\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 84, pp. 1--6.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2010/84\hfil Uniqueness and parameter dependence] {Uniqueness and parameter dependence of solutions of fourth-order four-point nonhomogeneous BVPs} \author[J.-P. Sun, X.-Y. Wang\hfil EJDE-2010/84\hfilneg] {Jian-Ping Sun, Xiao-Yun Wang} % in alphabetical order \address{Jian-Ping Sun \newline Department of Applied Mathematics, Lanzhou University of Technology\\ Lanzhou, Gansu 730050, China} \email{jpsun@lut.cn} \address{Xiao-Yun Wang \newline Department of Applied Mathematics, Lanzhou University of Technology\\ Lanzhou, Gansu 730050, China} \email{catherine699@163.com} \thanks{Submitted September 21, 2009. Published June 18, 2010.} \thanks{Supported by grant 10801068 from the National Natural Science Foundation of China} \subjclass[2000]{34B08, 34B10} \keywords{Nonhomogeneous; fourth-order; four-point; Sturm-Liouville; \hfill\break\indent boundary-value problem; positive solution; uniqueness; dependence on parameter} \begin{abstract} In this article, we investigate the fourth-order four-point nonhomogeneous Sturm-Liouville boundary-value problem \begin{gather*} u^{(4)}(t)=f(t,u(t)),\quad t\in [0,1], \\ \alpha u(0)-\beta u'(0)=\gamma u(1)+\delta u'(1)=0, \\ au''(\xi _1)-bu'''(\xi _1)=-\lambda ,\quad cu''(\xi _2)+du'''(\xi _2)=-\mu , \end{gather*} where $0\leq \xi _1<\xi _2\leq 1$ and $\lambda$ and $\mu $ are nonnegative parameters. We obtain sufficient conditions for the existence and uniqueness of positive solutions. The dependence of the solution on the parameters $\lambda$ and $\mu$ is also studied. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Recently, nonhomogeneous boundary-value problems (BVPs for short) have received much attention from many authors. For example, Ma \cite{m1,m2} and Kong and Kong \cite{k1,k2,k3} studied some second-order multi-point nonhomogeneous BVPs. In particular, Kong and Kong \cite{k3} considered the following second-order BVP with nonhomogeneous multi-point boundary condition \begin{gather*} u''+a(t)f(u)=0,\quad t\in (0,1), \\ u(0)=\sum_{i=1}^ma_iu(t_i) +\lambda ,\quad u(1)=\sum_{i=1}^mb_iu( t_i)+\mu , \end{gather*} where $\lambda$ and $\mu $ are nonnegative parameters. They derived some conditions for the above BVP to have a unique solution and then studied the dependence of this solution on the parameters $\lambda $ and $\mu $. Sun \cite{s1} discussed the existence and nonexistence of positive solutions to a class of third-order three-point nonhomogeneous BVP. However, to the best of our knowledge, fewer results on fourth-order nonhomogeneous BVPs can be found in the literature. It is worth mentioning that the authors in \cite{o1} studied the multiplicity of positive solutions for some fourth-order two-point nonhomogeneous BVP by using a fixed point theorem of cone expansion/compression type. Being directly inspired by \cite{k3}, in this paper we are concerned with the nonhomogeneous Sturm-Liouville BVP consisting of the fourth-order differential equation \begin{equation} u^{(4)}(t)=f(t,u(t)),\quad t\in [0,1] \label{1.1} \end{equation} and the four-point boundary conditions \begin{gather} \alpha u(0)-\beta u'(0)=\gamma u(1)+\delta u'(1)=0, \label{1.2}\\ au''(\xi _1)-bu'''(\xi _1)=-\lambda ,\quad cu''(\xi _2)+du'''(\xi _2)=-\mu , \label{1.3} \end{gather} where $0\leq \xi _1<\xi _2\leq 1$ and $\lambda$ and $\mu $ are nonnegative parameters. We will use the following assumptions: \begin{itemize} \item[(A1)] $\alpha ,\beta ,\gamma ,\delta ,a,b,c$ and $d$ are nonnegative constants with $ \beta >0$, $\delta >0$, $\rho _{_1}:=\alpha \gamma +\alpha \delta +\gamma \beta >0$, $\rho _{_2}:=ad+bc+ac(\xi _2-\xi _1)>0$, $-a\xi _1+b>0$ and $c(\xi _2-1)+d>0$; \item[(A2)] $f(t,u): [0,1]\times [0,+\infty )\to[0,+\infty )$ is continuous and monotone increasing in $u$; \item[(A3)] There exists $0\leq \theta <1$ such that \[ f(t,ku)\geq k^\theta f(t,u)\quad \text{for all }t\in [0,1],\; k\in (0,1), \; u\in [0,+\infty ). \] \end{itemize} We prove the existence and uniqueness of a positive solution for the BVP \eqref{1.1}--\eqref{1.3} and study the dependence of this solution on the parameters $\lambda $ and $\mu $. \section{Preliminary lemmas} First, we recall some fundamental definitions. \begin{definition} \rm Let $X$ be a Banach space with a norm $\|\cdot \|$. \begin{itemize} \item[(1)] A nonempty closed convex set $P\subseteq X$ is said to be a cone if $\lambda P\subseteq P$ for all $\lambda \geq 0$ and $P\cap(-P)=\{\textbf{0}\ \}$, where $\textbf{0}$ is the zero element of $X$; \item[(2)] Every cone $P$ in $X$ defines a partial ordering in $X$ by $u\leq v\Leftrightarrow v-u\in P$; \item[(3)] A cone $P$ is said to be normal if there exists $M >0$ such that $\textbf{0} \leq u\leq v$ implies $\|u\| \leq M\|v\| $; \item[(4)] A cone $P$ is said to be solid if the interior $P^0$ of $P$ is nonempty. \end{itemize} Let $P$ be a solid cone in a real Banach space $X$, $T:P^0\to P^0$ be an operator and $0\leq \theta <1$. Then $T$ is called a $\theta$-concave operator if \[ T(ku)\geq k^\theta Tu\quad \text{for all $k\in (0,1)$, $u\in P^0$}. \] \end{definition} Next, we state a fixed point theorem, which is our main tool. \begin{lemma}[\cite{g1}] \label{lem2.2} Assume that $P$ is a normal solid cone in a real Banach space $X$, $0\leq \theta <1$ and $T:P^0\to P^0$ is a $\theta$-concave increasing operator. Then $T$ has a unique fixed point in $P^0$. \end{lemma} The following two lemmas are crucial for our main results. \begin{lemma} \label{lem2.3} Let $\rho _{_1}\neq 0$ and $\rho _{_2}\neq 0$. Then for any $h\in C[0,1]$, the BVP consisting of the equation \[ u^{(4)}(t)=h(t),\quad t\in [0,1] \] and the boundary conditions \eqref{1.2}--\eqref{1.3} has a unique solution \[ u(t)=\int_0^1G_1(t,s)\int_{\xi _1}^{\xi _2}G_2(s,\tau )h(\tau )d\tau ds+\lambda \Phi (t)+\mu \Psi (t),\quad t\in [0,1], \] where \begin{gather*} G_1(t,s)=\frac 1{\rho _{_1}} \begin{cases} (\alpha s+\beta )(\gamma +\delta -\gamma t), & 0\leq s\leq t\leq 1, \\ (\alpha t+\beta )(\gamma +\delta -\gamma s), & 0\leq t\leq s\leq 1, \end{cases} \\ G_2(t,s)=\frac 1{\rho _{_2}} \begin{cases} (a(s-\xi _1)+b)(c(\xi _2-t)+d), & s\leq t,\; \xi _1\leq s\leq \xi _2, \\ (a(t-\xi _1)+b)(c(\xi _2-s)+d), & t\leq s,\; \xi _1\leq s\leq \xi _2, \end{cases} \\ \Phi (t)=\frac 1{\rho _{_2}}\int_0^1(c(\xi _2-s)+d)G_1(t,s)ds,\quad t\in [0,1], \\ \Psi (t)=\frac 1{\rho _{_2}}\int_0^1(a( s-\xi _1)+b)G_1(t,s)ds,\quad t\in [0,1]. \end{gather*} \end{lemma} \begin{proof} Let \begin{equation} u''(t)=v(t),\quad t\in [0,1].\label{2.2} \end{equation} Then \begin{equation} v''(t)=h(t),\quad t\in [0,1]. \label{2.3} \end{equation} By \eqref{2.2} and \eqref{1.2}, we know that \begin{equation} u(t)=-\int_0^1G_1(t,s)v(s)ds,\quad t\in [0,1].\label{2.4} \end{equation} On the other hand, in view of \eqref{2.2} and \eqref{1.3}, we have \begin{equation} av(\xi _1)-bv^{'}(\xi _1)=-\lambda ,\text{ }cv(\xi _2)+dv^{'}(\xi _2)=-\mu .\label{2.5} \end{equation} So, it follows from \eqref{2.3} and \eqref{2.5} that \[ v(t)=-\int_{\xi _1}^{\xi _2}G_2(t,s) h(s)ds+\frac 1{\rho _{_2}}(c\lambda -a\mu )t+\frac 1{\rho _{_2}}((a\xi _1-b)\mu -(c\xi _2+d)\lambda ),\quad t\in [0,1], %\label{2.6} \] which together with \eqref{2.4} implies \[ u(t)=\int_0^1G_1(t,s)\int_{\xi _1}^{\xi _2}G_2(s,\tau )h(\tau )d\tau ds+\lambda \Phi (t)+\mu \Psi (t),\quad t\in [0,1]. \] \end{proof} \begin{lemma} \label{lem2.4} Assume {\rm (A1)}. Then \begin{itemize} \item[(1)] $G_1(t,s)>0$ for $t,s\in [0,1]$; \item[(2)] $G_2(t,s)>0$ for $t\in [0,1]$ and $s\in [\xi _1,\xi _2]$; \item[(3)] $\Phi (t)>0$ and $\Psi (t)>0$ for $t\in [0,1]$. \end{itemize} \end{lemma} \section{Main result} In the remainder of this article, the following notation will be used: \begin{itemize} \item[(1)] $(\lambda ,\mu )\to \infty $ if at least one of $\lambda $ and $\mu $ approaches $\infty $; \item[(2)] $(\lambda _1,\mu _1)>(\lambda _2,\mu _2)$ if $\lambda _1\geq \lambda _2$ and $\mu _1\geq \mu _2$ and at least one of them is strict; \item[(3)] $(\lambda _1,\mu _1)<(\lambda _2,\mu _2)$ if $\lambda _1\leq \lambda _2$ and $\mu _1\leq \mu _2$ and at least one of them is strict; \item[(4)] $(\lambda ,\mu )\to (\lambda _0,\mu_0)$ if $\lambda \to \lambda _0$ and $\mu \to \mu _0$. \end{itemize} Our main result is the following theorem. Here, for any $u\in C[0,1]$, we write $\|u\|=\max_{t\in [0,1]}|u(t)|$. \begin{theorem} \label{thm3.1} Assume {\rm (A1)-(A3)}. Then the BVP \eqref{1.1}-\eqref{1.3} has a unique positive solution $u_{\lambda ,\mu}(t)$ for any $(\lambda ,\mu )>(0,0)$. Furthermore, such a solution $u_{\lambda ,\mu }(t)$ satisfies the following three properties: \begin{itemize} \item[(P1)] $lim_{(\lambda ,\mu )\to \infty }\| u_{\lambda ,\mu }\| =\infty $; \item[(P2)] $u_{\lambda ,\mu }(t)$ is strictly increasing in $\lambda $ and $\mu $; i.e., \[ (\lambda _1,\mu _1)>(\lambda _2,\mu _2)>( 0,0)\Longrightarrow u_{\lambda _1,\mu _1}(t)>u_{\lambda _2,\mu _2}(t)\text{ on }[0,1]; \] \item[(P3)] $u_{\lambda ,\mu }(t)$ is continuous in $\lambda $ and $% \mu $; i.e., for any $(\lambda _0,\mu _0)>(0,0)$, \[ (\lambda ,\mu )\to (\lambda _0,\mu _0) \Longrightarrow \|u_{\lambda ,\mu }-u_{\lambda _0,\mu _0}\| \to 0. \] \end{itemize} \end{theorem} \begin{proof} Let $X=C[0,1]$. Then $(X,\|\cdot \|)$ is a Banach space, where $\|\cdot \| $ is defined as usual by the sup norm. Denote $P=\{ u\in X:u(t)\geq 0,\; t\in [0,1]\} $. Then $P$ is a normal solid cone in $X$ with $P^0=\{ u\in X\ |\text{ }u(t)>0,\ t\in [0,1]\} $. For any $(\lambda ,\mu)>(0,0)$, if we define an operator $ T_{\lambda,\mu }:P^0\to X $ as follows \begin{equation} T_{\lambda ,\mu }u(t)=\int_0^1G_1(t,s) \int_{\xi _1}^{\xi _2}G_2(s,\tau )f(\tau ,u( \tau ))d\tau ds+\lambda \Phi (t)+\mu \Psi(t), \label{3.1} \end{equation} then it is not difficult to verify that $u$ is a positive solution of the BVP \eqref{1.1}-\eqref{1.3} if and only if $u$ is a fixed point of $T_{\lambda ,\mu }$. Now, we prove that $T_{\lambda ,\mu }$ has a unique fixed point by using Lemma \ref{lem2.2} First, in view of Lemma \ref{lem2.4}, we know that $T_{\lambda ,\mu}:P^0\to P^0$. Next, we claim that $T_{\lambda ,\mu }:P^0\to P^0$ is a $\theta$-concave operator. In fact, for any $k\in (0,1)$ and $u\in P^0$, it follows from \eqref{3.1} and (A3) that \begin{align*} %\label{3.2} T_{\lambda ,\mu }(ku)(t) &= \int_0^1G_1(t,s)\int_{\xi _1}^{\xi _2}G_2( s,\tau )f(\tau ,ku (\tau ))d\tau ds+\lambda \Phi (t)+\mu \Psi (t) \\ &\geq k^\theta \int_0^1G_1(t,s)\int_{\xi _1}^{\xi _2}G_2(s,\tau )f(\tau ,u(\tau ) )d\tau ds+\lambda \Phi (t)+\mu \Psi (t) \\ &\geq k^\theta (\int_0^1G_1(t,s)\int_{\xi _1}^{\xi _2}G_2(s,\tau )f(\tau ,u(\tau ))d\tau ds+\lambda \Phi (t)+\mu \Psi (t)) \\ &= k^\theta T_{\lambda ,\mu }u(t),\quad t\in [0,1], \end{align*} which shows that $T_{\lambda ,\mu }$ is $\theta$-concave. Finally, we assert that $T_{\lambda ,\mu }:P^0\to P^0$ is an increasing operator. Suppose $u,v\in P^0$ and $u\leq v$. By \eqref{3.1} and (A2), we have \begin{align*} T_{\lambda ,\mu }u(t) &= \int_0^1G_1(t,s)\int_{\xi_1}^{\xi _2}G_2(s,\tau )f(\tau ,u(\tau ) )d\tau ds+\lambda \Phi (t)+\mu \Psi (t)\\ &\leq \int_0^1G_1(t,s)\int_{\xi _1}^{\xi _2}G_2(s,\tau )f(\tau ,v(\tau ))d\tau ds+\lambda \Phi (t)+\mu \Psi (t)\\ &= T_{\lambda ,\mu }v(t),\ t\in [0,1], \end{align*} which indicates that $T_{\lambda ,\mu }$ is increasing. Therefore, it follows from Lemma \ref{lem2.2} that $T_{\lambda ,\mu }$ has a unique fixed point $u_{\lambda ,\mu }\in P^0$, which is the unique positive solution of the BVP \eqref{1.1}-\eqref{1.3}. The first part of the theorem is proved. In the rest of the proof, we prove that the solution $u_{\lambda ,\mu }$ satisfies the properties (P1), (P2) and (P3). First, for $t\in [0,1]$, \begin{align*} u_{\lambda ,\mu }(t) &=T_{\lambda ,\mu }u_{\lambda ,\mu}(t) \\ &=\int_0^1G_1(t,s)\int_{\xi _1}^{\xi _2}G_2(s,\tau )f(\tau ,u_{\lambda ,\mu }( \tau ))d\tau ds+\lambda \Phi (t)+\mu \Psi(t), \end{align*} %\label{3.3} which together with $\Phi (t)>0$ and $\Psi (t)>0$ for $t\in [0,1]$ implies (P1). Next, we show (P2). Assume $(\lambda _1,\mu _1)>(\lambda _2,\mu _2)>(0,0)$. Let \[ \overline{\chi }=\sup \big\{ \chi >0:u_{\lambda _{1,}\mu _1}( t)\geq \chi u_{\lambda _{2,}\mu _2}(t),\ t\in [0,1]\big\} . \] Then $u_{\lambda _{1,}\mu _1}(t)\geq \overline{\chi }u_{\lambda _{2,}\mu _2}(t)$ for $t\in [0,1]$. We assert that $\overline{\chi }\geq 1$. Suppose on the contrary that $0<\overline{\chi }<1$. Since $T_{\lambda ,\mu }$ is a $\theta$-concave increasing operator, and for given $u\in P^0$, $T_{\lambda ,\mu }u$ is strictly increasing in $\lambda $ and $\mu$, we have \begin{align*} u_{\lambda _1,\mu _1}(t) &= T_{\lambda _1,\mu _1}u_{\lambda _1,\mu _1}(t)\geq T_{\lambda _1,\mu _1}(\overline{\chi } u_{\lambda _{2,}\mu _2})(t)\\ &>T_{\lambda _2,\mu _2}(\overline{\chi }u_{\lambda _{2,}\mu _2})(t)\\ &\geq (\overline{\chi })^\theta T_{\lambda _2,\mu _2}u_{\lambda _{2,}\mu _2}(t) =(\overline{\chi })^\theta u_{\lambda _{2,}\mu _2}(t)\\ &>\overline{\chi }u_{\lambda _{2,}\mu _2}(t),\quad t\in [0,1], \end{align*} which contradicts the definition of $\overline{\chi }$. Thus, we get $u_{\lambda _{1,}\mu _1}(t)\geq u_{\lambda _{2,}\mu _2}(t)$ for $t\in [0,1]$. And so, \begin{align*} u_{\lambda _1,\mu _1}(t) &= T_{\lambda _1,\mu _1}u_{\lambda _1,\mu _1}(t)\geq T_{\lambda _1,\mu _1}u_{\lambda _{2,}\mu_2}(t)\\ &> T_{\lambda _2,\mu _2}u_{\lambda _{2,}\mu _2}(t) =u_{\lambda _{2,}\mu _2}(t),\quad t\in [0,1], \end{align*} which indicates that $u_{\lambda ,\mu }(t)$ is strictly increasing in $\lambda $ and $\mu $. Finally, we show (P3). For any given $(\lambda _0,\mu_0)>(0,0)$, we first suppose $(\lambda,\mu )\to (\lambda _0,\mu _0)$ with $(\lambda_0/2, \mu_0/2)<(\lambda ,\mu ) <(\lambda _0,\mu _0)$. From (P2), we have \begin{equation} u_{\lambda ,\mu }(t)0:u_{\lambda,\mu}( t)\geq \sigma u_{\lambda _{0,}\mu _0}(t),\quad t\in [0,1]\} . \] Then $0<\overline{\sigma }<1$ and $u_{\lambda ,\mu }(t) \geq \overline{\sigma }u_{\lambda _0,\mu _0}(t)$ for $t\in [0,1]$. Define \[ \omega (\lambda ,\mu )=\begin{cases} \min \{ \frac \lambda {\lambda _0},\frac \mu {\mu _0}\}, &\text{if }\lambda _0\neq 0\text{ and }\mu _0\neq 0, \\ \frac \mu {\mu _0}, & \text{if }\lambda _0=0, \\ \frac \lambda {\lambda _0}, &\text{if }\mu _0=0, \end{cases} \] then $0<\omega (\lambda ,\mu )<1$ and \begin{align*} u_{\lambda ,\mu }(t) &= T_{\lambda ,\mu }u_{\lambda ,\mu }( t)\geq T_{\lambda ,\mu }(\overline{\sigma }u_{\lambda _0,\mu _0})(t)\\ &> \omega (\lambda ,\mu )T_{\lambda _0,\mu _0}(\overline{% \sigma }u_{\lambda _0,\mu _0})(t)\\ &\geq \omega (\lambda ,\mu )(\overline{\sigma }) ^\theta T_{\lambda _0,\mu _0}u_{\lambda _0,\mu _0}(t)\\ &= \omega (\lambda ,\mu )(\overline{\sigma })^\theta u_{\lambda _0,\mu _0}(t),\quad t\in [0,1], \end{align*} which together with the definition of $\overline{\sigma }$ implies \[ \omega (\lambda ,\mu )(\overline{\sigma })^\theta \leq \overline{\sigma }. \] Thus $\overline{\sigma }\geq (\omega (\lambda ,\mu )) ^{\frac 1{1-\theta }}$. And so, \begin{equation} u_{\lambda ,\mu }(t)\geq \overline{\sigma }u_{\lambda _0,\mu _0}(t)\geq (\omega (\lambda ,\mu ))^{\frac 1{1-\theta }}u_{\lambda _0,\mu _0}(t),\quad t\in [0,1]. \label{3.5} \end{equation} In view of \eqref{3.4} and \eqref{3.5}, we have \[ \|u_{\lambda _0,\mu _0}-u_{\lambda ,\mu }\| \leq ( 1-(\omega (\lambda ,\mu ))^{\frac 1{1-\theta }})\|u_{\lambda _0,\mu _0}\|, \] which together with the fact that $\omega (\lambda ,\mu )\to 1$ as $(\lambda ,\mu )\to (\lambda _0,\mu _0)$ shows that \[ \|u_{\lambda _0,\mu _0}-u_{\lambda ,\mu }\| \to 0\text{ as }(\lambda ,\mu )\to (\lambda _0,\mu _0). \] Similarly, we can also prove that \[ \|u_{\lambda _0,\mu _0}-u_{\lambda ,\mu }\| \to 0 \] as $(\lambda ,\mu )\to (\lambda _0,\mu _0)$ with $(\lambda ,\mu )>(\lambda _0,\mu _0)$. Hence, (P3) holds. The proof is complete. \end{proof} \begin{thebibliography}{0} \bibitem{g1} D. Guo, V. 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