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

\AtBeginDocument{{\noindent\small
Sixth Mississippi State Conference on Differential Equations and
Computational Simulations,
{\em Electronic Journal of Differential Equations},
Conference 15 (2007),  pp. 127--139.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{127}
\title[\hfilneg EJDE-2006/Conf/15\hfil
 A multi-point boundary-value problem]
{A non-resonant generalized multi-point boundary-value problem of
Dirichelet type involving a p-laplacian type operator}

\author[C. P. Gupta\hfil EJDE/Conf/15 \hfilneg]
{Chaitan P. Gupta}

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

\thanks{Published February 28, 2007.}
\subjclass[2000]{34B10, 34B15, 34L30, 34L90}
\keywords{Generalized multi-point boundary value problems;  non-resonance;
\hfill\break\indent
p-Laplace type operator; a priori estimates; topological degree}

\begin{abstract}
 We study the existence of solutions for the generalized multi-point
 boundary-value problem
 \begin{gather*}
 (\phi (x'))'=f(t,x,x')+e\quad 0<t<1, \\
 x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i),\quad
 x(1)=\sum_{j=1}^{n-2}b_jx(\tau _j),
 \end{gather*}
 in the non-resonance case.
 Our methods consist in using topological degree and some a priori estimates.
\end{abstract}

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

\section{Introduction}

Let $\phi $ be an odd increasing homeomorphism from $\mathbb{R}$
onto $\mathbb{R}$ satisfying $\phi (0)=0$, $f:[0,1]\times
\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a function
satisfying Carath\'{e}odory conditions and $e:[0,1]\to \mathbb{R}$
be a function in $L^{1}[0,1]$. Let $\xi _i$,$\tau _j\in (0,1)$,
$a_i$, $b_j\in \mathbb{R}$, $i=1,2,\dots,m-2$, $j=1,2,\dots,n-2$,
$0<\xi _{1}<\xi _{2}<\dots <\xi _{m-2}<1$,
 $0<\tau _{1}<\tau _{2}<\dots <\tau _{n-2}<1$ be given. We study the
problem of existence of solutions for the generalized multi-point
boundary-value problem
\begin{equation}
\begin{gathered}
(\phi (x'))'=f(t,x,x')+e, \quad 0<t<1,  \\
x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i), \quad
x(1)=\sum_{j=1}^{n-2}b_jx(\tau _j),
\end{gathered}   \label{1mbp}
\end{equation}
in the non-resonance case. We say that this problem is non-resonant if the
associated problem:
\begin{equation}
\begin{gathered}
(\phi (x'))'=0, \quad 0<t<1,   \\
x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i), \quad
x(1)=\sum_{j=1}^{n-2}b_jx(\tau _j),   \label{1hmbp}
\end{gathered}
\end{equation}
has the trivial solution as its only solution.  This is the case,
(see Proposition \ref{Prop0} below), if
\begin{equation*}
\Big(\sum_{i=1}^{m-2}a_i\xi _i\Big)\Big(1-\sum_{j=1}^{n-2}b_j\Big)\neq
\Big(1-\sum_{i=1}^{m-2}a_i\Big)\Big(\sum_{j=1}^{n-2}b_j\tau _j-1\Big).
\end{equation*}
This problem was studied by Gupta, Ntouyas, and Tsamatos in \cite{GNT4} and
by the author in \cite{GU6} when the homeomorphism $\phi $ from $\mathbb{R}$
onto $\mathbb{R}$ is the identity homeomorphism, i.e for second order
ordinary differential equations. The study of multi-point boundary value
problems for second order ordinary differential equations was initiated by
Il'in and Moiseev in \cite{I1,I2} motivated by the works of Bitsadze
and Samarskii on nonlocal linear elliptic boundary value problems,
\cite{b1,b2,bs} and has been the subject of many papers, see for
example,
\cite{FW1,FW2,GU1,GU2,GU3,GU4,GU5,GNT1,GNT2,GNT3,GT,L4,SE,TH2}.
More recently multipoint boundary value
problems involving a $p$-Lalacian type operator or the more general operator
$-(\phi (x'))'$ has been studied in
\cite{B,GGM1,GGM2,GGM3,GM,LG} to mention a few.

We present in Section 2 some a priori estimates for functions $x(t)$ that
satisfy the boundary conditions in \eqref{1mbp}. Our a priori estimates are
sharper versions of the corresponding estimates in \cite{GU6} and explicitly
utilize the non-resonance condition for the boundary value problem
\eqref{1mbp}.  In section 3, we present an existence theorem for the boundary
value problem \eqref{1mbp} using degree theory.

\section{A Priori Estimates}

We shall assume throughout that $\phi $ is an odd increasing homeomorphism
from $\mathbb{R}$ onto $\mathbb{R}$ satisfying $\phi (0)=0$. We shall also
assume that the homeomorphism $\phi $ satisfies the following conditions:

(a) For any constant $M>0$,
\begin{equation}
\limsup_{z\to \infty }\frac{\phi (Mz)}{\phi (z)}\equiv \alpha
(M)<\infty .   \label{cond1}
\end{equation}

(b) For any $\sigma $, $0\leq \sigma <1$,
\begin{equation}
\widetilde{\alpha }(\sigma )\equiv \limsup_{z\to \infty }\frac{
\phi (\sigma z)}{\phi (z)}<1.   \label{cond2}
\end{equation}

\begin{proposition} \label{Prop0}
The boundary-value problem \eqref{1hmbp} has only
the trivial solution if and only if
\begin{equation}
\Big(\sum_{i=1}^{m-2}a_i\xi _i\Big)\Big(1-\sum_{j=1}^{n-2}b_j\Big)\neq
\Big(1-\sum_{i=1}^{m-2}a_i\Big)\Big(\sum_{j=1}^{n-2}b_j\tau _j-1\Big).
\label{NRcond}
\end{equation}
\end{proposition}

\begin{proof} It is obvious that $x(t)=At+B$, $t\in [0,1]$, $A$, $B\in
\mathbb{R}$, is a general solution for the differential equation
\begin{equation*}
(\phi (x'))'=0, \quad 0<t<1,
\end{equation*}
in \eqref{1hmbp}. If, now, $x(t)=At+B$, $t\in [0,1]$, $A$, $B\in
\mathbb{R}$, is a solution to the boundary value problem \eqref{1hmbp} then
we must have
\[
B =\sum_{i=1}^{m-2}a_i(A\xi _i+B),  \quad
A+B =\sum_{j=1}^{n-2}b_j(A\tau _j+B).
\]
In other words $A$, $B$ must satisfy the system of equations
\begin{equation}
\begin{gathered}
A(\sum_{i=1}^{m-2}a_i\xi _i)+B(\sum_{i=1}^{m-2}a_i-1) =0, \\
A(\sum_{j=1}^{n-2}b_j\tau _j-1)+B(\sum_{j=1}^{n-2}b_j-1) =0.
\end{gathered}\label{eqn0}
\end{equation}
Now, the system of equations (\ref{eqn0}) has $A=0$, $B=0$ as the
only solution if and only if
\begin{equation*}
\det
\begin{pmatrix}
\sum_{i=1}^{m-2}a_i\xi _i & \sum_{i=1}^{m-2}a_i-1 \\
\sum_{j=1}^{n-2}b_j\tau _j-1 & \sum_{j=1}^{n-2}b_j-1
\end{pmatrix} \neq 0,
\end{equation*}
or
\begin{equation}
\Big(\sum_{i=1}^{m-2}a_i\xi
_i\Big)\Big(\sum_{j=1}^{n-2}b_j-1\Big)
-\Big(\sum_{i=1}^{m-2}a_i-1\Big)\Big(
\sum_{j=1}^{n-2}b_j\tau _j-1\Big)\neq 0.   \label{eqn1}
\end{equation}
It is now obvious that (\ref{eqn1}) is equivalent to \eqref{NRcond}. Hence
the boundary value problem \eqref{1hmbp} has only the trivial solution if
and only if the condition \eqref{NRcond} holds. This completes the proof of
the Proposition.
\end{proof}

 We shall assume in the following that $\xi _i$,
$\tau _j\in (0,1) $, $a_i$, $b_j\in \mathbb{R}$, $i=1, 2, \dots ,
m-2$, $j=1, 2, \dots , n-2$, $0<\xi _{1}<\xi _{2}<\dots <\xi
_{m-2}<1$, $0<\tau _{1}<\tau _{2}<\dots <\tau _{n-2}<1$ satisfy
the condition \eqref{NRcond}. We observe that when condition
\eqref{NRcond} holds at least one of $1-\sum_{i=1}^{m-2}a_i$,
$1-\sum_{j=1}^{n-2}b_j$ is non-zero. Now, for $a\in R$, we set
$a^{+}=\max (a,0)$, $a^{-}=\max (-a,0)$ so that $a=a^{+}-a^{-}$
and $|a|=a^{+}+a^{-}$. Next, in case $1-\sum_{i=1}^{m-2}a_i\neq
0$, we notice that
\begin{equation*}
\sigma _{1}\equiv \min \big\{\frac{\sum_{i=1}^{m-2}a_i^{+}}{
1+\sum_{i=1}^{m-2}a_i^{-}},\frac{1+\sum_{i=1}^{m-2}a_i^{-}}{
\sum_{i=1}^{m-2}a_i^{+}}\big\}\in [0,1)
\end{equation*}
is well-defined. Similarly, if $1-\sum_{j=1}^{n-2}b_j\neq 0$, we see that
\begin{equation*}
\sigma _{2}\equiv \min \{\frac{\sum_{j=1}^{n-2}b_j^{+}}{
1+\sum_{j=1}^{n-2}b_j^{-}},\frac{1+\sum_{j=1}^{n-2}b_j^{-}}{
\sum_{j=1}^{n-2}b_j^{+}}\}\in [0,1)
\end{equation*}
is well-defined. Accordingly, let us define
\begin{equation}
\sigma _{1} \equiv \begin{cases}
 \min \{\frac{\sum_{i=1}^{m-2}a_i^{+}}{
1+\sum_{i=1}^{m-2}a_i^{-}},\frac{1+\sum_{i=1}^{m-2}a_i^{-}}{
\sum_{i=1}^{m-2}a_i^{+}}\}\in [0,1)&\text{if }1-
\sum_{i=1}^{m-2}a_i\neq 0,    \\
1&\text{if }1-\sum_{i=1}^{m-2}a_i=0;
\end{cases} \label{defsigma1}
\end{equation}
and
\begin{equation}
\sigma _{2} \equiv
\begin{cases}
\min \{\frac{\sum_{j=1}^{n-2}b_j^{+}}{
1+\sum_{j=1}^{n-2}b_j^{-}},\frac{1+\sum_{j=1}^{n-2}b_j^{-}}{
\sum_{j=1}^{n-2}b_j^{+}}\}\in [0,1) &\text{if }1-
\sum_{j=1}^{n-2}b_j\neq 0,    \\
1 &\text{if }1-\sum_{j=1}^{n-2}b_j=0.
\end{cases}\label{defsigma2}
\end{equation}
The a priori estimate obtained in the following proposition is a sharpening
of the a priori estimate of Lemma 2 of \cite{GU6}. We repeat the details
given in Lemma 2 of \cite{GU6} for the sake of completeness.

\begin{proposition} \label{Prop1}
Let $\xi _i$,$\tau _j\in (0,1)$, $a_i$, $b_j\in \mathbb{R}$,
$i=1,2,\dots,m-2$, $j=1, 2, \dots , n-2$, $0<\xi _{1}<\xi
_{2}<\dots <\xi _{m-2}<1$, $0<\tau _{1}<\tau _{2}<\dots <\tau
_{n-2}<1$, with $(\sum_{i=1}^{m-2}a_i\xi
_i)(1-\sum_{j=1}^{n-2}b_j)\neq
(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j\tau _j-1)$ be given.
Also let the function $x(t)$ be such that $x(t)$, $x'(t)$ be
absolutely continuous on $[0,1]$ and
$x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i)$,
$x(1)=\sum_{j=1}^{n-2}b_jx(\tau _j)$. Then
\begin{equation}
\| x\| _{\infty }\leq M\| x'\| _{\infty},   \label{est0}
\end{equation}
where
\begin{align*}
M =\min \Big\{&\frac{1}{| \sum_{i=1}^{m-2}a_i| }
(\sum_{i=1}^{m-2}| a_i| \lambda _i+\frac{\sum_{i=1}^{m-2}|
a_i\xi _i| }{| 1-\sum_{i=1}^{m-2}a_i| }), \\
&\frac{1}{| \sum_{j=1}^{n-2}b_j| }(\sum_{j=1}^{n-2} | b_j|
\mu _j+\frac{\sum_{j=1}^{n-2}| b_j(1-\tau _j)| }{|
1-\sum_{j=1}^{n-2}b_j| }),1+\frac{\sum_{i=1}^{m-2}| a_i\xi
_i| }{| 1-\sum_{i=1}^{m-2}a_i| }, \\
&1+\frac{\sum_{j=1}^{n-2}| b_j(1-\tau _j)| }{|
1-\sum_{j=1}^{n-2}b_j| },\frac{1}{1-\sigma _{1}},\frac{1}{1-\sigma
_{2}}\Big\}
\end{align*}
with $\lambda _i=\max (\xi _i,1-\xi _i)$ for
$i=1,2\dots, m-2$,
$\mu _j=\max (\tau _j,1-\tau _j)$ for $j=1, 2, \dots, n-2$,
$\sigma _{1}$ as defined in (\ref{defsigma1}) and $\sigma _{2}$
as defined in (\ref{defsigma2}).
\end{proposition}

\begin{proof} We first observe that at least one of
$(1-\sum_{i=1}^{m-2}a_i)$,
$(1-\sum_{j=1}^{n-2}b_j)$ is non-zero, in view of our assumption
\begin{equation*}
\Big(\sum_{i=1}^{m-2}a_i\xi _i\Big)\Big(1-\sum_{j=1}^{n-2}b_j\Big)\neq
\Big(1-\sum_{i=1}^{m-2}a_i\Big)\Big(\sum_{j=1}^{n-2}b_j\tau _j-1\Big).
\end{equation*}
Accordingly, $M<\infty $. Next, we see from
$x(\xi _i)-x(0)=\int_{0}^{\xi_i}x'(s)ds$ for $i=1,2,\dots , m-2$ and the
assumption that
$x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i)$, that
\[
(1-\sum_{i=1}^{m-2}a_i)x(0)=\sum_{i=1}^{m-2}a_i\int_{0}^{\xi _i}x'(s)ds.
\] It then follows that
\begin{equation}
| x(0)| \leq \frac{\sum_{i=1}^{m-2}| a_i\xi _i| }{|
1-\sum_{i=1}^{m-2}a_i| }\| x'\| _{\infty }.  \label{est1}
\end{equation}
Also, since $x(t)=x(\xi _i)+\int_{\xi _i}^{t}x'(s)ds$, we see
that
\begin{equation*}
(\sum_{i=1}^{m-2}a_i)x(t)=\sum_{i=1}^{m-2}a_ix(\xi
_i)+\sum_{i=1}^{m-2}a_i\int_{\xi _i}^{t}x'(s)ds=x(0)+\sum_{i=1}^{m-2}a_i\int_{\xi _i}^{t}x'(s)ds.
\end{equation*}
We, now, use (\ref{est1}) to get
\begin{align*}
| \sum_{i=1}^{m-2}a_i| | x(t)|
&\leq | x(0)|
+\sum_{i=1}^{m-2}| a_i| | \int_{\xi _i}^{t}x'(s)ds|\\
&\leq \big(\frac{\sum_{i=1}^{m-2}| a_i\xi _i| }{|
1-\sum_{i=1}^{m-2}a_i| }+\sum_{i=1}^{m-2}\lambda _i| a_i|
\Big)\| x'\| _\infty .
\end{align*}
It is now immediate that
\begin{equation}  \label{est2}
\| x\| _\infty \leq \frac 1{| \sum_{i=1}^{m-2}a_i| }
\Big(
\frac{\sum_{i=1}^{m-2}| a_i\xi _i| }{| 1-\sum_{i=1}^{m-2}a_i| }
+\sum_{i=1}^{m-2}\lambda _i| a_i| \Big)\| x'\|
_\infty .
\end{equation}
Similarly, starting from $x(1)-x(\tau _j)=\int_{\tau _j}^1x'(s)ds$
and proceeding, as above, we obtain the estimate
\begin{equation}  \label{est3}
\| x\| _\infty \leq \frac 1{| \sum_{j=1}^{n-2}b_j| }
\Big(\frac{\sum_{j=1}^{n-2}| b_j(1-\tau _j)| }{|
1-\sum_{j=1}^{n-2}b_j| }+\sum_{j=1}^{n-2}\mu _j| b_j| \Big)\|
x'\| _\infty .
\end{equation}
If we next use the equation $x(t)=x(0)+\int_0^tx'(s)ds$ and the
estimate (\ref{est1}) we obtain
\begin{equation}  \label{est21}
\| x\| _\infty \leq \Big(\frac{\sum_{i=1}^{m-2}| a_i\xi _i|
}{| 1-\sum_{i=1}^{m-2}a_i| }+1\Big)\| x'\| _\infty.
\end{equation}
Similarly, starting from the equation $x(t)=x(1)-\int_t^1x'(s)ds$,
we obtain the estimate
\begin{equation}
\| x\| _{\infty }\leq (\frac{\sum_{j=1}^{n-2}| b_j(1-\tau
_j)| }{| 1-\sum_{j=1}^{n-2}b_j| }+1)\| x'\| _{\infty }.   \label{est31}
\end{equation}
Next, since $x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i)$ we see that
\begin{equation*}
x(0)+\sum_{i=1}^{m-2}a_i^{-}x(\xi _i)=\sum_{i=1}^{m-2}a_i^{+}x(\xi
_i).
\end{equation*}
It follows that there must exist $\chi _{1}$, $\chi _{2}$ in $[0,1]$ such
that
\begin{equation}
(1+\sum_{i=1}^{m-2}a_i^{-})x(\chi _{1})=(\sum_{i=1}^{m-2}a_i^{+})x(\chi
_{2}).   \label{est14}
\end{equation}
If, now, one of $x(\chi _{1})$, $x(\chi _{2})$ is zero, we see using one of
the two equations
\begin{equation}
x(t)=x(\chi _{k})+\int_{\tau _{k}}^{t}x'(s)ds, k=1,2\text{; }
t\in [0,1]  \label{est24}
\end{equation}
that
\begin{equation}
\| x\| _{\infty }\leq \| x'\| _{\infty }.   \label{est4}
\end{equation}
If both $x(\chi _{1})$, $x(\chi _{2})$ are non-zero and $1-
\sum_{i=1}^{m-2}a_i\neq 0$, so that $1+\sum_{i=1}^{m-2}a_i^{-}\neq
\sum_{i=1}^{m-2}a_i^{+}$, it is easy to see from (\ref{est14}) that
$x(\chi _{1})\neq x(\chi _{2})$. It then follows easily from (\ref{est14})
and (\ref{est24}) that
\begin{equation}
\| x\| _{\infty }\leq \frac{1}{1-\sigma _{1}}\|
x'\| _{\infty },   \label{est5}
\end{equation}
where
\[
\sigma _{1}=\min \{\frac{\sum_{i=1}^{m-2}a_i^{+}}{
1+\sum_{i=1}^{m-2}a_i^{-}},\frac{1+\sum_{i=1}^{m-2}a_i^{-}}{
\sum_{i=1}^{m-2}a_i^{+}}\}\in [0,1).
\]
Similarly, we see from $x(1)=\sum_{j=1}^{n-2}b_jx(\tau _j)$ that either
(\ref{est4}) holds or
\begin{equation}
\| x\| _{\infty }\leq \frac{1}{1-\sigma _{2}}\|
x'\| _{\infty },   \label{est6}
\end{equation}
where
\[
\sigma _{2}=\min \{\frac{\sum_{j=1}^{n-2}b_j^{+}}{
1+\sum_{j=1}^{n-2}b_j^{-}},\frac{1+\sum_{j=1}^{n-2}b_j^{-}}{
\sum_{j=1}^{n-2}b_j^{+}}\}\in [0,1).
\]
The proposition is now immediate from (\ref{est2}), (\ref{est3}),
(\ref{est21}), (\ref{est31}), (\ref{est4}), (\ref{est5}) and (\ref{est6})
and the definitions of $\sigma _{1}$, $\sigma _{2}$ as given in
(\ref{defsigma1}), (\ref{defsigma2}).
\end{proof}

The following lemma is needed in the next proposition.

\begin{lemma} \label{lemma1}
 Let us set
\begin{equation} \label{defA}
\begin{aligned}
A&=[(1-\sum_{i=1}^{m-2}a_i)(1-\sum_{j=1}^{n-2}b_j)]^{+}+
\sum_{j=1}^{n-2}[b_j(1-\tau _j)(1-\sum_{i=1}^{m-2}a_i)]^{+}
 \\
&\quad+\sum_{i=1}^{m-2}[a_i\xi _i(1-\sum_{j=1}^{n-2}b_j)]^{+}
\end{aligned}
\end{equation}
and
\begin{equation} \label{defB}
\begin{aligned}
B&=[(1-\sum_{i=1}^{m-2}a_i)(1-\sum_{j=1}^{n-2}b_j)]^{-}+
\sum_{j=1}^{n-2}[b_j(1-\tau _j)(1-\sum_{i=1}^{m-2}a_i)]^{-}\\
&\quad+\sum_{i=1}^{m-2}[a_i\xi _i(1-\sum_{j=1}^{n-2}b_j)]^{-}.
\end{aligned}
\end{equation}
Then $A\neq B$,
when the non-resonance assumption \eqref{NRcond} holds.
\end{lemma}

\begin{proof} We note that
\begin{align*}
&A-B\\
&=(1-\sum_{i=1}^{m-2}a_i)(1-\sum_{j=1}^{n-2}b_j)+
\sum_{j=1}^{n-2}b_j(1-\tau _j)(1-\sum_{i=1}^{m-2}a_i)
+\sum_{i=1}^{m-2}a_i\xi _i(1-\sum_{j=1}^{n-2}b_j)\\
&=1-\sum_{i=1}^{m-2}a_i-(1-\sum_{i=1}^{m-2}a_i)(
\sum_{j=1}^{n-2}b_j)+(\sum_{j=1}^{n-2}b_j)(1-\sum_{i=1}^{m-2}a_i) \\
&\quad -(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j\tau
_j)+(\sum_{i=1}^{m-2}a_i\xi _i)(1-\sum_{j=1}^{n-2}b_j) \\
&=1-\sum_{i=1}^{m-2}a_i-(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j
\tau _j)+(\sum_{i=1}^{m-2}a_i\xi _i)(1-\sum_{j=1}^{n-2}b_j) \\
&=(\sum_{i=1}^{m-2}a_i\xi
_i)(1-\sum_{j=1}^{n-2}b_j)-(1-\sum_{i=1}^{m-2}a_i)(
\sum_{j=1}^{n-2}b_j\tau _j-1)
\neq 0,
\end{align*}
in view of the non-resonance assumption \eqref{NRcond}. Hence $A\neq B$.
This completes the proof of the lemma.
\end{proof}

 Let us define
\begin{equation}
\sigma ^{\ast }=\min \{\frac{A}{B},\frac{B}{A}\}\in [0,1),
\label{eq0}
\end{equation}
where $A$, $B$ are as defined in Lemma \ref{lemma1}. Accordingly, we see
that
\begin{equation*}
\widetilde{\alpha }(\sigma ^{\ast })=\limsup_{z\to \infty }\frac{
\phi (\sigma ^{\ast }z)}{\phi (z)}<1,
\end{equation*}
in view of our assumption (\ref{cond2}). Let $\varepsilon >0$ be such that
$\widetilde{\alpha }(\sigma ^{\ast })+\varepsilon <1$ and the constant
$C_{\varepsilon }$ be such that
\begin{equation}
\phi (\sigma ^{\ast }z)\leq (\widetilde{\alpha }(\sigma ^{\ast
})+\varepsilon )\phi (z)+C_{\varepsilon },\quad
\text{for every }z\in \mathbb{R}.   \label{eq1}
\end{equation}

\begin{proposition}\label{Prop2}
 Let $\xi _i$,$\tau _j\in (0,1)$, $a_i$, $b_j\in \mathbb{R}$,
$i=1, 2, \dots , m-2$, $j=1, 2, \dots, n-2$, $0<\xi _{1}<\xi
_{2}<\dots <\xi _{m-2}<1$, $0<\tau _{1}<\tau _{2}<\dots <\tau
_{n-2}<1$, with $(\sum_{i=1}^{m-2}a_i\xi
_i)(1-\sum_{j=1}^{n-2}b_j)\neq
(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j\tau _j-1)$ be given.
Also let the function $x(t)$ be such that $x(t)$, $x'(t)$ be
absolutely continuous on $[0,1]$ with $(\phi (x'))'\in L^{1}(0,1)$
and $x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i)$,
$x(1)=\sum_{j=1}^{n-2}b_jx( \tau _j)$. Then
\begin{equation}
\| \phi (x')\| _{\infty }\leq \frac{1}{1-\widetilde{
\alpha }(\sigma ^{\ast })-\varepsilon }\| (\phi (x'))'\| _{L^{1}(0,1)}+\frac{C_{\varepsilon }}{1-\widetilde{
\alpha }(\sigma ^{\ast })-\varepsilon },   \label{est17}
\end{equation}
where $\varepsilon $ and $C_{\varepsilon }$ are as in \eqref{eq1}.
\end{proposition}

\begin{proof} For $i=1,2,\dots,m-2$ we see using mean value
theorem that there exist $\chi _i$ in $[0,1]$ such that
\begin{equation*}
x(\xi _i)-x(0)=\xi _ix'(\chi _i).
\end{equation*}
It then follows using $x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i)$ that
\begin{equation}
(1-\sum_{i=1}^{m-2}a_i)x(0)=\sum_{i=1}^{m-2}a_i\xi _ix'(\chi
_i).   \label{eq2}
\end{equation}
Again, for $j=1,2,\dots,n-2$ we see using
mean value theorem that there exist $\lambda _j$ in $[0,1]$ such that
\begin{equation*}
x(1)-x(\tau _j)=(1-\tau _j)x'(\lambda _j),
\end{equation*}
and we see using $x(1)=\sum_{j=1}^{n-2}b_jx(\tau _j)$ that
\begin{equation}
(\sum_{j=1}^{n-2}b_j-1)x(1)=\sum_{j=1}^{n-2}b_j(1-\tau _j)x'(\lambda _j).   \label{eq3}
\end{equation}
Also, we see that there exists a $\lambda \in [0,1]$ such that
\begin{equation}
x(1)-x(0)=x'(\lambda ).   \label{eq4}
\end{equation}
Now, we see from equations (\ref{eq2}), (\ref{eq3}), (\ref{eq4}) that
\begin{align*}
&(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j-1)x'(\lambda ) \\
&=(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j-1)(x(1)-x(0)) \\
&=(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j(1-\tau _j)
  x'(\lambda _j))-(\sum_{j=1}^{n-2}b_j-1)(\sum_{i=1}^{m-2}
  a_i\xi_ix'(\chi _i)).
\end{align*}
It follows that
\begin{align*}
&(1-\sum_{i=1}^{m-2}a_i)(1-\sum_{j=1}^{n-2}b_j)x'(\lambda
)+\sum_{j=1}^{n-2}b_j(1-\tau _j)(1-\sum_{i=1}^{m-2}a_i)x'(\lambda _j) \\
&+\sum_{i=1}^{m-2}a_i\xi _i(1-\sum_{j=1}^{n-2}b_j)x'(\chi_i)=0.
\end{align*}
Using, next, the intermediate value theorem we see that there exist
$\upsilon _{1}$, $\upsilon _{2}$ in $[0,1]$ such that
\begin{equation}
Ax'(\upsilon _{1})-Bx'(\upsilon _{2})=0,
\label{eq15}
\end{equation}
where $A$, $B$ are as defined in (\ref{defA}), (\ref{defB}). Suppose, now,
one of $x'(\upsilon _{1})$, $x'(\upsilon _{2})$ is zero.
We then see from one of the following equations
\begin{equation}
\phi (x'(t))=\phi (x'(\upsilon _{k}))+\int_{\upsilon
_{k}}^{t}(\phi (x'))'(s)ds,\quad k=1,2;\; t\in [0,1]  \label{eq5}
\end{equation}
that
\begin{equation}
\| \phi (x')\| _{\infty }\leq \| (\phi
(x'))'\| _{L^{1}(0,1)}.   \label{est7}
\end{equation}
Let us, next, suppose that both $x'(\upsilon _{1})$, $x'(\upsilon _{2})$
 are non-zero. Since, now, $A\neq B$, in view of Lemma \ref{lemma1}
we see from equation (\ref{eq15}) that
$x'(\upsilon _{1})\neq x'(\upsilon _{2})$.
We now use the equations
\begin{gather*}
\phi (x'(t)) =\phi (x'(\upsilon _{1}))+\int_{\upsilon
_{k}}^{t}(\phi (x'))'(s)ds
=\phi (\frac{B}{A}x'(\upsilon _{2}))+\int_{\upsilon
_{k}}^{t}(\phi (x'))'(s)ds,
\\
\phi (x'(t)) =\phi (x'(\upsilon _{2}))+\int_{\upsilon
_{k}}^{t}(\phi (x'))'(s)ds
=\phi (\frac{A}{B}x'(\upsilon _{1}))+\int_{\upsilon
_{k}}^{t}(\phi (x'))'(s)ds,
\end{gather*}
along with the definition of $\sigma ^{\ast }$, as given in \eqref{eq0},
(\ref{eq1}) and the estimate (\ref{est7}) to obtain the estimate
(\ref{est17}). This completes the proof of the proposition.
\end{proof}

\section{Existence Theorem}

 Let $\phi $ be an odd increasing homeomorphism from $\mathbb{R}$
onto $\mathbb{R}$ satisfying $\phi (0)=0$, $f:[0,1]\times
\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a function
satisfying Carath\'{e}odory conditions and $e:[0,1]\to \mathbb{R}$
be a function in $L^{1}[0,1]$. Let $\xi _i$, $\tau _j\in (0,1)$,
$a_i$, $b_j\in \mathbb{R}$, $i=1, 2,\dots,m-2$, $j=1,2,\dots,n-2$,
$0<\xi _{1}<\xi _{2}<\dots <\xi _{m-2}<1$, $0<\tau _{1}<\tau
_{2}<\dots <\tau _{n-2}<1$ with $(\sum_{i=1}^{m-2}a_i\xi
_i)(1-\sum_{j=1}^{n-2}b_j)\neq
(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j\tau _j-1)$.

\begin{theorem} \label{Thm1}
 Let $f:[0,1]\times \mathbb{R}\times \mathbb{R}\to
\mathbb{R}$ be a function satisfying Carath\'{e}odory's conditions
such that there exist non-negative functions $d_{1}(t)$,
$d_{2}(t)$, and $r(t)$ in $L^{1}(0,1)$ such that
\begin{equation*}
|f(t,u,v)|\leq d_{1}(t)\phi (|u|)+d_{2}(t)\phi (|v|)+r(t),
\end{equation*}
for a. e. $t\in [0,1]$ and all $u$, $v\in \mathbb{R}$. Suppose,
further,
\begin{equation}
\alpha (M)\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)}<1-\widetilde{\alpha }
(\sigma ^{\ast })  \label{existcond}
\end{equation}
where $M$ is as defined in Proposition \ref{Prop1}, $\alpha (M)$ is as
defined in \eqref{cond1}, $\sigma ^{\ast }$ and $\widetilde{\alpha }(\sigma
^{\ast })$ are as defined in \eqref{eq0}, \eqref{eq1}. Then, for every given
function $e(t)\in L^{1}[0,1]$, the boundary value problem \eqref{1mbp} has
at least one solution $x(t)\in $ $C^{1}[0,1]$.
\end{theorem}

\begin{proof}
We consider the family of boundary-value problems
\begin{equation}
\begin{gathered}
(\phi (x'))'=\lambda f(t,x,x')+\lambda e,
0<t<1, \lambda \in [0,1]   \\
x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i), \
x(1)=\sum_{j=1}^{n-2}b_jx(\tau _j).
\end{gathered} \label{f1mbp}
\end{equation}
Also, we define an operator
$\Psi :C^{1}[0,1]\times [0,1]\to C^{1}[0,1]$ by setting for
$(x,\lambda )\in C^{1}[0,1]\times [0,1]$
\begin{equation}
\begin{aligned}
\Psi (x,\lambda )(t)
&=x(0)+\int_{0}^{t}\phi ^{-1}(\phi (x'(0))
 +\lambda \int_{0}^{s}(f(\tau ,x(\tau ),x'(\tau ))+e(\tau))d\tau )ds   \\
&\quad +(x(0)-\sum_{i=1}^{m-2}a_ix(\xi
_i))+t(x(1)-\sum_{j=1}^{n-2}b_jx(\tau _j))
\end{aligned} \label{opdef}
\end{equation}
Let us, suppose that $x(t)\in C^{1}[0,1]$ is a solution to the operator
equation, for some $\lambda \in [0,1]$,
\begin{equation}
\begin{aligned}
x &= \Psi (x,\lambda )   \\
&= x(0)+\int_{0}^{t}\phi ^{-1}(\phi (x'(0))+\lambda
\int_{0}^{s}(f(\tau ,x(\tau ),x'(\tau ))+e(\tau ))d\tau )ds  \\
&\quad +(x(0)-\sum_{i=1}^{m-2}a_ix(\xi
_i))+t(x(1)-\sum_{j=1}^{n-2}b_jx(\tau _j))
\end{aligned} \label{eq6}
\end{equation}
Evaluating this equation  at $t=0$ we see that $x(t)$ satisfies
the boundary condition
\begin{equation*}
x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i).
\end{equation*}
Next, we differentiate the equation (\ref{eq6}) with respect to $t$ to get
\begin{equation}
x'(t) =\phi ^{-1}(\phi (x'(0))+\lambda
\int_{0}^{t}(f(\tau ,x(\tau ),x'(\tau ))+e(\tau ))d\tau )
+x(1)-\sum_{j=1}^{n-2}b_jx(\tau _j).   \label{eq7}
\end{equation}
Evaluating, now, the equation (\ref{eq7}) at $t=0$ we see that $x(t)$
satisfies the boundary condition
\begin{equation*}
x(1)=\sum_{j=1}^{n-2}b_jx(\tau _j),
\end{equation*}
and on differentiating the equation (\ref{eq7}) with respect to $t$ we get
\begin{equation*}
(\phi (x'))'=\lambda f(t,x,x')+\lambda e, \quad
0<t<1,\; \lambda \in [0,1].
\end{equation*}
Thus we see that if $x(t)\in C^{1}[0,1]$ is a solution to the operator
equation $x=\Psi (x,\lambda )$ for some $\lambda \in [0,1]$ then
$x(t) $ is a solution to the boundary value problems (\ref{f1mbp}) for the
corresponding $\lambda \in [0,1]$. Conversely, it is easy to see that
if $x(t)\in C^{1}[0,1]$ is a solution to the boundary value problems (\ref
{f1mbp}) for some $\lambda \in [0,1]$ then $x(t)\in C^{1}[0,1]$ is a
solution to the operator equation $x=\Psi (x,\lambda )$ for the
corresponding $\lambda \in [0,1]$.

Next, it is easy to show, following standard arguments, that $\Psi
:C^{1}[0,1]\times [0,1]\to C^{1}[0,1]$ is a completely
continuous operator.

We shall next show that there is a constant $R>0$, independent of $\lambda
\in [0,1]$, such that if $x(t)\in C^{1}[0,1]$ is a solution to (\ref
{eq6}), equivalently to the boundary value problems (\ref{f1mbp}), for some
$\lambda \in [0,1]$ then $\|x\|_{C^{1}[0,1]}<R$.

We note first that if $x(t)\in C^{1}[0,1]$ satisfies
\begin{equation}
x=\Psi (x,0),   \label{eq8}
\end{equation}
then $x(t)=0$ for all $t\in [0,1]$. Indeed, from the definition of
$\Psi $ or from the boundary value problem (\ref{f1mbp}), it follows that
$x(t)=x(0)+x'(0)t$. It then follows from the two boundary conditions
in (\ref{f1mbp}) and the non-resonance assumption \eqref{NRcond} that
$x(0)=x'(0)=0$, implying $x(t)=0$ for all $t\in [0,1]$.

We shall assume, in the following, that $\lambda \in (0,1]$. We shall also
assume that $\sigma ^{\ast }$, as defined in \eqref{eq0} is positive, since
the proof for the case $\sigma ^{\ast }=0$ is simpler. Let us choose
$\varepsilon >0$ such that $\widetilde{\alpha }(\sigma ^{\ast })+\varepsilon
<1$ and
\begin{equation}
(\alpha (M)+\varepsilon )\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)}<1-
\widetilde{\alpha }(\sigma ^{\ast })-\varepsilon ,   \label{eq9}
\end{equation}
which is possible to do, in view of our assumption (\ref{existcond}). Here
$M $ is as defined in Propostion \ref{Prop1} and $\alpha (M)$ is as defined
in \eqref{cond1} so that for the $\varepsilon >0$, chosen above, there
exists a constant $C_{\varepsilon }^{1}>0$ such that
\begin{equation}
\phi (Mz)\leq (\alpha (M)+\varepsilon )\phi (z)+C_{\varepsilon }^{1},\quad
\text{for every }z\in \mathbb{R}.   \label{eq10}
\end{equation}
Also, from Proposition \ref{Prop2} we see that there is a constant
$C_{\varepsilon }^{2}>0$, for the chosen $\varepsilon >0$, such that
\begin{equation}
\phi (\|x'\|_{\infty })\leq \frac{1}{1-\widetilde{\alpha }(\sigma
^{\ast })-\varepsilon }\|(\phi (x'))'\|_{L^{1}(0,1)}+C_{\varepsilon }^{2}.
 \label{eq11}
\end{equation}
We, now, see from the equation in (\ref{f1mbp}), using our assumptions on
the function $f$, Proposition \ref{Prop1}, and estimates (\ref{eq10}),
 (\ref{eq11}) that
\begin{align*}
&\|(\phi (x'))'\|_{L^{1}(0,1)} \\
&\leq \phi (\|x\|_{\infty })\|d_{1}\|_{L^{1}(0,1)}+\phi (\|x'\|_{\infty
 })\|d_{2}\|_{L^{1}(0,1)}+\|r\|_{L^{1}(0,1)}+\|e\|_{L^{1}(0,1)} \\
&\leq \phi (M\|x'\|_{\infty })\|d_{1}\|_{L^{1}(0,1)}+\phi
 (\|x'\|_{\infty})\|d_{2}\|_{L^{1}(0,1)}+\|r\|_{L^{1}(0,1)}+\|e\|_{L^{1}(0,1)} \\
&\leq ((\alpha (M)+\varepsilon)\|d_{1}\|_{L^{1}(0,1)}
  +\|d_{2}\|_{L^{1}(0,1)})\phi (\|x'\|_{\infty})+\|r\|_{L^{1}(0,1)}
  +\|e\|_{L^{1}(0,1)} \\
&\quad  +C_{\varepsilon }^{1}\|d_{1}\|_{L^{1}(0,1)} \\
&\leq \frac{(\alpha (M)+\varepsilon
)\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)}}{1-\widetilde{\alpha }(\sigma
^{\ast })-\varepsilon }\|(\phi (x'))'\|_{L^{1}(0,1)}+C\varepsilon ,
\end{align*}
where $C\varepsilon =\|r\|_{L^{1}(0,1)}+\|e\|_{L^{1}(0,1)}+C_{\varepsilon
}^{1}\|d_{1}\|_{L^{1}(0,1)}+C_{\varepsilon }^{2}[(\alpha (M)+\varepsilon
)\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)}]$.
It, now, follows from (\ref{eq9}) that there exists a constant $R_{0}$,
independent of $\lambda \in [0,1]$, such that if
$x(t)\in C^{1}[0,1]$ is a solution to the
boundary value problems (\ref{f1mbp}) for some $\lambda \in [0,1]$
then
\begin{equation*}
\|(\phi (x'))'\|_{L^{1}(0,1)}\leq R_{0}.
\end{equation*}
This combined with (\ref{eq11}) and (\ref{est0}) give that there
exists a constant $R>0$ such that
\begin{equation*}
\|x\|_{C^{1}[0,1]}<R.
\end{equation*}
This then implies that $\deg _{LS}(I-\Psi (\cdot ,\lambda ),B(0,R),0)$ is
well-defined for all $\lambda \in [0,1]$, where $B(0,R)$ is the ball
with center $0$ and radius $R$ in $C^{1}[0,R]$.

Let, now, $X$ denote the two-dimensional subspace of $C^{1}[0,1]$ given by
\begin{equation}
X=\{A+Bt\text{ }|\text{ for }A, B\in \mathbb{R}\}.
\label{eq12}
\end{equation}
Let us define the isomorphism $i:\mathbb{R}^{2}\to X$ by
\begin{equation}
i\begin{pmatrix}
A \\
B\end{pmatrix}
=i_{\begin{pmatrix}
A \\
B\end{pmatrix}} \in X, \quad \text{for }
\begin{pmatrix}
A \\
B \end{pmatrix} \in \mathbb{R}^{2},   \label{eq13}
\end{equation}
where
\begin{equation}
i_{\begin{pmatrix}
A \\
B\end{pmatrix}} (t)
=A+Bt, \quad \text{for }t\in [0,1].   \label{eq14}
\end{equation}
Also, we define a $2\times 2$ matrix
\begin{equation}
\mathbb{A}=\begin{pmatrix}
-(1-\sum_{i=1}^{m-2}a_i) & \sum_{i=1}^{m-2}a_i\xi _i \\
-(1-\sum_{j=1}^{n-2}b_j) & -(1-\sum_{j=1}^{n-2}b_j\tau _j)
\end{pmatrix}.   \label{eq16}
\end{equation}
We note that
\[\det \mathbb{A}=(1-\sum_{i=1}^{m-2}a_i)(1-
\sum_{j=1}^{n-2}b_j\tau _j)+(\sum_{i=1}^{m-2}a_i\xi
_i)(1-\sum_{j=1}^{n-2}b_j)\neq 0,
\]
in view of the non-resonance assumption \eqref{NRcond}.

Next, we define a function $G:\mathbb{R}^{2}\to \mathbb{R}^{2}$
by setting
\begin{equation}
\begin{aligned}
G\begin{pmatrix}
A \\
B \end{pmatrix}
&=\mathbb{A}\cdot \begin{pmatrix}
A \\
B \end{pmatrix} \\
&=\begin{pmatrix}
-A(1-\sum_{i=1}^{m-2}a_i)+B(\sum_{i=1}^{m-2}a_i\xi _i) \\
-A(1-\sum_{j=1}^{n-2}b_j)-B(1-\sum_{j=1}^{n-2}b_j\tau _j)
\end{pmatrix}
\quad \text{for }
\begin{pmatrix}
A \\
B \end{pmatrix} \in \mathbb{R}^{2}.
\end{aligned} \label{eq17}
\end{equation}
We note that for $v(t)=A+Bt\in X$ we have
\begin{equation*}
(I-\Psi (\cdot ,0))(v)=i_{G
\begin{pmatrix}
A \\
B \end{pmatrix}}
\end{equation*}
and it follows that
\begin{equation*}
G=i^{-1}\circ ((I-\Psi (\cdot ,0))|_{X}\circ i.
\end{equation*}
Now, we see from the homotopy invariance property of the Leray-Schauder
degree that
\begin{align*}
\deg _{LS}(I-\Psi (\cdot ,1),B(0,R),0)
&=\deg _{LS}(I-\Psi (\cdot ,0),B(0,R),0) \\
&=\deg _{B}(I-\Psi (\cdot ,0)|_{X},X\cap B(0,R),0) \\
&=\deg _{B}(G,\mathbb{B}(0,R),0),
\end{align*}
where $\mathbb{B}(0,R)$ denotes the ball of radius $R$ in
$\mathbb{R}^{2}$ with center at the origin. Finally, we have that
\begin{equation*}
\deg _{B}(G,\mathbb{B}(0,R),0)=
\begin{cases}
1,& \text{if }\det \mathbb{A}>0 \\
-1, &\text{if }\det \mathbb{A}<0.
\end{cases}
\end{equation*}
Accordingly, we see from the non-resonance assumption \eqref{NRcond} i.e.
\[
\det \mathbb{A=(}1-\sum_{i=1}^{m-2}a_i)(1-\sum_{j=1}^{n-2}b_j\tau
_j)+(\sum_{i=1}^{m-2}a_i\xi _i)(1-\sum_{j=1}^{n-2}b_j)\neq 0
\]
that
$\deg _{LS}(I-\Psi (\cdot ,1),B(0,R),0)\neq 0$ and there is $x(t)\in
B(0,R)\subset C^{1}[0,1]$ that satisfies
\begin{equation*}
x=\Psi (x,1),
\end{equation*}
equivalently $x(t)$ is a solution to the boundary value \eqref{1mbp}. This
completes the proof of the theorem.
\end{proof}

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\end{document}