\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 144, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{7mm}} \begin{document} \title[\hfilneg EJDE-2011/144\hfil Existence of three positive solutions] {Existence of three positive solutions for a system of nonlinear third-order ordinary differential equations} \author[N. Nyamoradi\hfil EJDE-2011/144\hfilneg] {Nemat Nyamoradi} \address{Nemat Nyamoradi \newline Department of Mathematics, Faculty of Sciences\\ Razi University, 67149 Kermanshah, Iran} \email{nyamoradi@razi.ac.ir} \thanks{Submitted September 11, 2010. Published November 1, 2011.} \thanks{Supported the Razi University} \subjclass[2000]{34L30, 34B18, 34B27} \keywords{Positive solution; boundary value problem; fixed point theorem} \begin{abstract} In this work, we use the Leggett-Williams fixed point theorem, we prove the existence of at least three positive solutions of a boundary-value problem for system of third-order ordinary differential equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Third-order differential equations arise in a variety of different areas of applied mathematics and physics. In recent years, boundary-value problems (BVPs for short) have included, as special cases, multi-point BVPs considered by many authors (see \cite{Anderson1,Anderson2,Guo2,Sun,Yao1,Yu} and references therein). Naturally, further study in this specific field is on BVPs for systems of ordinary differential equations. However, to our knowledge, various results for systems of second and third order differential equations have been established (see \cite{Hu,Li,Wei} and references therein). Guo et al. \cite{Guo2} obtained some existence results for positive solutions for the BVP \begin{gather*} u'''(t)+a(t)f(u(t))=0 \quad 0 0$, or $u(t) > 0$, $v(t) \geq 0$, for all $t \in (0,1)$. For the convenience of the reader, we present here the Leggett-Williams fixed point theorem. Given a cone $K$ in a real Banach space $E$, a map $\alpha$ is said to be a nonnegative continuous concave (resp. convex) functional on $K$ provided that $\alpha : K \to [0,+ \infty)$ is continuous and \begin{gather*} \alpha (tx + (1-t)y) \geq t \alpha (x) + (1-t) \alpha (y), \\ \text{(resp. } \alpha (tx + (1-t)y) \leq t \alpha (x) + (1-t) \alpha (y)), \end{gather*} for all $x,y \in K$ and $t \in [0,1]$. 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 \[ P_r = \{ x \in K | \|x\| < r\}, \quad P(\alpha, a, b) = \{ x \in K |a \leq \alpha (x), \|x\| \leq b\}. \] \begin{theorem}[Leggett-Williams fixed point theorem] \label{thm1} Let $A : \overline{P_c} \to \overline{P_c}$ be a completely continuous operator and let $\alpha$ be a nonnegative continuous concave functional on $K$ such that $\alpha (x) \leq \|x\|$ for all $x \in \overline{P_c}$. Suppose there exist $0 < a < b < d \leq c$ such that \begin{itemize} \item[(A1)] $ \{x \in P(\alpha, b, d) :\alpha(x) > b\} \neq \emptyset $, and $\alpha (Ax) > b $ for $x \in P(\alpha, b, d)$; \item[(A2)] $\| Ax \| < a $ for $\| x \| \leq a$; and \item[(A3)] $\alpha (Ax) > b $ for $x \in P(\alpha, b, c)$ with $\| Ax \| > d$. \end{itemize} Then $A$ has at least three fixed points $x_1$, $x_2$ and $x_3$ and such that $\|x_1\| < a$, $b < \alpha (x_2)$ and $\|x_3\| > a$, with $\alpha (x_3) < b$. \end{theorem} Inspired and motivated by the works mentioned above, in this work we consider the existence of positive solutions to \eqref{e1}. We shall first give a new form of the solution, and then determine the properties of the Green's function for associated linear boundary-value problems; finally, by employing the Leggett-Williams fixed point theorem, some sufficient conditions guaranteeing the existence of a positive solution. The rest of the article is organized as follows: in Section 2, we present some preliminaries that will be used in Section 3. The main results and proofs will be given in Section 3. Finally, in Section 4, we shall give an example to illustrate our main result. \section{Preliminaries} In this section, we present some notations and preliminary lemmas that will be used in the proof of the main result. Obviously, $(u(t),v(t)) \in C^3([0,1], (0,+ \infty)) \times C^3([0,1], (0,+ \infty))$ is a solution of \eqref{e1} if and only if $(u(t),v(t))$ is a solution of the system of integral equations \begin{gather} u(t)= \int_0^1 G(t,s)a(s)f(s, u(s),v(s))ds , \label{e2}\\ v(t)= \int_0^1 G(t,s)b(s)h(s,u(s), v(s))ds , \label{e3} \end{gather} where $G(t,s)$ is the Green's function \cite{Guo} defined as follows: \[ G(t,s)= \frac{1}{2(1- \beta \eta)}\begin{cases} (2ts-s^2)(1- \beta \eta) + t^2s(\beta -1), & s \leq \min \{\eta,t \} \\ t^2(1- \beta \eta) + t^2s(\beta -1), & t \leq s \leq \eta\\ (2ts-s^2)(1- \beta \eta) + t^2(\beta \eta - s), & \eta \leq s \leq t, \\ t^2 (1-s), & \max \{\eta,t \} \leq s\,. \end{cases} \] We need some properties of function $G(t,s)$ in order to discuss the existence of positive solutions. For convenience, we define \begin{eqnarray} g(s)= \frac{1 + \alpha}{1 - \beta \eta} s(1- s) \quad s \in [0,1]. \end{eqnarray} For the Green's function $G(t,s)$, we have the following two lemmas \cite{Guo}. \begin{lemma} \label{lem1} Let $0 < \eta < 1$ and $1 < \beta < 1/\eta$. Then for any $(t,s) \in [0,1]\times [0,1] $, we have $0\leq G(t,s)\leq g(s)$. \end{lemma} \begin{lemma} \label{lem2} Let $0 < \eta < 1$ and $1 < \beta < 1/\eta$. Then for any $(t,s) \in [\eta/\beta,\eta] \times [0,1] $, we have $$ \lambda g(s)\leq G(t,s), $$ where $0 < \lambda = \eta^2 \min \{\beta -1, 1\} /(2\beta^2 (1 + \beta))< 1$. \end{lemma} In this article, we assume that the following conditions are satisfied \begin{equation} \label{e5} 0 < \int_0^1 g(s) a(s) ds < + \infty, \quad 0 < \int_0^1 g(s) b(s) ds < + \infty. \end{equation} Also, we use the following notation \begin{gather*} M_1 = \max_{0 \leq t \leq 1} \int_0^1 G(t,s) a(s) ds , \quad M_2 = \max_{0 \leq t \leq 1} \int_0^1 G(t,s) b(s) ds,\\ m_1 = \min_{\eta/\beta \leq t \leq \eta} \int_{\eta/\beta }^\eta G(t,s) a(s) ds , \quad m_2 = \min_{\eta/\beta \leq t \leq \eta} \int_{\eta/\beta }^\eta G(t,s) b(s)ds. \end{gather*} Clearly, we see that $0 < m_i < M_i$; for $i= 1,2$. Let $E = C([0,1], \mathbb{R}) \times C([0,1], \mathbb{R})$ endowed with the norm $\|(u,v)\|:= \|u\| + \|v\|$, where $\|u\| = \max_{0 \leq t \leq 1} |u(t)|$, $\|v\| = \max_{0 \leq t \leq 1} |v(t)|$, and define \[ %\label{e6} K=\{(u,v) \in E; u(t) \geq 0, v(t) \geq 0, t \in [0,1], \min_{\eta/\beta \leq t \leq \eta}( u(t) + v(t)) \geq \gamma \|(u,v)\|\}. \] It is obvious that $E$ is a Banach space and $K$ is a cone in $E$. Define operator $T : E \to E$ as \begin{equation} \label{e7} T (u , v)(t) = (A(u , v)(t), B(u , v)(t)), \quad \forall t \in (0,1), \end{equation} where \begin{gather} A(u , v)(t)=\int_0^1 G(t,s)a(s)f(s, u(s),v(s))ds, \label{e8}\\ B(u , v)(t)= \int_0^1 G(t,s)b(s)f(s, u(s),v(s))ds. \label{e9} \end{gather} \begin{lemma} \label{lem3} For $T$ and $K$ as above, $T (K) \subset K$. \end{lemma} \begin{proof} For any $(u , v) \in K$, from properties of $G(t,s)$, $A(u , v)(t) \geq 0, B(u , v)(t) \geq 0, t \in [0,1]$, and it follows from \eqref{e8}, \eqref{e9} and Lemma \ref{lem1} that \begin{equation} \label{e10} \begin{gathered} \|A(u , v)\|\leq \int_0^1 g(s)a(s)f(s, u(s),v(s))ds,\\ \|B(u , v)\|\leq \int_0^1 g(s)b(s)f(s, u(s),v(s))ds, \end{gathered} \end{equation} Thus, for any $(u , v) \in K$, by Lemma \ref{lem2} and the above inequality, \begin{align*} \min_{\eta/\beta \leq t \leq \eta}A(u , v)(t) &= \min_{\eta/\beta \leq t \leq \eta} \int_0^1 G(t,s)a(s)f(s, u(s),v(s))ds,\\ &\geq \lambda \int_0^1 g(s)a(s)f(s, u(s),v(s))ds \\ &\geq \lambda \|A(u , v)\|. \end{align*} In the same way, for any $(u , v) \in K$, we have \[ \min_{\eta/\beta \leq t \leq \eta}B(u , v)(t) \geq \lambda \|B(u,v)\|. \] Therefore, \begin{align*} &\min_{\eta/\beta \leq t \leq \eta}(A(u , v)(t) + B(u , v)(t) \geq \lambda \|A(u , v)\| + \lambda \|B(u,v)\|) \\ &= \lambda \|(A(u , v) , B(u,v))\|. \end{align*} From the above, we conclude that $T(u,v) = (A(u , v) , B(u , v)) \in K$, that is, $T (K) \subset K$. The proof is complete. \end{proof} It is clear that the existence of a positive solution for \eqref{e1} is equivalent to the existence of a nontrivial fixed point of $T$ in $K$. \section{Main results} In this section, we discuss the existence of a positive solution \eqref{e1}. We define the nonnegative continuous concave functional on $K$ by \[ \alpha(u,v)= \min_{\eta/\beta \leq t \leq \eta}(u(t) ,v(t)). \] It is obvious that, for each $(u , v) \in K, \alpha(u,v) \leq \|(u,v)\|$. In this section, we assume that $p_i, i= 1, 2$, are two positive numbers satisfying $\frac{1}{p_1} + \frac{1}{p_2} \leq 1$. Also, we use the following assumptions: There exist nonnegative numbers $a, b, c$ such that $0 < a < b \leq \min \{\lambda, \frac{m_1}{p_1 M_1}, \frac{m_2}{p_2 M_2}\}c$, and $f(t, u, v)$, $h(t, u, v)$ satisfy the following conditions: \begin{itemize} \item[(H1)] $f(t, u, v) < \frac{1}{p_1} \cdot \frac{c}{M_1}$, $h(t, u, v) < \frac{1}{p_2} \cdot \frac{c}{M_2}$, for all $t \in [0,1]$, $u+ v\in [0,c]$; \item[(H2)] $f(t, u, v) < \frac{1}{p_1} \cdot \frac{a}{M_1}$, $h(t, u, v) < \frac{1}{p_2} \cdot \frac{a}{M_2}$, for all $t \in [0,1]$, $u+ v\in [0,a]$; \item[(H3)] $f(t, u, v) > b/m_1$ or $h(t, u, v) > b/m_2 $, for all $t \in [0,1]$, $u + v \in [b, b/\lambda]$. \end{itemize} \begin{theorem} \label{thm2} Assume \eqref{e5} and {\rm (H1)--(H3)}. Then \eqref{e1} has at least three positive solutions $(u_1,v_1), (u_2, v_2), (u_3, v_3)$ such that $\|(u_1, v_1)\| < a$, $b < \min_{[\eta/\beta,\eta]}(u_2(t) + v_2(t))$, and $\|(u_3,v_3)\| > a$, with $\min_{\eta/\beta \leq t \leq \eta}(u_3(t) + v_3(t)) < b$. \end{theorem} \begin{proof} First, we show that $T : \overline{P_c} \to \overline{P_c}$ is a completely continuous operator. If $(u,v) \in \overline{P_c}$, by condition (H1), we have \begin{align*} \|T(u,y)\|&= \max_{0 \leq t \leq 1} |A(u,v)(t)| + \max_{0 \leq t \leq 1} |B(u,v)(t)|\\ &= \max_{0 \leq t \leq 1}\int_0^1 G(t,s)a(s)f(s,u(s),v(s))ds \\ &\quad +\max_{0 \leq t \leq 1}\int_0^1 G(t,s)b(s)h(s,u(s),v(s))ds\\ &\leq \frac{1}{p_1} \cdot \frac{c}{M_1} \max_{0 \leq t \leq 1}\int_0^1 G(t,s)a(s)ds + \frac{1}{p_2} \cdot \frac{c}{M_2} \max_{0 \leq t \leq 1}\int_0^1 G(t,s)b(s)ds \\ &\leq \frac{1}{p_1} \cdot \frac{c}{M_1} \cdot M_1 + \frac{1}{p_2} \cdot \frac{c}{M_2} \cdot M_2 \leq c. \end{align*} Therefore, $\|T(u,y)\| \leq c$, that is, $T : \overline{P_c} \to \overline{P_c}$. The operator $T$ is completely continuous by an application of the Ascoli-Arzela theorem. In the same way, the condition (H2) implies that the condition (A2) of Theorem \ref{thm1} is satisfied. We now show that condition (A1) of Theorem \ref{thm1} is satisfied. Clearly, $\{(u,v) \in P(\alpha, b,b/\lambda) | \alpha(u,v) > b\} \neq \emptyset$. If $(u,v) \in P(\alpha, b, b/\lambda)$, then $b \leq u(s) + v(s) \leq \frac{b}{\lambda}, s \in [\eta/\beta,\eta]$. By condition (H3), we obtain \begin{align*} \alpha (T(u,v)(t)) &= \min_{\eta/\beta \leq t \leq \eta} (A(u,v)(t) + B(u,v)(t))\\ &\geq \min_{\eta/\beta \leq t \leq \eta} \int_{\eta/\beta }^\eta G(t,s)a(s)f(s,u(s),v(s))ds\\ &\quad +\min_{\eta/\beta \leq t \leq \eta} \int_{\eta/\beta }^\eta G(t,s)b(s)h(s,u(s),v(s))ds \\ &\geq \frac{b}{m_1} \min_{\eta/\beta \leq t \leq \eta} \int_{\eta/\beta }^\eta G(t,s)a(s)ds = \frac{b}{m_1} \cdot m_1 = b. \end{align*} Therefore, condition (A3) of Theorem \ref{thm1} is satisfied. Finally, we show that the condition (A3) of Theorem \ref{thm1} is satisfied. If $(u,v) \in P(\alpha, b,c)$, and $\|T(u,v)\| > \frac{b}{\lambda}$, then \[ \alpha (T(u,v)(t))= \min_{\eta/\beta \leq t \leq \eta} (A(u,v)(t) + B(u,v)(t)) \geq \lambda \|T(u,v)\| > b. \] Therefore, the condition (A3) of Theorem \ref{thm1} is also satisfied. By Theorem \ref{thm1}, there exist three positive solutions $(u_1, v_1), (u_2, v_2), (u_3, v_3)$ such that $\|(u_1, v_1)\| < a, b < \min_{\eta/\beta \leq t \leq \eta}(u_2(t) + v_2(t))$, and $\|(u_3, v_3)\| > a$, with $\min_{\eta/\beta \leq t \leq \eta}(u_3(t) + v_3(t)) < b$. \end{proof} \section{Application} Consider the system of nonlinear third-order ordinary differential equations \begin{equation} \label{e11} \begin{gathered} u'''(t) + f(t,u(t),v(t))=0 0< t < 1,\\ v'''(t) + h(t,u(t), v(t)), 0< t < 1,\\ u(0) = u'(0)=0, \quad u'(1)= \frac{3}{2} u'(\frac{1}{2}),\\ v(0) = v'(0)=0,\quad v'(1)= \frac{3}{2} v'(\frac{1}{2}), \end{gathered} \end{equation} where \begin{align*} &f(t, u, v) = h(t, u, v) \\ &= \begin{cases} \frac{t}{100} + \frac{1}{200} (u + v)^2, & t \in [0, 1],\; 0 \leq u + v \leq 1,\\ \frac{t}{100} + 240 [(u + v)^2 - (u + v)] + \frac{1}{200}, & t \in [0, 1], \; 1 < u + v < 2,\\ \frac{t}{100} + 30 [10 \log_2{(u + v)} + 3(u + v)] + \frac{1}{200}, & t \in [0, 1], \; 2 \leq u + v \leq 4\\ \frac{t}{100} + \frac{\sqrt{u + v}}{2} + \frac{191801}{200}, & t \in [0, 1], \; 4 < u + v < + \infty. \end{cases} \end{align*} It is easy to check that $g(s) = 10 s (1 - s)$, for all $s \in [0,1]$, $0 < \int_0^1 g(s) a(s) ds < + \infty, 0 < \int_0^1 g(s) b(s) ds < + \infty$ hold. Choose $p_1 = p_2 = 2$. Then by direct calculations, we can obtain that $M_1 = M_2 = 5/3$, $m_1 = m_2 = 13/2916$. So we choose $a = 1$, $b = 2$, $c = 3500$. It is easy to check that $f, h$ satisfy the conditions (H1)--(H3). So system \eqref{e11} has at least three positive solutions $(u_1, v_1), (u_2, v_2), (u_3,v_3)$ such that $\|(u_1, v_1)\| < 1, 2 < \min_{\frac{1}{3} \leq t \leq \frac{1}{2}}(u_2(t) + v_2(t))$, and $\|(u_3, v_3)\| > 1$, with $\min_{\frac{1}{3} \leq t \leq \frac{1}{2}}(u_3(t) + v_3(t)) <2$. \subsection*{Acknowledgments} The author would like to thank the anonymous referee for his/her valuable suggestions and comments. \begin{thebibliography}{99} \bibitem{Anderson1} D.\ R.\ Anderson; \emph{Green's function for a third- order generalized right focal problem}, Math. Anal. Appl. 288 (2003) 1-14. \bibitem{Anderson2} D.\ R.\ Anderson, J.\ M.\ Davis; \emph{Multiple solutions and eigenvalues for third-order right focal boundary-value problems}, J. Math. Anal. 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