\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
Seventh Mississippi State - UAB Conference on Differential Equations and
Computational Simulations,
{\em Electronic Journal of Differential Equations},
Conf. 17 (2009),  pp. 81--94.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{81}
\title[\hfilneg EJDE-2009/Conf/17\hfil
A third-order m-point boundary-value problem]
{A third-order m-point boundary-value problem of Dirichlet type
involving a p-Laplacian type operator}

\author[C. P. Gupta\hfil EJDE/Conf/17 \hfilneg]
{Chaitan P. Gupta}

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

\thanks{Published April 15, 2009.}
\subjclass[2000]{34B10, 34B15, 34L30} 
\keywords{m-point boundary value problems; $p$-Laplace type operator;
 non-resonance; resonance; topological degree}

\begin{abstract}
 Let $\phi $, be an odd increasing homeomorphisms from $\mathbb{R}$ onto
 $\mathbb{R}$ satisfying $\phi (0)=0$, and let
 $f:[0,1]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ be a
 function satisfying Caratheodory's conditions.
 Let $\alpha _{i}\in {\mathbb{R}}$, $\xi _{i}\in (0,1)$, $i=1,\dots ,m-2$,
 $0<\xi _{1}<\xi _{2}<\dots <\xi _{m-2}<1$ be given. We are interested in the
 existence of solutions for the $m$-point boundary-value problem:
 \begin{gather*}
 (\phi (u''))'=f(t,u,u',u''), \quad t\in (0,1), \\
 u(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i}), \quad
 u''(0)=0,
 \end{gather*}
 in the resonance and non-resonance cases. We say that this problem is at
 \emph{resonance} if the associated problem
 \[
 (\phi (u''))'=0, \quad t\in (0,1),
 \]
 with the above boundary conditions has a non-trivial solution.
 This is the case if and only if
 $\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}=1$. Our results use topological degree
 methods. In the non-resonance case; i.e., when
 $\sum_{i=1}^{m-2}\alpha_{i}\xi _{i}\neq 1$ we note that the sign of
 degree for the relevant operator depends on the sign of
  $\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}-1$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

In this paper we consider the boundary-value problem
\begin{equation}
\begin{gathered}
(\phi (u''))'=f(t,u,u',u''), \quad t\in (0,1),  \\
u(0)=0, \quad u(1)=\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i}), \quad
u''(0)=0,
\end{gathered} \label{P}
\end{equation}
where $\phi $ is an odd increasing homeomorphism from $\mathbb{R}$ onto
$\mathbb{R}$ with $\phi (0)=0$ and the function
$f:[0,1]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$
is Caratheodory. Also $\alpha_{i}\in {\mathbb{R}}$, $\xi _{i}\in (0,1)$,
for $i=1,2,\dots m-2$, are such
that $0<\xi _{1}<\xi _{2}<\dots <\xi _{m-2}<1$.

We say that \eqref{P} is at \emph{resonance}, if the associated
multi-point boundary-value problem
\begin{equation}
\begin{gathered}
(\phi (u''))'=0, \quad t\in (0,1),
\\
u(0)=0, u(1)=\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i}),\quad  u''(0)=0,
\end{gathered}\label{HP}
\end{equation}
has a non-trivial solution.

We are interested here in the existence of solutions for the $m$-point
 boundary-value problem \eqref{P} in the resonance and in the
non-resonance cases.

The study of multipoint second-order boundary-value problems for
$\phi (u)\equiv u$ was initiated by Il'in and Moiseev in
\cite{I1,I2} and has been the subject of many papers, see for example
\cite{FW1, FW2,GU1,GU2,GU3,GU4,GNT1,GNT,GT,L1,L2,L3,L4,rayun}.

More recently multipoint second-order boundary-value problems
containing the $p$-Laplace operator or the more general operator
$-(\phi (u'))'$ complemented with linear boundary conditions, have
been studied in \cite{B,GGM1,GM,LG,WA1,WA2}.

Problem \eqref{P} is at resonance if and only if $\sum_{i=1}^{m-2}\alpha
_{i}\xi _{i}=1$, having $u(t)=\rho t$ as a non-trivial solution, where $\rho
\in \mathbb{R}$ is an arbitrary constant.

Our aim in this paper is to obtain existence of solutions for problem (\ref
{P}), by using topological degree arguments. Thus, in section 2, we first
derive a deformation lemma that is needed when problem \eqref{P} is at
resonance.

In section 3 an existence theorem for problem \eqref{P} is derived from this
lemma. Finally in section 4 we consider problem \eqref{P} when it is
non-resonant. The crucial point here is to prove that the Leray Schauder
degree of a certain operator is different from zero which is shown to be an
explicit consequence of the non-resonance condition, i.e.,
$\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}\neq 1$. In addition we obtain the
interesting property that the degree of the operator changes sign when
$\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}$ goes from being less than one to being
greater than one.

We shall denote by $C[0,1]$ (resp. $C^{1}[0,1]$, $C^{2}[0,1]$) the classical
space of continuous (resp. continuously differentiable, twice continuously
differentiable) real-valued functions on the interval $[0,1]$. The norm in
$C[0,1]$ is denoted by $| \cdot | _{\infty }$. Also, we shall denote by
$L^{1}(0,1)$ the space of real-valued (equivalence classes of) functions
whose absolute value is Lebesgue integrable on $(0,1)$. The Brouwer and
Leray-Schauder degree shall be respectively denoted by $\deg _{B}$ and $\deg
_{LS}$.

\section{\label{defor-lemma}A deformation lemma for the resonance case}

We begin this section by formulating a general deformation lemma for the
solvability of the boundary-value problem \eqref{P} in the resonance case.

Let $f^{\ast }:[0,1]\times \mathbb{R}\times \mathbb{R\times R}\times
[ 0,1]\mapsto \mathbb{R}$ be a  function satisfying Caratheodory's
conditions; i.e., (i) for all $(s,r,q,\lambda )\in \mathbb{R}\times
\mathbb{R} \times \mathbb{R}\times [ 0,1]$ the function $f^{\ast
}(\cdot ,s,r,q,\lambda )$ is measurable on $[0,1]$, (ii) for a.e.
$t\in [ 0,1]$ the function $f^{\ast }(t,\dots ,\cdot )$ is
continuous on $\mathbb{R}\times \mathbb{R\times R}\times [ 0,1]$,
and (iii) for each $R>0$ there exists a Lebesgue integrable function
$\rho _{R}:[0,1]\mapsto \mathbb{R}$ such that $|f^{\ast
}(t,s,r,q,\lambda )|\leq \rho _{R}(t)$ for a.e. $t\in [ 0,1]$ and
all $(s,r,q,\lambda )\in \mathbb{R}\times \mathbb{R\times R}\times [
0,1]$ with $| s| \leq R$, $| r| \leq R$, and $| q| \leq R$. We
suppose that $f(t,s,r,q)=f^{\ast }(t,s,r,q,1)$ is the given function
in problem \eqref{P}.

We, now, introduce an operator ${\mathfrak{B}}(u,\lambda ):C^{2}[0,1]\times
[ 0,1]\mapsto \mathbb{R}$ defined for $(u,\lambda )\in
C^{2}[0,1]\times [ 0,1]$ by
\begin{equation}
\begin{aligned}
\mathfrak{B}(u,\lambda )
&=\lambda \Big(u(1)-\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i})\Big)\\
&\quad+(1-\lambda )\Big(\int_{0}^{1}\int_{0}^{s}f^{\ast }(\tau
,u(\tau
),u'(\tau ),u''(\tau ),\lambda )d\tau ds  \\
&\quad -\sum_{i=1}^{m-2}\alpha _{i}\int_{0}^{\xi _{i}}\int_{0}^{s}f^{\ast }(\tau
,u(\tau ),u'(\tau ),u''(\tau ),\lambda )d\tau ds
\Big).
\end{aligned} \label{eqB}
\end{equation}
 For $\lambda \in [ 0,1]$ we consider the family of boundary-value
problems:
\begin{equation}
\begin{gathered}
(\phi (u''))'=\lambda f^{\ast }(t,u,u',u'',\lambda ), \quad t\in (0,1),  \\
u(0)=0, \quad u''(0)=0, \quad {\mathfrak{B}}(u,\lambda )=0.
\end{gathered} \label{Plambda}
\end{equation}
Let $\Omega \subset C^{2}[0,1]$ be a bounded open set. Let us set for
$\rho \in \mathbb{R}$, $i_{\rho }(t)=\rho t$, for $t\in [ 0,1]$, and
\begin{equation*}
X=\{i_{\rho }: \rho \in \mathbb{R}\},
\end{equation*}
then $X$ is a one dimensional subspace of $C^{2}[0,1]$. Defining
$i:{\mathbb{R}}\mapsto X$ by $i(\rho )=i_{\rho }$ it is clear that $i$ is
an isomorphism from ${\mathbb{R}}$ onto $X$.

Next let us define $F:X\mapsto \mathbb{R}$ by
\begin{equation*}
F(i_{\rho })=\int_{0}^{1}\int_{0}^{s}f^{\ast }(\tau ,\rho \tau ,\rho
,0,0)d\tau ds-\sum_{i=1}^{m-2}\alpha _{i}\int_{0}^{\xi
_{i}}\int_{0}^{s}f^{\ast }(\tau ,\rho \tau ,\rho ,0,0)d\tau ds,
\end{equation*}
and set $\mathcal{F}=F\circ i$, then $\mathcal{F}:{\mathbb{R}}\mapsto {
\mathbb{R}}$ is continuous, and is given by
\begin{equation*}
\mathcal{F}(\rho )=\int_{0}^{1}\int_{0}^{s}f^{\ast }(\tau ,\rho \tau ,\rho
,0,0)d\tau ds-\sum_{i=1}^{m-2}\alpha _{i}\int_{0}^{\xi
_{i}}\int_{0}^{s}f^{\ast }(\tau ,\rho \tau ,\rho ,0,0)d\tau ds.
\end{equation*}
We have the following lemma.

\begin{lemma} \label{DefLemma}
Assume that
\begin{itemize}
\item[(i)] for $\lambda \in (0,1)$ the boundary-value problem (\ref{Plambda}) has
no solution $u\in \partial \Omega $,

\item[(ii)] the equation $\mathcal{F}(\rho)= 0$ has no solution for any $\rho$ with
$i_{\rho }(t)\in \partial \Omega \cap X$, and

\item[(iii)] the Brouwer degree $\deg _{B}(F,\Omega \cap X,0)\neq 0$.
\end{itemize}
Then the boundary-value problem \eqref{P} has at least one solution in
$\overline{\Omega }$.
\end{lemma}

\begin{proof}
If the boundary-value problem \eqref{P} has a solution in
$\partial \Omega $, then there is nothing to prove. Accordingly, let us
assume that the boundary-value problem \eqref{P} has no solution in
$\partial \Omega $. This assumption combined with assumption (i) implies that
the boundary-value problem (\ref{Plambda}) has no solution $u\in \partial
\Omega $ for $\lambda \in (0,1]$.

Let us define an operator $\Psi ^{\ast }:C^{2}[0,1]\times [
0,1]\mapsto C^{2}[0,1]$ by setting for $(u,\lambda )\in C^{2}[0,1]\times
[ 0,1]$
\begin{equation}
\begin{aligned}
\Psi ^{\ast }(u,\lambda )(t)
&=\int_{0}^{t}\Big(u'(0)+\int_{0}^{s}\phi ^{-1}\Big(\lambda \int_{0}^{r}f^{\ast }(\tau
,u(\tau ),u'(\tau ),u''(\tau ),\lambda )d\tau
\Big)dr\Big)ds\\
&\quad +t{\mathfrak{B}}(u,\lambda ),
\end{aligned} \label{Eq2}
\end{equation}
where ${\mathfrak{B}}(u,\lambda )$ is as defined in equation (\ref{eqB}).

We note from our assumptions that the function $f^{\ast }$ satisfies
Caratheodory's conditions so that for $(u,\lambda )\in $ $C^{2}[0,1]\times
[ 0,1]$, $f^{\ast }(t,u(t),u'(t),u''(t),\lambda )\in L^{1}(0,1)$. Accordingly, the function $s\in [
0,1]\mapsto \int_{0}^{s}f^{\ast }(\tau ,u(\tau ),u'(\tau
),u''(\tau ),\lambda )d\tau $ is absolutely continuous on
$[0,1]$. Since, now, the integrand in (\ref{Eq2}) is continuous on $[0,1]$ we
see that the operator $\Psi ^{\ast }$ is well defined.

Next, let us suppose that $u(t)$ be a solution to the boundary-value problem
(\ref{Plambda}) for some $\lambda \in [ 0,1]$. We, then, see by
integrating the equation in (\ref{Plambda}) and using the boundary
conditions in (\ref{Plambda}) that $u(t)$ satisfies the equation
\begin{equation*}
u(t)=\Psi ^{\ast }(u,\lambda )(t), t\in [ 0,1],
\end{equation*}
along with
\begin{equation*}
u(0)=0, u''(0)=0, {\mathfrak{B}}(u,\lambda )=0.
\end{equation*}
Conversely, let us suppose that for some $\lambda \in [ 0,1]$, $u(t)$,
$t\in [ 0,1]$, satisfies the equation
\begin{equation}
u(t)=\Psi ^{\ast }(u,\lambda )(t).  \label{Eq3}
\end{equation}
We first see from the equation (\ref{Eq3}) and the definition of $\Psi
^{\ast }(u,\lambda )$ that
\begin{equation*}
u(0)=0.
\end{equation*}
Next, we obtain, by differentiating the equation (\ref{Eq3}) that
\begin{equation}
u'(t)=u'(0)+\int_{0}^{t}\phi ^{-1}\Big(\lambda
\int_{0}^{r}f^{\ast }(\tau ,u(\tau ),u'(\tau ),u''(\tau ),\lambda )d\tau \Big)dr
+{\mathfrak{B}}(u,\lambda ),
t\in [ 0,1].  \label{Eq4}
\end{equation}
Evaluating  (\ref{Eq4}) at $t=0$ we see that
\begin{equation*}
{\mathfrak{B}}(u,\lambda )=0.
\end{equation*}
Again, we obtain, by differentiating  (\ref{Eq4}) that
\begin{equation}
u''(t)=\phi ^{-1}\Big(\lambda \int_{0}^{t}f^{\ast }(\tau
,u(\tau ),u'(\tau ),u''(\tau ),\lambda )d\tau
\Big).  \label{eq41}
\end{equation}
Evaluating the equation (\ref{eq41}) at $t=0$ we see that
\begin{equation*}
u''(0)=0.
\end{equation*}
Also, equation (\ref{eq41}) further implies that $\phi (u''(t))$ is absolutely continuous on $[0,1]$ and
\begin{equation*}
(\phi (u''(t)))'=\lambda f^{\ast }(t,u(t),u'(t),u''(t),\lambda ),
t\in [ 0,1].
\end{equation*}
Thus $u(t)$, $t\in (0,1)$, is a solution to the boundary-value problem (\ref
{Plambda}). We have, accordingly, proved that $u(t)$, $t\in (0,1)$, is a
solution to the boundary-value problem (\ref{Plambda}) if and only if
$u(t)$, $t\in [ 0,1]$, is a solution to the equation (\ref{Eq3}).

We observe that it is easy to show, using standard arguments, that
$\Psi^{\ast }:C^{2}[0,1]\times [ 0,1]\mapsto C^{2}[0,1]$ is a
completely continuous operator. If, now, $u(t)\in \partial \Omega $ is a
solution to the boundary-value problem \eqref{P} then we are done.
Accordingly, let us assume that the boundary-value problem \eqref{P} has no
solution on $\partial \Omega $. Since, now, $f^{\ast }(t,s,r,q,1)=f(t,s,r,q)$
for all $(t,s,r,q)\in [ 0,1]\times \mathbb{R}\times \mathbb{R}\times
\mathbb{R}$ we see that the assumption (i) of the lemma implies that
\begin{equation*}
u\neq \Psi ^{\ast }(u,\lambda )\quad \text{for all }u\in \partial \Omega
\text{ and }\lambda \in (0,1].
\end{equation*}
We, next, assert that $u\neq \Psi ^{\ast }(u,0)$ for all $u\in
\partial \Omega $. Indeed, let $u\in \partial \Omega $ be such that
$u=\Psi ^{\ast }(u,0)$. It then follows from the definition of $\Psi
^{\ast }$, as given in (\ref{Eq2}), that $u(t)=\rho t=i_{\rho }(t)$,
with $\rho =u'(0)+{\mathfrak{B}}(u,0)$, $u'(t)=\rho
+{\mathfrak{B}}(u,0) $, $u''(0)=0$, ${\mathfrak{B}}(u,0)=0 $, $u\in
\partial \Omega \cap X$, and
\begin{align*}
{\mathfrak{B}}(u,0)
&= \int_{0}^{1}\int_{0}^{s}f^{\ast }(\tau ,u(\tau
),u'(\tau ),u''(\tau ),0)d\tau ds \\
&\quad -\sum_{i=1}^{m-2}\alpha _{i}\int_{0}^{\xi _{i}}\int_{0}^{s}f^{\ast }(\tau
,u(\tau ),u'(\tau ),u''(\tau ),0)d\tau ds \\
&= \int_{0}^{1}\int_{0}^{s}f^{\ast }(\tau ,\rho \tau ,\rho ,0,0)d\tau
ds-\sum_{i=1}^{m-2}\alpha _{i}\int_{0}^{\xi _{i}}\int_{0}^{s}f^{\ast }(\tau
,\rho \tau ,\rho ,0,0)d\tau ds \\
&= \mathcal{F}(\rho )=0.
\end{align*}
But this contradicts the assumption (ii) of the lemma. We thus get that
\begin{equation*}
u\neq \Psi ^{\ast }(u,\lambda )\quad \text{for all }u\in \partial \Omega \text{
and }\lambda \in [ 0,1].
\end{equation*}
Thus $\deg _{LS}(I-\Psi ^{\ast }(\cdot ,\lambda ),\Omega ,0)$ is well
defined for all $\lambda \in [ 0,1]$. By the homotopy invariance
property of Leray-Schauder degree we obtain immediately that
\begin{equation}
\deg _{LS}(I-\Psi ^{\ast }(\cdot ,1),\Omega ,0)
= \deg _{LS}(I-\Psi ^{\ast}(\cdot ,0),\Omega ,0)
= \deg _{B}(I-\Psi ^{\ast }(\cdot ,0)| _{X},\Omega _{0},0),
\label{eq5}
\end{equation}
where, $\Omega _{0}=\Omega \cap X$. Now since for $v\in X$
\begin{equation*}
\big(I-\Psi ^{\ast }(\cdot ,0)\big)v=-i_{F(v)},
\end{equation*}
we have
\begin{equation*}
\deg _{LS}(I-\Psi ^{\ast }(\cdot ,1),\Omega ,0)=\deg _{B}(-i_{F(\cdot
)},\Omega _{0},0)=-\deg _{B}(i_{F(\cdot )},\Omega _{0},0).
\end{equation*}
Since, $i^{-1}\circ i_{F(\cdot )}\circ i=\mathcal{F}$, we obtain by using a
standard formula in degree theory that
\begin{equation*}
\deg _{B}(i_{F(\cdot )},\Omega _{0},0))=\deg _{B}(\mathcal{F},i^{-1}(\Omega
_{0}),0)).
\end{equation*}
Hence, by assumption $(iii)$ of the lemma, it follows that $\deg
_{LS}(I-\Psi ^{\ast }(\cdot ,1),\Omega ,0)\neq 0$. Thus, the mapping $\Psi
\equiv \Psi ^{\ast }(\cdot ,1):C^{2}[0,1]\mapsto C^{2}[0,1]$ has at
least one fixed-point in $\overline{\Omega }$ and hence the boundary value
problem \eqref{P} has at least one solution in $\overline{\Omega }$. This
completes the proof of the lemma.
\end{proof}

\section{Existence Theorems}

We shall assume that for any constants $\Lambda \geq 0$, $A>0$ with
$\Lambda<A$ it holds that
\begin{equation}
\tilde{\alpha}(A,\Lambda )\equiv \limsup_{z\rightarrow \infty }\frac{\phi (
\frac{A+\Lambda }{A-\Lambda }z+c)}{\phi (z)}<\infty .  \label{A-cond}
\end{equation}
We need the following lemma in the proof of our existence theorems.

\begin{lemma}\label{Lemma1}
 Let $g:[0,1]\mapsto \mathbb{R}$ be a strictly
increasing (resp. strictly decreasing) function on $[0,1]$. Then the
function $G:(0,1]\hookrightarrow \mathbb{R}$ defined for $t\in
(0,1]$ by
\begin{equation*}
G(t)=\frac{1}{t}\int_{0}^{t}g(s)ds
\end{equation*}
is strictly increasing (resp. decreasing) function on $(0,1]$. In
particular, $\int_{0}^{1}g(s)ds-\frac{1}{t}\int_{0}^{t}g(s)ds>0$
(resp. $<0$) for every $t\in (0,1)$. Moreover, given
$\alpha _{i}\geq 0$, $\xi _{i}\in (0,1)$, $i=1,2,\cdot \cdot \cdot ,m-2$
with $\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}=1$
we have $\int_{0}^{1}g(s)ds-\sum_{i=1}^{m-2}\alpha
_{i}\int_{0}^{\xi _{i}}g(s)ds>0$ (resp. $<0$).
\end{lemma}

\begin{proof}
 Let us suppose that $g$ is a strictly increasing
function on $[0,1]$. Now we see that
\[
G'(t) = \frac{g(t)}{t}-\frac{1}{t^{2}}\int_{0}^{t}g(s)ds
= \frac{1}{t^{2}}(\int_{0}^{t}(g(t)-g(s))ds
>0,
\]
for every $t\in (0,1]$. Accordingly, $G$ is strictly increasing on $(0,1]$
and $\int_{0}^{1}g(s)ds-\frac{1}{t}\int_{0}^{t}g(s)ds>0$ for every $t\in
(0,1)$. Finally, we see that
\begin{align*}
& \int_{0}^{1}g(s)ds-\sum_{i=1}^{m-2}\alpha _{i}\int_{0}^{\xi _{i}}g(s)ds \\
&= \sum_{i=1}^{m-2}\alpha _{i}\xi _{i}(\int_{0}^{1}g(s)ds-\frac{1}{\xi _{i}}
\int_{0}^{\xi _{i}}g(s)ds)>0.
\end{align*}
Similarly $G$ is strictly decreasing on $(0,1]$ and $\int_{0}^{1}g(s)ds-
\sum_{i=1}^{m-2}\alpha _{i}\int_{0}^{\xi _{i}}g(s)ds<0$ when $g$ is a
strictly decreasing function on $[0,1]$. This completes the proof of the
lemma.
\end{proof}

\begin{theorem} \label{thm3}
Let $f:[0,1]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}
\mapsto \mathbb{R}$ in the boundary-value problem \eqref{P} be a
continuous function and satisfies the following conditions:
\begin{itemize}
\item[(i)] there exist non-negative functions $d_{1}(t)$, $d_{2}(t)$, $d_{3}(t)$,
and $r(t)$ in $L^{1}(0,1)$ such that
\[
|f(t,u,v,w)|\leq d_{1}(t)\phi (|u|)+d_{2}(t)\phi (|v|)
 +d_{3}(t)\phi (|w|)+r(t),
\]
for all $t\in [ 0,1]$, $u$, $v$, $w\in \mathbb{R}$,

\item[(ii)] there exist constants $\Lambda \geq 0$, $B\geq 0$, $A>0$ with
$\Lambda<A$ and a $v_{0}>0$ such that for all $v$ with $|v|>v_{0}$, all
$t\in[ 0,1]$ and all $u$, $w\in \mathbb{R}$ one has
\begin{equation*}
|f(t,u,v,w)|\geq -\Lambda |u|+A|v|-\Lambda |w|-B,
\end{equation*}

\item[(iii)] there exists an $R>0$ such that for all $\rho $, with $|\rho |>R$,
either
\begin{gather*}
\rho f(t,\rho t,\rho ,0)>0,\text{ for all }t\in [ 0,1],\quad\text{or}\\
\rho f(t,\rho t,\rho ,0)<0,\text{ for all }t\in [ 0,1].
\end{gather*}
\end{itemize}
Suppose, further, that
\begin{equation}
\tilde{\alpha}(A,\Lambda)(\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)})
+\|d_{3}\|_{L^{1}(0,1)}<1.  \label{cond1}
\end{equation}
Then, given $\alpha _{i}\geq 0$, $\xi _{i}\in (0,1)$, $i=1,2,\cdot \cdot
\cdot ,m-2$ with $\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}=1$ the boundary value
problem \eqref{P} has at least one solution in $u(t)\in C^{2}[0,1]$.
\end{theorem}

\begin{proof}
We first choose an $\varepsilon >0$ be such that
\begin{equation*}
(\tilde{\alpha}(A,\Lambda )+\varepsilon
)(\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)})+\|d_{3}\|_{L^{1}(0,1)}<1,
\end{equation*}
which is possible to do, in view of (\ref{cond1}).
We consider the family of boundary-value problems:
\begin{equation}
\begin{gathered}
(\phi (u''(t)))'=\lambda f(t,u(t),u'(t),u''(t)), \quad
t\in (0,1), \lambda \in [0,1],  \\
u(0)=0,  \quad
{\mathfrak{B}}(u,\lambda )=0,  \quad
u''(0)=0,
\end{gathered} \label{EQ1}
\end{equation}
where ${\mathfrak{B}}(u,\lambda )$ is as defined in (\ref{eqB}). Let
$u(t) $ be a solution to the boundary-value problem (\ref{EQ1})
for some $\lambda \in (0,1)$. Then either there exists a
$t_{0}\in [0,1]$ such that
\begin{equation}
|u'(t_{0})|\leq v_{0}  \label{Eq31}
\end{equation}
or $|u'(t)|>v_{0}$ for all $t\in [ 0,1]$. In case,
$|u'(t)|>v_{0}$ for all $t\in [ 0,1]$, we claim that there
exists a $\tau _{0}\in [ 0,1]$ such that $f(\tau _{0},u(\tau
_{0}),u'(\tau _{0}),u''(\tau _{0}))=0$. Indeed, let
us suppose that $f(t,u(t),u'(t),u''(t))\neq 0$ for
all $t\in [ 0,1]$. It then follows from the continuity of
$f(t,u(t),u'(t),u''(t))$ on the interval $[0,1]$
either $f(t,u(t),u'(t),u''(t))>0$ for all $t\in
[ 0,1]$ or $f(t,u(t),u'(t),u''(t))<0$ for all
$t\in [ 0,1]$. Let us first suppose that $f(t,u(t),u'(t),u''(t))>0$
for all $t\in [ 0,1]$. It then follows
from the boundary condition in (\ref{Eq3}) that
\begin{equation}
\begin{aligned}
&\lambda \Big[\int_{0}^{1}\Big(u'(0)+\int_{0}^{s}\phi ^{-1}
\Big(\lambda \int_{0}^{r}f(\tau ,u(\tau ),u'(\tau ),u''(\tau ))d\tau \Big)dr\Big)
ds  \\
&-\sum_{i=1}^{m-2}\alpha _{i}\int_{0}^{\xi _{i}}\Big(u'(0)+\int_{0}^{s}\phi ^{-1}\Big(\lambda \int_{0}^{r}f(\tau ,u(\tau
),u'(\tau ),u''(\tau ))d\tau \Big)dr\Big)ds
\Big]  \\
&+(1-\lambda )\Big[\int_{0}^{1}\int_{0}^{r}f(\tau ,u(\tau ),u'(\tau ),u''(\tau ))
d\tau ds\\
& -\sum_{i=1}^{m-2}\alpha
_{i}\int_{0}^{\xi _{i}}\int_{0}^{r}f(\tau ,u(\tau ),u'(\tau
),u''(\tau ))d\tau dr\Big]\\
&=0.
\end{aligned} \label{EQ4}
\end{equation}
We, next, see that the functions
\begin{gather*}
\int_{0}^{t}\Big(u'(0)+\int_{0}^{s}\phi ^{-1}\Big(\lambda
\int_{0}^{r}f(\tau ,u(\tau ),u'(\tau ),u''(\tau
))d\tau \Big)dr\Big)ds, \\
\int_{0}^{s}\int_{0}^{r}f(\tau ,u(\tau ),u'(\tau ),u''(\tau ))d\tau dr
\end{gather*}
are strictly increasing functions on $(0,1]$, in view of our assumption
\begin{equation*}
f(t,u(t),u'(t),u''(t))>0
\end{equation*}
for all $t\in [ 0,1]$. We then get from Lemma \ref{Lemma1} and (\ref
{EQ4}) that $0>0$, a contradiction. Similarly, the supposition
$f(t,u(t),u'(t),u''(t))<0$ for all $t\in [ 0,1]$
leads to the contradiction $0<0$. Hence, there must exist a
$\tau _{0}\in[ 0,1]$ such that
\begin{equation}
f(\tau _{0},u(\tau _{0}),u'(\tau _{0}),u''(\tau
_{0}))=0,  \label{Eq5}
\end{equation}
proving the claim. We next see from (\ref{Eq5}) and assumption (ii) that
\begin{equation}
|u'(\tau _{0})|\leq \frac{B}{A}+\frac{\Lambda }{A}\|u\|_{\infty }+
\frac{\Lambda }{A}\|u''\|_{\infty }.  \label{Eq32}
\end{equation}
Thus we see from (\ref{Eq31}) and (\ref{Eq32}) that there exists a
$\tau _{1}\in [ 0,1]$ (either $t_{0}$ or $\tau _{0}$) such that
\begin{equation}
|u'(\tau _{1})|\leq v_{0}+\frac{B}{A}+\frac{\Lambda }{A}
\|u\|_{\infty }+\frac{\Lambda }{A}\|u''\|_{\infty }.
\label{Eq33}
\end{equation}
It then follows from the equation $u'(t)=u'(\tau
_{1})+\int_{\tau _{1}}^{t}u''(s)ds$ and (\ref{Eq33}) that
\begin{equation}
\|u'\|_{\infty }\leq \frac{A+\Lambda }{A-\Lambda }\|u''\|_{\infty }
+\frac{Av_{0}+B}{A-\Lambda }.  \label{Eq34}
\end{equation}
Next, we see by integrating the equation in (\ref{EQ1}) from $0$ to $t\in
[ 0,1]$ and noting $u''(0) =0$, that
\begin{equation}
\phi (u''(t))=\lambda \int_{0}^{t}f(\tau ,u(\tau ),u'(\tau
),u''(\tau ))d\tau .  \label{Eq51}
\end{equation}
It now follows from equations (\ref{Eq51}), (\ref{Eq33}) using assumption
(i), the fact that $u(0)=0$ implies $\|u\|_{\infty }\leq \|u'\|_{\infty }$ that
\begin{align*}
&\phi (|u''(t)|) \\
&\leq \phi (\|u\|_{\infty})\|d_{1}\|_{L^{1}(0,1)}
  +\phi (\|u'\|_{\infty })\|d_{2}\|_{L^{1}(0,1)}+\phi (\|u''\|_{\infty })\|d_{3}\|_{L^{1}(0,1)}+\|r\|_{L^{1}(0,1)} \\
&\leq (\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)})\phi (\frac{A+\Lambda
}{A-\Lambda }\|u''\|_{\infty }+\frac{Av_{0}+B}{A-\Lambda }) \\
&\quad +\|d_{3}\|_{L^{1}(0,1)})\phi (\|u''\|_{\infty
})+\|r\|_{L^{1}(0,1)} \\
&\leq ((\tilde{\alpha}(A,\Lambda )+\varepsilon
)(\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)})+\|d_{3}\|_{L^{1}(0,1)})\phi
(\|u''\|_{\infty }) \\
&\quad +C_{\varepsilon
}(\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)})+\|r\|_{L^{1}(0,1)},
\end{align*}
and hence
\begin{equation}
\begin{aligned}
\phi (\|u''\|_{\infty })
&\leq ((\tilde{\alpha}(A,\Lambda )+\varepsilon
)(\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)})
+\|d_{3}\|_{L^{1}(0,1)})\phi (\|u''\|_{\infty })  \\
&\quad +C_{\varepsilon}(\|d_{1}\|_{L^{1}(0,1)}
+\|d_{2}\|_{L^{1}(0,1)})+\|r\|_{L^{1}(0,1)}.
\end{aligned}
\label{Eq6}
\end{equation}
It now follows from (\ref{cond1}), the estimates (\ref{Eq6}),
(\ref{Eq34}) and $\|u\|_{\infty }\leq \|u'\|_{\infty }$ that there
exists an $R_{0}>R$ , where $R$ is as in assumption (iii), such that
the family of boundary value problems (\ref{EQ1}) have no solution
on the boundary of a bounded open set $\Omega
=B(0,\widetilde{R})\subset C^{2}[0,1]$, for every $\widetilde{R}\geq
R_{0}$. Accordingly, we see that the family of boundary value
problems (\ref{EQ1}) satisfy condition (i) of Lemma \ref{DefLemma}.
Next, we see from assumption (iii) and Lemma \ref{Lemma1} for all
$\rho $, $|\rho |>R$, that
\begin{equation*}
\int_{0}^{1}\int_{0}^{s}f(\tau ,\rho \tau ,\rho ,0)d\tau
ds-\sum_{i=1}^{m-2}\alpha _{i}\int_{0}^{\xi _{i}}\int_{0}^{s}f(\tau ,\rho
\tau ,\rho ,0)d\tau ds
\end{equation*}
is strictly positive or strictly negative.  Accordingly, we see that
$f^{\ast }(t,u,v,w,\lambda )=f(t,u,v,w)$ satisfies the condition (ii) of
Lemma \ref{DefLemma}.

Finally, we again see from assumption (iii), the continuity in $\rho \in
\mathbb{R}$ of the function
\begin{equation*}
\psi (\rho )=\int_{0}^{1}\int_{0}^{s}f(\tau ,\rho \tau ,\rho ,0)d\tau
ds-\sum_{i=1}^{m-2}\alpha _{i}\int_{0}^{\xi _{i}}\int_{0}^{s}f(\tau ,\rho
\tau ,\rho ,0)d\tau ds
\end{equation*}
and the assumption that $\widetilde{R}>R$, that $F(i_{\widetilde{R}}(t))$
and $F(i_{-\widetilde{R}}(t))$ have opposite signs. It follows immediately
that $F(i_{\rho }(t))=0$ for an odd number of $\rho \in (-\widetilde{R},
\widetilde{R})$ which implies that the Brouwer degree $\deg _{B}(F,\Omega
\cap X,0)\neq 0$. Thus the condition (iii) of Lemma \ref{DefLemma} is also
satisfied. \ Thus it follows from Lemma \ref{DefLemma} that the boundary
value problem \eqref{P} has at least one solution in $\overline{\Omega }$.
This completes the proof of the theorem.
\end{proof}

\section{A result for the non-resonance case}

In this section we will consider problem \eqref{P} in the
non-resonance case. Problem \eqref{P} is in the non-resonance case
if problem (\ref{HP}) has only the trivial solution. This holds if
and only if the $\alpha _{i}$, $\xi _{i}$ satisfy
$\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}\neq 1$. We assume henceforth
that $\alpha _{i}$, $\xi _{i}$ satisfy this condition. Notice that
we do not assume a sign condition on the $\alpha _{i}'s$. In
addition, we shall assume that for any $\sigma$, $0<\sigma <1$, it
holds that
\begin{equation}
\tilde{\alpha}(\sigma )=\limsup_{z\rightarrow \infty }\frac{\phi (\frac{1}{
1-\sigma }z)}{\phi (z)}<\infty .  \label{EQ8}
\end{equation}

Let us set $\xi _{m-1}=1$, $\alpha _{m-1}=-1$, $\sigma _{ij}=\alpha _{i}(\xi
_{i}-\xi _{j})$ for $i\neq j$ and $\sigma _{jj}=\sum_{i=1}^{m-1}\alpha
_{i}\xi _{j}$ for $i, j = 1, 2, \cdot,\cdot,\cdot,m-1$. We note that the
assumption $\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}\neq 1$ is equivalent to
$\sum_{i=1}^{m-1}\alpha _{i}\xi _{i}\neq 0$. Also, for each $j = 1, 2,
\cdot,\cdot,\cdot,m-1$ we have
\begin{equation*}
\sum_{i=1}^{m-1}\sigma _{ij}=\sum_{i=1,i\neq j}^{m-1}\sigma _{ij}+\sigma
_{jj}=\sum_{i=1,i\neq j}^{m-1}\alpha _{i}(\xi _{i}-\xi
_{j})+\sum_{i=1}^{m-1}\alpha _{i}\xi _{j}=\sum_{i=1}^{m-1}\alpha _{i}\xi
_{i}\neq 0.
\end{equation*}
It follows that
\[
\sum_{i=1}^{m-1}(\sigma _{ij})^{+} \neq \sum_{i=1}^{m-1}(\sigma _{ij})^{-},
\]
for $j = 1, 2, \cdot,\cdot,\cdot,m-1$, where for $\alpha \in
\mathbb{R}$, $\alpha^{+}= \max(\alpha,0)$ and $\alpha^{-} = \max (
-\alpha, 0)$. Let us set
\begin{equation}
\sigma ^{\ast }=
\begin{cases}
\min \{\frac{\sum_{i=1}^{m-1}(\sigma _{ij})^{+}}{\sum_{i=1}^{m-1}(\sigma
_{ij})^{-}},\frac{\sum_{i=1}^{m-1}(\sigma _{ij})^{-}}{\sum_{i=1}^{m-1}(
\sigma _{ij})^{+}}\}& \text{if }\sum_{i=1}^{m-1}(\sigma
_{ij})^{+}\neq 0 \text{ and}\\
    &\quad \sum_{i=1}^{m-1}(\sigma _{ij})^{-}\neq 0
\text{ for all }j, \\
0,& \text{otherwise.}
\end{cases}
\label{EQ9}
\end{equation}
Note that $0\leq \sigma ^{\ast }<1$. The main result of this section is the
following theorem.

\begin{theorem}\label{nonresonant}
 Let $f:[0,1]\times \mathbb{R}\times\mathbb{R}\times\mathbb{R}\mapsto
\mathbb{R}$ be a function satisfying Caratheodory's conditions such that the
following condition holds:
\\
there exist non-negative functions $d_{1}(t)$, $d_{2}(t)$,
$d_{3}(t)$, and $r(t)$ in $L^{1}(0,1)$ such that
\[
| f(t,u,v,w)| \leq d_{1}(t)\phi (| u| )+d_{2}(t)\phi (| v|) \\
+d_{3}(t)\phi (| w| )+r(t),
\]
for a. e. $t\in [ 0,1]$ and all $u,v,w\in \mathbb{R}$. Suppose,
further,
\begin{equation}
\tilde{\alpha}(\sigma ^{\ast })(\|d_{1}\|_{L^{1}(0,1)}+\|
d_{2}\| _{L^{1}(0,1)})+\| d_{3}\| _{L^{1}(0,1)}<1\text{,
}  \label{cond1m}
\end{equation}
where $\sigma ^{\ast }$ is as defined in (\ref{EQ9}) and $\tilde{\alpha}$ is
as defined in (\ref{EQ8}).

Then, the boundary-value problem \eqref{P} has at least one solution $u\in
C^{2}[0,1]$.
\end{theorem}

We need the following variant of an a priori estimate from \cite{GT0} in
the proof of Theorem \ref{nonresonant} and present this in the following
lemma.

\begin{lemma}\label{APestimate}
Let $u\in C^{1}[0,1]$, be such that $u''\in L^{\infty }(0,1)$ and satisfies
\begin{equation*}
u(0)=0, \quad u(1)=\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i}),
\end{equation*}
with $\sum \alpha _{i}\xi _{i}\neq 1$. If
$\sum_{i=1}^{m-1}(\sigma_{ij})^{+}\neq 0$, and
$\sum_{i=1}^{m-1}(\sigma _{ij})^{-}\neq 0$ for all $j$, then
\begin{equation}
\| u'\| _{\infty }\leq \frac{1}{1-\sigma ^{\ast }}
\| u''\| _{\infty }.  \label{APEstimate}
\end{equation}
If one of $\sum_{i=1}^{m-1}(\sigma _{ij})^{+}$, $\sum_{i=1}^{m-1}(\sigma
_{ij})^{-}$ is zero for some $j=1,2,\dots  ,m-1$, then
$u'(\eta _{0})=0$ for some $\eta _{0}\in [ 0,1]$, and
\begin{equation}
\| u'\| _{\infty }\leq \| u''\| _{\infty }.  \label{mana10}
\end{equation}
\end{lemma}

\begin{proof}
 We first, note, that the assumption
\begin{equation*}
u(1)=\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i})
\end{equation*}
is equivalent to
\begin{equation*}
\sum_{i=1}^{m-1}\alpha _{i}u(\xi _{i})=0,
\end{equation*}
with $\xi _{m-1}=1$, $\alpha _{m-1}=-1$ and the non-resonant condition
$\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}\neq 1$ is equivalent to
$\sum_{i=1}^{m-1}\alpha _{i}\xi _{i}\neq 0$.

Next, for each $j = 1, 2, \cdot,\cdot,\cdot,m-1$ we have $ u(\xi
_{j})=\xi _{j}u'(\eta _{jj})$ for some $\eta _{jj}\in [ 0,1]$. Also
for $i, j = 1, 2, \cdot,\cdot,\cdot,m-1$ with $i\neq j$ we have
$u(\xi _{i})-u(\xi _{j})=u'(\eta _{ij})(\xi _{i}-\xi _{j})$ for some
$\eta _{ij}\in [ 0,1]$. Accordingly,
\begin{align*}
\sum_{i=1,i\neq j}^{m-1}\alpha _{i}u'(\eta _{ij})(\xi _{i}-\xi _{j})
&=\sum_{i=1,i\neq j}^{m-1}\alpha _{i}(u(\xi _{i})-u(\xi_{j}))\\
&=-\sum_{i=1}^{m-1}\alpha _{i}u(\xi _{j})=-\sum_{i=1}^{m-1}\alpha
_{i}\xi _{j}u'(\eta _{jj}),
\end{align*}
 using the mean-value theorem and the
assumptions $u(0)=0$, $\sum_{i=1}^{m-1}\alpha _{i}u(\xi _{i})=0$
(equivalently, $u(1)=\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i})$). We thus get
$\sum_{i=1}^{m-1}\sigma _{ij}u'(\eta _{ij})=0$, and hence
$\sum_{i=1}^{m-1}(\sigma _{ij})^{+}u'(\eta
_{ij})=\sum_{i=1}^{m-1}(\sigma _{ij})^{-}u'(\eta _{ij})$. So there
must exist $\chi _{j}^{1}$ and $\chi _{j}^{2}$ in $[0,1]$ such that
\begin{equation}
\big(\sum_{i=1}^{m-1}(\sigma _{ij})^{+}\Big)u'(\chi
_{j}^{1})=\Big(\sum_{i=1}^{m-1}(\sigma _{ij})^{-}\Big)u'(\chi _{j}^{1})
.  \label{EQ100}
\end{equation}
If one of $\sum_{i=1}^{m-1}(\sigma _{ij})^{+}$,
$\sum_{i=1}^{m-1}(\sigma _{ij})^{-}$ is zero for some
$j=1,2,\dots ,m-1$ then it
follows from (\ref{EQ100}) that there is an $\eta _{0}\in [ 0,1]$
(indeed one of $\chi _{j}^{1}$ or $\chi _{j}^{2}$) such that
$u'(\eta _{0})=0$ and the estimate (\ref{mana10}) is immediate.

Next, suppose that $\sum_{i=1}^{m-1}(\sigma _{ij})^{+}\neq 0$ and
$\sum_{i=1}^{m-1}(\sigma _{ij})^{-}\neq 0$ for every
$j=1,2,\dots ,m-1$. Then either $u'(\chi _{j}^{1})=u'(\chi
_{j}^{1})=0$ for some $j=1,2,\dots  ,m-1$, in which case the
estimate (\ref{mana10}) is immediate, or $u'(\chi _{j}^{1})\neq
u'(\chi _{j}^{1})$ for every $j=1,2,\dots  ,m-1$. It
follows that there exist $\eta _{1},\eta _{2}\in [ 0,1]$ with
$u'(\eta _{1})\neq u'(\eta _{2})$ such that
\begin{equation}
u'(\eta _{1})=\sigma ^{\ast }u'(\eta _{2}).  \label{EQ101}
\end{equation}
The estimate (\ref{APEstimate}) is now immediate from (\ref{EQ8}), (\ref
{EQ101}) and the equation
\begin{equation*}
u'(t)=u'(\eta _{1})+\int_{\eta _{1}}^{t}u''ds.
\end{equation*}
This completes the proof of the lemma.
\end{proof}

\begin{proof}[Proof of Theorem \ref{nonresonant}]
 We consider the family of boundary-value problems:
\begin{equation}
\begin{gathered}
(\phi (u''(t)))'=\lambda f(t,u(t),u'(t),u''(t)), \quad
t\in (0,1), \lambda \in [0,1],  \\
u(0)=0,   \quad
u(1)=\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i}),  \quad
u''(0)=0.
\end{gathered} \label{NRlambda}
\end{equation}
Also, we define an operator
$\Psi ^{\ast }:C^{2}[0,1]\times [0,1]\mapsto C^{2}[0,1]$ by setting for
$(u,\lambda )\in C^{2}[0,1]\times[ 0,1]$
\begin{align*}
\Psi ^{\ast }(u,\lambda )
&= \int_{0}^{t}\Big(u'(0)+\int_{0}^{s}\phi ^{-1}\Big(\lambda
\int_{0}^{r}f^{\ast }(\tau,u(\tau ),u'(\tau ),u''(\tau ),\lambda )d\tau \Big)dr\Big)ds \\
&\quad +t\Big(u(1)-\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i})\Big).
\end{align*}
Following standard arguments, it can be proved that $\Psi ^{\ast }$
is a completely continuous operator. Furthermore reasoning in an
entirely similar way as we did in the proof of Lemma \ref{DefLemma}
it can be proved that $u$ is a solution to the family of
boundary-value problems (\ref{NRlambda}) if and only if $u$ is a
fixed point for the operator $\Psi ^{\ast }(\cdot,\lambda )$; i.e.,
$u$ satisfies
\begin{equation*}
u=\Psi ^{\ast }(u,\lambda ).
\end{equation*}
We will show next that there is a constant $R>0$ independent of
$\lambda \in [ 0,1]$ such that if $u$ satisfies (\ref{NRlambda})
for some $\lambda \in [ 0,1]$ then $\|u\|_{C^{2}[0,1]}<R$.

We note first that if $u$ satisfies
\begin{equation*}
u=\Psi ^{\ast }(u,0),
\end{equation*}
then we must have $u=0$. Indeed from the definition of $\Psi ^{\ast }$ or
from problem (\ref{NRlambda}), it follows that $u(t)=\rho t$ with $\rho
=u'(0)=u'(t)$, for all $t\in [ 0,1]$. Then from the
second boundary condition in (\ref{NRlambda}), and the assumption
$\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}\neq 1$, we find that $\rho =0$, implying
that $u(t)=0$ for all $t\in [ 0,1].$

In the rest of the argument we will assume that $\lambda \in (0,1]$. Also we
will suppose that $\sigma ^{\ast }>0$ since the proof for the case
$\sigma ^{\ast }=0$ is simpler.

Let us choose $\varepsilon >0$ such that
\begin{equation}
(\tilde{\alpha}(\sigma ^{\ast })+\varepsilon
)(\|d_{1}\|_{L^{1}(0,1)}+\| d_{2}\| _{L^{1}(0,1)})+\|
d_{3}\| _{L^{1}(0,1)}<1,  \label{EQ11}
\end{equation}
which can be done in view of the assumption (\ref{cond1m}). Next, we
have from the definition of $\tilde{\alpha}$, as given in
(\ref{EQ8}), that there exists a constant $C_{\varepsilon }^{1}$
such that
\begin{equation}
\phi (\frac{1}{1-\sigma ^{\ast }}z)\leq (\tilde{\alpha}(\sigma ^{\ast
})+\varepsilon )\phi (z)+C_{\varepsilon }^{1}\text{, for all }z.
\label{EQ12}
\end{equation}
Let, now, $u$ be a solution of the family of boundary-value problems (\ref
{NRlambda}). Then $u\in C^{2}[0,1]$ with $\phi (u''(t))$
absolutely continuous on $[0,1]$ and satisfies
\begin{equation*}
u(0)=0, u(1)=\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i}), u''(0)=0.
\end{equation*}
We, now, use the estimates
\begin{equation}
\|u\|_{\infty }\leq \|u'\|_{\infty }, \| u'\| _{\infty }
\leq \frac{1}{1-\sigma ^{\ast }}\| u''\| _{\infty },
\phi (\| u''\| _{\infty })\leq \| (\phi (u''))'\| _{L^{1}(0,1)}  \label{EQ13n}
\end{equation}
and the inequality (\ref{EQ12}) to get
\begin{align*}
&\| (\phi (u''))'\| _{L^{1}(0,1)}\\
&\leq \phi (\| u\| _{\infty })\| d_{1}\|
 _{L^{1}(0,1)}+\phi (\| u'\| _{\infty })\|
 d_{2}\| _{L^{1}(0,1)}  \\
&\quad +\phi (\| u''\| _{\infty })\|
 d_{3}\| _{L^{1}(0,1)}+\| r\| _{L^{1}(0,1)}  \\
&\leq (\| d_{1}\| _{L^{1}(0,1)}+\| d_{2}\|_{L^{1}(0,1)})
 \phi (\frac{1}{1-\sigma ^{\ast }}\|u''\|_{\infty })  \\
&\quad +\phi (\| u''\| _{\infty })\|
 d_{3}\| _{L^{1}(0,1)}+\| r\| _{L^{1}(0,1)}  \\
&\leq \Big(\tilde{\alpha}(\sigma ^{\ast })+\varepsilon \Big)
 (\| d_{1}\| _{L^{1}(0,1)}+\| d_{2}\|
  _{L^{1}(0,1)})\phi (\|u''\|_{\infty })+\|
 d_{3}\| _{L^{1}(0,1)}\phi (\| u''\|
 _{\infty })+C_{\varepsilon }  \\
&\leq { [(}\tilde{\alpha}(\sigma ^{\ast })+\varepsilon \Big)
(\| d_{1}\| _{L^{1}(0,1)}+\| d_{2}\|
_{L^{1}(0,1)})+\| d_{3}\| _{L^{1}(0,1)}{ ]}\|
(\phi (u''))'\| _{L^{1}(0,1)}+C_{\varepsilon },
\end{align*} %\label{EQ15}
where
\begin{equation*}
C_{\varepsilon }=\| r\| _{L^{1}(0,1)}+C_{\varepsilon
}^{1}(\| d_{1}\| _{L^{1}(0,1)}+\| d_{2}\|
_{L^{1}(0,1)}).
\end{equation*}
It, now, follows from (\ref{EQ11}) that there exists a constant $R_{0}>0$,
independent of $\lambda \in (0,1]$ such that if $u$ is a solution of the
family of boundary-value problems (\ref{NRlambda}) then
\begin{equation*}
\| (\phi (u''))'\| _{L^{1}(0,1)})\leq R_{0}.
\end{equation*}
This, combined with (\ref{EQ13n}) gives that there exist a constant $R>0$
such that
\begin{equation*}
\|u\|_{C^{2}[0,1]}<R.
\end{equation*}
This in turn implies that $\deg _{LS}(I-\Psi ^{\ast }(\cdot ,\lambda
),B(0,R),0)$ is well defined for all $\lambda \in [ 0,1]$, where
$B(0,R)$ is the ball with center 0 and radius $R$ in $C^{2}[0,1]$.

In what follows we will use the notation of section \ref{defor-lemma}, thus
$X$ will denote the one dimensional subspace of $C^{2}[0,1]$ given by
$X=\{i_{\rho }: \rho \in {\mathbb{R}}\}$, $i_{\rho }(t)=\rho t$ and
$i:{\mathbb{R}}\mapsto X$ is the isomorphism from ${\mathbb{R}}$ onto $X$ given
by $i(\rho )=i_{\rho }$. Let us define the function
$G:{\mathbb{R}}\mapsto {\mathbb{R}}$ by
\begin{equation}
G(\rho )=\Big(\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}-1\Big)\rho ,
\label{func-G}
\end{equation}
 for $w\in X$, $w(t)=\rho t$ for some $\rho \in {\mathbb{R}}$.
Now, since
\begin{equation*}
(I-\Psi ^{\ast }(\cdot ,0))(w)=i_{G(\rho )},
\end{equation*}
it is easy to see that
\begin{equation*}
G=i^{-1}\circ (I-\Psi ^{\ast }(\cdot ,0))|_{X}\circ i,
\end{equation*}
and hence, by the homotopy invariance property of Leray-Schauder degree, it
follows that
\begin{gather*}
\deg _{LS}(I-\Psi ^{\ast }(\cdot ,1),B(0,R),0) = \deg _{LS}(I-\Psi ^{\ast
}(\cdot ,0),B(0,R),0) \\
\deg _{B}(I-\Psi ^{\ast }(\cdot ,0)|_{X},X\cap B(0,R),0) = \deg
_{B}(G,(-R,R),0).
\end{gather*}
Thus taking into account (\ref{func-G}), we obtain the interesting formulas
for the degree
\begin{equation*}
\deg _{LS}(I-\Psi ^{\ast }(\cdot ,1),B(0,R),0)=
\begin{cases}
1 &\text{if }\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}>1 \\
-1&\text{if }\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}<1.
\end{cases}
\end{equation*}
Hence if $\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}\neq 1$ we have that $\deg
_{LS}(I-\Psi ^{\ast }(\cdot ,1),B(0,R),0)\neq 0$ and there is a $u\in B(0,R)$
that satisfies
\begin{equation*}
u=\Psi ^{\ast }(\cdot ,1),
\end{equation*}
equivalently $u$ is a solution to the boundary-value problem (\ref
{nonresonant}). This completes the proof of the theorem.
\end{proof}

\begin{thebibliography}{00}

\bibitem{B} {Cuan-zhi, Bai and Fang, Jin-xuan}; \emph{Existence of
multiple positive solutions for nonlinear $m$-point boundary-value problems},
Applied Mathematics and Computation, \textbf{140 }(2003), 297-305.

\bibitem{FW1} {Feng, W. and Webb, J. R. L.} \emph{Solvability of
three-point boundary-value problems at resonance}, Nonlinear Analysis T.M.A.
\textbf{30} (1997) 3227-3238 .

\bibitem{FW2} {Feng, W. and Webb, J. R. L.}; \emph{Solvability of $m$
-point boundary-value problems with nonlinear growth}, J. Math. Anal. Appl.
212 (1997) 467--480.

\bibitem{GGM1} {Garcia-Huidobro, M.; Gupta, C. P. and Manasevich, R.};
\emph{Solvability for a Non-linear Three-Point Boundary Value problem
with p-Laplacian-Like Operator at Resonance }. Abstract Analysis and
Applications, Vol. 6, No. 4, (2001) pp. 191-213.

\bibitem{GGM2} {Garcia-Huidobro, M.; Gupta, C. P. and
Manasevich, R.} \emph{ An m-point boundary-value problem of Neumann type for a
p-Laplacian like operator}, Nonlinear Analysis \textbf{56} (2004) 1071-1089.


\bibitem{GM} {Garcia-Huidobro, M. and Manasevich, R.}; \emph{A three
point boundary-value problem containing the operator
$-(\phi(u'))'$}, Discrete and Continuous Dynamical Systems, Proceedings of the
fourth international conference on dynamical systems and differential
equations, Wilmington, (2003), 313-319.

\bibitem{GMZ1} {Garcia-Huidobro M., Manasevich R.
and Zanolin, F.}; \emph{Strongly Nonlinear Second Order ODE's with
Unilateral Conditions}, Differential and Integral Equations \textbf{6}
(1993) 1057-1078 .

\bibitem{GU1} {Gupta, C. P.}; \emph{Solvability of a
three-point boundary-value problem for a second order ordinary differential
equation}, Journal of Mathematical Analysis and Applications \textbf{168}
(1992) 540-551.

\bibitem{GU2} {Gupta, C. P.}; \emph{A note on a
second order three-point boundary-value problem }, Journal of Mathematical
Analysis and Applications \textbf{186} (1994) 277-281.

\bibitem{GU3} {Gupta, C. P.}; \emph{A second order
m-point boundary-value problem at resonance}, Nonlinear Analysis T.M.A.,
\textbf{24} (1995) 1483-1489.

\bibitem{GU4} {Gupta, C. P.}; \emph{Existence
theorems for a second order m-point boundary-value problem at resonance}.
International Jour. Math. \& Math. Sci. 18(1995) pp. 705-710

\bibitem{GNT1} {Gupta, C. P.; Ntouyas, S.; Tsamatos, P. Ch.};
 \emph{On an m-point boundary-value problem for second order
ordinary differential equations}, Nonlinear Analysis T.M.A. \textbf{23}
(1994) 1427-1436.

\bibitem{GNT} {Gupta, C. P.; Ntouyas, S.; Tsamatos, P. Ch.};
 \emph{Solvability of an m-point boundary-value problem for second
order ordinary differential equations}, Journal of Mathematical Analysis and
Applications \textbf{189} (1995) 575-584.

\bibitem{GT0} {Gupta, Chaitan P.; Trofimchuk, Sergej I.};
 \emph{Solvability of a multi-point boundary-value problem and related
a priori estimates}, Geoffrey J. Butler Memorial Conference in Differential
Equations and Mathematical Biology (Edmonton, AB, 1996). Canad. Appl. Math.
Quart. \textbf{6} (1998), no. 1, 45--60.

\bibitem{GT} Gupta, C. P. and Trofimchuk, S.; \emph{Solvability of
multi point boundary-value problem of Neumann type}, Abstract Analysis and
Applications \textbf{4} (1999) 71-81.

\bibitem{I1} {Il'in ,V. A. and Moiseev, E. I.}; \emph{Nonlocal
boundary-value problem of the first kind for a Sturm Liouville operator in
its differential and difference aspects}, Differential Equations \textbf{23}
(1987) 803-810.

\bibitem{I2} {Il'in, V. A. and Moiseev, E. I.}; \emph{Nonlocal
boundary-value problem of the second kind for a Sturm Liouville operator},
Differential Equations \textbf{23} (1987) 979-987.

\bibitem{L1} Liu, B.; \emph{Solvability of multi point boundary
value problem at resonance (I)}, Indian Journal of Pure and Applied
Mathematics \textbf{33} (2002) 475-494.

\bibitem{L2} Liu, B.; \emph{Solvability of multi point boundary
value problem at resonance (II)}, Applied Mathematics and Computations
\textbf{136} (2003) 353-377.

\bibitem{L3}  Liu, B. and Yu, J.; \emph{Solvability of multi point
boundary-value problem at resonance (III)}, Applied Mathematics and
Computations \textbf{129} (2002) 119-143.

\bibitem{L4}  Liu, B.; \emph{Solvability of multi point boundary
value problem at resonance (IV)}, Applied Mathematics and Computations
\textbf{143} (2003) 275-299.

\bibitem{LG} Liu, Y. and Ge, W.; \emph{Multiple positive
solutions to a three-point boundary value problem with $p$-Laplacian},
Journal of Mathematical Analysis and Applications \textbf{277} (2003)
293-302.

\bibitem{rayun} Ma, Rayun and Casta\~{n}eda, Nelson;
\emph{Existence of Solutions of Nonlinear $m$-point Boundary Value Problems},
Journal of Mathematical Analysis and Applications \textbf{256 }(2001)
556-567.

\bibitem{[SE]} Sedziwy, S.; \emph{Multipoint
boundary-value problems fro a second order differential equations}, Journal
of Mathematical Analysis and Applications \textbf{236} (1999) 384-398.

\bibitem{TH2} Thompson, H. B. and Tisdell, C.;
\emph{Three-point boundary-value problems for second-order, ordinary,
differential equations}, Math. Comput. Modelling \textbf{34} (2001), no.
3-4, 311--318.

\bibitem{WA1} Wang, J.-Y and Jiang, D.;
 \emph{A unified approach to some two-point, three-point, and four-point boundary
value problems with Carath\'{e}odory functions}, Journal of Mathematical
Analysis and Applications \textbf{211} (1997) 223-232.

\bibitem{WA2} Wang, J.-Y. and Zheng, D.-W.;
\emph{On the existence of positive solutions to a three-point boundary
value problem for the one-dimensional $p$-Laplacian}, ZAAM \textbf{77}
(1997) 477-479.


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