\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 75, 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} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/75\hfil Three positive solutions] {Three positive solutions for $m$-point boundary-value problems with one-dimensional $p$-Laplacian} \author[D. Bai, H. Feng\hfil EJDE-2011/75\hfilneg] {Donglong Bai, Hanying Feng} % in alphabetical order \address{Donglong Bai \newline Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, Hebei, China} \email{baidonglong@yeal.net} \address{Hanying Feng \newline Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, Hebei, China} \email{fhanying@yahoo.com.cn} \thanks{Submitted April 5, 2011. Published June 16, 2011.} \thanks{Supported by grants 10971045 from the NNSF of China, and A2009001426 from the \hfill\break\indent HEBNSF of China} \subjclass[2000]{34B10, 34B15, 34B18} \keywords{Multipoint boundary value problem; positive solution; \hfill\break\indent Avery-Peterson fixed point theorem; one-dimensional $p$-Laplacian} \begin{abstract} In this article, we study the multipoint boundary value problem for the one-dimensional $p$-Laplacian $$(\phi_p(u'))'+ q(t)f(t,u(t),u'(t))=0,\quad t\in (0,1),$$ subject to the boundary conditions $$u(0)=\sum_{i=1}^{m-2} a_iu(\xi_i),\quad u'(1)=\beta u'(0).$$ Using a fixed point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The interesting point is that the nonlinear term $f$ involves the first derivative of the unknown function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article, we study the existence of multiple positive solutions to the boundary value problem (BVP for short) for the one-dimensional $p$-Laplacian \begin{gather}\label{e1.1} (\phi_p(u'))'+ q(t)f(t,u(t),u'(t))=0, \quad t\in (0,1),\\ \label{e1.2} u(0)=\sum_{i=1}^{m-2}\ a_iu(\xi_i),\quad u'(1)=\beta u'(0), \end{gather} where $\phi_p(s)=|s|^{p-2}s$, $p>1$, $\xi_i\in(0,1)$ with $0<\xi_1<\xi_2<\dots<\xi_{m-2}<1$. use the following assumptions: \begin{itemize} \item[(H1)] $a_i\in [0,1)$ satisfies $\sum_{i=1}^{m-2} a_i<1$, $\beta\in(0,1)$; \item[(H2)] $f\in C([0,1]\times [0,+\infty)\times\mathbb{R}, [0,+\infty))$; \item[(H3)] $q\in L^1[0,1]$ is nonnegative on $(0,1)$ and $q$ is not identically zero on any subinterval of $(0,1)$. Furthermore, $q$ satisfies $0<\int_0^1q(t)dt<\infty$. \end{itemize} Multipoint boundary value problems of ordinary differential equations arise in a variety of areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross-section and composed of N parts of different densities can be set up as a multipoint boundary value problem (see \cite{mo}). The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev \cite{il}. Since then there has been much current attention focused on the study of nonlinear multipoint boundary value problems, see \cite{ag,bz,fh1,fh2,fh3,fm,gu1,li,lu,md,su,wf,zh}. However, to the best knowledge of the authors, no work has been done for \eqref{e1.1}, \eqref{e1.2}. The aim of this paper is to fill this gap in the relevant literature. Karakostas \cite{ka} proved the existence of positive solutions for the multipoint boundary-value problem $$x''(t)-sign(1-\alpha)q(t)f(x,x')x'=0, \ t\in (0,1),$$ with one of the following sets of boundary conditions: $$x(0)=0, \quad x'(1)=\alpha x'(0),$$ or $$x(1)=0, \quad x'(1)=\alpha x'(0),$$ where $\alpha>0$, $\alpha\neq1$. By using indices of convergence of the nonlinearities at $0$ and at $+\infty$, the author provide a priori upper and lower bounds for the slope of the solutions. Ma \cite{ma} proved the existence of positive solutions for the multipoint boundary-value problem \begin{gather*} x''(t)-q(t)f(x,x')x'=0, \quad t\in (0,1),\\ x(0)=\sum_{i=1}^{n-2} b_ix(\xi_i),\quad x'(1)=\alpha x'(0), \end{gather*} where $\xi_i\in(0, 1)$ with $0<\xi_1<\xi_2<\dots<\xi_{m-2}<1$, $b_i\in[0,1)$, $\alpha>1$. They provided sufficient conditions for the existence of multiple positive solutions to the above BVP by applying the fixed point theorem in cones. Motivated by these results, our purpose of this paper is to show the existence of at least three positive solutions to multipoint BVP \eqref{e1.1} and \eqref{e1.2}. \section{Preliminaries} For the convenience of readers, we provide some background material from the theory of cones in Banach spaces. We also state in this section the Avery-Peterson's fixed point theorem. \begin{definition} \label{def2.1} \rm Let $E$ be a real Banach space over $\mathbb{R}$. A nonempty closed set $P \subset E$ is said to be a cone provide that \begin{itemize} \item[(i)] $au+bv\in P$ for all $u, v\in P$ and all $a\geq0, b\geq0$; and \item[(ii)] $u, -u \in P$ implies $u=0$. \end{itemize} Every cone $P\subset E$ induces an ordering in $E$ given by $x\leq y$ if and only if $y-x\in P$. \end{definition} \begin{definition} \label{def2.2} \rm The map $\alpha$ is said to be a nonnegative continuous concave functional on a cone $P$ of a real Banach space $E$ provided that $\alpha: P\to [0, \infty)$ is continuous and $$\alpha(tx+(1-t)y)\geq t\alpha(x)+(1-t)\alpha(y)$$ for all $x,y\in P$ and $0 \leq t \leq 1$. Similarly, we say the map $\gamma$ is a nonnegative continuous convex functional on a cone $P$ of a real Banach space $E$ provided that $\gamma: P\to [0, \infty)$ is continuous and $$\gamma(tx+(1-t)y)\leq t\gamma(x)+(1-t)\gamma(y)$$ for all $x,y\in P$ and $0 \leq t\leq1$. \end{definition} Let $\gamma$ and $\theta$ be nonnegative continuous convex functionals on a cone $P$, $\alpha$ be a nonnegative continuous concave functional on a cone $P$, and $\psi$ a nonnegative continuous functional on a cone $P$. Then for positive real numbers $a, b, c, d$, we define the following convex sets: \begin{gather*} P(\gamma, d)=\{u\in P: \gamma(u)b\} \neq \phi$and$\alpha(Tu)>b$for all$u$in$P(\gamma, \theta,\alpha, b, c, d)$; \item[(S2)]$\alpha(Tu)>b$for$u\in P(\gamma, \alpha, b, d)$with$\theta(Tu)>c$; \item[(S3)]$0 \not\in R(\gamma, \psi, a, d)$and$\psi(Tu) \max\{\frac{1}{\xi_1},\frac{1}{1-\xi_{m-2}} \}$such that$0<\int_{1/k}^{1-(1/k)}q(t)dt<\infty$. Let the nonnegative continuous concave functional$\alpha$, the nonnegative continuous convex functionals$\theta, \gamma$, and the nonnegative continuous functional$\psi$be defined on the cone$P$by \begin{gather*} \gamma(u)=\max _{0\leq t\leq 1}|u'(t)|,\quad \psi(u)=\theta(u)=\max _{0\leq t\leq 1}|u(t)|,\\ \alpha(u)=\min_{1/k\leq t\leq 1-(1/k) } |u(t)| \quad\text{for } u\in P. \end{gather*} \begin{lemma} \label{lem3.1} Assume that {\rm (H1)--(H3)} hold. Then, for any$x\in C^{+}[0,1]=:\{x\in C^1[0,1]:x(t)\geq 0\}, \begin{gather}\label{e3.1} (\phi_p(u'))'+ q(t)f(t,x(t),x'(t))=0,\quad t \in (0,1), \\ \label{e3.2} u(0)=\sum_{i=1}^{m-2}\ a_iu(\xi_i),\ u'(1)=\beta u'(0), \end{gather} has the unique solution \label{e3.3} \begin{aligned} u(t)&=\int_0^{t}\phi^{-1}_p\Big(\int_s^1 q(\tau)f(\tau,x(\tau),x'(\tau))d\tau\\ &\quad +\frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_0^1 q(\tau)f(\tau,x(\tau),x'(\tau))d\tau\Big)ds\\ &\quad +\frac{1}{1- \sum_{i=1}^{m-2}a_i}\sum_{i=1}^{m-2}a_i\int_0^{\xi_i} \phi^{-1}_p\Big(\int_s^1 q(\tau)f(\tau,x(\tau),x'(\tau))d\tau \\ &\quad + \frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_0^1 q(\tau)f(\tau,x(\tau),x'(\tau))d\tau\Big)ds. \end{aligned} \end{lemma} \begin{proof} For anx\in C^{+}[0,1]$, suppose$uis a solution of \eqref{e3.1}, \eqref{e3.2}. By integration of \eqref{e3.1}, it follows that \begin{gather*} u'(t)=\phi^{-1}_p\Big(\phi_p(u'(0)) -\int_0^{t}q(\tau)f(\tau,x(\tau),x'(\tau))d\tau\Big),\\ u(t)=u(0)+\int_0^{t}\phi^{-1}_p\Big(\phi_p(u'(0)) -\int_0^{s}q(\tau) f(\tau,x(\tau),x'(\tau))d\tau\Big)ds. \end{gather*} Using the boundary condition \eqref{e3.2}, we can easily show that \begin{align*} u(t) &=\int_0^{t}\phi^{-1}_p\Big(\int_s^1 q(\tau)f(\tau,x(\tau),x'(\tau))d\tau\\ &\quad +\frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_0^1 q(\tau)f(\tau,x(\tau),x'(\tau))d\tau\Big)ds\\ &\quad +\frac{1}{1-\sum_{i=1}^{m-2}a_i} \sum_{i=1}^{m-2}a_i\int_0^{\xi_i} \phi^{-1}_p\Big(\int_s^1 q(\tau)f(\tau,x(\tau),x'(\tau))d\tau\\ &\quad +\frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_0^1 q(\tau)f(\tau,x(\tau),x'(\tau))d\tau\Big)ds. \end{align*} One the other hand, it is easy to verify that ifu$is the solution of \eqref{e3.3}, then$u$is a solution of \eqref{e3.1} and \eqref{e3.2}. \end{proof} \begin{lemma} \label{lem3.2} Assume that {\rm (H1)--(H3)} hold. If$x\in C^{+}[0,1]$, then the unique solution$u(t)$of \eqref{e3.1} and \eqref{e3.2} is concave and$u(t)\geq 0$,$u'(t)\geq 0$,$t\in[0,1]$. \end{lemma} \begin{proof} From the fact that$(\phi_p(u'))'(t)=-q(t)f(t,x(t),x'(t))\leq 0$, we have$\phi_p(u'(t))$is nonincreasing. It follows that$u'(t)$is also nonincreasing. Thus, we know that the graph of$u(t)$is concave down on$(0,1)$. Then the concavity of$u$together with boundary condition$u'(1)=\beta u'(0)$implies that$u'(t)\geq0$for$t\in[0,1]$. From$u'(t)\geq0$, we know that$u(\xi_i)\geq u(0)$, for$i=1,2,\dots,m-2$. This implies $$u(0)=\sum _{i=1}^{m-2}a_iu(\xi_i)\geq\sum _{i=1}^{m-2}a_iu(0).$$ By$1-\sum _{i=1}^{m-2}a_i>0$, it is obvious that$u(0)\geq0$. Hence, we know that$u(t)\geq0$,$t\in[0,1]$. \end{proof} \begin{lemma} \label{lem3.3} If$u\in P$, then$\max _{0\leq t\leq 1}|u(t)| \leq\overline{M}\max _{0\leq t\leq 1}|u'(t)|$, where$\overline{M}=1+\frac{\sum _{i=1}^{m-2} a_i\xi_i}{1-\sum _{i=1}^{m-2}a_i}$. \end{lemma} \begin{proof} For$u\in P$, by the concavity of$u$and that$u'(t)\geq0, one arrives at $$u(1)-u(0)\leq u'(0)=\max _{0\leq t\leq 1}|u'(t)|.$$ On the other hand, \begin{align*} (1-\sum_{i=1}^{m-2}a_i)u(0) &=u(0)-\sum_{i=1}^{m-2}a_iu(0) =\sum_{i=1}^{m-2}a_iu(\xi_i)-\sum_{i=1}^{m-2}a_iu(0)\\ &=\sum_{i=1}^{m-2}a_i(u(\xi_i)-u(0)) =\sum_{i=1}^{m-2}a_i\xi_iu'(\eta_i), \end{align*} where\eta_i\in(0, \xi_i)$. So $$u(0)= \frac{\sum_{i=1}^{m-2}a_i\xi_iu'(\eta_i)}{1-\sum _{i=1}^{m-2}a_i} \leq\frac{\sum_{i=1}^{m-2}a_i\xi_i}{1-\sum _{i=1}^{m-2}a_i}\max _{0\leq t\leq 1}|u'(t)|.$$ Thus one has $$\max _{0\leq t\leq 1}|u(t)|=u(1)\leq\Big(1+\frac{\sum _{i=1}^{m-2}a_i\xi_i}{1-\sum _{i=1}^{m-2}a_i}\Big)u'(0)=\overline{M}\max _{0\leq t\leq 1}|u'(t)|.$$ With Lemma \ref{lem3.3} and the concavity of$u$, for all$u\in P$, we obtain $$\label{e3.4} \frac{1}{k}\theta(u)\leq\alpha(u)\leq\theta(u) =\psi(u), \quad \|u\|=\max\{\theta(u),\gamma(u)\}\leq\overline{M}\gamma(u).$$ \end{proof} \begin{lemma} \label{lem3.4} Define an operator$T: P\to P, \label{e3.5} \begin{aligned} (Tu)(t) &=\int_0^{t}\phi^{-1}_p\Big(\int_s^1 q(\tau)f(\tau,u(\tau),u'(\tau))d\tau {l}\\ &\quad +\frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_0^1 q(\tau)f(\tau,u(\tau),u'(\tau))d\tau\Big)ds\\ &\quad +\frac{1}{1-\sum_{i=1}^{m-2}a_i} \sum_{i=1}^{m-2}a_i\int_0^{\xi_i} \phi^{-1}_p\Big(\int_s^1 q(\tau)f(\tau,u(\tau),u'(\tau))d\tau \\ &\quad + \frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_0^1 q(\tau)f(\tau,u(\tau),u'(\tau))d\tau\Big)ds. \end{aligned} ThenT:\ P\to P$is completely continuous. \end{lemma} \begin{proof} According to the definition of$T$and Lemma \ref{lem3.2}, it is easy to show that$T(P)\subset P$. By similar arguments in \cite{ka, wf},$T : P\to P$is completely continuous. \end{proof} \section{Existence of positive solutions} We are now ready to apply the Avery-Peterson's fixed point theorem to the operator$T$to give sufficient conditions for the existence of at least three positive solutions to \eqref{e1.1}, \eqref{e1.2}. For convenience we introduce following notation. Let \begin{gather*} K=\frac{4k^{2}(1- \sum_{i=1}^{m-2}a_i)}{(k\beta+\beta+3k-1) (1- \sum_{i=1}^{m-2}a_i)+(k^{2}+k) \sum_{i=1}^{m-2}a_i [2\xi_i-(1-\beta)\xi_i^{2}]}, \\ L=\phi_p^{-1}\Big(\frac{1}{1-\phi_p(\beta)}\int_0^1q(\tau)d\tau\Big), \\ \begin{split} M&=\int_{1/k}^{1-(1/k)}\phi_p^{-1} \Big(\int_s^{1-(1/k)}q(\tau) d\tau +\frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_{1/k} ^{1-(1/k)}q(\tau)d\tau\Big)ds \\ &\quad +\frac{1}{1-\sum_{i=1}^{m-2}a_i} \sum_{i=1}^{m-2}a_i \int_{1/k}^{\xi_i}\phi_p^{-1} \Big(\int_s^{1-(1/k)}q(\tau) d\tau\\ &\quad +\frac{\phi_p(\beta)}{1-\phi_p(\beta)} \int_{1/k}^{1-(1/k)}q(\tau) d\tau\Big)ds, \end{split}\\ \begin{split} N&=\int_0^1\phi_p^{-1}\Big(\int_s^1q(\tau) d\tau+\frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_0^1q(\tau) d\tau\Big)ds\\ &\quad +\frac{1}{1-\sum_{i=1}^{m-2}a_i} \sum_{i=1}^{m-2}a_i \int_0^{\xi_i}\phi_p^{-1}\Big(\int_s^1q(\tau) d\tau+\frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_0^1q(\tau) d\tau\Big)ds. \end{split} \end{gather*} \begin{theorem} \label{thm4.1} Assume that {\rm (H1)--(H3)} hold. Let$0b, \end{align*} \begin{align*} \theta(u_0) &=\max _{0\leq t\leq 1}|u_0(t)|=u_0(1)\\ &=\frac{2k^{2}b\Big((1- \sum_{i=1}^{m-2}a_i)(\beta+1)+\sum_{i=1}^{m-2} a_i[2\xi_i-(1-\beta)\xi_i^{2}]\Big)}{(k\beta+\beta+3k-1) (1- \sum_{i=1}^{m-2}a_i)+(k^{2}+k) \sum_{i=1}^{m-2}a_i[2\xi_i-(1-\beta)\xi_i^{2}]}\\ &b\} \neq \phi$. Thus for$ u\in P(\gamma, \theta,\alpha, b, kb, d)$, there is$b\leq u(t)\leq kb$,$0\leq u'(t)\leq d$, for$1/k\leq t\leq 1-1/k$. Hence by condition (A2) of this theorem, one has$f(t,u,u'(t))\geq\phi_p(kb/M)$, for$t\in [1/k,1-1/k]$. Noting$(Tu)(1)\geq0$from Lemma \ref{lem3.2} and combining the conditions on$\alpha$and$P, one arrives at \begin{align*} \alpha(Tu) &=\min _{1/k \leq t \leq 1-(1/k)}|(Tu)(t)|\geq \frac{1}{k}\max_{0\leq t \leq 1}|(Tu)(t)|= \frac{1}{k}(Tu)(1)\\ &=\frac{1}{k}\int_0^1\phi^{-1}_p\Big(\int_s^1 q(\tau)f(\tau,u(\tau),u'(\tau))d\tau\\ &\quad +\frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_0^1 q(\tau)f(\tau,u(\tau),u'(\tau))d\tau\Big)ds\\ &\quad+ \frac{1}{k(1- \sum_{i=1}^{m-2}a_i)} \sum_{i=1}^{m-2}a_i\int_0^{\xi_i} \phi^{-1}_p\Big(\int_s^1 q(\tau)f(\tau,u(\tau),u'(\tau))d\tau\\ &\quad + \frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_0^1 q(\tau)f(\tau,u(\tau),u'(\tau))d\tau\Big)ds \\ &>\frac{1}{k}\int_{1/k}^{1-(1/k)}\phi^{-1}_p \Big(\int_s^{1-(1/k)} q(\tau)f(\tau,u(\tau),u'(\tau))d\tau\\ &\quad+ \frac{\phi_p(\beta)}{1-\phi_p(\beta)} \int_{1/k}^{1-(1/k)} q(\tau)f(\tau,u(\tau),u'(\tau))d\tau\Big)ds\\ &\quad +\frac{1}{k(1-\sum_{i=1}^{m-2}a_i)} \sum_{i=1}^{m-2}a_i\int_{1/k}^{\xi_i} \phi^{-1}_p\Big(\int_s^{1-(1/k)} q(\tau)f(\tau,u(\tau),u'(\tau))d\tau \\ &\quad+\frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_{1/k}^{1-(1/k)} q(\tau)f(\tau,u(\tau),u'(\tau))d\tau\Big)ds\\ &\geq \frac{1}{k}\int_{1/k}^{1-(1/k)}\phi^{-1}_p \Big(\int_s^{1-(1/k)} q(\tau)d\tau\phi_p(\frac{kb}{M})\\ &\quad +\frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_{1/k}^{1-(1/k)} q(\tau)d\tau\phi_p(\frac{kb}{M})\Big)ds\\ &\quad +\frac{1}{k(1-\sum_{i=1}^{m-2}a_i)} \sum_{i=1}^{m-2}a_i\int_{1/k}^{\xi_i} \phi^{-1}_p\Big(\int_s^{1-(1/k)}q(\tau)d\tau\phi_p(\frac{kb}{M})\\ &\quad + \frac{\phi_p(\beta)}{1-\phi_p(\beta)}\int_{1/k}^{1-(1/k)} q(\tau)d\tau\phi_p(\frac{kb}{M})\Big)ds \\ &= \frac{1}{k}\frac{kb}{M}M =b. \end{align*} Therefore,\alpha(Tu)>b$, for all$u\in P(\gamma, \theta,\alpha, b, kb, d)$. Consequently, condition (S1) in Theorem \ref{thm2.1} holds. (3) We now prove (S2) in Theorem \ref{thm2.1} holds. With \eqref{e3.4} we have $$\alpha(Tu)\geq \frac{1}{k}\theta(Tu)>\frac{1}{k}kb=b,$$ for$u\in P(\gamma,\alpha,b,d)$with$\theta(Tu)>kb$. Hence, condition (S2) in Theorem \ref{thm2.1} is satisfied. (4) Finally, we show that (S3) in Theorem \ref{thm2.1} is satisfied. Since$\psi(0)=06. \end{cases}  Choose $a=2/3$, $b=1$, $k=6$, $d=3\times 10^{21}$, we note that $K=4/5$, $L=2\sqrt{3}/3$, $\overline{M}=7/3$, $M\doteq 1.185$, $N\doteq 2.281$. Consequently, $f(t,u,v)$ satisfies \begin{itemize} \item[(1)] $f(t,u,v)<2.407\times 10^{35}<\phi_3(d/L)\doteq 6.75\times 10^{42}$ for $(t,u,v)\in [0,1] \times [0,7\times 10^{21}]\times [0,3\times 10^{21}]$; \item[(2)] $f(t,u,v)>1.8\times 10^{4}>\phi_3(kb/M)\doteq25.637$ for $(t,u,v)\in [1/6,5/6]\times [1,6]\times [0,3\times 10^{21}]$; \item[(3)] $f(t,u,v)<0.063<\phi_3(a/N)\doteq 0.085$ for $(t,u,v)\in [0,1] \times [0,2/3]\times [0,3\times 10^{21}]$. \end{itemize} Then all conditions of Theorem \ref{thm4.1} hold. 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