\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 176, 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 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/176\hfil Multiple positive solutions] {Multiple positive solutions for a nonlinear 3n-th order three-point boundary-value problem} \author[K. L. S. Devi, K. R. Prasad, \hfil EJDE-2010/176\hfilneg] {K. L. Saraswathi Devi, Kapula R. Prasad} % in alphabetical order \address{K. L. Saraswathi Devi \newline Department of Applied Mathematics, Andhra University, Visakhapatnam, 530 003, India \newline Department of Mathematics, Ch. S. D. St. Theresa's Degree College for Women, Eluru, 534 003, India} \email{saraswathikatneni@gmail.com} \address{Kapula Rajendra Prasad \newline Department of Applied Mathematics, Andhra University, Visakhapatnam, 530 003, India} \email{rajendra92@rediffmail.com} \thanks{Submitted June 23, 2010. Published December 17, 2010.} \subjclass[2000]{34B18, 34A10} \keywords{Boundary value problem; multiple positive solutions; fixed point; cone} \begin{abstract} In this article we establish the existence of at least three positive solutions for 3n-th order three-point boundary value problem by using five functional fixed point theorem. We also establish the existence of at least $2m-1$ positive solutions of the problem for arbitrary positive integer $m$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} The general theory of differential equations has emerged as an important area of investigation due to its powerful and versatile applications to almost all areas of science, engineering and technology. Much interest has been developed since last decade regarding the study of existence of positive solutions to the boundary value problems as they are arising in the branches of applied mathematics, physics and technological problems. In this article, we prove the existence of multiple positive solutions of $3n^{th}$ order ordinary differential equation $$\label{e11} {(-1)}^ny^{(3n)}= f(y(t), y^{(3)}(t), y^{(6)}(t),\dots,y^{(3(n-1))}(t)),\quad t\in[t_1,t_3],$$ satisfying the general three point boundary conditions $$\label{e12} \begin{gathered} \alpha_{3i-2,1}y^{(3i-3)}(t_1)+\alpha_{3i-2,2}y^{(3i-2)}(t_1) +\alpha_{3i-2,3}y^{(3i-1)}(t_1)=0,\\ \alpha_{3i-1,1}y^{(3i-3)}(t_2)+ \alpha_{3i-1,2}y^{(3i-2)}(t_2)+\alpha_{3i-1,3}y^{(3i-1)}(t_2)=0,\\ \alpha_{3i,1}y^{(3i-3)}(t_3)+\alpha_{3i,2}y^{(3i-2)}(t_3) +\alpha_{3i,3}y^{(3i-1)}(t_3)=0, \end{gathered}$$ where the coefficients $\alpha_{3i-j,1},\alpha_{3i-j,2},\dots,\alpha_{3i-j,3}$ for $j=0,1,2$ and $i=1,\dots, n-1$, are real constants. Boundary-value problems of the form \eqref{e11}-\eqref{e12} constitute a natural extension of third order three-point boundary-value problems studied in many papers with simple boundary conditions. Here we refer to Graef, Yang \cite{gra1}, Eloe and Henderson \cite{pweaj}, Yang \cite{by}, Anderson\cite{dra}, Anderson and Davis \cite{dra1}, Guo, Sun and Zhao \cite{li} and references there in. Recently Prasad and Murali \cite{prasad} studied the multiple positive solutions for nonlinear third order general three-point boundary-value problem. For convenience we adopt the notation Let \begin{gather*} \beta_{ij}=\alpha_{3i-3+j,1}t_j+\alpha_{3i-3+j,2}, \quad \gamma_{ij}=\alpha_{3i-3+j,1}t_j^2+2\alpha_{3i-3+j,2}t_j+2\alpha_{3i-3+j,3},\\ l_{ij}=\alpha_{3i-3+j,1}s^2-2\beta_{ij}s+\gamma_{ij}, \end{gather*} and define $m_{i_{kj}}=\frac{\alpha_{3i-2+k,1}\gamma_{ij}-\alpha_{3i-2+j,1} \gamma_{ik}}{2(\alpha_{3i-2+k,1}\beta_{ij}-\alpha_{3i-2+j,1}\beta_{ik})},\quad M_{i_{kj}}=\frac{\beta_{3i-2+k,1} \gamma_{ij}-\beta_{ij}\gamma_{ik}}{(\alpha_{3i-2+k,1}\beta_{ij} -\alpha_{3i-2+j,1}\beta_{ik})}$ for $k=1,2,3$, $j=1,2,3$. Also let $m=\max{ \{m_{i_{12}},m_{i_{13}},m_{i_{23}}}\}$, $$M_i=\min\Big\{m_{i_{23}}+\sqrt{m_{i_{23}}^2-M_{i_{23}}},\, m_{i_{13}} +\sqrt{m_{i_{13}}^2-M_{i_{13}}}\Big\}$$ and $$d_i=[\alpha_{3i-2,1}(\beta _{i2}\gamma_{i3}-\beta _{i3}\gamma_{i2)}-\beta_{i1}(\alpha_{3i-1,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i2}) +\gamma_{i1}(\alpha_{3i-1,1}\beta_{i3}-\alpha_{3i,1}\beta_{i2})].$$ We assume the following conditions throughout this paper: \begin{itemize} \item[(A1)] $f:\mathbb{R}^n\to \mathbb{R}^+$ is continuous; \item[(A2)] $\alpha_{3i-2,1}>0$, $\alpha_{3i-1,1}>0$ and $\alpha_{3i,1} >0$ for $1\leq i\leq n$ are real constants, $\frac{\alpha_{3i-2,2}}{\alpha_{3i-2,1}} <\frac{\alpha_{3i-1,2}}{\alpha_{3i-1,1}}<\frac{\alpha_{3i,2}}{\alpha_{3i,1}}$. \item[(A3)] $m_i\leq t_1\leq t_2\leq t_3\leq M_i$, $2\alpha_{3i-1,3}\alpha_{3i-1,1}> \alpha_{3i-1,2}^2$,\\ $2\alpha_{3i-2,3}\alpha_{3i-2,1} < \alpha_{3i-2,2}^2$, $2\alpha_{3i,3} \alpha_{3i,3} > \alpha_{3i,2}^2$. \item[(A4)] $m_{i_{23}}^2 >M_{i_{23}}$, $m_{i_{12}}^2 < M_{i_{12}}$, $m_{i_{13}}^2 > M_{i_{13}}$ and $d_i>0$. \end{itemize} The rest of the paper is organized as follows. In Section 2, we construct the Green's function for the homogeneous boundary value problem corresponding to \eqref{e11}-\eqref{e12} and estimate the bounds for the Green's function. In Section 3, we establish the existence of at least three positive solutions for \eqref{e11}-\eqref{e12}, using five functional fixed point theorem. We also establish the existence of at least $2m-1$ positive solutions of \eqref{e11}-\eqref{e12}, for arbitrary positive integer $m$. \section{The Green's Function and Bounds} In this section, we construct the Green's function for the homogeneous boundary value problem corresponding to \eqref{e11}-\eqref{e12} and estimate the bounds of the Greens function. We prove certain lemmas which are needed to establish our main results. Let $G_i(t,s)$ be the Green's function for the homogeneous problem $$\label{e21} -y'''=0,\quad t\in [t_1,t_3]$$ satisfying the general three point boundary conditions \eqref{e12}. First we establish results on the related third order homogeneous boundary-value problem \eqref{e21} and \eqref{e12}. \begin{lemma}\label{l21} The homogeneous boundary-value problem \eqref{e21} and \eqref{e12} has only the trivial solution if and only if $d_i=[\alpha_{3i-2,1}(\beta _2\gamma_3-\beta _3\gamma_2)-\beta_1(\alpha_{3i-1,1}\gamma_{3}-\alpha_{3i,1}\gamma_{2}) +\gamma _1(\alpha_{3i-1,1}\beta_3-\alpha_{3i,1}\beta_2)] \neq0$ for $1\leq i\leq n$. \end{lemma} \begin{proof} On application of boundary conditions \eqref{e12} to the general solution of \eqref{e21}, it can be established. \end{proof} \begin{lemma}\label{l22} For $1 \leq i\leq n$, the Green's function for the homogeneous boundary value problem \eqref{e21} and \eqref{e12} is \label{e22} G_i(t,s)= \begin{cases} G_{i_1}(t,s), & t_1< s0$, for$(t,s)\in[t_1,t_3]\times[t_1,t_3]$. \end{lemma} \begin{proof} For$(t,s)\in[t_1,t_3]\times[t_1,t_3]$,$G_{i}(t,s)$as stated in \eqref{e22}, if we consider sequentially, from {\rm (A2)--(A4)}, we obtain $$\label{e23} G_{i}(t,s)>0, \quad\text{for }(t,s)\in[t_1,t_3]\times[t_1,t_3].$$ \end{proof} \begin{lemma}\label{l24} Assume the conditions {\rm (A1)--(A4)} are satisfied. Then, for$1\leq i\leq n$, the Green's function$G_{i}(t,s)$given by \eqref{e22} satisfies $$G_{i}(t,s)\leq\max\big\{G_{i}(t_1,s), G_{i}(s,s),G_{i}(t_3,s)\big\}.$$ \end{lemma} \begin{proof} This can be proved by proceeding sequentially with the branches of$G_{i}(t,s)$in \eqref{e22}. \textbf{Case 1.} For$t_{1}0,\quad\text{for } 1\leq j \leq n, \\ L_j=\int_{t_2}^{t_3}\| G_j(s,s)\| ds>0, \quad\text{for } 1\leq j\leq n. \end{gather*} \end{lemma} Using Lemma \ref{l25} and induction on $n$, we can easily establish the proof of the above lemma. Let $C=\{v\mid v:[t_1,t_3]\to \mathbb{R} \text{ is continuous function } \}$. For each $1\leq j\leq n-1$, define the operator $T_j:C\to C$ by $$(T_jv)(t)=\int_{t_1}^{t_3}H_j(t,s)v(s)ds,\quad t\in [t_1,t_3].$$ By the construction of $T_j$, and the properties of $H_j(t,s)$, it is clear that \begin{gather*} (-1)^j(T_jv)^{(3j)}(t)=v(t),\quad t\in [t_1,t_3],\\ \alpha_{3i-2,1}{T_jv}^{(3i-3)}(t_1) +\alpha_{3i-2,2}{T_jv}^{(3i-2)}(t_1) +\alpha_{3i-2,3}{T_jv}^{(3i-1)}(t_1)=0,\\ \alpha_{3i-1,1}{T_jv}^{(3i-3)}(t_2)+ \alpha_{3i-1,2}{T_jv}^{(3i-2)}(t_2) +\alpha_{3i-1,3}{T_jv}^{(3i-1)}(t_2)=0,\\ \alpha_{3i,1}{T_jv}^{(3i-3)}(t_3)+\alpha_{3i,2}{T_jv}^{(3i-2)}(t_3) +\alpha_{3i,3}{T_jv}^{(3i-1)}(t_3)=0, \end{gather*} for $i=1,2,\dots, j-1$. Hence, we see that \eqref{e11}-\eqref{e12} has a solution if and only if the following boundary-value problem has a solution \begin{gather}\label{e26} v^{(3)}(t)+f(T_{n-1}v(t),T_{n-2}v(t),\dots,T_1v(t),v(t))=0, \quad t\in[t_1,t_3], \\ \label{e27} \begin{gathered} \alpha_{3i-2,1}{v}^{(3i-3)}(t_1) +\alpha_{3i-2,2}{v}^{(3i-2)}(t_1) +\alpha_{3i-2,3}{v}^{(3i-1)}(t_1)=0,\\ \alpha_{3i-1,1}{v}^{(3i-3)}(t_2)+ \alpha_{3i-1,2}{v}^{(3i-2)}(t_2)+\alpha_{3i-1,3}{v}^{(3i-1)}(t_2)=0,\\ \alpha_{3i,1}{v}^{(3i-3)}(t_3)+\alpha_{3i,2}{v}^{(3i-2)}(t_3) +\alpha_{3i,3}{v}^{(3i-1)}(t_3)=0. \end{gathered} \end{gather} for $i=1,2,\dots, j-1$. Indeed, if $y$ is a solution of \eqref{e11}-\eqref{e12}, then $v(t)=y^{3 (n-1)}(t)$ is a solution of \eqref{e26}-\eqref{e27}. Conversely, if $v$ is a solution of \eqref{e26}-\eqref{e27}, then $y(t)=T_{n-1}v(t)$ is a solution of \eqref{e11}-\eqref{e12}. In fact, $y(t)$ is represented as $$y(t)=\int_{t_1}^{t_3}H_n(t,s)v(s)ds,$$ where $$v(s)=\int_{t_1}^{t_3}G_1(s,\tau)f(T_{n-1}v(\tau),T_{n-2}v(\tau), \dots,T_1v(\tau),v(\tau))d \tau.$$ is a solution of \eqref{e11}-\eqref{e12}. \section{Existence of multiple positive solutions} In this section, we establish the existence of multiple positive solutions for \eqref{e11}-\eqref{e12}, by using five functional fixed point theorem which is Avery generalization of the Leggett-Williams fixed point theorem. And then, we establish $2m-1$ positive solutions for an arbitrary positive integer $m$. Let $B$ be a real Banach space with cone $P$. A map $\alpha:P\to [0,\infty)$ is said to be a nonnegative continuous concave functional on $P$ if $\alpha$ is continuous and $$\alpha(\lambda x+(1-\lambda )y)\geq \lambda \alpha(x) +(1-\lambda )\alpha(y),$$ for all $x, y\in P$ and $\lambda \in [0,1]$. Similarly, we say that a map $\beta:P\to [0, \infty)$ is said to be a nonnegative continuous convex functional on $P$ if $\beta$ is continuous and $$\beta(\lambda x+(1-\lambda )y)\leq \lambda \beta(x) +(1-\lambda )\beta(y),$$ for all $x,y\in P$ and $\lambda \in [0,1]$. Let $\gamma, \beta, \theta$ be nonnegative continuous convex functional on $P$ and $\alpha, \psi$ be nonnegative continuous concave functionals on $P$, then for nonnegative numbers $h', a', b', d'$ and $c'$, we define the following convex sets \begin{gather*} P(\gamma,c')=\{y\in P|\gamma(y)a'\}\neq \emptyset$and$\alpha(Ty)>a'$\\ for$y\in P(\gamma,\theta,\alpha,a',b',c')$; \item[(B2)]$\{y\in Q(\gamma,\beta,\psi,h',d',c')|\beta(y)a'$provided$y\in P(\gamma,\alpha,a',c') $with$\theta(Ty)>b'$; \item[(B4)]$\beta(Ty)b'/M\overline K$for all$(|u_{n-1}|,|u_{n-2}|,\dots,|u_1|,|u_0|)$in$\prod_{j=n-1}^1[\frac{b'm_n\overline L'L}{M},\frac{c'KK_j}{M}]\times [b',\frac{b'}{M}]$; \item[(D3)]$f(u_{n-1},u_{n-2},\dots,u_1,u_0)b'\} \neq \emptyset,\\ \{v\in Q(\gamma,\beta,\psi, Ma',a',c'):\beta(v)b'/M, we obtain \begin{align*} \alpha(Tv)&=\min_{t\in [t_2, t_3]}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))d s \\ &\geq M\int_{t_1}^{t_3}G_1(s,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\ &\geq M\max_{t\in[t_1, t_3]}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\ &\geq M\max_{t\in [t_2, t_3]}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\ &=M\theta(Tv)>b'. \end{align*} Finally, we show that (B4) holds. Letv\in Q(\gamma,\beta,a',c')$with$\psi(Tv)\frac{b_l}{K \overline M} for all $(|u_{n-1}|,|u_{n-2}|,\dots,|u_1|,|u_0|)$ in $\prod_{j=n-1}^1[\frac{b_lm_n\overline L'L}{M},\frac{b_{m-1}KK_j}{M}]\times [b_l,\frac{b_l}{M}]$, $1\leq l \leq m-1$. Then \eqref{e11}-\eqref{e12} has at least $2m-1$ positive solutions in $\overline{P}_{a_{m}}$. \end{theorem} \begin{proof} We use induction on $m$. First, for $m=1$, we know from \eqref{e41} that $T:\overline{P}_{a_{1}}\to P_{a_{1}}$, then, it follows from Schauder fixed point theorem that \eqref{e11}-\eqref{e12} has at least one positive solution in $\overline{P}_{a_{1}}$. Next, we assume that this conclusion holds for $m=k$. In order to prove that this conclusion holds for $m=k+1$, we suppose that there exist numbers $a_i(1\leq i\leq k+1)$ and $b_j(1\leq j\leq k)$ with $0\frac{b_l}{K \overline M} for all$(|u_{n-1}|,|u_{n-2}|,\dots,|u_1|,|u_0|)$in$\prod_{j=n-1}^1[\frac{b_lm_n\overline L'L}{M},\frac{b_{m-1}KK_j}{M}]\times [b_l,\frac{b_l}{M}]$,$1\leq l \leq k$. By assumption, Problem \eqref{e11}-\eqref{e12} has at least$2k-1$positive solutions$u_i(i=1,2,\dots,2k-1)$in$\overline{P}_{a_{k}}$. At the same time, it follows from Theorem \ref{t32}, and \eqref{e43} and \eqref{e44} that \eqref{e11}-\eqref{e12} has at least three positive solutions$u,v$and$w$in$\overline{P}_{a_{k+1}}$such that,$\| u\|a_k,\min_{t \in [t_2, t_3]}w(t)