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\markboth{\hfil A non-resonant multi-point boundary-value problem \hfil EJDE/Conf/10}
{EJDE/Conf/10 \hfil Chaitan P. Gupta \hfil}

\begin{document}
\setcounter{page}{143}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Fifth Mississippi State Conference on Differential Equations and
Computational Simulations, \newline
Electronic Journal of Differential Equations,
Conference 10, 2003, pp 143--152. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 A non-resonant multi-point boundary-value problem for a $p$-Laplacian
 type operator
%
\thanks{ {\em Mathematics Subject Classifications:} 34B10, 34B15, 34G20.
\hfil\break\indent
{\em Key words:}
 Multi-point boundary-value problem, three-point boundary-value problem,
\hfil\break\indent
 $p$-Laplacian, Leray Schauder Continuation theorem,
 Caratheodory's conditions.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Published February 28, 2003. } }

\date{}
\author{Chaitan P. Gupta}
\maketitle

\begin{abstract}
 Let $\phi $ be an odd increasing homeomorphism from $\mathbb{R}$ onto
 $\mathbb{R}$ with $\phi (0)=0$,
 $f:[0$,$1]\times \mathbb{R}^{2}\to \mathbb{R}$ be a
 function satisfying Caratheodory's conditions and $e(t)\in L^{1}[0,1]$.
 Let $\xi_{i}\in (0,1)$, $a_{i}\in \mathbb{R}$, $i=1,2, \dots , m-2$,
 $\sum_{i=1}^{m-2}a_{i}\neq 1$,
 $0<\xi_{1}<\xi_{2}<\dots<\xi_{m-2}<1$ be given.
 This paper is concerned with the problem of existence of a solution for
 the multi-point boundary-value problem
 \begin{gather*}
 (\phi (x'(t)))'=f(t,x(t),x'(t))+e(t),\quad 0<t<1, \\
 x(0)=0, \quad \phi (x'(1))=\sum_{i=1}^{m-2}a_{i}\phi(x'(\xi_{i})).
 \end{gather*}
 This paper gives conditions for the existence of a solution for
 the above boundary-value problem using some new Poincar\'{e} type
 a priori estimates. In the case $\phi (t)\equiv t$ for $t\in \mathbb{R}$,
 this problem was studied earlier by Gupta, Trofimchuk in
 \cite{gt3} and by Gupta, Ntouyas and Tsamatos in \cite{gnt1}. We
 give a priori estimates needed for this problem
 that are similar to a priori estimates obtained by Gupta, Trofimchuk in
 \cite{gt3}. We then obtain existence theorems for the above multi-point
 boundary-value problem using the a priori estimates and
 Leray-Schauder continuation theorem.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\numberwithin{equation}{section}

\section{Introduction}

Let $\phi $ be an odd increasing homeomorphism from $\mathbb{R}$ onto
$\mathbb{R}$ with $\phi (0)=0$,
$f:[0,1]\times \mathbb{R}^{2}\to \mathbb{R}$ be a function satisfying
Caratheodory's conditions, $e:[0,1]\mapsto \mathbb{R}$ be a function in
$L^{1}[0,1]$, $a_{i}\in \mathbb{R}$, $\xi_{i}\in (0,1)$, $i=1,2,\dots,
m-2$, $\sum_{i=1}^{m-2}a_{i}\neq 1$, $0<\xi_{1}<\xi_{2}<\dots<\xi_{m-2}<1$
be given. We study the problem of existence of solutions for the $m$-point
boundary-value problem
\begin{equation}
\begin{gathered}
(\phi (x'(t)))'=f(t,x(t),x'(t))+e(t), \quad 0<t<1, \\
x(0)=0, \quad \phi (x'(1))=\sum_{i=1}^{m-2}a_{i}\phi (x'(\xi_{i})).
\end{gathered} \label{mbp}
\end{equation}
This problem was studied earlier in the case $\phi (t)\equiv t$ for
$t\in \mathbb{R}$, by Gupta, Trofimchuk in \cite{gt3} and by Gupta, Ntouyas and
Tsamatos in \cite{gnt1}. Gupta, Ntouyas and Tsamatos had studied
the problem (\ref{mbp}) when all of the $a_{i}\in \mathbb{R}$,
$i=1,2,\dots, m-2$, had the same sign by first studying the three-point
boundary-value problem, for a given $\alpha \in \mathbb{R}$, $\alpha \neq 1$,
$\eta\in (0,1)$,
\begin{equation}
\begin{gathered}
x''(t)=f(t,x(t),x'(t))+e(t),\quad 0<t<1,\\
x(0)=0, \quad x'(1)=\alpha x'(\eta ),
\end{gathered}\label{3bp}
\end{equation}
while Gupta, Trofimchuk in \cite{gt3} studied the problem
(\ref{mbp}) when the $a_{i}\in \mathbb{R}$, $i=1, 2, \dots, m-2$, do not
necessarily have the same sign.

We also study the three-point boundary-value problem analogue of (\ref{mbp}),
for a given $\alpha \in \mathbb{R}$, $\alpha \neq 1$, $\eta \in (0,1)$,
\begin{equation}
\begin{gathered}
(\phi (x'(t)))'=f(t,x(t),x'(t))+e(t),\quad  0<t<1, \\
x(0)=0, \quad \phi (x'(1))=\alpha \phi (x'(\eta )).
\end{gathered} \label{phi3bp}
\end{equation}

The purpose of this paper is to obtain conditions for the
existence of a solution for the boundary-value problem
(\ref{mbp}), using new estimates and inequalities involving a
function $x(t)$, its derivative $x'(t)$, the functions $\phi
(x'(t))$ and its derivative $(\phi (x'(t)))'$. These results are
motivated by the so called {\it nonlocal }boundary-value problem
studied by Il'in and Moiseev in \cite{VA}. We may mention that the
reason for studying the three-point boundary-value problem
(\ref{phi3bp}) is that in this case we obtain a better existence
theorem using a priori estimates involving $L_{2}$ norm.

We use the classical spaces $C[0,1]$, $C^{k}[0,1]$, $L^{k}[0,1]$, and
$L^{\infty }[0,1]$ of continuous, $k$-times continuously
differentiable, measurable real-valued functions whose $k$-th
power of the absolute value is Lebesgue integrable on $[0,1]$,
or measurable functions that are essentially bounded on $[0,1]$.
We also use the Sobolev spaces $W_{\phi }^{2,k}(0,1)$, $k=1,2$
defined by
\[
W_{\phi }^{2,k}(0,1)=\big\{x:[0,1]\to R :
x,x'\text{ abs. cont. on }[0,1], (\phi (x'(t)))'\in
L^{k}[0,1]\big\}
\]
with its usual norm. We denote the norm in $L^{k}[0,1]$ by
$\|\cdot\|_{k}$, and the norm in $L^{\infty }[0,1]$ by
$\|\cdot\|_{\infty }$.

\section{A Priori Estimates}

Let $a_{i}\in \mathbb{R}$, $\xi_{i}\in (0,1)$, $i=1, 2, \dots, m-2$,
$0<\xi_{1}<\xi_{2}<\dots<\xi_{m-2}<1$, with
$\alpha =\sum_{i=1}^{m-2}a_{i}\neq 1$ be given.
Let $x(t)\in W_{\phi }^{2,1}(0,1)$ be such that $x(0)=0$, $\phi
(x'(1))=\sum_{i=1}^{m-2}a_{i}\phi (x'(\xi_{i}))$ be given. We are
interested in obtaining a priori estimates of the form
$\| \phi (x'(t))\|_{\infty }\leq C\| (\phi(x'(t)))'\|_{1}$. The
following theorem gives such an estimate.
We recall that for $a\in \mathbb{R}$, $a_{+}=\max \{a,0\}$,
 $a_{-}=\max \{-a,0\}$ so that $a=a_{+}-a_{-}$
and $|a| =a_{+}+a_{-}$.

\begin{theorem} \label{ONE}
 Let $a_{i}\in \mathbb{R}$, $\xi_{i}\in (0,1)$,
$i=1, 2, \dots, m-2$, $0<\xi_{1}<\xi_{2}<\dots<\xi_{m-2}<1$, with
$\alpha =\sum_{i=1}^{m-2}a_{i}\neq 1$
be given. Then for $x(t)\in W_{\phi }^{2,1}(0,1)$ with $x(0)=0$,
$\phi (x'(1))=\sum_{i=1}^{m-2}a_{i}\phi (x'(\xi_{i}))$ we have
\begin{equation}
\| \phi (x'(t))\|_{\infty }\leq \frac{1}{1-\tau }
\| (\phi (x'(t)))'\|_{1} \label{est1}
\end{equation}
where either $\tau =0$ or
\[
\tau =\min \Big\{\frac{\sum_{i=1}^{m-2}(a_{i})_{+}}{%
\sum_{i=1}^{m-2}(a_{i})_{-}+1},\frac{\sum_{i=1}^{m-2}(a_{i})_{-}+1}{%
\sum_{i=1}^{m-2}(a_{i})_{+}}\Big\}.
\]
\end{theorem}

\paragraph{Proof} We see that the assumption $\phi
(x'(1))=\sum_{i=1}^{m-2}a_{i}\phi (x'(\xi_{i}))$ implies
\[
\phi (x'(1))+\sum_{i=1}^{m-2}(a_{i})_{-}\phi (x'(\xi
_{i}))=\sum_{i=1}^{m-2}(a_{i})_{+}\phi (x'(\xi_{i}))
\]
and thus there exist $\lambda_{1}$, $\lambda_{2}\in [
0,1]$ such that
\begin{equation}
\big(1+\sum_{i=1}^{m-2}(a_{i})_{-}\big)\phi (x'(\lambda
_{1}))=\sum_{i=1}^{m-2}(a_{i})_{+}\phi (x'(\lambda_{2}))\text{. }
\label{EQ3}
\end{equation}
If, now, either $x'(\lambda_{1})=0$ or $x'(\lambda_{2})=0 $, so
that either $\phi (x'(\lambda_{1}))=0$ or $\phi (x'(\lambda
_{2}))=0$, then we clearly have
\begin{equation}
\| \phi (x'(t))\|_{\infty }\leq \| (\phi
(x'(t)))'\|_{1}.  \label{est4}
\end{equation}
Suppose, now, that $x'(\lambda_{1})\neq 0$ and $x'(\lambda
_{2})\neq 0$, so that $\phi (x'(\lambda_{1}))\neq 0$ and $\phi
(x'(\lambda_{2}))\neq 0$. It then follows easily from equation
(\ref{EQ3}) that $\phi (x'(\lambda_{1}))\neq \phi (x'(\lambda
_{2}))$, in view of the assumption $\alpha
=\sum_{i=1}^{m-2}a_{i}\neq 1$. It then follows from equation
(\ref{EQ3}), the estimate (\ref{est4}) and the equations
\begin{gather*}
\phi (x'(t)) =\phi (x'(\lambda_{1}))+\int_{\lambda_{1}}^{t}(\phi (x'(s)))'ds, \\
\phi (x'(t)) =\phi (x'(\lambda_{2}))+\int_{\lambda
_{2}}^{t}(\phi (x'(s)))'ds,
\end{gather*}
that
\begin{equation}
\begin{gathered}
\| \phi (x'(t))\|_{\infty }\leq \frac{1}{1-\tau }%
\| (\phi (x'(t)))'\|_{1}  \nonumber \\
\text{ with }\tau =\min \{\frac{\sum_{i=1}^{m-2}(a_{i})_{+}}{%
\sum_{i=1}^{m-2}(a_{i})_{-}+1},\frac{\sum_{i=1}^{m-2}(a_{i})_{-}+1}{%
\sum_{i=1}^{m-2}(a_{i})_{+}}\}.
\end{gathered}\label{est3}
\end{equation}
This completes the proof of the theorem. \hfill$\square$

\paragraph{Remark} % 1
We note that if $a_{i}\leq 0$ for every $i=1, 2, \dots, m-2$,
 then $\tau =0$. Also, if $a_{i}\geq 0$ for every
$i=1,2,\dots, m-2$ so that
$\alpha=\sum_{i=1}^{m-2}a_{i}=\sum_{i=1}^{m-2}(a_{i})_{+}\geq 0$
then $\tau =\min \{\alpha ,1/\alpha \}\in [0,1)$ since
$\alpha \neq 1$, by assumption.

The following theorem gives a
better estimate for the three-point boundary-value problem in the
case of the $L^{2}$ norm.

\begin{theorem} \label{TWO}
 Let $\alpha \in \mathbb{R}$, $\alpha \neq 1$, $\eta \in (0,1)$ be given.
Let $x(t)\in W_{\phi }^{2,2}(0,1)$ be such that
$\phi (x'(1))=\alpha \phi (x'(\eta ))$. Then
\begin{equation}
\| \phi (x'(t))\|_{2}\leq C(\alpha ,\eta )\| (\phi
(x'(t)))'\|_{2}, \label{est5}
\end{equation}
where
\begin{gather*}
C(\alpha ,\eta ) =\begin{cases}
\min \{\sqrt{F(\alpha ,\eta )}, 2/\pi\} & \text{if  }\alpha \leq 0, \\
\sqrt{F(\alpha ,\eta )},  &\text{if }\alpha >0,
\end{cases}\\
F(\alpha ,\eta ) =\frac{1}{2(\alpha -1)^{2}}[\alpha ^{2}(1-\eta
)^{2}+(\alpha ^{2}-2\alpha )\eta ^{2}+1].
\end{gather*}
\end{theorem}

\paragraph{Proof}
If $\alpha \leq 0$, we note from $\phi (x'(1))=\alpha \phi(x'(\eta ))$
that there exists an $\xi \in (\eta ,1)$ such that $\phi (x'(\xi ))=0$.
It follows from the Wirtinger's inequality \cite[Theorem 256]{hardy} that
\begin{equation}
\| \phi (x'(t))\|_{2}\leq \frac{2}{\pi }\| (\phi (x'(t)))'\|_{2}.
\label{Eq0}
\end{equation}
Next, we note, again, from $\phi (x'(1))=\alpha \phi (x'(\eta ))$
that for $0<t<1$,
\begin{equation} \label{Eq1}
\phi (x'(t))=\int_{0}^{t}(\phi (x'(s)))'ds +\frac{\alpha
}{1-\alpha }\int_{0}^{\eta }(\phi (x'(s)))'ds -\frac{1}{1-\alpha
}\int_{0}^{1}(\phi (x'(s)))'ds.
\end{equation}
Accordingly, we have for $t\in [0,\eta]$
\begin{eqnarray}
\phi (x'(t)) &=&\int_{0}^{t}(\phi (x'(s)))'ds+%
\frac{\alpha }{1-\alpha }\int_{0}^{\eta }(\phi (x'(s)))'ds-%
\frac{1}{1-\alpha }\int_{0}^{1}(\phi (x'(s)))'ds \nonumber \\
&=&\int_{0}^{t}(1+\frac{\alpha }{1-\alpha }-\frac{1}{1-\alpha
})(\phi
(x'(s)))'ds  \nonumber \\
&&+\int_{t}^{\eta }(\frac{\alpha }{1-\alpha }-\frac{1}{1-\alpha
})(\phi (x'(s)))'ds-\frac{1}{1-\alpha }\int_{\eta }^{1}(\phi
(x'(s)))'ds  \nonumber \\
&=&-\int_{t}^{\eta }(\phi (x'(s)))'ds-\frac{1}{1-\alpha }%
\int_{\eta }^{1}(\phi (x'(s)))'ds, \label{EQ1}
\end{eqnarray}
and, for $t\in [ \eta ,1]$
\begin{eqnarray}
\phi (x'(t)) &=&\int_{0}^{t}(\phi (x'(s)))'ds+%
\frac{\alpha }{1-\alpha }\int_{0}^{\eta }(\phi (x'(s)))'ds-
\frac{1}{1-\alpha }\int_{0}^{1}(\phi (x'(s)))'ds \nonumber
\\
&=&\int_{0}^{\eta }(1+\frac{\alpha }{1-\alpha }-\frac{1}{1-\alpha
})(\phi (x'(s)))'ds+\int_{\eta }^{t}(1-\frac{1}{1-\alpha })(\phi
(x'(s)))'ds  \nonumber \\
&&-\frac{1}{1-\alpha }\int_{t}^{1}(\phi (x'(s)))'ds
\nonumber \\
&=&-\int_{\eta }^{t}\frac{\alpha }{1-\alpha }(\phi
(x'(s)))'ds-\frac{1}{1-\alpha }\int_{t}^{1}(\phi (x'(s)))'ds.
\label{EQ2}
\end{eqnarray}
Let us, now, define a function $K:[0,1]\times [0,1]\to \mathbb{R}$ by
\begin{equation}
K(t,s)=\begin{cases}
-\chi_{[ t,\eta ]}(s)-\frac{1}{1-\alpha }\chi_{[ \eta ,1]}(s),
&\text{for }t\in [ 0,\eta ], s\in [ 0,1],\\
-\frac{\alpha }{1-\alpha }\chi_{[ \eta ,t]}(s)-\frac{1}{1-\alpha }
\chi_{[ t,1]}(s), &\text{for }t\in [\eta ,1], s\in [ 0,1].
\end{cases} \label{EQ4}
\end{equation}
Now, we see from equations (\ref{EQ1}), (\ref{EQ2}) that
\begin{equation}
\phi (x'(t))=\int_{0}^{1}K(t,s)(\phi (x'(s)))'ds, \quad \text{for
}t\in [0,1],  \label{EQ5}
\end{equation}
and
\begin{equation}
\| \phi (x'(t))\|_{2}^{2}\leq
(\int_{0}^{1}\int_{0}^{1}(K(t,s))^{2}\,ds\,dt) \|
(\phi(x'(s)))'\|_{2}^{2}.  \label{EQ6}
\end{equation}
Now, it is easy to see that
\begin{equation}
\int_{0}^{1}\int_{0}^{1}(K(t,s))^{2}\,ds\,dt
=\frac{1}{2(\alpha-1)^{2}}[\alpha ^{2}(1-\eta )^{2}
+(\alpha ^{2}-2\alpha )\eta^{2}+1].  \label{EQ7}
\end{equation}
For $\alpha \leq 0$ the estimate (\ref{est5}) is now immediate from
(\ref{Eq0}), (\ref{EQ6}), (\ref{EQ7}) and for $\alpha >0$, $\alpha \neq 1$,
by (\ref{EQ6}), (\ref{EQ7}). This completes the proof of the
theorem. \hfill$\square$

\paragraph{Remark} %2
 It is easy to see that $C(-0.1,\eta )=2/\pi$, for
all $\eta \in (0,1)$. Indeed, $\sqrt{F(-0.1,\eta )}\geq 0.648986183$ and
$2/\pi\approx 0.6366197724$. Also
$C(-2,1/3)=\sqrt{11/54}$ and $C(-2,15/16)=2/\pi$,
since $\sqrt{F(-2,15/16)}=\sqrt{1030}/48>2/\pi$.

\section{Existence Theorems}

\paragraph{Definition}
A function $f:[0,1]\times \mathbb{R}^{2}\to \mathbb{R}$ satisfies
Caratheodory's conditions if
 \begin{itemize}
\item[(i)] For each $(x$,$y)\in \mathbb{R}^{2}$, the function
$t\in [0,1]\to f(t,x,y)\in \mathbb{R}$ is measurable on $[0,1]$
\item[(ii)] for a.e. $t\in [0,1]$, the function
$(x,y)\in \mathbb{R}^{2}\to f(t,x,y)\in \mathbb{R}$ is continuous on
$\mathbb{R}^{2}$
\item[(iii)] for each $r>0$, there exists $\alpha_{r}(t)\in L^{1}[0,1]$
such that $| f(t,x,y)|\leq \alpha_{r}(t)$ for a.e. $t\in [0,1]$ and all
$(x,y)\in \mathbb{R}^{2}$ with $\sqrt{x^{2}+y^{2}}\leq r$.
\end{itemize}

\begin{theorem} \label{one}
Let $f:[0,1]\times \mathbb{R}^{2}\to \mathbb{R}$ be a function
satisfying Caratheodory's conditions. Assume that there exist functions
$p(t) $, $q(t)$, $r(t)$ in $L^{1}(0,1)$ such that
\begin{equation}
| f(t,x_{1},x_{2})| \leq p(t)\phi (| x_{1}|)+q(t)\phi (| x_{2}| )+r(t)  \label{cond1}
\end{equation}
for a.e. $t\in [ 0,1]$ and all $(x_{1},x_{2})\in \mathbb{R}^{2}$.
Also let $a_{i}\in \mathbb{R}$, $\xi_{i}\in (0,1)$, $i=1,2,\dots, m-2$,
$0<\xi_{1}<\xi_{2}<\dots<\xi_{m-2}<1$, with
$\alpha=\sum_{i=1}^{m-2}a_{i}\neq 1$ be given.
Then the boundary-value problem (\ref{mbp}) has at least one solution in
$C^{1}[0,1]$ provided
\begin{equation}
\| p(t)\|_{1}+\| q(t)\|_{1}+\tau<1. \label{cond2}
\end{equation}
where $\tau$ is as defined in Theorem \ref{ONE}.
\end{theorem}

\paragraph{Proof}
It is easy to see that the boundary-value problem
(\ref{mbp}) is equivalent to the fixed point problem
\begin{equation}
x(t)=\int_{0}^{t}\phi ^{-1}\Big(\int_{0}^{s}[f(\tau ,x(\tau ),x'(\tau
))+e(\tau )]d\tau +A\Big)ds,   \label{EQ8}
\end{equation}
where
\begin{eqnarray*}
A&=&\sum_{i=1}^{m-2}(\frac{a_{i}}{1-\sum_{i=1}^{m-2}a_{i}})\int_{0}^{\xi
_{i}}[f(\tau ,x(\tau ),x'(\tau ))+e(\tau )]d\tau \\
&&-\frac{1}{1-\sum_{i=1}^{m-2}a_{i}}\int_{0}^{1}[f(\tau ,x(\tau
),x'(\tau ))+e(\tau )]d\tau .
\end{eqnarray*}
It is standard to check that the mapping
\[
x(t)\in C^{1}[0,1]\mapsto \int_{0}^{t}\phi
^{-1}(\int_{0}^{s}[f(\tau ,x(\tau ),x'(\tau ))+e(\tau )]d\tau
+A)ds\in C^{1}[0,1],
\]
is a compact mapping. We apply the Leray-Schauder Continuation
theorem (see, e.g. \cite{mawhin}) to obtain the existence of a
solution for (\ref{EQ8}) or equivalently to the boundary-value problem
(\ref{mbp}).

To do this, it suffices to verify that
the set of all possible solutions of the family of equations
\begin{equation}
\begin{gathered}
(\phi (x'(t)))'=\lambda f(t,x(t),x'(t))+\lambda e(t),\quad 0<t<1, \\
x(0)=0, \quad \phi (x'(1))=\sum_{i=1}^{m-2}a_{i}\phi (x'(\xi_{i})),
\end{gathered} \label{fbp}
\end{equation}
is, a priori, bounded in $C^{1}[0,1]$ by a constant independent of
$\lambda \in [ 0,1]$.
We observe that if $x\in W_{\phi
}^{2,1}(0,1)$, with $x(0)=0$,
$\phi (x'(1))=\sum_{i=1}^{m-2}a_{i}\phi (x'(\xi_{i}))$ then
$x(t)=\int_{0}^{t}x'(s)ds$. Hence, $| x(t)| \leq \|x'\|_{\infty }$ for
$t\in [ 0,1]$ and
\[
\|\phi (x'(t))\|_{\infty }\leq \frac{1}{1-\tau }\|
(\phi (x'(t)))'\|_{1},
\]
where $\tau $ is as defined in
Theorem \ref{ONE}. Also, it is easy to see that
$\phi (\|x'\|_{\infty })\leq \| \phi (x'(t))\|_{\infty }$.

Let, now, $x(t)$ be a solution of (\ref{fbp}) for some $\lambda
\in[ 0,1]$, so that $x\in W_{\phi }^{2,1}(0,1)$ with $x(0)=0,\phi
(x'(1))=\sum_{i=1}^{m-2}a_{i}\phi (x'(\xi_{i}))$. We then get from
the equation in (\ref{fbp}) and Theorem \ref{ONE} t
hat
\begin{align*}
\| \phi (x'(t))\|_{\infty}
&\leq \frac{\lambda}{1-\tau}\| f(t,x(t),x'(t))+e(t)\|_{1} \\
&\leq \frac{1}{1-\tau }(\| p(t)\phi (| x(t)|)
+q(t)\phi (| x'(t)|)+r(t)\|_{1}+\|e(t)\|_{1}) \\
&\leq \frac{1}{1-\tau }\big(\{\| p(t)\|_{1}+\|
q(t)\|_{1}\}\| \phi (| x'(t)| )\|_{\infty }
+\| r(t)\|_{1}+\| e(t)\|_{1}\big) \\
&\leq \frac{1}{1-\tau }(\| p(t)\|_{1}+\|q(t)\|_{1})\|
\phi (x'(t))\|_{\infty }+\frac{1}{1-\tau }(\| r(t)\|_{1}+\| e(t)\|_{1}).
\end{align*}
It follows from the assumption (\ref{cond2}) that there is a
constant $c$, independent of $\lambda \in [ 0,1]$, such that
\[
\| x\|_{\infty }\leq \| x'\|_{\infty}\leq c.
\]
It is now immediate that the set of solutions of the family of equations
(\ref{fbp}) is, a priori, bounded in $C^{1}[0,1]$ by a constant,
independent of $\lambda \in [ 0,1]$.
This completes the proof of the theorem. \hfill$\square$

\paragraph{Remark} %3
Suppose that the the odd increasing homeomorphism $\phi $ in
Theorem \ref{one} is $k$-homogeneous, in the sense that
$\phi (tx)=t^{k}\phi (x)$ for
$t\geq 0$ and $x\in $ $\mathbb{R}$. Then the existence condition
\ref{cond2} in Theorem \ref{one} becomes
\[
\| t^{k}p(t)\|_{1}+\|q(t)\|_{1}+\tau <1.
\]

\begin{theorem} \label{two}
Let $f:[0,1]\times \mathbb{R}^{2}\to \mathbb{R}$ be a function
satisfying Caratheodory's conditions. Assume that there exist functions $%
p(t) $, $q(t)$, $r(t)$ in $L^{2}(0,1)$ such that
\begin{equation}
| f(t,x_{1},x_{2})| \leq p(t)\phi (| x_{1}|)
+q(t)\phi (| x_{2}| )+r(t)  \label{cond10}
\end{equation}
for a.e. $t\in [ 0,1]$ and all $(x_{1},x_{2})\in \mathbb{R}^{2}$.
Also let $\alpha \neq 1$, and $\eta \in (0$,$1)$ be given. Then
for any given $e(t)$ in $L^{2}(0,1)$ the boundary-value problem
(\ref{phi3bp}) has at least one solution in $C^{1}[0,1]$ provided
\begin{equation}
\frac{1}{1-\tau }\| p\|_{2}+C(\alpha ,\eta )\|q\|_{2}<1,  \label{cond11}
\end{equation}
where $C(\alpha ,\eta )$ is as in Theorem \ref{TWO}.
\end{theorem}

\paragraph{Proof}
As in the proof of Theorem \ref{one} it suffices to prove
that the set of all possible solutions of the family of
equations
\begin{equation}
\begin{gathered}
(\phi (x'(t)))'=\lambda f(t,x(t),x'(t))+\lambda e(t),\quad 0<t<1, \\
x(0)=0, \quad \phi (x'(1))=\alpha \phi (x'(\eta )),
\end{gathered} \label{fbp2}
\end{equation}
is, a priori, bounded in $C^{1}[0$,$1]$ by a constant independent of
$\lambda \in [ 0$,$1]$. For an $x\in W_{\phi }^{2,2}(0$,$1)$, with
$x(0)=0$, and $\phi (x'(1))=\alpha \phi (x'(\eta ))$ we use Theorem
\ref{ONE} and Theorem \ref{TWO}, above, to note that
\begin{eqnarray*}
\| \phi (| x(t)| )\|_{2} &\leq &\phi (\|x\|_{\infty })
\leq \phi (\| x'\|_{\infty})
\leq \| \phi (x'(t))\|_{\infty }  \nonumber \\
&\leq &\frac{1}{1-\tau }\| (\phi (x'(t)))'\|_{1}
\leq \frac{1}{1-\tau }\| (\phi (x'(t)))'\|_{2}
\end{eqnarray*}
and
\begin{equation}
\| \phi (x'(t))\|_{2} \leq C(\alpha ,\eta )\|(\phi (x'(t)))'\|_{2}.
\label{cond12}
\end{equation}
Now, for a solution $x$ of the family of equations (\ref{fbp2}) for some $%
\lambda \in [ 0$,$1]$ we have
\begin{align*}
\| (\phi (x'(t)))'\|_{2}
&\leq \lambda \|f(t,x(t),x'(t))+e(t)\|_{2} \\
&\leq \| p(t)\phi (| x(t)| )+q(t)\phi (|x'(t)| )+r(t)\|_{2}+\| e\|_{2} \\
&\leq \| p\|_{2}\| \phi (| x(t)|)\|_{2}
  +\| q\|_{2}\| \phi (|x'(t)|)\|_{2}+\| r\|_{2}+\| e\|_{2} \\
&\leq (\frac{1}{1-\tau }\| p\|_{2}+C(\alpha ,\eta)\| q\|_{2})
\| (\phi (x'(t)))'\|_{2}+\| r(t)\|_{2}+\| e\|_{2},
\end{align*}
in view of estimate (\ref{cond12}), for a solution $x$ of the
family of equations (\ref{fbp2}) for some $\lambda \in [
0$,$1]$. It then
follows from (\ref{cond11}) that there is a constant $c$ independent of $%
\lambda \in [ 0,1]$ such that
\[
\| (\phi (x'(t)))'\|_{2}\leq c,
\]
for a solution $x$ of the family of equations (\ref{fbp2}) for
some $\lambda \in [ 0$,$1]$. Finally, we see, using Theorem
\ref{ONE} that
\[
\phi (\| x\|_{\infty })\leq \phi
(\| x'\|_{\infty })\leq \frac{1}{1-\tau }\|
(\phi (x'(t)))'\|_{1}\leq \frac{1}{1-\tau }\|
(\phi (x'(t)))'\|_{2}
\]
and accordingly, the set of solutions of the family of equations (\ref{fbp2})
is, a priori, bounded in $C^{1}[0,1]$ by a constant independent of
$\lambda \in [ 0,1]$. This completes the proof of Theorem
\ref{two}. \hfill$\square$

\begin{example} \label{ex6} \rm
Let $\alpha \leq 0$ and $\eta \in (0,1)$ be given and $A\in \mathbb{R}$.
For $e(t)\in L^{1}(0,1)$, we consider the three point boundary-value problem
\begin{equation}
\begin{gathered}
(\phi (x'(t)))'=t^{\frac{1}{2}}\phi (| x(t)|
)+At\phi (| x'(t)| )+e(t), 0<t<1,   \\
x(0)=0, \phi (x'(1))=\alpha \phi (x'(\eta )).
\end{gathered}\label{eq8}
\end{equation}
We apply Theorem \ref{one} to obtain a condition for the existence
of a solution for the three-point boundary-value problem (\ref{eq8}).
Here $p(t)=t^{1/2}$, $q(t)=At$ and $\tau =0$. Now,
$\|p(t)\|_{1}=2/3$ and $\| q(t)\|_{1}=\frac{1}{2}| A| $.
Now, if
\[
\frac{2}{3}+\frac{1}{2}| A| <1,
\]
or, equivalently $| A| <2/3$,
then Theorem \ref{one} implies the existence of a solution for the
three-point boundary-value problem (\ref{eq8}).
\end{example}

\begin{example} \label{exA}
Let $\alpha =-2$, $\eta =\frac{1}{3}$ and $A\in \mathbb{R}$.
For $e(t)\in L^{2}(0,1)$, we, next, consider the three point
boundary-value problem
\begin{equation}
\begin{gathered}
(\phi (x'(t)))'=t^{\frac{1}{4}}\phi (| x(t)| )
+At^{-\frac{1}{4}}\phi (| x'(t)| )+e(t),\quad  0<t<1, \\
x(0)=0, \quad \phi (x'(1))=\alpha \phi (x'(\eta )).
\end{gathered} \label{eq9}
\end{equation}
We apply Theorem \ref{two} to obtain a condition for the existence
of a solution for the three-point boundary-value problem (\ref{eq9}).
Here $p(t)=t^{1/4}$, $q(t)=At^{-1/4}$. Now,
$\|p(t)\|_{2}=\sqrt{2/3}$ and
$\| q(t)\|_{2}=\sqrt{2}| A|$. Now the existence condition required
to apply Theorem \ref{two}
\begin{equation}
\sqrt{\frac{2}{3}}+\sqrt{2}C(\alpha ,\eta )| A| <1.
\label{eq10}
\end{equation}
Since we have $C(-2,\frac{1}{3})=\sqrt{\frac{11}{54}}$, we get from (\ref%
{eq10})
\[
\sqrt{\frac{2}{3}}+\sqrt{\frac{22}{54}}| A| <1.
\]
Accordingly, we see from Theorem \ref{two} that a solution for the
three-point bound\-ary-value problem (\ref{eq9}) exists if
 $| A| <\sqrt{54/22}(1-\sqrt{2/3})=.287\,49$.
\end{example}


\begin{example} \label{ex1} \rm
Let $\alpha =-2$, $\eta =1/3$ and $A\in \mathbb{R}$.  For
$e(t)\in L^{2}(0,1)$, we, next, consider the three point boundary-value
problem
\begin{equation}
\begin{gathered}
(\phi (x'(t)))'=t^{\frac{15}{32}}\phi (| x(t)| )+At\phi
(| x'(t)| )+e(t),\quad 0<t<1, \\
x(0)=0, \quad \phi (x'(1))=\alpha \phi (x'(\eta )).
\end{gathered} \label{eq11}
\end{equation}
We apply Theorem \ref{two} to obtain a condition for the existence
of a solution for the three-point boundary-value problem (\ref{eq11}).
 Here $p(t)=t^{15/32}$, $q(t)=At$. Now,
 $\| p(t)\|_{2}=4/\sqrt{31}$ and $\| q(t)\|_{2}=|A|/\sqrt{3} $.
Now the existence condition required to apply
Theorem \ref{two}
\begin{equation}
\frac{4}{\sqrt{31}}+\frac{1}{\sqrt{3}}C(\alpha ,\eta )| A| )<1\text{.%
}  \label{eq12}
\end{equation}
Since, $C(-2,1/3)=\sqrt{11/54}$ and we get from (\ref{eq12}%
)
\[
\frac{4}{\sqrt{31}}+\sqrt{\frac{11}{162}}| A| <1,
\]
which implies
\[
| A| <\text{ }\sqrt{\frac{162}{11}}(1-\frac{4}{\sqrt{31}}%
)=\allowbreak 1.\,\allowbreak 080\,6.
\]
Now, to apply Theorem \ref{one} we see that
\[
\|p(t)\|_{1}=\int_{0}^{1}t^{\frac{15}{32}}dt=\frac{32}{47}
\]
and $\| q(t)\|_{1}=\frac{1}{2}|A|$. Accordingly, we see
using Theorem \ref{one} a solution for the three-point boundary-value
problem (\ref{eq11}) exists if
\[
\frac{32}{47}+\frac{1}{2}| A| <1,
\]
or, equivalently, if
\[
| A| <2(1-\frac{32}{47})=\frac{30}{47}=0.6383.
\]
We thus see that Theorem \ref{two} gives a better result than Theorem \ref%
{one}.
\end{example}

\paragraph{Remark} %4
Note that if we take for $p>1$, the odd
increasing homeomorphism $\phi :\mathbb{R}\to \mathbb{R}$ defined by
\[
\phi (t)=| t| ^{p-2}t \quad \text{for }t\in \mathbb{R},
\]
then Theorems \ref{one}, \ref{two} give existence theorems for the
analogous three-point boundary-value problems for the
one-dimensional analogue of the p-Laplacian. However, Theorems
\ref{one}, \ref{two} apply to more general
differential operators than a p-Laplacian, since Theorems \ref{one},
\ref{two} do not require the homeomorphism $\phi $ to be homogeneous
as happens to be the case for the p-Laplacian.

\begin{thebibliography}{0} \frenchspacing

\bibitem{gnt1} C. P. Gupta; S. K. Ntouyas and P. Ch. Tsamatos;
{\it Existence Results for m-Point Boundary Value Problems,}
Differential Equations Dynam. Systems, Vol. 2 (1994), 289-298.

\bibitem{gt3} C. P. Gupta, S. I. Trofimchuk, {\it Solvability of a
multi-point boundary-value problem of Neumann type,} Abstr. Anal.
\& Appl., 4, \# 2, (1999) 71-81..

\bibitem{hardy} G. H. Hardy, J. E. Littlewood, and G. Polya,
{\it Inequalities,} Cambridge University Press, London/New York,
1967.

\bibitem{mawhin} J. Mawhin, {\it Topological degree methods in nonlinear
boundary value problems,} in ''NSF-CBMS Regional Conference Series
in Math.'' No. 40, Amer. Math. Soc., Providence, RI, 1979.

\bibitem{VA} V.A. Il'in, and E. I. Moiseev, {\it Nonlocal boundary value
problem of the second kind for a Sturm-Liouville operator,}
Differential Equations, Vol. 23, No. 8, (1988), pp. 979-987.

\end{thebibliography}

\noindent\textsc{Chaitan P. Gupta}\\
Department of Mathematics, 084 \\
University of Nevada, Reno \\
Reno, NV 89557, USA \\
email: gupta@unr.edu

\end{document}