\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 110, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/110\hfil Positive solutions] {Positive solutions for nonlinear second-order $m$-point boundary-value problems} \author[J. Jiang, L. Liu \hfil EJDE-2009/110\hfilneg] {Jiqiang Jiang, Lishan Liu} % in alphabetical order \address{Jiqiang Jiang \newline School of Mathematical Sciences, Qufu Normal University \\ Qufu 273165, Shandong, China} \email{qfjjq@mail.qfnu.edu.cn, qfjjq@163.com} \address{Lishan Liu \newline School of Mathematical Sciences, Qufu Normal University \\ Qufu 273165, Shandong, China} \email{lls@mail.qfnu.edu.cn, Tel. 86-537-4456234, Fax 86-537-4455076} \thanks{Submitted December 26, 2008. Published September 10, 2009.} \thanks{Supported by grants: 10771117 from the National Natural Science Foundation of China, \hfill\break\indent 20060446001 from the State Ministry of Education Doctoral Foundation of China, and \hfill\break\indent Y2007A23 from the Natural Science Foundation of Shandong Province of China.} \subjclass[2000]{34B15, 34B25} \keywords{Positive solution; singular; $m$-point boundary-value problem; cones} \begin{abstract} By constructing a special cone and applying the fixed index theory in the cone, we prove the existence of positive solutions for a class of singular $m$-point boundary-value problems. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} This paper considers the existence of positive solutions for the second-order $m$-point boundary-value problem \begin{gather} \label{e1.1} (p(t)x'(t))'-q(t)x(t)+f(t,x(t))=0 ,\quad t\in(0,1) ,\\ \label{e1.2} ax(0)-bp(0)x'(0)=\sum_{i=1}^{m-2}\alpha_i x(\xi_i),\quad cx(1)+dp(1)x'(1)=\sum_{i=1}^{m-2}\beta_i x(\xi_i), \end{gather} where $a,c\in[0,+\infty)$, $b,d\in(0,+\infty)$ with $ac+ad+bc>0$, $\xi_i\in(0,1)$, $\alpha_i,\beta_i\in[0,+\infty)$ for $i\in\{1,2,\dots,m-2\}$ are given constants, $p\in C^1([0,1],(0,+\infty))$, $q\in C([0,1],(0,+\infty))$ and $f\in C((0,1)\times(0,+\infty),[0,+\infty))$, $f(t,x)$ is allowed to be singular at $t=0$, $t=1$ and $x=0$. If $p\equiv1$, $q\equiv0$, $\alpha_i,\beta_i=0$, (for $i=1,2,\dots,m-2)$, then \eqref{e1.1}-\eqref{e1.2} reduces to the two-point boundary-value problem \begin{gather} \label{e1.3} x''(t)+f(t,x(t))=0 ,\quad t\in(0,1) ,\\ \label{e1.4} ax(0)-bx'(0)=0,\quad cx(1)+dx'(1)=0, \end{gather} which has been intensively studied; see \cite{h2,l1}. In \cite{l2}, by using the fixed index theory in a cone, positive solutions were obtained for differential systems \begin{gather*} -x''(t)=f(t,y) ,\quad t\in(0,1) ,\\ -y''(t)=g(t,x) ,\quad t\in(0,1) ,\\ \alpha_1x(0)-\beta_1x'(0)=\gamma_1x(1)+\delta_1x'(1)=0,\\ \alpha_2y(0)-\beta_2y'(0)=\gamma_2y(1)+\delta_2y'(1)=0, \end{gather*} where $\alpha_i,\beta_i,\gamma_i,\delta_i\geq0$ and $\rho_i=\alpha_i\gamma_i+\alpha_i\gamma_i+\gamma_i\beta_i>0$ ($i=1,2$), $f(t,y)$ and $g(t,x)$ may be singular at $t=0$, $t=1$ and $x=0$, $y=0$, respectively. In recent years, singular multi-point boundary-value problems have been extensively studied and many optimal results have been obtained, see \cite{l1,w1,x1,z1,z2} and references therein. In addition, many papers investigated the existence of solutions for the nonsingular multi-point boundary-value problems, for example, \cite{f1,h1,h2,s1}. Recently, Ma \cite{m1}, Ma and Thompson \cite{m2} obtained excellent results about the existence of positive solutions for the more general $m$-point boundary-value problem \eqref{e1.1}-\eqref{e1.2}, but in the above papers there are no studies for singularity of the nonlinearity $f(t,x)$ at the point $x = 0$. Recently, by using Nonlinear Alternative of Leray-Schauder with the properties of the associated vector field at the $(u,u')$ plane, Galanis and Palamides \cite{g1} studied the problem $$ -[\phi_p(u')]'=q(t)f(t,u(t)),\quad 00$ \item[(ii)] $Tu\neq\lambda u$, for all $u\in\partial\Omega \cap P$, $\lambda\in(0,1]$, \end{itemize} then $i(T,\Omega\cap P,P)=0$. \end{lemma} \section{Preliminaries} Let $E=C[0,1]$ be a real Banach space, with the norm $\|x\|=\max_{t\in[0,1]}|x(t)|$ for $x\in C[0,1]$. Let $P=\{x\in E: x(t)\geq 0,t\in[0,1]\}$. Clearly $P$ is a cone in $E$. The function $x$ is said to be a positive solution of \eqref{e1.1}-\eqref{e1.2} if $x(t)$ is positive solution on $(0,1)$ and satisfies the differential equation \eqref{e1.1} and the boundary conditions \eqref{e1.2}. The following lemmas play an important role when proving our main results. \begin{lemma}[\cite{m1,m2}] \label{lem2.1} Assume \begin{itemize} \item[(H1)] $p\in C^1([0,1], (0, +\infty))$, $q\in C([0,1], (0, +\infty))$. \end{itemize} Let $\psi$ and $\phi$ be the solutions of the linear problems \begin{gather} \label{e2.1} (p(t)\psi'(t))'(t)-q(t)\psi(t)=0, \quad t\in (0,1), \\ \label{e2.2} \psi(0)=b,\quad p(0)\psi'(0)= a, \end{gather} and \begin{gather} \label{e2.3} (p(t)\phi'(t))'(t)-q(t)\phi(t)=0, \quad t\in (0,1), \\ \label{e2.4} \phi(1)=d,\quad p(1)\phi'(1)= -c, \end{gather} respectively. Then \begin{itemize} \item[(i)] $\psi$ is strictly increasing on $[0,1]$, and $\psi(t)>0$ on $[0,1]$; \item[(ii)] $\phi$ is strictly decreasing on $[0,1]$, and $\phi(t)>0$ on $[0,1]$. \end{itemize} \end{lemma} As in \cite{m2}, set $$ \Delta= \det \begin{pmatrix} -\sum_{i=1}^{m-2} \alpha_i\psi(\xi_i) & \rho-\sum_{i=1}^{m-2} \alpha_i\phi(\xi_i) \\ \rho-\sum_{i=1}^{m-2} \beta_i\psi(\xi_i) & -\sum_{i=1}^{m-2} \beta_i\phi(\xi_i) \end{pmatrix}, \quad \rho= p(t)\det\begin{pmatrix} \phi(t) & \psi(t) \\ \phi'(t) & \psi'(t) \end{pmatrix}. $$ Then, by Liouville's formula, we have $$ \rho= p(0)\det\begin{pmatrix} \phi(0) & \psi(0) \\ \phi'(0) & \psi'(0) \end{pmatrix}=\text{constant}. $$ Define \begin{equation} \label{e2.5} G(t,s)= \frac{1}{\rho} \begin{cases} \phi(t)\psi(s), & 0\leq s\leq t\leq 1,\\ \phi(s)\psi(t), & 0\leq t\leq s\leq 1. \end{cases} \end{equation} It is easy to see that \begin{equation} \label{e2.6} 0\leq G(t,s)\leq G(s,s),\quad 0\leq s,\; t\leq 1. \end{equation} \begin{remark} \label{rmk2.1}\rm By \eqref{e2.5} and Lemma \ref{lem2.1}, for any $t\in [0,1]$, we have $$ \frac{G(t,s)}{G(s,s)} =\begin{cases}\frac{\phi(t)}{\phi(s)}, & 0\leq s\leq t\leq 1,\\[3pt] \frac{\psi(t)}{\psi(s)}, & 0\leq t\leq s\leq 1, \end{cases} \geq \begin{cases} \frac{d}{\phi(0)}, & 0\leq s\leq t\leq 1,\\[3pt] \frac{b}{\psi(1)},& 0\leq t\leq s\leq 1. \end{cases} $$ \end{remark} Let $\gamma= \min\{\frac{d}{\phi(0)},\ \frac{b}{\psi(1)}\}$, then $ G(t,s)\geq\gamma G(s,s)$, for $t,s\in [0,1]$. \begin{remark} \label{rmk2.2}\rm Since $\gamma= \min\{\frac{d}{\phi(0)}, \frac{b}{\psi(1)}\}$, according to the monotonicity of $\psi(t)$, we have $\gamma\leq\frac{b}{\psi(1)} =\frac{\psi(0)}{\psi(1)}\leq\frac{\psi(t)}{\psi(1)}$, so $\psi(t)\geq\gamma\psi(1)$, for $t\in [0,1]$. Similarly, by the monotonicity of $\phi(t)$, we have $\gamma\leq\frac{d}{\phi(0)} =\frac{\phi(1)}{\phi(0)}\leq\frac{\phi(t)}{\phi(0)}$, so $\phi(t)\geq\gamma\phi(0)$, for $t\in [0,1]$. \end{remark} \begin{lemma}[\cite{m1,m2}] \label{lem2.2} Assume {\rm (H1)} and that $\Delta\neq 0$. Then for any $y\in L[0,1]$, the problem \begin{gather} \label{e2.7} (p(t)x'(t))'(t)-q(t)x(t)+y(t)=0, \quad t\in (0,1), \\ \label{e2.8} ax(0)-bp(0)x'(0)= \sum_{i=1}^{m-2}\alpha_i x(\xi_i),\quad cx(1)+dp(1)x'(1)= \sum_{i=1}^{m-2}\beta_i x(\xi_i), \end{gather} has a unique solution \begin{equation} \label{e2.9} x(t)=\int_0^1G(t,s)y(s)ds +A(y)\psi(t)+B(y) \phi(t), \end{equation} where \begin{gather} \label{e2.10} A(y)=\frac{1}{\Delta} \det\begin{pmatrix} \sum_{i=1}^{m-2} \alpha_i\int_0^1G(\xi_i,s)y(s)ds & \rho-\sum_{i=1}^{m-2} \alpha_i\phi(\xi_i) \\ \sum_{i=1}^{m-2} \beta_i\int_0^1G(\xi_i,s)y(s)ds & -\sum_{i=1}^{m-2} \beta_i\phi(\xi_i) \end{pmatrix} \\ \label{e2.11} B(y)=\frac{1}{\Delta} \det\begin{pmatrix} -\sum_{i=1}^{m-2} \alpha_i\psi(\xi_i) & \sum_{i=1}^{m-2} \alpha_i\int_0^1G(\xi_i,s)y(s)ds \\ \rho-\sum_{i=1}^{m-2} \beta_i\psi(\xi_i) & \sum_{i=1}^{m-2} \beta_i\int_0^1G(\xi_i,s)y(s)ds \end{pmatrix}. \end{gather} \end{lemma} \begin{lemma}[\cite{m1,m2}] \label{lem2.3} Assume {\rm (H1)} and \begin{itemize} \item[(H2)] $\Delta<0$, $\rho-\sum_{i=1}^{m-2}\alpha_i\phi(\xi_i)>0$, $\rho-\sum_{i=1}^{m-2}\beta_i\psi(\xi_i)>0$. \end{itemize} Then for $y\in L[0,1]$ with $y\geq0$, the unique solution $x$ of \eqref{e2.7}-\eqref{e2.8} satisfies $x(t)\geq0$, for $t\in[0,1]$. \end{lemma} Let $Q=\{x\in P :x(t)\geq\gamma\|x\|\}$. It is obvious that $Q$ is a subcone of $P$. With Lemma \ref{lem2.2}, Problem \eqref{e1.1}-\eqref{e1.2} has a positive solution $x=x(t)$ if and only if $x\in Q\backslash\{\theta\}$ is a solution of the nonlinear integral equation \begin{equation} \label{e2.12} x(t)=\int_0^1G(t,s)f(s,x(s))ds +A(f(s,x(s)))\psi(t)+B(f(s,x(s))) \phi(t), \end{equation} where $f$ satisfies the condition \begin{itemize} \item[(H3)] $f\in C((0,1)\times(0,+\infty),[0,+\infty))$ and there exist $h\in C((0,1),[0,+\infty))$, $g\in C((0,+\infty),[0,+\infty))$ satisfying that for any $t\in (0,1)$, $u\in (0,+\infty)$ implies \begin{gather*} f(t,u)\leq h(t)g(u),\quad t\in(0,1),\; u\in(0,+\infty),\\ 0<\int_0^1G(s,s)h(s)ds<+\infty. \end{gather*} \end{itemize} Define an operator $T:Q\backslash \{\theta\}\to P$ by \begin{equation} \label{e2.13} (Tx)(t)=\int_0^1G(t,s)f(s,x(s))ds +A(f(s,x(s)))\psi(t)+B(f(s,x(s)))\phi(t). \end{equation} It is easy to prove that the existence of solutions to \eqref{e1.1}-\eqref{e1.2} is equivalent to the existence of solutions to \eqref{e2.12}. That is, the existence of a fixed point of operator $T$. To overcome the singularity, we consider the following approximating equation of \eqref{e2.13} with the boundary conditions \eqref{e1.2}. \begin{equation} \label{e2.14} (T_nx)(t)=\int_0^1G(t,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\psi(t)+B(f_n(s,x(s)))\phi(t), \end{equation} where $n$ is a positive integer and \begin{equation} \label{e2.15} f_n(t,x)=f(t,\max\{\frac{1}{n},x\}). \end{equation} \begin{remark} \label{rmk2.3}\rm By (H3), there exists $\tau\in(0,\frac{1}{2})$ such that $$ 0<\int_{\tau}^{1-\tau}G(s,s)h(s)ds<+\infty. $$ \end{remark} \begin{lemma} \label{lem2.4} Assume {\rm (H1)-(H3)}. Then $T_n: P\to P$ is completely continuous for any fixed natural number $n$. \end{lemma} \begin{proof} First it is easy to see that $T_n$ maps $P$ into $P$. Then we prove that $T_n$ maps bounded sets into bounded sets. Suppose $D\subset P$ is an arbitrary bounded set. Then there exists a constant $M_1>0$ such that $\| x\|\leq M_1$ for any $x\in D$. By (H1), for any $x \in D$ and $s \in [0,1]$, we have $$\begin{aligned} |(T_nx)(t)| &=\int_0^1G(t,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\psi(t)+B(f_n(s,x(s)))\phi(t)\\ &\leq \int_0^1G(s,s)h(s)g\big(\max\{\frac{1}{n},x(s)\}\big)ds\\ &\quad +A\Big(h(s)g\big(\max\{\frac{1}{n},x(s)\}\big)\Big)\psi(1)\\ &\quad +B\Big(h(s)g\big(\max\{\frac{1}{n},x(s)\}\big)\Big)\phi(0)\\ &\leq M_2\Big[\int_0^1G(s,s)h(s)ds+A\left(h(s)\right)\psi(1) +B\left(h(s)\right)\phi(0)\Big]\\ &\leq M_2(1+A\psi(1) +B\phi(0))\int_0^1G(s,s)h(s)ds, \end{aligned}$$ where $M_2=\sup_{x\in[\frac{1}{n},\frac{1}{n}+M_1]}g(x)$, \begin{gather} \label{e2.16} A=\frac{1}{\Delta} \det\begin{pmatrix} \sum_{i=1}^{m-2} \alpha_i & \rho-\sum_{i=1}^{m-2} \alpha_i\phi(\xi_i) \\ \sum_{i=1}^{m-2} \beta_i & -\sum_{i=1}^{m-2} \beta_i\phi(\xi_i) \\ \end{pmatrix}, \\ \label{e2.17} B=\frac{1}{\Delta} \det \begin{pmatrix} -\sum_{i=1}^{m-2} \alpha_i\psi(\xi_i) & \sum_{i=1}^{m-2} \alpha_i \\ \rho-\sum_{i=1}^{m-2} \beta_i\psi(\xi_i) & \sum_{i=1}^{m-2} \beta_i\\ \end{pmatrix}. \end{gather} Therefore, $T_n(D)$ is uniformly bounded. Now we show that $T_n(D)$ is equicontinuous on $[0,1]$. For any $\varepsilon>0$, since $G(t,s),\psi(t)$ and $\phi(t)$ are uniformly continuous on $[0,1]\times[0,1]$ and $[0,1]$, respectively. There exists $\delta>0$ such that for any $t_1,t_2\in[0,1],|t_1-t_2|<\delta$ implies that \begin{gather*} |G(t_1,s)-G(t_2,s)|<\frac{\varepsilon\min_{0\leq s\leq1}G(s,s)}{3M_2\int_0^1G(s,s)h(s)ds}, \\ |\psi(t_1)-\psi(t_2)|<\frac{\varepsilon}{3M_2A\int_0^1G(s,s)h(s)ds}, \\ |\phi(t_1)-\phi(t_2)|<\frac{\varepsilon}{3M_2B\int_0^1G(s,s)h(s)ds}. \end{gather*} Consequently, for any $x\in D$, $t_1,t_2\in[0,1]$, $|t_1-t_2|<\delta$, we have \begin{align*} &|T_nx(t_1)-T_nx(t_2)|\\ &\leq\int_0^1|G(t_1,s)-G(t_2,s)|f_n(s,x(s))ds\\ &\quad +A(f_n(s,x(s)))|\psi(t_1)-\psi(t_2)| +B(f_n(s,x(s)))|\phi(t_1)-\phi(t_2)| \\ &\leq \int_0^1|G(t_1,s)-G(t_2,s)|h(s) g\big(\max\{\frac{1}{n},x(s)\}\big)ds\\ &\quad +A\Big(h(s)g\big(\max\{\frac{1}{n},x(s)\}\big)\Big) |\psi(t_1)-\psi(t_2)|\\ &\quad +B\Big(h(s)g\big(\max\{\frac{1}{n},x(s)\}\big)\Big) |\phi(t_1)-\phi(t_2)|\\ &\leq M_2\int_0^1|G(t_1,s)-G(t_2,s)|h(s)ds\\ &\quad +A(h(s)M_2)|\psi(t_1)-\psi(t_2)| +B(h(s)M_2)|\phi(t_1)-\phi(t_2)|\\ &\leq M_2\int_0^1|G(t_1,s)-G(t_2,s)|h(s)ds\\ &\quad +M_2A|\psi(t_1)-\psi(t_2)|\int_0^1G(s,s)h(s)ds\\ &\quad +M_2B|\phi(t_1)-\phi(t_2)|\int_0^1G(s,s)h(s)ds\\ &<\frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3} =\varepsilon. \end{align*} Thus, $T_n(D)$ is equicontinuous on $[0,1]$. According to Ascoli-Arzela Theorem, $T_n(D)$ is a relatively compact set. In the end, we show $T_n$ is continuous. Suppose $x_m,x\in D,x_m\to x \ (m\to +\infty)$. Then there exists a constant $M_3>0$ such that $\|x\|\leq M_3$, $\|x_m\|\leq M_3$ ($m=1,2,\dots$). Since $f_n(t,x)$ is uniformly continuous on $[0,1]\times D$ for any fixed natural number $n$, hence, $$ \lim_{m\to+\infty}f_n(t,x_m(t))=f_n(t,x(t)),\quad \text{uniformly on } t\in[0,1]. $$ According to the Lebesgue dominated convergence theorem, $$ \lim_{m\to+\infty}\int_0^1G(s,s)|f_n(s,x_m(s))-f_n(s,x(s))|ds=0. $$ Thus for the above $\varepsilon>0$, there exists a natural number $M$, such that $m>M$ implies that \begin{equation} \label{e2.18} \int_0^1G(s,s)|f_n(s,x_m(s))-f_n(s,x(s))|ds <\frac{\varepsilon}{1+A\psi(1)+B\phi(0)}. \end{equation} From \eqref{e2.18}, we obtain that for $m>M$, \begin{align*} &\|T_nu_m-T_nu\|\\ &= \max_{0\leq t\leq1}\Big[\int_0^1G(t,s)f_n(s,x_m(s))ds +A(f_n(s,x_m(s)))\psi(t)+B(f_n(s,x_m(s)))\phi(t) \\ &\quad -\int_0^1G(t,s)f_n(s,x(s))ds -A(f_n(s,x(s)))\psi(t)-B(f_n(s,x(s)))\phi(t)\Big]\\ &\leq \int_0^1G(s,s)|f_n(s,x_m(s))-f_n(s,x(s))|ds\\ &\quad +A\big(|f_n(s,x_m(s))-f_n(s,x(s))|\big)\psi(1)\\ &\quad +B(|f_n(s,x_m(s))-f_n(s,x(s))|)\phi(0)\\ &\leq \big(1+A\psi(1)+B\phi(0)\big)\int_0^1G(s,s)|f_n(s,x_m(s)) -f_n(s,x(s))|ds<\varepsilon. \end{align*} Therefore, $T_n:P\to P$ is continuous. Thus $T_n:P \to P$ is a completely continuous operator. \end{proof} \begin{lemma} \label{lem2.6} $T_n(Q)\subset Q$. \end{lemma} \begin{proof} For any $x\in Q$, (H2) and (H3) imply $(T_nx)(t)\geq0$. From \eqref{e2.6}, \eqref{e2.14} and the monotonicity of $\psi(t)$ and $\phi(t)$, we have $$ (T_nx)(t)\leq\int_0^1G(s,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\psi(1)+B(f_n(s,x(s)))\phi(0), $$ which implies \begin{equation} \label{e2.19} \|T_nx\|\leq\int_0^1G(s,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\psi(1)+B(f_n(s,x(s)))\phi(0). \end{equation} By Remarks \ref{rmk2.1} and \ref{rmk2.2}, we have \begin{equation} \label{e2.20} \begin{aligned} (T_nx)(t) &=\int_0^1G(t,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\psi(t)+B(f_n(s,x(s)))\phi(t)\\ &\geq \gamma\int_0^1G(s,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\gamma\psi(1)+B(f_n(s,x(s)))\gamma\phi(0)\\ &\geq\gamma\Big[\int_0^1G(s,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\psi(1)+B(f_n(s,x(s)))\phi(0)\Big]. \end{aligned} \end{equation} Then, \eqref{e2.19} and \eqref{e2.20} yield $$ (T_nx)(t)\geq\gamma\|T_nx\|. $$ Hence $T_nx\in Q$. \end{proof} \section{Main results} In this section, we present our main results as follows. \begin{theorem} \label{thm3.1} Suppose that {\rm (H1)--(H3)} hold and there exist numbers $R>0$ and $L>0$ such that \begin{gather} \label{e3.1} \int_0^1G(s,s)h(s)ds<\frac{R}{\widetilde{M}(1+A\psi(1)+B\phi(0))}, \\ \label{e3.2} L\gamma^2\int_\tau^{1-\tau}G(s,s)ds>1,\quad \liminf_{x\to+\infty}\min_{\tau\leq t\leq1-\tau}\frac{f(t,x)}{x}>L. \end{gather} Then \eqref{e1.1}-\eqref{e1.2} has at least one positive solution, where $\widetilde{M}=\max_{u\in[\gamma R,1+R]}g(u)$, $\gamma$ is defined in Remark \ref{rmk2.1} and $A,B$ are defined by \eqref{e2.16} and \eqref{e2.17}, respectively. \end{theorem} \begin{proof} Firstly, we shall prove that when $n$ is sufficiently large, we have \begin{equation} \label{e3.3} T_nx\neq\lambda x,\quad x\in\partial Q_R,\; \lambda\geq1, \end{equation} where $Q_R=\{x\in Q:\|x\|0$. In fact, if there exists $x_0\in \partial Q_R$, and $\lambda_0\geq 1$ such that $\lambda_0x_0=T_nx_0$, then $x_0(t)\leq T_nx_0(t)$ for $t\in[0,1]$ and any $n$. Choose a sufficiently large $n$ satisfying $n>\frac{1}{\gamma R}$. Then we have \begin{equation} \label{e3.4} \begin{aligned} x_0(t)&\leq(T_nx_0)(t)\\ &=\int_0^1G(t,s)f_n(s,x_0(s))ds +A(f_n(s,x_0(s)))\psi(t)+B(f_n(s,x_0(s)))\phi(t)\\ &\leq\int_0^1G(s,s)f_n(s,x_0(s))ds +A(f_n(s,x_0(s)))\psi(1)+B(f_n(s,x_0(s)))\phi(0)\\ &\leq(1+A\psi(1)+B\phi(0))\int_0^1G(s,s)f_n(s,x_0(s))ds\\ &\leq(1+A\psi(1)+B\phi(0))\int_0^1G(s,s)h(s) g\big(\max\{\frac{1}{n},x_0(s)\}\big)ds\\ &\leq(1+A\psi(1)+B\phi(0))\widetilde{M}\int_0^1G(s,s)h(s)dsR_1$ implies \begin{equation} \label{e3.5} f(t,x)>Lx,\quad t\in[\tau,1-\tau]. \end{equation} Choose $R'>\{R,\gamma^{-1}R_1\}$. When $n$ being sufficiently large we can claim that \begin{equation} \label{e3.6} T_nx\neq\lambda x,\quad \forall x\in\partial Q_{R'},\; \lambda\in(0,1], \end{equation} where $Q_R'=\{x\in Q:\|x\|\frac{1}{\gamma R'}$. Therefore, by \eqref{e3.5} we have \begin{align*} x_1(t)&\geq (T_nx_1)(t)\\ &=\int_0^1G(t,s)f_n(s,x_1(s))ds +A(f_n(s,x_1(s)))\psi(t)+B(f_n(s,x_1(s)))\phi(t)\\ &\geq\int_0^1G(t,s)f_n(s,x_1(s))ds\\ &\geq\gamma\int_\tau^{1-\tau}G(s,s)f_n(s,x_1(s))ds\\ &\geq L\gamma\int_\tau^{1-\tau}G(s,s)x_1(s)ds \\ &\geq LR'\gamma^2\int_\tau^{1-\tau}G(s,s)ds. \end{align*} This is a contradiction to $x_1\in\partial Q_{R'}$. Consequently, \eqref{e3.6} holds. Furthermore, for each $x\in\partial Q_{R}$, \begin{align*} \|T_nx\|&\geq \int_0^1G(t,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\psi(t)+B(f_n(s,x(s)))\phi(t)\\ &\geq\int_0^1G(t,s)f_n(s,x(s))ds\\ &\geq\gamma\int_\tau^{1-\tau}G(s,s)f_n(s,x(s))ds\\ &\geq L\gamma\int_\tau^{1-\tau}G(s,s)x(s)ds \\ &\geq LR'\gamma^2\int_\tau^{1-\tau}G(s,s)ds. \end{align*} So $\inf_{x\in\partial Q_{R'}}\|T_nx\|>0$. Thus from Lemma \ref{lem1.2}, $i(T_n,Q_{R'},Q)=0$. By the additivity of fixed point index, we know that $$ i(T_n,Q_{R'}\setminus\overline{Q}_R,Q)=i(T_n,Q_{R'},Q)-i(T_n,Q_R,Q)=-1. $$ As a result, there exist $x_n\in Q_{R'}\setminus\overline{Q}_R$ satisfying $T_nx_n=x_n$ provided that $n$ is sufficiently large. Without loss of generality, suppose $T_nx_n=x_n$, $n\geq n_0$. Let $D=\{x_n\}_{n\geq n_0}$ be the sequence of solutions to \eqref{e2.14}. It is not difficult to prove that $D$ is uniformly bounded. Next we show $\{x_n\}_{n\geq n_0}$ is equicontinuous on $[0,1]$. It is obvious that we only need to prove $\lim_{t\to0+}(x_n(t)-x_n(0))=0$, $\lim_{t\to1-}(x_n(t)-x_n(1))=0$ uniformly with respect to $n\geq n_0$ and $D$ is equicontinuous on $[\sigma,1-\sigma]\subset(0,1)$ for $\sigma\in(0,1/2)$. Now we prove that \begin{equation} \label{e3.7} \lim_{t\to0+}(x_n(t)-x_n(0))=0, \quad \text{uniformly with respect to } n\geq n_0. \end{equation} According to \eqref{e2.12}, \begin{align} &\big|x_n(t)-x_n(0)\big| \notag \\ &=\Big|\int_0^1G(t,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\psi(t)+B(f_n(s,x(s)))\phi(t) \notag \\ &\quad -\int_0^1G(0,s)f_n(s,x(s))ds -A(f_n(s,x(s))\psi(0)-B(f_n(s,x(s))\phi(0)\Big| \notag \\ &=\frac{1}{\rho}\phi(t)\int_0^t\psi(s)f_n(s,x(s))ds +\frac{1}{\rho}(\psi(t)-\psi(0))\int_t^1\phi(s)f_n(s,x(s))ds \label{e3.8} \\ &\quad -\frac{1}{\rho}\phi(0)\int_0^t\psi(s)f_n(s,x(s))ds +A(f_n(s,x(s)))(\psi(t)-\psi(0)) \notag \\ &\quad +B(f_n(s,x(s)))(\phi(t)-\phi(0)) \notag \\ &\leq \frac{1}{\rho}\phi(t)\int_0^t\psi(s)h(s)g \big(\max\{\frac{1}{n},x(s)\}\big)ds \notag \\ &\quad +\frac{1}{\rho}(\psi(t)-\psi(0)) \int_t^1\phi(s)h(s)g\big(\max\{\frac{1}{n},x(s)\}\big)ds \notag\\ &\quad-\frac{1}{\rho}\phi(0)\int_0^t\psi(s)h(s) g\big(\max\{\frac{1}{n},x(s)\}\big)ds \notag \\ &\quad +A\Big(h(s)g\big(\max\{\frac{1}{n},x(s)\}\big)\Big) (\psi(t)-\psi(0)) \notag\\ &+B\Big(h(s)g\big(\max\{\frac{1}{n},x(s)\}\big)\Big)(\phi(0)-\phi(t)) \notag \\ &\leq \frac{1}{\rho}M_4\phi(t)\int_0^t\psi(s)h(s)ds+ \frac{1}{\rho}M_4(\psi(t)-\psi(0))\int_t^1\phi(s)h(s)ds \notag\\ &\quad +\frac{1}{\rho}\phi(0)M_4\int_0^t\psi(s)h(s)ds +AM_4(\psi(t)-\psi(0))\int_0^1G(s,s)h(s)ds \notag\\ &\quad +BM_4(\phi(0)-\phi(t))\int_0^1G(s,s)h(s)ds. \notag \end{align} Since $\{x_n(t)\}_{n\geq n_0}$ is uniformly bounded, it follows that $\{g(x_n)\}_{n\geq n_0}$ is bounded. Therefore, there exists a constant $M_4$ such that $\|g(x_n)\|0$, by the absolutely continuity of integral function and (H3), there exists $\delta_1\in(0,\frac{1}{2})$ such that $t_1,t_2\in[0,1],|t_1-t_2|<\delta_1$ implies \begin{equation} \label{e3.13} \big|\int_{t_1}^{t_2}G(s,s)h(s)ds\big|<\varepsilon. \end{equation} Therefore, from \eqref{e2.5} and \eqref{e3.13}, we have \begin{gather*} \frac{1}{\rho}\phi(t)\int_0^t\psi(s)h(s)ds\leq\int_0^tG(s,s)h(s)ds <\varepsilon,\quad t\in(0,\delta_1], \\ \frac{1}{\rho}\phi(0)\int_0^t\psi(s)h(s)ds \leq\frac{\phi(0)}{\phi(\delta_1)}\int_0^tG(s,s)h(s)ds<\varepsilon, \quad t\in(0,\delta_1]; \end{gather*} i.e., \eqref{e3.9} and \eqref{e3.10} hold. \begin{align*} &\frac{1}{\rho}(\psi(t)-\psi(0))\int_t^1\phi(s)h(s)ds\\ &\leq \frac{1}{\rho}\psi(t)\int_t^{\delta_1}\phi(s)h(s)ds+\frac{1}{\rho}(\psi(t)-\psi(0)) \int^1_{\delta_1}\phi(s)h(s)ds\\ &\leq \int_t^{\delta_1} G(s,s)h(s)ds+\frac{\psi(t)-\psi(0)}{\psi(\delta_1)} \int_{\delta_1}^1G(s,s)h(s)ds\leq2\varepsilon. \end{align*} That is, \eqref{e3.11} holds. By \eqref{e3.9}-\eqref{e3.12}, \eqref{e3.7} holds. Since \begin{align*} &|x_n(t)-x_n(1)| \\ &= \Big|\int_0^1G(t,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\psi(t)+B(f_n(s,x(s)))\phi(t)\\ &\quad -\int_0^1G(1,s)f_n(s,x(s))ds -A(f_n(s,x(s))\psi(1)-B(f_n(s,x(s))\phi(1)\Big|\\ &\leq \frac{1}{\rho}(\phi(t)-\phi(1))\int_0^t\psi(s)f_n(s,x(s))ds+\frac{1}{\rho}\psi(t)\int_t^1\phi(s)f_n(s,x(s))ds\\ &\quad -\frac{1}{\rho}\phi(1)\int_t^1\psi(s)f_n(s,x(s))ds+A(f_n(s,x(s)))(\psi(t)-\psi(1))\\ &\quad +B(f_n(s,x(s)))(\phi(t)-\phi(1))\\ &\leq \frac{1}{\rho}M_4(\phi(t)-\phi(1))\int_0^t\psi(s)h(s)ds+ \frac{1}{\rho}M_4\psi(t)\int_t^1\phi(s)h(s)ds\\ &\quad +\frac{1}{\rho}\phi(1)M_4\int_0^t\psi(s)h(s)ds +AM_4(\psi(1)-\psi(t))\int_0^1G(s,s)h(s)ds\\ &\quad +BM_4(\phi(t)-\phi(1))\int_0^1G(s,s)h(s)ds, \end{align*} similar to the above, we can easily prove that \begin{equation} \label{e3.14} \lim_{t\to1-}(x_n(t)-x_n(1))=0, \quad \text{uniformly with respect to } n\geq n_0. \end{equation} Next we prove that $D$ is equicontinuous on $[\sigma,1-\sigma]$ for any $\sigma\in(0,1/2)$. In fact, for $n\geq n_0$, $t_1,t_2\in[\sigma,1-\sigma]$ with $t_2>t_1$, we have \begin{equation} \label{e3.15} \begin{aligned} &|x_n(t_2)-x_n(t_1)|\\ &= \Big|\int_0^1G(t_2,s)f_n(s,x(s))ds +A(f_n(s,x(s)))\psi(t_2)+B(f_n(s,x(s)))\phi(t_2) \\ &\quad -\int_0^1G(t_1,s)f_n(s,x(s))ds -A(f_n(s,x(s)))\psi(t_1)-B(f_n(s,x(s)))\phi(t_1)\Big|\\ &\leq \frac{1}{\rho}(\phi(t_2)-\phi(t_1))\int_0^{t_1}\psi(s)f_n(s,x(s))ds +\frac{1}{\rho}\phi(t_2)\int_{t_1}^{t_2}\psi(s)f_n(s,x(s))ds\\ &\quad +\frac{1}{\rho}(\psi(t_2)-\psi(t_1))\int_{t_2}^1\phi(s)f_n(s,x(s))ds- \frac{1}{\rho}\psi(t_1)\int_{t_1}^{t_2}\phi(s)f_n(s,x(s))ds\\ &\quad +A(f_n(s,x(s)))(\psi(t_2)-\psi(t_1))+B(f_n(s,x(s))) (\phi(t_2)-\phi(t_1)). \end{aligned} \end{equation} By \eqref{e2.5} and the monotonicity of $\psi(t)$ and $\phi(t)$, we have \begin{gather} \label{e3.16} \frac{1}{\rho}\int_0^{t_1}\psi(s)h(s)ds\leq\frac{1}{d} \int_0^1G(s,s)h(s)ds, \\ \label{e3.17} \frac{1}{\rho}\int_{t_2}^1\phi(s)h(s)ds\leq\frac{1}{b} \int_0^1G(s,s)h(s)ds, \\ \label{e3.18} \frac{1}{\rho}\phi(t_2)\int_{t_1}^{t_2}\psi(s)h(s)ds \leq\int_{t_1}^{t_2}G(s,s)h(s)ds, \\ \label{e3.19} \frac{1}{\rho}\psi(t_1)\int_{t_1}^{t_2}\phi(s)h(s)ds \leq\int_{t_1}^{t_2}G(s,s)h(s)ds. \end{gather} By \eqref{e3.15}-\eqref{e3.19}, we have \begin{align*} %\label{e3.20} &|x_n(t_2)-x_n(t_1)|\\ &\leq M_4(\frac{1}{b}+A)(\psi(t_2)-\psi(t_1))\int_0^1G(s,s)h(s)ds \\ &\quad +M_4(\frac{1}{d}+B)(\phi(t_1)-\phi(t_2))\int_0^1G(s,s)h(s)ds +2M_4\int_{t_1}^{t_2}G(s,s)h(s)ds. \end{align*} By the above inequality, (H3), \eqref{e3.9}-\eqref{e3.12}, and continuity of $\psi(t)$, $\phi(t)$, $D$ is equicontinuous on $[\sigma,1-\sigma]$. From the above proof, we can know $D$ is equicontinuous on $[0,1]$. It follows from Ascoli-Arzela's theorem that the sequence $\{x_n\}_{n\geq n_0}$ has a subsequence which uniformly converges on $[0,1]$. Without loss of generality, we assume that $\{x_n\}$ itself uniformly converges to $x$ on $[0,1]$. According to the Lebesgue's dominated theorem, we know that $x$ is the positive solution of \eqref{e1.1}-\eqref{e1.2}. \end{proof} \begin{thebibliography}{00} \bibitem{f1} W. Feng; \emph{On a $m$-point nonlinear boundary-value problem}, Nonlinear Anal. 30 (1997) 5369-5374. \bibitem{g1} G. N. Galanis, A. P. Palamides; \emph{Positive solutions of three-point boundary-value problems for p-Laplacian singular differential equations}, Electron. J. Differential Equations 2005:106 (2005), 1-18. \bibitem{g2} D. Guo, V. lakshmikantham; Nonlinear Problems in Abstract Cone, Academic Press, New York, 1988. \bibitem{h1} X. He, W. Ge; \emph{Triple solutions for second-order three-point boundary-value problems}, J. Math. Anal. Appl. 268 (2002) 256-265. \bibitem{h2} J. Henderson, H. Wang; \emph{Positive solutions for nonlinear eigenvalue problems},\ J. Math. Anal. Appl. 208 (1997) 252-259. \bibitem{l1} K. Lan, J. R. L. Webb; \emph{Positive solutions for semilinear differential equations with singularities}, J. Differential Equations 148 (1998) 407-421. \bibitem{l2} L. Liu, B. Liu, Y. Wu; \emph{Positive solutions of singular boundary-value problems for nonlinear differential systems}, Appl. Math. Comput. 186 (2007) 1163-1172. \bibitem{m1} R. Ma; \emph{Multiple positive solutions for nonlinear $m$-point boundary-value problems},\ Appl. Math. Comput. 148 (2004) 249-262. \bibitem{m2} R. Ma, B. Thompson; \emph{Positive solutions for nonlinear $m$-point eigenvalue problems}, J. Math. Anal. Appl. 297 (2004) 24-37. \bibitem{s1} Y. Sun; \emph{Positive solutions of nonlinear second-order $m$-point boundary-value problem}, Nonlinear Anal. 61 (2005) 1283-1294. \bibitem{w1} Z. Wei, C. Pang; \emph{Positive solutions of some singular $m$-point boundary-value problems at non-resonance}, Appl. Math. Comput. 171 (2005) 433-449. \bibitem{x1} X. Xu; \emph{Positive solutions for singular $m$-point boundary-value problems with positive parameter}, J. Math. Anal. Appl. 291 (2004) 352-367. \bibitem{z1} G. Zhang, J. Sun; \emph{Positive solutions of $m$-point boundary-value problems}, J. Math. Anal. Appl. 291 (2004) 406-418. \bibitem{z2} G. Zhang, J. Sun; \emph{Multiple positive solutions of singular second-order $m$-point boundary-value problems}, J. Math. Anal. Appl. 317 (2006) 442-447. \end{thebibliography} \end{document}