\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 163, pp. 1--9.\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-2009/163\hfil Three positive solutions] {Three positive solutions for a system of singular generalized Lidstone problems} \author[J. Xu, Z. Yang\hfil EJDE-2009/163\hfilneg] {Jiafa Xu, Zhilin Yang} % in alphabetical order \address{Jiafa Xu \newline Department of Mathematics, Qingdao Technological University, No 11, Fushun Road, Qingdao, China} \email{xujiafa292@sina.com} \address{Zhilin Yang \newline Department of Mathematics, Qingdao Technological University, No 11, Fushun Road, Qingdao, China} \email{zhilinyang@sina.com} \thanks{Submitted October 7, 2009. Published December 21, 2009.} \thanks{Supported by grants 10871116 and 10971179 from the NNSF of China} \subjclass[2000]{34A34, 34B18, 45G15, 47H10} \keywords{Singular generalized Lidstone problem; positive solution; cone; \hfill\break\indent concave functional} \begin{abstract} In this article, we show the existence of at least three positive solutions for the system of singular generalized Lidstone boundary value problems \begin{gather*} (-1)^m x^{(2m)}=a(t)f_1(t,x,-x'',\dots,(-1)^{m-1}x^{(2m-2)},y,-y'',\\ \dots,(-1)^{n-1}y^{(2n-2)}), \\ (-1)^n y^{(2n)}=b(t)f_2(t,x,-x'',\dots,(-1)^{m-1}x^{(2m-2)},y,-y'',\\ \dots,(-1)^{n-1}y^{(2n-2)}), \\ a_1 x^{(2i)}(0)-b_1 x^{(2i+1)}(0)=c_1x^{(2i)}(1)+d_1 x^{(2i+1)}(1)=0,\\ a_2y^{(2j)}(0)-b_2y^{(2j+1)}(0)=c_2y^{(2j)}(1)+d_2y^{(2j+1)}(1)=0. \end{gather*} The proofs of our main results are based on the Leggett-Williams fixed point theorem. Also, we give an example to illustrate our results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} In this article, we study the existence positive solutions for the following system of singular generalized Lidstone boundary value problems $$\label{eq1} \begin{gathered} (-1)^m x^{(2m)}=a(t)f_1(t,x,-x'',\dots,(-1)^{m-1}x^{(2m-2)},y,-y'', \dots,(-1)^{n-1}y^{(2n-2)}),\\ (-1)^n y^{(2n)}=b(t)f_2(t,x,-x'',\dots,(-1)^{m-1}x^{(2m-2)},y,-y'', \dots,(-1)^{n-1}y^{(2n-2)}),\\ a_1 x^{(2i)}(0)-b_1 x^{(2i+1)}(0)=c_1x^{(2i)}(1)+d_1 x^{(2i+1)}(1)=0 \quad (i=0,1,\dots,m-1),\\ a_2y^{(2j)}(0)-b_2y^{(2j+1)}(0)=c_2y^{(2j)}(1)+d_2y^{(2j+1)}(1)=0\quad (j=0,1,\dots,n-1). \end{gathered}$$ where $m, n\geq 1$, $a(t),b(t) \in C((0, 1), [0,+\infty))$, $a(t)$ and $b(t)$ are allowed to be singular at $t = 0$ and/or $t = 1$; $f_i\in C([0,1]\times \mathbb{R}^{m+n}_+,\mathbb{R}_+)(\mathbb{R}_+:=[0,+\infty))$; $a_i,b_i,c_i,d_i\in \mathbb{R}_+$ with $\rho_i := a_ic_i+a_id_i+b_ic_i> 0$, $i=1,2$. Singular boundary value problems for ordinary differential equation describe many phenomena in applied mathematics and physical sciences, which can be found in the theory of nonlinear diffusion generated by nonlinear sources and in the thermal ignition of gases, see \cite{Anderson,Lin}. Very recently, increasing attention is paid to question of positive solutions for systems of second-order or higher order singular differential equations, see for example \cite{Kang,Li,Liu_1,Liu} and references therein. In \cite{Kang}, by using Leggett-Williams fixed point theorem \cite{Leggett}, Kang et al obtained at least three positive solutions to the following singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. \label{kang} \begin{gathered} u''(t)+a_1(t)f_1(t,u(t),v(t))=0, \quad 0 0$;$\int_0^1 u(s)d\phi_1(s)$and$\int_0^1 v(s)d\phi_1(s)$denote the Riemann-Stieltjes integrals,$i = 1, 2$. Motivated by \cite{Kang}, we deal with the system of singular generalized Lidstone problems \eqref{eq1}. To overcome the difficulties of \eqref{eq1} resulting from the derivatives of even orders, as in \cite{Yang}, we use the method of order reduction to transform \eqref{eq1} into an equivalent system of integro-integral equations, then prove the existence of positive solutions for the resulting system, thereby establishing that of positive solutions for \eqref{eq1}(see the main result in Section 3). The features of this paper mainly include the following aspects. Firstly, our study is on systems of singular generalized Lidstone problems. Secondly,$a$and$b$are allowed to be singular at$t = 0$and/or$t=1$. Finally, the system contains two equations, which can be of different orders. Thus the results presented here are different from those in \cite{Kang,Li,Liu_1,Liu}. The remaining of this paper is organized as follows: Section 2 gives some preliminary results. The main result is stated and proved in Section 3, then followed by an example to illustrate the validity of our main result. \section{Preliminaries} Given a cone$K$in a real Banach space$E$, a map$\alpha$is said to be a nonnegative continuous concave functional on$K$provided that$\alpha : K \to [0,+\infty)$is continuous and $\alpha(tx+(1-t)y)\geq t\alpha(x)+(1-t)\alpha(y)$ for all$x, y \in K$and$0 \leq t \leq 1$. Let$0 < a < b$be given and let$\alpha$be a nonnegative continuous concave functional on$K$. Define the convex sets$P_r$and$P(\alpha, a, b)$by \begin{gather*} P_r:=\{x\in K:\|x\|b\}\neq \emptyset$, and $\alpha(Tx)>b$ for $x\in P(\alpha,b,d)$, \item[(C2)] $\|Tx\|b$ for $x\in P(\alpha,b,c)$ with $\|Tx\|>d$. \end{itemize} Then $T$ has at least three fixed points $x_1,x_2$, and $x_3$ such that $\|x_1\|a$ with $\alpha(x_3)0$, we let $$\label{k} k(t,s):=\frac{1}{\rho} \begin{cases} (b+as)(c+d-ct), & 0\leq s\leq t\leq 1,\\ (b+at)(c+d-cs), & 0\leq t\leq s\leq 1, \end{cases}$$ By definition, $k\in C([0,1]\times [0,1],\mathbb{R}_+)$ has the following properties: \begin{itemize} \item[(i)] $k(t,s)\leq k(s,s)$ for all $t,s\in [0,1]$. \item[(ii)] For any $\theta\in (0,1/2)$, there exists $\gamma\in (0,\min\{\frac{\theta a+ b}{a+b},\frac{\theta c+ d}{c+d}\} ]$ such that $k(t,s)\geq \gamma k(s,s),\forall t\in[\theta,1-\theta],s\in [0,1].$ \end{itemize} \begin{lemma} \label{lem2} Let $f\in C[0,1],h(t)\in C(0,1)$ and $\int_0^1 k(s,s)h(s)ds<+\infty$. The boundary value problem \begin{gather*} -u''=h(t)f(t),\\ au(0)-bu'(0)=0,\\ cu(1)+du'(1)=0, \end{gather*} has a unique solution $u(t)=\int_0^1 k(t,s)h(s)f(s)ds,$ where $k(t,s)$ is given by \eqref{k}. \end{lemma} Let \begin{gather}\label{k1} k_1(t,s):=\frac{1}{\rho_1} \begin{cases} (b_1+a_1s)(c_1+d_1-c_1t), & 0\leq s\leq t\leq 1,\\ (b_1+a_1t)(c_1+d_1-c_1s), & 0\leq t\leq s\leq 1, \end{cases}\\ \label{g1} g_1(t,s):=\frac{1}{\rho_2} \begin{cases} (b_2+a_2s)(c_2+d_2-c_2t), & 0\leq s\leq t\leq 1,\\ (b_2+a_2t)(c_2+d_2-c_2s), & 0\leq t\leq s\leq 1. \end{cases} \end{gather} For $i,j=2,\dots$, define $$k_i(t,s):=\int_0^1 k_1(t,\tau)k_{i-1}(\tau,s)d\tau,\quad g_j(t,s):=\int_0^1 g_1(t,\tau)g_{j-1}(\tau,s)d\tau$$ and the operators $A_i: C[0,1]\to C[0,1]$ and $B_j: C[0,1]\to C[0,1]$ by \begin{gather*} (A_iu)(t):=\int_0^1 k_i(t,s)u(s)ds,\quad i=1,2,\dots,m,\\ (B_jv)(t):=\int_0^1 g_j(t,s)v(s)ds,\quad j=1,2,\dots,n. \end{gather*} For $1\leq i\leq m-1$ and $1\leq j \leq n-1$, let \begin{gather*} \xi_i:=\min_{\theta\leq t\leq 1-\theta}\int_0^1 k_i(t,s)ds,\quad \eta_i:=\max_{0\leq t\leq 1}\int_0^1 k_i(t,s)ds,\\ \mu_j:=\min_{\theta\leq t\leq 1-\theta}\int_0^1 g_j(t,s)ds,\quad \nu_j:=\max_{0\leq t\leq 1}\int_0^1 g_j(t,s)ds,\\ \xi:=\min_{1\le i\le m-1,1\le j\le n-1}\{\xi_i,\mu_j\},\quad \eta:=\max_{1\le i\le m-1,1\le j\le n-1}\{\eta_i,\nu_j\}. \end{gather*} \section{Main result} Let $u(t)=(-1)^{m-1}x^{(2m-2)}$ and $v(t)=(-1)^{n-1}y^{(2n-2)}$. It is easy to see that \eqref{eq1} is equivalent to the system of integro-ordinary differential equations \begin{align*} -u''(t)&= a(t)f_1\Big(t,\int_0^1 k_{m-1}(t,s)u(s)ds,\dots,\int_0^1 k_1(t,s)u(s)ds,u(t),\\ &\quad \int_0^1 g_{n-1}(t,s)v(s)ds,\dots,\int_0^1 g_1(t,s)v(s)ds,v(t)\Big),\\ -v''(t)&= b(t)f_2\Big(t,\int_0^1 k_{m-1}(t,s)u(s)ds,\dots,\int_0^1 k_1(t,s)u(s)ds,u(t),\\ &\quad \int_0^1 g_{n-1}(t,s)v(s)ds,\dots,\int_0^1 g_1(t,s)v(s)ds,v(t)\Big),\\ \end{align*} subject to the boundary conditions $a_1u(0)-b_1u'(0)= c_1u(1)+d_1u'(1)=0 ,\quad a_2v(0)-b_2v'(0)= c_2v(1)+d_2v'(1)=0 .$ Furthermore, the above system is equivalent to the system \label{sys} \begin{aligned} u(t)&= \int_0^1 k_1(t,s)a(s)f_1\Big(s,\int_0^1 k_{m-1}(s,\tau)u(\tau)d\tau,\dots,\int_0^1 k_1(s,\tau)u(\tau)d\tau,u(s),\\ &\quad \int_0^1g_{n-1}(s,\tau)v(\tau)d\tau,\dots,\int_0^1 g_1(s,\tau)v(\tau)d\tau,v(s)\Big)ds,\\ v(t)&= \int_0^1 g_1(t,s)b(s)f_2\Big(s,\int_0^1 k_{m-1}(s,\tau)u(\tau)d\tau,\dots,\int_0^1 k_1(s,\tau)u(\tau)d\tau,u(s),\\ \quad & \int_0^1 g_{n-1}(s,\tau)v(\tau)d\tau,\dots,\int_0^1 g_1(s,\tau)v(\tau)d\tau,v(s)\Big)ds. \end{aligned} Let $E := C([0, 1],\mathbb{R})\times C([0, 1],\mathbb{R})$ endowed with the norm $\|(u, v)\|:= \|u\|+\|v\|$, where $\|u\|:= \max_{0\leq t\leq 1} |u(t)|$, and define the cone $K\subset E$ by $K := \Big\{(u, v)\in E : u(t) \geq 0, v(t) \geq 0, t \in [0, 1], \; \min_{\theta \leq t\leq 1-\theta} (u(t) + v(t)) \geq \gamma \|(u, v)\| \Big\}.$ Clearly, $(E,\|\cdot\|)$ is a real Banach space and $P$ is a cone on $E$. Define the operator $T:K\to K$ by $T(u,v)(t):=(T_1(u,v)(t),T_2(u,v)(t)),$ where \begin{align*} T_1(u,v)(t) & := \int_0^1 k_1(t,s)a(s)f_1\Big(s,(A_{m-1}u)(s),\dots,(A_1u)(s),\\ &\quad u(s),(B_{n-1}v)(s),\dots,(B_1v)(s),v(s)\Big)ds, \\ T_2(u,v)(t) & := \int_0^1 g_1(t,s)b(s)f_2\Big(s,(A_{m-1}u)(s),\dots,(A_1u)(s),\\ &\quad u(s),(B_{n-1}v)(s),\dots,(B_1v)(s),v(s)\Big)ds. \end{align*} Now $f\in C([0,1]\times \mathbb{R}^{m+n}_+,\mathbb{R}_+)$ and $g\in C([0,1]\times \mathbb{R}^{m+n}_+,\mathbb{R}_+)$ imply that $T: K\to K$ is a completely continuous operator. In our setting, the existence of positive solutions for \eqref{sys} is equivalent to that of positive fixed points of $T$. \begin{lemma} The operator $T$ maps $K$ into $K$. \end{lemma} \begin{proof} If $(u,v) \in K$, then \begin{align*} & (T_1(u,v)(t))''\\ &=-a(t)f_1\Big(t,(A_{m-1}u)(t),\dots,(A_1u)(t),u(t), (B_{n-1}v)(t),\dots,(B_1v)(t),v(t)\Big)\\ & \leq 0, \\ &(T_2(u,v)(t))''\\ &=- b(t)f_2\Big(t,(A_{m-1}u)(t),\dots,(A_1u)(t),u(t), (B_{n-1}v)(t),\dots,(B_1v)(t),v(t)\Big)\\ & \leq 0. \end{align*} So $T_1(u,v)(t)$ and $T_2(u,v)(t)$ are concave on [0,1]. If $(u, v) \in K$, then from the properties of $k_1(t, s)$ and $g_1(t, s)$, we have \begin{align*} \|T_1(u,v)(t)\| & \leq \int_0^1 k_1(s,s) a(s)f_1\Big(s,(A_{m-1}u)(s),\dots,(A_1u)(s),\\ &\quad u(s), (B_{n-1}v)(s),\dots,(B_1v)(s),v(s)\Big)ds, \end{align*} and \begin{align*} \|T_2(u,v)(t)\| & \leq \int_0^1 g_1(s,s)b(s)f_2\Big(s,(A_{m-1}u)(s),\dots,(A_1u)(s),\\ &\quad u(s), (B_{n-1}v)(s),\dots,(B_1v)(s),v(s)\Big)ds. \end{align*} On the other hand, \begin{align*} \min_{\theta\leq t\leq 1-\theta}T_1(u,v)(t) & \geq \gamma \int_0^1 k_1(s,s)a(s) f_1\Big(s,(A_{m-1}u)(s),\dots,(A_1u)(s),u(s),\\ &\quad (B_{n-1}v)(s),\dots,(B_1v)(s),v(s)\Big)ds\\ & \geq \gamma\|T_1(u,v)(t)\|, \end{align*} and \begin{align*} \min_{\theta\leq t\leq 1-\theta}T_2(u,v)(t) & \geq \gamma \int_0^1 g_1(s,s)b(s) f_2\Big(s,(A_{m-1}u)(s), \dots,(A_1u)(s),u(s),\\ &\quad (B_{n-1}v)(s),\dots,(B_1v)(s),v(s)\Big)ds\\ & \geq \gamma\|T_2(u,v)(t)\|. \end{align*} Combining the preceding inequalities, we arrive at \begin{align*} \min_{\theta\leq t\leq 1-\theta} (T_1(u,v)(t)+T_2(u,v)(t)) & \geq \min_{\theta\leq t\leq 1-\theta}(T_1(u,v)(t)) +\min_{\theta\leq t\leq 1-\theta}(T_2(u,v)(t))\\ & \geq \gamma (\|T_1(u,v)(t)\|+\|T_2(u,v)(t)\|). \end{align*} This completes the proof. \end{proof} In this paper, we use the following assumptions: \begin{itemize} \item[((H1)] $a(t)$ and $b(t)$ do not vanish identically on any subinterval of $(0, 1)$, and there exists $t_0 \in (0, 1)$ such that $a(t_0) > 0$, $b(t_0) > 0$ and $0 < \int_0^1 k_1(s, s)a(s)ds < +\infty$, $0 < \int_0^1 g_1(s, s)b(s)ds < +\infty$. \end{itemize} Finally, we define the nonnegative continuous concave functional $\alpha(u,v):=\min_{\theta\leq t\leq 1-\theta}(u(t)+v(t)).$ We observe here that, for each $(u, v) \in K$, $\alpha(u, v) \leq \|(u, v)\|$. Let \begin{gather*} \widetilde{\xi}_1:=\min_{\theta\leq t\leq 1-\theta} \int_0^1 k_1(t,s)a(s)ds,\quad \widetilde{\eta}_1:=\max_{0\leq t\leq 1}\int_0^1 k_1(t,s)a(s)ds,\\ \widetilde{\mu}_1:=\min_{\theta\leq t\leq 1-\theta}\int_0^1 g_1(t,s)b(s)ds, \quad \widetilde{\nu}_1:=\max_{0\leq t\leq 1}\int_0^1 g_1(t,s)b(s)ds\,. \end{gather*} \begin{theorem}\label{thm1} Let $1\le i\le m-1$, and $1\le j\le n-1$.Assume there exist nonnegative numbers $a, b, c$ such that $0 < a < b \leq \min\{\gamma, \widetilde{\xi}_{1}/\widetilde{\eta}_{1}, \xi_i/\eta_i, \widetilde{\mu}_1/\widetilde{\nu}_1,\mu_j/\nu_j\}c$, and $f (t, x_1,\dots,x_{m-1},x_m,y_{1}, \dots,y_{n-1},y_n)$, $g(t, x_1,\dots,x_{m-1},x_m,y_{1}, \dots,y_{n-1},y_n)$ satisfy the following growth conditions: \begin{itemize} \item[(H2)] $f_1 (t, x_1,\dots,x_{m-1},x_m,y_{1}, \dots,y_{n-1},y_n)\leq c/2\widetilde{\eta}_1$,\\ $f_2 (t, x_1,\dots,x_{m-1},x_m,y_{1}, \dots,y_{n-1},y_n)\leq c/2\widetilde{\nu}_1$, for all $t\in [0,1]$, $x_i+y_j\in [0,\eta c]$, $x_m+y_n \in[0,c]$. \item[(H3)] $f_1 (t, x_1,\dots,x_{m-1},x_m,y_{1}, \dots,y_{n-1},y_n)< a/2\widetilde{\eta}_1$,\\ $f_2 (t, x_1,\dots,x_{m-1},x_m,y_{1}, \dots,y_{n-1},y_n)< a/2\widetilde{\nu}_1$, for all $t\in [0,1]$, $x_i+y_j\in [0,\eta a]$, $x_m+y_n \in[0,a]$. \item[(H4)] $f_1 (t, x_1,\dots,x_{m-1},x_m,y_{1}, \dots,y_{n-1},y_n)\geq b/2\widetilde{\xi}_1$,\\ $f_2 (t, x_1,\dots,x_{m-1},x_m,y_{1}, \dots,y_{n-1},y_n)\geq b/2\widetilde{\mu}_1$, for all $t\in [\theta,1-\theta], x_i+y_j\in [\xi b,\eta b/\gamma]$, $x_m+y_n \in[b,b/\gamma]$. \end{itemize} Then \eqref{eq1} has at least three positive fixed points $(u_1, v_1)$, $(u_2, v_2)$, $(u_3, v_3)$ such that $\|(u_1, v_1)\| < a$, $b < \min_{\theta\leq t\leq 1-\theta} (u_2(t) + v_2(t))$, and $\|(u_3, v_3)\|> a$, with $\min_{\theta\leq t\leq 1-\theta} (u_3(t) + v_3(t)) < b$. \end{theorem} \begin{proof} We note first that $T : \overline{P_c} \to \overline{P_c}$ is a completely continuous operator. If $(u, v)\in \overline{P_c}$, then $\|(u, v)\| \leq c$. For $1 \leq i \leq m-1$ and $1 \leq j \leq n-1$, $(A_iu)(t)+(B_jv)(t)=\int_0^1 k_i(t,s)u(s)ds +\int_0^1 g_j(t,s)v(s)ds\leq \eta_i\|u\|+\nu_j\|v\|\leq \eta c.$ From (H2), we have \begin{align*} \|T(u, v)(t)\| &=\max_{0\leq t\leq 1} |T_1(u, v)(t)| +\max _{0\leq t\leq 1} |T_2(u, v)(t)|\\ & =\max _{0\leq t\leq 1} \int_0^1 k_1(t,s)a(s)f_1\Big(s,(A_{m-1}u)(s),\dots,(A_1u)(s),\\ &\quad u(s), (B_{n-1}v)(s),\dots,(B_1v)(s),v(s)\Big)ds\\ &\quad + \max _{0\leq t\leq 1}\int_0^1 g_1(t,s)b(s)f_2\Big(s,(A_{m-1}u)(s),\dots,(A_1u)(s),\\ &\quad u(s), (B_{n-1}v)(s),\dots,(B_1v)(s),v(s)\Big)ds\\ & \leq c/2\widetilde{\eta}_1 \max _{0\leq t\leq 1} \int_0^1k_1(t,s)a(s)ds+c/2\widetilde{\nu}_1 \max _{0\leq t\leq 1} \int_0^1g_1(t,s)b(s)ds\leq c. \end{align*} Therefore, $T : \overline{P_c} \to \overline{P_c}$. In an analogous argument, one can verify that (H3) implies condition (C2) of Lemma \ref{lem1}. Clearly, $\{(u,v)\in P(\alpha,b,b/\gamma):\alpha(u,v)>b\}\neq \emptyset$. If $(u, v) \in P(\alpha, b, b/\gamma )$, then $b \leq u(t) + v(t) \leq b/\gamma$, $t \in [\theta, 1 - \theta]$. For $1 \leq i \leq m-1$ and $1 \leq j \leq n-1$, we have \begin{gather*} (A_iu)(t)+(B_jv)(t)=\int_0^1 k_i(t,s)u(s)ds +\int_0^1 g_j(t,s)v(s)ds\geq \xi b,\\ (A_iu)(t)+(B_jv)(t)=\int_0^1 k_i(t,s)u(s)ds +\int_0^1 g_j(t,s)v(s)ds\leq \eta b/\gamma. \end{gather*} By (H4), \begin{align*} \alpha(T(u,v)(t)) &=\min_{\theta\leq t\leq 1-\theta}(T_1(u,v)(t)+T_2(u,v)(t))\\ & \geq\min_{\theta\leq t\leq 1-\theta}T_1(u,v)(t) +\min_{\theta\leq t\leq 1-\theta}T_2(u,v)(t)\\ & \geq \min_{\theta\leq t\leq 1-\theta} \int_0^1 k_1(t,s)a(s) f_1\Big(s,(A_{m-1}u)(s),\dots,(A_1u)(s),\\ &\quad u(s), (B_{n-1}v)(s),\dots,(A_1v)(s),v(s)\Big)ds\\ &\quad + \min_{\theta\leq t\leq 1-\theta} \int_0^1 g_1(t,s) b(s)f_2\Big(s,(A_{m-1}u)(s),\dots,(A_1u)(s),\\ &\quad u(s), (B_{n-1}v)(s),\dots,(B_1v)(s),v(s)\Big)ds\\ & \geq b/2\widetilde{\xi}_1\min_{\theta\leq t\leq 1-\theta} \int_0^1 k_1(t,s)a(s)ds+b/2\widetilde{\mu}_1 \min_{\theta\leq t\leq 1-\theta} \int_0^1 g_1(t,s)b(s)ds\\ &=b. \end{align*} Therefore, condition (C1) of Lemma \ref{lem1} is satisfied. Finally, we show that (C3) is also satisfied. If $(u,v)\in P(\alpha,b,c)\text{ and }\|T(u,v)\|> b/\gamma$, then $\alpha(T(u, v)(t)) = \min_{\theta\leq t\leq 1-\theta} (T_1(u, v)(t) + T_2(u, v)(t)) \geq \gamma \|T(u, v)\| > b.$ Therefore, (C3) of Lemma \ref{lem1} is also satisfied. By Lemma \ref{lem1}, the system \eqref{eq1} has at least three positive fixed points $(u_1, v_1), (u_2, v_2), (u_3, v_3)$ such that $\|(u_1,v_1)\| < a$, $b < \min_{\theta\leq t\leq 1-\theta} (u_2(t) + v_2(t))$, and $\|(u_3, v_3)\|> a$, with $\min_{\theta\leq t\leq 1-\theta} (u_3(t) + v_3(t)) < b$. \end{proof} \subsection*{An example} Consider a system of nonlinear second-order and fourth-order ordinary differential equations (with $m=1, n=2$): $$\label{system} \begin{gathered} -x''=a(t)f_1(t,x,y,-y'')=0,\\ y^{(4)}=b(t)f_2(t,x,y,-y'')=0,\\ x(0)-x'(0)=x(1)+ x'(1)=0,\\ y(0)-y'(0)=y(1)+y'(1)=0,\\ y''(0)-y'''(0)=y''(1)+y'''(1)=0. \end{gathered}$$ Let $u:=x$ and $v:=-y''$, then the problem \eqref{system} is equivalent to the following system of nonlinear integral equations $$\label{equivalentsystem} \begin{gathered} u(t)=\int_0^1 k_1(t,s)a(s)f_1(s,u(s),\int_0^1 g_1(s,\tau)v(\tau)d\tau,v(s)),\\ v(t)=\int_0^1 g_1(t,s)b(s)f_2(s,u(s),\int_0^1 g_1(s,\tau)v(\tau)d\tau,v(s)), \end{gathered}$$ where $k_1(t,s)=g_1(t,s):=\frac{1}{3}\begin{cases} (1+s)(2-t), 0\le s\le t\le 1,\\ (1+t)(2-s), 0\le t\le s\le 1. \end{cases}$ We choose $a(t):=(1-t)^{-1/2}$, $b(t):=t^{-1/2}$, $\theta:=1/4$, $f_1(t,x_1,x_2,x_3)$ equals $\begin{cases} \frac{t}{100}+\frac{1}{10}(x_1+x_3)^2, & t\in [0,1],\; 0\leq x_1+x_3\leq 2,\; x_2\ge 0,\\ \frac{t}{100}+6[(x_1+x_3)^2-2(x_1+x_3)]+\frac{2}{5}, & t\in [0,1],\; 2< x_1+x_3< 4,\; x_2\ge 0,\\ \frac{t}{100}+20\log_2(x_1+x_3)+2(x_1+x_3)+\frac{2}{5}, & t\in [0,1],\; 4\leq x_1+x_3\leq 16,\; x_2\ge 0,\\ \frac{t}{100}+\frac{1}{4}(x_1+x_3)+\frac{542}{5}, & t\in [0,1],\; x_1+x_3>16,x_2\ge 0. \end{cases}$ and $f_2(t,x_1,x_2,x_3)$ equals $\begin{cases} \frac{t}{100}+\frac{1}{10}(x_1+x_3)^2, & t\in [0,1],\; 0\leq x_1+x_3\leq 2,\; x_2\ge 0,\\ \frac{t}{100}+\frac{13}{8}[(x_1+x_3)^2-2(x_1+x_3)]+\frac{2}{5}, & t\in [0,1],\; 2< x_1+x_3< 4,\; x_2\ge 0,\\ \frac{t}{100}+\frac{5}{2}\log_2(x_1+x_3)+2(x_1+x_3)+\frac{2}{5}, & t\in [0,1],\; 4\leq x_1+x_3\leq 16,\; x_2\ge 0,\\ \frac{t}{100}+\frac{1}{4}(x_1+x_3)+\frac{192}{5}, & t\in [0,1],\; x_1+x_3>16,x_2\ge 0. \end{cases}$ Then by direct calculation, we obtain $\xi\approx 0.4120,\quad \eta=0.5,\quad \widetilde{\xi}_1 \approx 0.0417,\quad \widetilde{\mu}_1 \approx 0.9273,\quad \widetilde{\eta}_1\approx 0.8889,\quad \widetilde{\nu}_1\approx 1.1111 .$ It is easy to check that (H1) holds. Choose $\gamma=\frac{1}{4}$, $a=1$, $b=4$, $c=800$. Also, it is easy to verify that $f_1$ and $f_2$ satisfy conditions (H2)-(H4). So the system \eqref{system} has at least three positive solutions $(u_1, v_1), (u_2, v_2), (u_3, v_3)$ such that $\|(u_1, v_1)\| < 1$, $4 < \min_{ 1/ 4\leq t\leq 3 /4 }(u_2(t) + v_2(t))$, and $\|(u_3, v_3)\|> 1$, with $\min_{1/ 4\leq t\leq 3 /4 } (u_3(t) + v_3(t)) < 4$. \begin{thebibliography}{00} \bibitem{Anderson} N. Anderson, A. M. Arthurs; \emph{Analytic bounding functions for diffusion problems with Michaelis-Menten kinetics}, Bull. Math. Biol. 47 (1985) 145-153. \bibitem{Kang} P. Kang, Z. Wei; \emph{Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations}, Nonlinear Analysis 70 (2009) 444-451. \bibitem{Leggett} R. W. Leggett, L. R. Williams; \emph{Multiple positive fixed points of nonlinear operators on ordered Banach spaces}, Indiana Univ. Math. J. 28 (1979) 673-688. \bibitem{Lin} S. H. Lin; \emph{Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics}, J. Theor. Biol. 60 (1976) 449-457. \bibitem{Li} Y. Li, Y. Guo, G. Li; \emph{Existence of positive solutions for systems of nonlinear third-order differential equations}, Commun Nonlinear Sci Numer Simulat 14 (2009) 3792-3797. \bibitem{Liu_1} L. Liu, P. Kang, Y. Wu; \emph{Benchawan Wiwatanapataphee, Positive solutions of singular boundary value problems for systems of nonlinear fourth order differential equations}, Nonlinear Analysis 68 (2008) 485-498. \bibitem{Liu} L. Liu, B. Liu, Y. Wu; \emph{Positive solutions of singular boundary value problems for nonlinear differential systems}, Applied Mathematics and Computation 186 (2007) 1163-1172. \bibitem{Yang} Z. Yang; \emph{Existence and uniqueness of positive solutions for a higher order boundary value problem}, Computers and Mathematics with Applications 54 (2007) 220-228. \end{thebibliography} \end{document}