\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 125, 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 (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/125\hfil Existence of solutions] {Existence of solutions to third-order $m$-point boundary-value problems} \author[J. P. Sun, H. E Zhang\hfil EJDE-2008/125\hfilneg] {Jian-Ping Sun, Hai-E Zhang} % 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{Hai-E Zhang \newline Department of Basic Teaching, Tangshan College\\ Tangshan, Hebei 063000, China} \email{ninthsister@tom.com} \thanks{Submitted March 25, 2008. Published September 4, 2008.} \thanks{Supported by grant 2007GS05333 from the NSF of Gansu Province of China} \subjclass[2000]{34B10, 34B15} \keywords{Third-order $m$-point boundary-value problem; Carath\'eodory; \hfill\break\indent Leray-Schauder continuation principle} \begin{abstract} This paper concerns the third-order $m$-point boundary-value problem \begin{gather*} u'''(t)+f(t,u(t),u'(t),u''(t))=0 ,\quad \text{a.e. } t\in (0,1), \\ u(0)=u'(0)=0, \quad u''(1)=\sum _{i=1}^{m-2}k_{i}u''(\xi_{i}), \end{gather*} where $f:[0,1]\times \mathbb{R}^{3}\to \mathbb{R}$ is $L_p$-Carath\'eodory, $1\leq p<+\infty$, $0=\xi_0<\xi _1<\dots <\xi _{m-2}<\xi_{m-1}=1$, $k_i\in \mathbb{R}$ ($i=1,2,\dots ,m-2$) and $\sum_{i=1}^{m-2}k_i\neq 1$. Some criteria for the existence of at least one solution are established by using the well-known Leray-Schauder Continuation Principle. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Third-order differential equations arise in a variety of areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three layer beam, electromagnetic waves or gravity driven flows and so on \cite{1}. Recently, third-order two-point or three-point boundary-value problems (BVPs for short) have received much attention \cite{9,10,2,3,5,4,6,7,11,12,8}. In particular, for two-point BVPs, Yao and Feng \cite{8} employed the upper and lower solution method to prove the existence of solutions for the problem $$\label{e1.1} \begin{gathered} u'''(t)+f(t,u(t))=0,\quad 0\leq t\leq 1, \\ u(0)=u'(0)=u'(1)=0. \end{gathered}$$ El-Shahed \cite{3} considered the existence of at least one positive solution for the problem \label{e1.2} \begin{gathered} u'''(t)+\lambda a(t)f(u(t))=0,\quad 00$such that if$u=\lambda Tu$for$\lambda \in (0,1)$, then$\Vert u\Vert \leq R$. Then$T$has a fixed point. \end{theorem} In the remainder of this section, we introduce some fundamental definitions. \begin{definition} \rm We say that a map$f:[0,1]\times \mathbb{R}^n\to \mathbb{R}$,$(t,x)\mapsto f(t,x)$is$L_p$-Carath\'eodory, if the following conditions are satisfied: \begin{enumerate} \item for each$x\in \mathbb{R}^n$, the mapping$t\mapsto f(t,x)$is Lebesgue measurable; \item for a.e.$t\in [0,1]$, the mapping$x\mapsto f(t,x)$is continuous on$\mathbb{R}^n$; \item for each$r>0$, there exists an$\alpha _r\in L_p[0,1]$such that for a.e.$t\in [0,1]$and every$x$with$|x|\leq r$,$|f(t,x)|\leq \alpha _r(t)$. \end{enumerate} \end{definition} Let$X=C^2[0,1]$. For$x\in X$, we use the norm$\| x\| =\max \{\| x\| _\infty ,\| x'\| _\infty ,\| x''\| _\infty \} $, where$\| x\| _\infty =\max_{t\in [0,1] }| x(t)| $. We denote the usual Lebesgue norm in$L_p[0,1]$by$\| \cdot \| _p$and the space of absolutely continuous functions on the interval$[0,1]$by$AC[0,1]. We also use the Sobolev space \begin{align*} W^{3,p}[0,1]=\big\{&u:[0,1]\to \mathbb{R}:u,u',u''\in AC[0,1],\; u(0)=u'(0)=0,\\ &u''(1)=\sum_{i=1}^{m-2}k_iu''(\xi _i),u'''\in L_p[0,1]\big\}. \end{align*} \section{Main results} \begin{lemma} \label{lem2.1} Lety\in L_p[0,1]$. Then the BVP $$\label{e2.1} \begin{gathered} u'''(t)+y(t)=0,\quad\text{a.e. }t\in (0,1), \\ u(0)=u'(0)=0,\quad u''(1)=\sum_{i=1}^{m-2}k_iu''(\xi _i) \end{gathered}$$ has a unique solution $u(t)=\int_0^1G_0(t,s)y(s)ds,$ which satisfies $u'(t)=\int_0^1G_1(t,s)y(s)ds, \quad u''(t)=\int_0^1G_2(t,s)y(s)ds,$ where, for$j=1,2,\dots ,m-1$, \begin{gather} \label{e2.2} G_0(t,s)=\begin{cases} \frac{\sum_{i=1}^{j-1}k_i}{2(1-\sum_{i=1}^{m-2}k_i)}t^2+ts-\frac 12s^2, & s\leq t,\; \xi _{j-1}t,\quad \xi _{j-1}t,\; \xi _{j-1}t,\; \xi _{j-1}t,\; \xi _{j-1}1$ and $p=1$. \noindent \textbf{Case 1: $p>1$.} Let $\frac 1p+\frac 1q=1$. Then by H\"older's inequality, $|u^{(i)}(t)|\leq \int_0^1| G_i(t,s)| | y(s)| ds\leq \| G_i(t,\cdot ) \| _q\| y\| _p\leq \max_{0\leq t\leq 1}\| G_i(t,\cdot )\| _q\| y\| _p,$ for $t\in [0,1]$, $i=0,1,2$. In view of Lemma \ref{lem2.2}, we have $\| G_i(t,\cdot )\| _q^q=\int_0^1| G_i( t,s)| ^qds\leq \int_0^1A_i^qds=A_i^q,\quad t\in [ 0,1],$ which implies that $\max_{0\leq t\leq 1}\| G_i(t,\cdot )\| _q\leq A_i$. So, $\Vert u^{(i)}\Vert _\infty \leq A_i\| y\| _p,\quad i=0,1,2.$ \noindent\textbf{Case 2: $p=1$.} By Lemma \ref{lem2.2}, we have $|u^{(i)}(t)|\leq \int_0^1| G_i(t,s)| | y(s)| ds\leq A_i\int_0^1| y(s) | ds=A_i\| y\| _1,$ for $t\in [0,1]$, $i=0,1,2$, which shows that $\Vert u^{(i)}\Vert _\infty \leq A_i\| y\| _1,\quad i=0,1,2.$ The proof is complete.\end{proof} Now, if we define the integral operator $T:X\to X$ by $Tu(t)=\int_0^1G_0(t,s)f(s,u(s),u'(s),u''(s))ds,\quad t\in [0,1],$ then it is obvious that if $u$ is a fixed point of $T$ in $X$, then $u$ is a solution of \eqref{e1.6}. \begin{lemma} \label{lem2.4} The mapping $T:X\to X$ is compact. \end{lemma} \begin{proof} At first, since $T$ is so-called the Hammerstein operator and $f$ is a $L_p$-Carath\'eodory function, we know that $T$ is continuous. Now, let $D\subset X$ be a bounded set, we will prove that $T(D)$ is relatively compact in $X$. Suppose that $\{ w_k\}_{k=1}^\infty \subset T(D)$ is an arbitrary sequence. Then there is $\{ u_k\} _{k=1}^\infty \subset D$ such that $T(u_k)=w_k$. Set $r=\sup_{u\in D} \| u\| .$ Since $f:[0,1]\times \mathbb{R}^3\to \mathbb{R}$ is $L_p$-Carath\'eodory, there exists $\alpha _r\in L_p[0,1]$ such that $| f(t,u_k(t),u_k'(t),u_k''(t))| \leq \alpha _r(t),\quad \text{a.e. }t\in [0,1],\quad k\in \mathbb{N}.$ Since the proof is similar for $p=1$, we only prove the case when $p>1$. First, it follows from H\"older's inequality and Lemma \ref{lem2.2} that \begin{align*} | w_k(t)| &= | Tu_k(t)| \\ &= \big| \int_0^1G_0(t,s)f(s,u_k(s),u_k'(s),u_k''(s))ds\big| \\ &\leq \int_0^1| G_0(t,s)| | f(s,u_k(s),u_k'(s),u_k''(s))| ds \\ &\leq \int_0^1| G_0(t,s)| \alpha _r(s) ds \\ &\leq \max_{t\in [0,1]}\| G_0(t,\cdot)\| _q\| \alpha _r\| _p \\ &\leq A_0\| \alpha _r\| _p,\quad t\in [0,1], \end{align*} which implies that $\{ w_k\} _{k=1}^\infty$ is uniformly bounded. Similarly, we get \begin{align*} | w_k'(t)| &= | Tu_k'(t)| \\ &= \big| \int_0^1G_1(t,s)f(s,u_k(s),u_k'(s),u_k''(s))ds\big| \\ &\leq \max_{t\in [0,1]}\| G_1(t,\cdot )\| _q\| \alpha _r\| _p \\ &\leq A_1\| \alpha _r\| _p,\quad t\in [0,1], \end{align*} which shows that $\{w_k'\} _{k=1}^\infty$ is also uniformly bounded. Therefore, $\{w_k\} _{k=1}^\infty$ is equicontinuous. By the Arzela-Ascoli theorem, $\{w_k\}_{k=1}^\infty$ has a convergent subsequence. Without loss of generality, we may assume that $\{w_k\} _{k=1}^\infty$ converges on $[0,1]$. Next, for all $t\in [0,1]$, we have \begin{align*} | w_k''(t)| &= | Tu_k''(t)| \\ &= \big| \int_0^1G_2(t,s)f(s,u_k(s),u_k'(s),u_k''(s))ds\big| \\ &\leq \max_{t\in [0,1]}\| G_2(t,\cdot )\| _q\| \alpha _r\| _p \\ &\leq A_2\| \alpha _r\| _p, \end{align*} that is to say, $\{w_k''\} _{k=1}^\infty$ is uniformly bounded, and so $\{w_k'\} _{k=1}^\infty$ is equicontinuous. As a result, without loss of generality, we may put that $\{w_k'\} _{k=1}^\infty$ is also convergent. Finally, for any $\varepsilon >0$, we can choose $\delta =\varepsilon^q/\| \alpha _r\| _p^q$ such that for any $k\in N$, $t_1$, $t_2\in [0,1]$ and $| t_2-t_1| <\delta$, \begin{align*} | w_k''(t_2)-w_k''(t_1)| &= | Tu_k''(t_2) -Tu_k''(t_1)| \\ &= \big| \int_{t_1}^{t_2}f(s,u_k(s),u_k'(s),u_k''(s))ds\big| \\ &\leq | t_2-t_1| ^{1/q}\| \alpha _r\|_p<\varepsilon , \end{align*} which shows that $\{w_k''\} _{k=1}^\infty$ is equicontinuous. Again, by the Arzela-Ascoli theorem, we know that $\{w_k''\} _{k=1}^\infty$ has a convergent subsequence. We establish that $\{w_k\} _{k=1}^\infty$ has a convergent subsequence in $X$. \end{proof} Now, we apply the Leray-Schauder Continuation Principle to obtain the existence of at least one solution for \eqref{e1.6}. \begin{theorem} \label{thm2.5} Assume that there exist $\alpha _0,\alpha _1,\alpha _2$ and $\delta \in L_p[0,1]$ such that \begin{gather} \label{e2.7} | f(t,x_0,x_1,x_2)| \leq \sum_{i=0}^2\alpha _i(t)x_i+\delta (t),\quad\text{a.e. }t\in (0,1)\,,\\ \label{e2.8} \sum_{i=0}^2A_i\| \alpha _i\| _p<1, \end{gather} where $A_i$ ($i=0,1,2$) is defined as in Lemma \ref{lem2.2}. Then \eqref{e1.6} has at least one solution. \end{theorem} \begin{proof} To complete the proof, it suffices to verify that the set of all possible solutions of the BVP $$\label{e2.9} \begin{gathered} u'''(t)+\lambda f(t,u(t),u'(t),u''(t))=0,\quad\text{a.e. }t\in (0,1),\\ u(0)=u'(0)=0,\quad u''(1)=\sum_{i=1}^{m-2}k_iu''(\xi _i) \end{gathered}$$ is a priori bounded in $X$ by a constant independent of $\lambda \in [0,1]$. Suppose that $u\in W^{3,p}[0,1]$ is a solution of \eqref{e2.9}. Then it follows from \eqref{e2.7}, Lemma \ref{lem2.2} and Lemma \ref{lem2.3} that \begin{align*} \Vert u'''\Vert _p &= \lambda \Vert f(t,u,u',u'')\Vert _p \\ &\leq \Vert f(t,u,u',u'')\Vert _p \\ &\leq \sum_{i=0}^2\Vert \alpha _iu^{(i)}\Vert _p+\Vert \delta \Vert _p \\ &\leq \sum_{i=0}^2\Vert \alpha _i\Vert _p\Vert u^{(i) }\Vert _\infty +\Vert \delta \Vert _p \\ &\leq \sum_{i=0}^2A_i\| \alpha _i\| _p\Vert u'''\Vert _p+\Vert \delta \Vert _p, \end{align*} which implies $\Vert u'''\Vert _p\leq \frac{\| \delta \| _p }{1-\sum_{i=0}^2A_i\| \alpha _i\| _p},$ and so, \begin{align*} \Vert u\Vert &= \max \{\| u\| _\infty ,\Vert u'\Vert _\infty ,\Vert u''\Vert _\infty \} \\ &\leq \max \{A_{0,}A_1,A_2\} \Vert u'''\Vert _p \\ &\leq \frac{A_{1}\| \delta \|_p}{1-\sum_{i=0}^2A_i\| \alpha _i\| _p}. \end{align*} It is now immediate from Theorem \ref{thm1.1} that $T$ has at least one fixed point, which is a desired solution of \eqref{e1.6}. \end{proof} \begin{thebibliography}{99} \bibitem{9} D. R. 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