_{3}(t)\leq \frac{b}{\lambda},\max_{0\leq t\leq 1} |x'_{3}(t)|\leq L_{2}. \end{gathered} \label{e3.10}$$ \end{theorem} \begin{proof} Problem \eqref{e1.5}-\eqref{e1.6} has a solution $x=x(t)$ if and only if $x$ solves the operator equation $$x(t)=\int^{1}_{0}G(t,s)p(s)f(s,x(s),x'(s))ds+A \phi_{1}(t)=(Tx)(t), \quad 0b. So \{x\in \overline{P}(\gamma,L_{2};\theta,c;\alpha,b)|\alpha(x)>b)\} \neq \emptyset. If x\in P(\gamma,L_{2};\theta,c;\alpha,b), we have b\leq x(t)\leq \frac{b}{\lambda},|x'(t)| \frac{b}{m}.$$ By the definition of $\alpha$ and the cone $P$, \begin{align*} \alpha(Tx)&=\min_{\delta\leq t\leq 1-\delta}[\int^{1}_{0} G(t,s)p(s)f(s,x(s),x'(s))ds \\ &\quad +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}(t)}{1-\sum^{m-2}_{i=1} \alpha_{i}\phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)f(s,x(s),x'(s))ds]\\ &> \frac{b}{m}\min_{0\leq t\leq 1}[\int^{1-\delta}_{\delta}G(t,s)p(s)ds +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}(t)}{1-\sum^{m-2}_{i=1} \alpha_{i}\phi_{1}(\xi_{i})}\int^{1-\delta}_{\delta}G(\xi_{i},s)p(s)ds]\\ &\leq \frac{b}{m}[\min_{0\leq t\leq 1}\int^{1-\delta}_{\delta}G(t,s)p(s)ds +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}(\delta)}{1-\sum^{m-2}_{i=1} \alpha_{i}\phi_{1}(\xi_{i})}\int^{1-\delta}_{\delta}G(\xi_{i},s)p(s)ds]\\ &\geq \frac{b}{m}m=b. \end{align*} Then $\alpha(Tx)>b$, for all $x\in \overline{P}(\gamma,L_{2};\theta,c;\alpha,b)$. This shows that condition (S1) of lemma \ref{lem2.1} is also satisfied. Finally we show (S3) holds too. Suppose $x\in \overline{P}(\gamma,L_{2};\theta,r_{2};\alpha,b)$ with $\theta (Tx)>\frac{b}{\lambda}$. Then,by the definition of $\alpha$ and $Tx\in P$, we have $$\alpha(Tx)=\min_{\delta\leq t\leq 1-\delta}|(Tx)(t)| \geq \lambda\cdot \max_{0\leq t\leq 1}|(Tx)(t)| =\lambda\cdot\theta(Tx) =\lambda\cdot\frac{b}{\lambda}=b.$$ So condition (S3) of lemma \ref{lem2.1} is satisfied. Therefore, Lemma \ref{lem2.1} yields that problem \eqref{e1.5}-\eqref{e1.6} has at least three positive solutions $x_{1},x_{2},x_{3}$ in $\overline{P}(\gamma,L_{2};\theta,r_{2})$ and \eqref{e3.10} is satisfied. \end{proof} \begin{remark} \label{rmk4.1} \rm In Lemma \ref{lem2.1}, we need only $T:\overline{P}(\gamma,L_{2};\theta,r_{2})\to \overline{P}(\gamma,L_{2}; \theta,r_{2})$; therefore, condition (A1) can be substituted with the weaker condition \begin{itemize} \item[(C1)] $f \in C([0,1]\times [0,r_{2}] \times [-L_{2},L_{2}],[0,+\infty))$. \end{itemize} From the proof of Theorem \ref{thm3.1}, it is easy to see that, if conditions like (A1)-(A4) are appropriate combined, we can obtained an arbitrary number of positive solutions for this problem. \end{remark} \begin{corollary} \label{coro3.1} Suppose condition (A1) is satisfied and there exist constants $0 \frac{b_{i}}{m}$, $(t,u,v)\in [\delta,1-\delta]\times [b_{i},\frac{b_{i}}{\lambda}] \times [-L_{i+1},-L_{i+1}],1\leq i\leq n-1$. \end{itemize} \end{corollary} \begin{proof} When $n=1$, it follows from condition (E1) that $$T:\overline{P}(\gamma,L_{1};\theta,r_{1})\to P(\gamma,L_{1};\theta,r_{1}) \subseteq \overline{P}(\gamma,L_{1};\theta,r_{1}).$$ Then by Schauder's fixed-point theorem, $T$ has at least one fixed point $x_{1}$ in $P(\gamma,L_{1};\theta,r_{1})$. When $n=2$, it is clear that Theorem \ref{thm3.1} holds. Then we can obtain at least three positive solutions $x_{2},x_{3},x_{4}$. Along this way, we can complete the proof by the induction method. \end{proof} \section{Positive solutions of \eqref{e1.5}, \eqref{e1.7}} In this section we study problem \eqref{e1.5}, \eqref{e1.7}. The method and existence results are remarkable analogous to those in section 3. First, we give some Lemmas. Suppose $\phi_{3}$ is the unique solution of the linear boundary-value problem \begin{gather} \phi_{3}''(t)+a(t)\phi_{3}'(t)+b(t)\phi_{1}(t)=0,\quad t\in (0,1),\label{e4.1}\\ \phi_{3}'(0)=0,\quad \phi_{3}(1)=1. \label{e4.2} \end{gather} satisfying $$\label{H6} 0<\sum^{m-2}_{i=1}\alpha_{i}\phi_{3}(\xi_{i})<1.$$ Then problem \begin{gather} x''(t)+a(t)x'(t)+b(t)x(t)+y(t)=0,t\in(0,1)\label{e4.3}\\ x'(0)=0,\hspace*{4em}x(1)=\sum^{m-2}_{i=1} \alpha_{i} x(\xi_{i}),\label{e4.4} \end{gather} is equivalent to integral equation $$x(t)=\int^{1}_{0}G_{1}(t,s)p(s)y(s)ds+A_{1} \phi_{3}(t),\label{e4.5}$$ where \begin{gather} A_{1}=\frac{\sum^{m-2}_{i=1}\alpha_{i}}{1-\sum^{m-2}_{i=1}\alpha_{i}\phi_{3}(\xi_{i})} \int^{1}_{0}G_{1}(\xi_{i},s)p(s)y(s)ds,\label{e4.6} \\ p(t)=\exp(\int^{t}_{0}a(s)ds),\quad \rho_{1}=-\phi_{3}(0)\phi'_{2}(0), \label{e4.7} \\ G_{1}(t,s)=\frac{1}{\rho_{1}} \begin{cases} \phi_{3}(t)\phi_{2}(s)&{1\geq s\geq t\geq 0}\\ \phi_{3}(s)\phi_{2}(t)&{0\leq s\leq t\leq 1}, \end{cases} \nonumber\\ x(t)\geq 0\quad \text{if }y(t)\geq 0. \nonumber \end{gather} Let (H3), \eqref{H6} be satisfied, substituting $\phi_{1}(t)$ with $\phi_{3}(t)$, we can obtain a similar result as in lemma \ref{lem3.3}. Let \begin{gather*} M_{1}=\max_{0\leq t\leq 1} \int^{1}_{0}G_{1}(t,s)p(s)ds +\frac{\sum^{m-2}_{i=1}\alpha_{i}}{1-\sum^{m-2}_{i=1} \alpha_{i}\phi_{3}(\xi_{i})}\int^{1}_{0}G_{1}(\xi_{i},s)p(s)ds;\\ N_{1}=\max_{0\leq t\leq 1}|\int^{1}_{0}\frac{\partial G_{1}(t,s)}{\partial t}p(s)ds +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{3}'(t)}{1-\sum^{m-2}_{i=1} \alpha_{i}\phi_{3}(\xi_{i})}\int^{1}_{0}G_{1}(\xi_{i},s)p(s)ds|;\\ m_{1}=\min_{\delta\leq t\leq 1-\delta}\int^{1-\delta}_{\delta} G_{1}(t,s)p(s)ds +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{3}(\delta)}{1-\sum^{m-2}_{i=1} \alpha_{i}\phi_{3}(\xi_{i})}\int^{1-\delta}_{\delta}G_{1}(\xi_{i},s)p(s)ds. \end{gather*} Analogous to Theorem \ref{thm3.1}, using results established above, it is not difficult to show that problem \eqref{e1.5},\eqref{e1.7} has at least three positive solutions. \begin{theorem} \label{thm4.1} Suppose conditions (H3), \eqref{H6}, (C1) are satisfied and there exist constants $r_{2}\geq \frac{b}{\lambda_{1}}>b>r_{1}>0$, $L_{2}\geq L_{1}>0$ such that $\frac{b}{m_{1}}<\min\{\frac{r_{2}}{M_{1}},\frac{L_{2}}{N_{1}}\}$ and the following assumptions hold: \begin{itemize} \item[(A5)] $f(t,u,v)< \min\{r_{1}/M_{1},L_{1}/N_{1}\},(t,u,v)\in [0,1] \times [0,r_{1}]\times [-L_{1},L_{1}]$; \item[(A6)] $f(t,u,v)> b/m_{1},(t,u,v)\in [\delta,1-\delta]\times [b,b/\lambda_{1}]\times [-L_{2},L_{2}]$; \item[(A7)] $f(t,u,v)\leq \min\{r_{2}/M_{1},L_{2}/N_{1}\},(t,u,v) \in [0,1]\times [0,r_{2}]\times [-L_{2},L_{2}]$. \end{itemize} Then problem \eqref{e1.5}, \eqref{e1.7} has at least three positive solutions $x_{1},x_{2},x_{3}$ satisfying $$\label{e4.8} \begin{gathered} \max_{0\leq t\leq1}x_{1}(t)\leq r_{1}, \quad \max_{0\leq t\leq1}|x'_{1}(t)|\leq L_{1};\\ b<\min_{\delta\leq t\leq 1-\delta}x_{2}(t)\leq \max_{0\leq t\leq 1} x_{2}(t)\leq r_{2}, \quad \max_{0\leq t\leq 1}|x'_{2}(t)|\leq L_{2};\\ \max_{0\leq t\leq 1} x_{3}(t)\leq \frac{b}{\lambda_{1}}, \quad \max_{0\leq t\leq 1} |x'_{3}(t)|\leq L_{2}. \end{gathered}$$ \end{theorem} Further we can establish following multiplicity results of problem \eqref{e1.5}, \eqref{e1.7}. \begin{corollary} \label{coro4.1} Suppose condition (A1) is satisfied and there exist constants $0 \frac{b_{i}}{m_{1}}$, $(t,u,v)\in [\delta,1-\delta]\times [b_{i},\frac{b_{i}}{\lambda_{1}}] \times [-L_{i+1},-L_{i+1}]$, $1\leq i\leq n-1$ \end{itemize} \end{corollary} \section{Examples} In this section we present an example to illustrate our main results. Consider the boundary-value problem \begin{gathered} x''(t)-x(t)+f(t,x(t),x'(t))=0,\quad 05} \end{cases} \] Considering lemma \ref{lem3.1}, \ref{lem3.2}, we obtain \begin{gather*} \phi_{1}(t)=\frac{e^{1+t}-e^{1-t}}{e^{2}-1},\quad \phi_{1}'(0)=\frac{2e}{e^{2}-1}, \\ \phi_{2}(t)=\frac{e^{2-t}-e^{t}}{e^{2}-1}, \quad p(t)=1,\quad \delta=\frac{1}{4},\quad \lambda=\frac{e^{\frac{5}{4}}-e^{\frac{3}{4}}}{e^{2}-1}, \\ G(t,s)=\frac{1}{2e(e^{2}-1)} \begin{cases} (e^{1+t}-e^{1-t})(e^{2-s}-e^{s})&{s\geq t}\\ (e^{1+s}-e^{1-s})(e^{2-t}-e^{t})&{s\leq t},\\ \end{cases} \end{gather*} \begin{align*} M&=\max_{0\leq t\leq 1} \int^{1}_{0}G(t,s)ds +\frac{e^{1/2}}{1-e^{1/2}\phi_{1}(1/2)}\int^{1}_{0}G(1/2,s)ds\\ &= \frac{(e+1-2e^{1/2})(1+e^{1/2}+e^{3/2})}{e+1}. \end{align*} \begin{align*} m&=\min_{\frac{1}{4}\leq t\leq \frac{3}{4}} \int_{\frac{1}{4}}^{\frac{3}{4}}G(t,s)ds+ \frac{e^{1/2}\phi_{1}(1/4)}{1-e^{1/2}\phi_{1}(1/2)} \int_{\frac{1}{4}}^{\frac{3}{4}}G(1/2,s)ds \\ &=\frac{1}{2}-e^{1/2}+\frac{e^{7/4}-e^{5/4}}{e-1}. \end{align*} \begin{align*} N&=\max_{0\leq t\leq 1}|\int^{1}_{0}\frac{\partial G(t,s)}{\partial t}ds+ \frac{e^{1/2}\phi_{1}'(t)}{1-e^{1/2}\phi_{1}(1/2)}\int^{1}_{0}G(1/2,s)ds|\\ &=\frac{(e^{1/2}-1)(e^{2}+1)}{e-1}. \end{align*} Choose $r_{1}=1$, $r_{2}=1000$, $b=4$, $L_{1}=10,L_{2}=1000$, then $\min\{\frac {r_{1}}{M},\frac{L_{1}}{N}\}=\frac{1}{M}$, $\min\{\frac{r_{2}}{M},\frac{L_{2}}{N}\}=\frac{1000}{N}$. We can check that conditions (C1), (H3), \eqref{H5} are satisfied and that $f(t,u,v)$ satisfies \begin{gather*} f(t,u,v)<\frac{1}{M},\quad\mbox{for } (t,u,v)\in [0,1]\times [0,1]\times [-10,10];\\ f(t,u,v)>\frac{4}{m}, \quad\mbox{for } (t,u,v)\in [\frac{1}{4},\frac{3}{4}]\times [4,\frac{4}{\lambda}] \times [-1000,1000];\\ f(t,u,v)<\frac{1000}{N},\quad\mbox{for } (t,u,v)\in [0,1]\times [0,1000]\times [-1000,1000]. \end{gather*} Then all assumptions of Theorem \ref{thm3.1} hold. Thus, \eqref{e5.1} has at least three positive solutions $x_{1},x_{2},x_{3}$ satisfying \begin{gather*} \max_{0\leq t\leq 1}x_{1}(t)\leq 1,\quad \max_{0\leq t\leq 1}|x'_{1}(t)|\leq 10;\\ 4<\min_{\frac{1}{4}\leq t\leq \frac{3}{4}}x_{2}(t)\leq \max_{0\leq t\leq 1}x_{2}(t)\leq 1000,\quad \max_{0\leq t\leq 1}|x'_{2}(t)|\leq 1000;\\ \max_{0\leq t\leq 1}x_{3}(t)\leq \frac{4}{\lambda}, \quad \max_{0\leq t\leq 1}|x'_{3}(t)|\leq 1000\,. \end{gather*} \begin{remark} \label{rmk5.1} \rm We see that the nonlinear term involves the first order derivative and can it change sign. The early results for multiplicity of positive solutions, to the author's best knowledge, are not applicable to the problem above. Meanwhile, as the nonlinear term does not satisfy the suplinear or sublinear condition even if the nonlinear term is $f(t,u,v)=f(u)$, we can not obtain even one positive solution of this problem from \cite{m4}. \end{remark} \begin{thebibliography}{99} \bibitem{a1} R. I. Avery, C. J. Chyan, J. 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