\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
\begin{equation} \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}
\end{equation}
El-Shahed \cite{3} considered the existence of at least one positive
solution for the problem
\begin{equation} \label{e1.2}
\begin{gathered}
u'''(t)+\lambda a(t)f(u(t))=0,\quad 0<t<1, \\
u(0)=u'(0)=0,\quad \alpha u'(1)+\beta u''(1)=0
\end{gathered}
\end{equation}
by using the Guo-Krasnoselskii fixed point theorem. Hopkins and Kosmatov
\cite{6} obtained the existence of at least one solution for the problem
\begin{equation} \label{e1.3}
\begin{gathered}
u'''(t)=f( t,u(t),u'(t),u''(t)) ,\quad \text{a.e. }t\in (0,1), \\
u(0)=u'(0)=u''(1)=0.
\end{gathered}
\end{equation}
Their main tool was the Leray-Schauder Continuation Principle. For
three-point BVPs, Anderson \cite{9} studied the existence and
multiplicity of positive solutions for the problem
\begin{equation} \label{e1.4}
\begin{gathered}
x'''(t)=f(t,x(t)),\quad t_1\leq t\leq t_3, \\
x(t_1)=x'(t_2)=0,\quad \gamma x(t_3)+\delta x''(t_3)=0
\end{gathered}
\end{equation}
by using the Guo-Krasnoselskii and Leggett-Williams fixed point theorems.
Guo, Sun and Zhao \cite{4} considered the existence of at least one positive
solution to the problem
\begin{equation} \label{e1.5}
\begin{gathered}
u'''(t)+a(t)f(u(t))=0,\quad t\in (0,1), \\
u(0)=u'(0)=0,\quad u'(1)=\alpha u'(\eta ).
\end{gathered}
\end{equation}
The main tool was the Guo-Krasnoselskii fixed point theorem.

Although there are many excellent works on third-order two-point or
three-point BVPs, a little work has been done for more general
third-order $m$-point BVP \cite{13,14}. Moreover, almost all of the
existing literatures assumed that the nonlinear term was continuous.

Motivated  by the above-mentioned  works, in this paper we
investigate the third-order $m$-point BVP
\begin{equation} \label{e1.6}
\begin{gathered}
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_iu''(\xi _i).
\end{gathered}
\end{equation}
Throughout this paper, we assume that $f:[0,1]\times \mathbb{R}^3\to \mathbb{R}$
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
$\sum_{i=1}^{m-2}k_i\neq 1$.
Firstly, Green's function for associated linear BVP is constructed.
Secondly, some useful properties of the Green's function are
obtained. Finally, existence results of at least one solution for
the BVP \eqref{e1.6} are established by applying the well-known
Leray-Schauder Continuation Principle \cite{15}, which we state here
for convenience of the reader.

\begin{theorem} \label{thm1.1}
Let $X$ be a Banach space and $T:X\to X$ be a compact map. Suppose
that there exists an $R>0$ 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}
Let $y\in L_p[0,1]$. Then the BVP
\begin{equation} \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}
\end{equation}
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}<s\leq \xi _j, \\[4pt]
\frac{1-\sum_{i=j}^{m-2}k_i}{2(1-\sum_{i=1}^{m-2}k_i)}t^2,
& s>t,\quad \xi _{j-1}<s\leq \xi _j,
\end{cases}
\\
 \label{e2.3}
G_1(t,s)=\begin{cases}
\frac{\sum_{i=1}^{j-1}k_i}{1-\sum_{i=1}^{m-2}k_i}t+s,
& s\leq t,\; \xi _{j-1}<s\leq \xi _j, \\[4pt]
\frac{1-\sum_{i=j}^{m-2}k_i}{1-\sum_{i=1}^{m-2}k_i}t,
&s>t,\; \xi _{j-1}<s\leq \xi _j,
\end{cases}
\\
 \label{e2.4}
G_2(t,s)=\begin{cases}
\frac{\sum_{i=1}^{j-1}k_i}{1-\sum_{i=1}^{m-2}k_i},
&s\leq t,\; \xi _{j-1}<s\leq \xi _j, \\[4pt]
\frac{1-\sum_{i=j}^{m-2}k_i}{1-\sum_{i=1}^{m-2}k_i},
&s>t,\; \xi _{j-1}<s\leq \xi _j,
\end{cases}
\end{gather}
are called Green function. Here, if $l'<l$, then we let
$\sum_{i=l}^{l'}k_i=0$.
\end{lemma}

\begin{proof}
In view of  \eqref{e2.1} and the boundary condition
$u''(1)=\sum_{i=1}^{m-2}k_iu''(\xi _i)$, we have
\[
u''(t)=-\int_0^ty(s)ds+\frac
1{1-\sum_{i=1}^{m-2}k_i}\int_0^1y(s)ds-\frac
1{1-\sum_{i=1}^{m-2}k_i}\sum_{i=1}^{m-2}k_i\int_0^{\xi
_i}y(s)ds.
\]
If $0\leq t\leq \xi _1$, then
\[
u''(t)=\int_t^{\xi _1}y(s)
ds+\sum_{j=2}^{m-2}\int_{\xi _{j-1}}^{\xi _j}\frac{
1-\sum_{i=j}^{m-2}k_i}{1-\sum_{i=1}^{m-2}k_i}y(s)
ds+\int_{\xi _{m-2}}^1\frac 1{1-\sum_{i=1}^{m-2}k_i}y(s)
ds,
\]
which together with the boundary conditions $u(0)=u'(0)=0$ imply
\begin{align*}
u'(t)
&=\int_0^tsy(s)ds+\int_t^{\xi
_1}ty(s)ds+\sum_{j=2}^{m-2}\int_{\xi _{j-1}}^{\xi _j}
\frac{1-\sum_{i=j}^{m-2}k_i}{1-\sum_{i=1}^{m-2}k_i}ty(s)ds\\
&\quad +\int_{\xi _{m-2}}^1\frac 1{1-\sum_{i=1}^{m-2}k_i}ty(s)ds
\end{align*}
and
\begin{align*}
u(t)&=\int_0^t(ts-\frac 12s^2)y(s)ds+\int_t^{\xi
_1}\frac 12t^2y(s)ds\\
&\quad +\sum_{j=2}^{m-2}k_i\int_{\xi
_{j-1}}^{\xi _j}\frac{1-\sum_{i=j}^{m-2}k_i}{2(
1-\sum_{i=1}^{m-2}k_i)}t^2y(s)ds
+\int_{\xi _{m-2}}^1\frac 1{2(1-\sum_{i=1}^{m-2}k_i)}t^2y(s)ds.
\end{align*}
If $\xi _{l-1}<t\leq \xi _l\quad (l=2,3,\dots ,m-2)$, then
\begin{align*}
u''(t)
&= \sum_{j=2}^{l-1}\int_{\xi_{j-1}}^{\xi _j}
 \frac{\sum_{i=1}^{j-1}k_i}{1-\sum_{i=1}^{m-2}k_i}y(s)ds
+\int_{\xi _{l-1}}^t\frac{\sum_{i=1}^{l-1}k_i}{1-\sum_{i=1}^{m-2}k_i}y(s)ds\\
&\quad +\int_t^{\xi _l}
 \frac{1-\sum_{i=l}^{m-2}k_i}{1-\sum_{i=1}^{m-2}k_i}y(s)ds
 +\sum_{j=l+1}^{m-2}\int_{\xi _{j-1}}^{\xi _j}\frac{
 1-\sum_{i=j}^{m-2}k_i}{1-\sum_{i=1}^{m-2}k_i}y(s)ds\\
&\quad +\int_{\xi _{m-2}}^1\frac 1{1-\sum_{i=1}^{m-2}k_i}y(s)ds,
\end{align*}
which together with the boundary conditions $u(0)=u'(0)=0$ imply
\begin{align*}
u'(t)&= \int_0^{\xi _1}sy(s)ds
 +\sum_{j=2}^{l-1}\int_{\xi _{j-1}}^{\xi _j}\Big(\frac{
 \sum_{i=1}^{j-1}k_i}{1-\sum_{i=1}^{m-2}k_i}t+s\Big)y(s)ds\\
&\quad +\int_{\xi _{l-1}}^t\big(\frac{\sum_{i=1}^{l-1}k_i}{
  1-\sum_{i=1}^{m-2}k_i}t+s\Big)y(s)ds
  +\int_t^{\xi _l}\frac{1-\sum_{i=l}^{m-2}k_i}{1-\sum_{i=1}^{m-2}k_i}ty(s)ds\\
&\quad+\sum_{j=l+1}^{m-2}\int_{\xi _{j-1}}^{\xi _j}\frac{
  1-\sum_{i=j}^{m-2}k_i}{1-\sum_{i=1}^{m-2}k_i}ty(s)ds
 +\int_{\xi _{m-2}}^1\frac 1{1-\sum_{i=1}^{m-2}k_i}ty(s)ds
\end{align*}
and
\begin{align*}
&u(t)\\
&= \int_0^{\xi _1}(ts-\frac 12s^2)y(
s)ds+\sum_{j=2}^{l-1}k_i\int_{\xi _{j-1}}^{\xi _j}\big(\frac{
\sum_{i=1}^{j-1}k_i}{2(1-\sum_{i=1}^{m-2}k_i)}
t^2+ts-\frac 12s^2\Big)y(s)ds \\
&\quad +\int_{\xi _{l-1}}^t\Big(\frac{\sum_{i=1}^{l-1}k_i}{2(
1-\sum_{i=1}^{m-2}k_i)}t^2+ts-\frac 12s^2\Big)y(
s)ds+\int_t^{\xi _l}\frac{1-\sum_{i=l}^{m-2}k_i}{2(
1-\sum_{i=1}^{m-2}k_i)}t^2y(s)ds \\
&\quad +\sum_{j=l+1}^{m-2}\int_{\xi _{j-1}}^{\xi _j}\frac{
1-\sum_{i=j}^{m-2}k_i}{2(1-\sum_{i=1}^{m-2}k_i)}
t^2y(s)ds+\int_{\xi _{m-2}}^1\frac 1{2(
1-\sum_{i=1}^{m-2}k_i)}t^2y(s)ds.
\end{align*}
Similarly, if $\xi _{m-2}<t\leq 1$,  then we get
\begin{align*}
u''(t)&=\sum_{j=2}^{m-2}\int_{\xi
_{j-1}}^{\xi _j}\frac{\sum_{i=1}^{j-1}k_i}{1-\sum
_{i=1}^{m-2}k_i}y(s)ds+\int_{\xi _{m-2}}^t\frac{
\sum_{i=1}^{m-2}k_i}{1-\sum_{i=1}^{m-2}k_i}y(s)ds\\
&\quad +\int_t^1\frac 1{1-\sum_{i=1}^{m-2}k_i}y(s)ds,
\end{align*}
\begin{align*}
u'(t)&= \int_0^{\xi _1}sy(s)
ds+\sum_{j=2}^{m-2}\int_{\xi _{j-1}}^{\xi _j}\Big(\frac{
\sum_{i=1}^{j-1}k_i}{1-\sum_{i=1}^{m-2}k_i}t+s\Big)y(s)ds\\
&\quad +\int_{\xi _{m-2}}^t\Big(\frac{\sum_{i=1}^{m-2}k_i}{
1-\sum_{i=1}^{m-2}k_i}t+s\Big)y(s)ds
 +\int_t^1\frac 1{1-\sum_{i=1}^{m-2}k_i}ty(s)ds
\end{align*}
and
\begin{align*}
&u(t)\\
&= \int_0^{\xi _1}(ts-\frac 12s^2)y(
s)ds+\sum_{j=2}^{m-2}k_i\int_{\xi _{j-1}}^{\xi _j}\Big(\frac{
\sum_{i=1}^{j-1}k_i}{2(1-\sum_{i=1}^{m-2}k_i)}
t^2+st-\frac 12s^2\Big)y(s)ds \\
&\quad +\int_{\xi _{m-2}}^t\Big(\frac{\sum_{i=1}^{m-2}k_i}{2(
1-\sum_{i=1}^{m-2}k_i)}t^2+st-\frac 12s^2\Big)y(
s)ds+\int_t^1\frac 1{2(1-\sum_{i=1}^{m-2}k_i)}t^2y(s)ds.
\end{align*}
Summing up, we obtain the relationships:
\[
u^{(i)}(t)=\int_0^1G_i(t,s)y(s)ds,\quad t\in [0,1],\; i=0,1,2.
\]
\end{proof}

\begin{lemma} \label{lem2.2}
Let
\[
A_0=\frac{\sum_{i=1}^{m-2}| k_i|+\max \big\{
| 1-\sum_{i=1}^{m-2}k_i|,\quad 1\big\}}
{2| 1-\sum_{i=1}^{m-2}k_i| }, \quad
A_1=A_2=2A_0.
\]
Then the Green functions $G_i(t,s)\quad (i=0,1,2)$
satisfy
\begin{equation} \label{e2.5}
| G_i(t,s)| \leq A_i,\quad (t,s)\in [0,1]\times [0,1].
\end{equation}
\end{lemma}

\begin{proof}
Since the proof of \eqref{e2.5} is very similar for $i=0,1,2$, we
only prove the case when $i=0$. In fact, for $j=1,2,\dots ,m-1$,
\[
| G_0(t,s)| \leq
\begin{cases}
\frac{\sum_{i=1}^{j-1}| k_i| }{2|
 1-\sum_{i=1}^{m-2}k_i| }t^2+| ts-\frac 12s^2| \\
\leq  \frac{\sum_{i=1}^{m-2}| k_i| }{2|1-\sum_{i=1}^{m-2}k_i| }t^2
 +\frac 12t^2\leq A_0,
&s\leq t, \; \xi _{j-1}<s\leq \xi _j, \\[4pt]
\frac{1+\sum_{i=j}^{m-2}| k_i| }{2|
 1-\sum_{i=1}^{m-2}k_i| }t^2
 \leq\frac{1+\sum_{i=1}^{m-2}| k_i| }{2|
 1-\sum_{i=1}^{m-2}k_i| }t^2\leq A_0,
& s>t,\; \xi _{j-1}<s\leq \xi _j\,.
\end{cases}
\]
\end{proof}

\begin{lemma} \label{lem2.3}
Let $y\in L_p[0,1]$. Then the unique solution of  \eqref{e2.1} satisfies
\begin{equation} \label{e2.6}
\Vert u^{(i)}\Vert _\infty \leq A_i\| y\| _p,\quad i=0,1,2,
\end{equation}
where $A_i$ ($i=0,1,2$) is defined as in Lemma \ref{lem2.2}.
\end{lemma}

\begin{proof}
We divide the proof into two cases: $p>1$ 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
\begin{equation} \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}
\end{equation}
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}

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\end{document}