\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 112, pp. 1--7.\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 2003 Texas State University-San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/112\hfil Multi-point boundary-value problems]
{Multi-point boundary-value problems for the p-Laplacian at resonance}

\author[Xiaohong Ni  \& Weigao Ge\hfil EJDE--2003/112\hfilneg]
{Xiaohong Ni \& Weigao Ge}  

\address{Department of Applied Mathematics,
Beijing Institute of Technology, Beijing 100081, China}
\email[Xiaohong  Ni]{nixiaohong1126@hotmail.com}

\date{}
\thanks{Submitted July 8, 2003. Published November 5, 2003.}
\thanks{Research supported by National Natural Science Foundation of China}
\subjclass[2000]{34B10, 34B15}
\keywords{Resonance, multi-point boundary value problem, p-Laplacian}


\begin{abstract}
   In this paper, we consider multi-point boundary-value problems 
   for the  p-laplacian at resonance. Applying an extension of 
   Mawhin's continuation theorem, we show the existence  of 
   at least one solution.
\end{abstract}

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


\section{Introduction}

 Let $\ f:[0,1]\times \mathbb{R}^2\to \mathbb{R}$ be a
 continuous function and  $ 0<\xi_1<\xi_2<\dots<\xi_{n-2}<1$,
 $0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$,
 $\alpha_j\in \mathbb{R}$ ($1\leq j\leq n-2$), $\beta_i\in \mathbb{R}$
 ($1\leq i\leq m-2$) be given.
  In this paper, we study the differential equation
$$
 (\phi_p(x'))'=f(t,x(t),x'(t)) \eqno(1.1)
$$
  subject to one of the following boundary-value conditions:
$$
x'(0)=0,\quad \phi_p(x'(1))=\sum_{i=1}^{m-2}\beta_i\phi_p(x'(\eta_i))\eqno(1.2)
$$
or
$$
\phi_p(x'(0))=\sum_{j=1}^{n-2}\alpha_j\phi_p(x'(\xi_j)),\quad x'(1)=0\eqno(1.3)
$$
where $\phi_p(s)=|s|^{p-2}s$, $p>1$,
$\sum_{i=1}^{m-2}\beta_i=\sum_{j=1}^{n-2}\alpha_j=1$.

Mawhin \cite{g1} proved a continuation theorem for the abstract
equation $Lx=Nx$ with $L$ being a non-invertible linear operator.
Since then, there has been much attention on the study of
boundary-value problems at resonance (cf \cite{b2,m1,p1}).
Recently, Liu \cite{l1,l2,l3} considered differential equation
(1.1) with $p=2$ and obtained some results. However, the
p-laplacian operator is not linear. Our approach is different and
based on an extension of Mawhin's continuation theorem in
\cite{g2}. We provide sufficient conditions to guarantee the
existence of solution for (1.1), (1.2) and for (1.1), (1.3).
Moreover, some examples are given to demonstrate our results.

For the convenience of the reader, we first recall some
notation.

  Let $X$ and $Z$ be Banach spaces with norms $\|\cdot\|_X$ and $ \|\cdot\|_Z$,
respectively. A continuous operator $M:X\cap \mathop{\rm dom} M\to Z$ is said to be
quasi-linear if
\begin{itemize}
 \item[(i)] $\mathop{\rm Im}M=M(X\cap \mathop{\rm dom}M)$  is a closed subset of $Z$;
 \item[(ii)] $\mathop{\rm Ker}M=\{x\in X\cap \mathop{\rm dom}M: Mx=0\}$
  is linearly homeomorphic to $\mathbb{R}^n$, $n<\infty$.
\end{itemize}
 Let $P:X\to X_1 $ and $Q:Z\to Z_1 $ be two projectors such that
 $\mathop{\rm Im}P=\mathop{\rm Ker}M$, $\mathop{\rm Ker}Q=\mathop{\rm Im}M$
 and $X=X_1\oplus X_2$, $Z=Z_1\oplus Z_2$,
 where $X_1=\mathop{\rm Ker}M$, $Z_2=\mathop{\rm Im}M$ and $X_2$, $Z_1$ are
respectively the complement space of $ X_1 $ in $X$, $ Z_2$  in $Z$. If
$ \Omega$ is an open and bounded  subset of $X$ such that
$\mathop{\rm dom}M\cap \Omega\neq \Phi$, the continuous operator
$N_{\lambda}:\overline{\Omega}\to Z, \lambda\in[0,1]$ will
 be called admissibly compact in $\overline \Omega$ with respect to $M$ if
\begin{itemize}
 \item[(iii)] There is a subset $Z_1$ of $Z$ with $\dim Z_1=\dim X_1$ and an
 operator $K:\mathop{\rm Im}M\to X_2$ with $K\theta=\theta $ such that
 for $\lambda\in[0,1]$,
 $$
 (I-Q)N_{\lambda}(\bar{\Omega})\subset \mathop{\rm Im}M\subset(I-Q)Z,\eqno(1.4)
$$
$$(I-Q)N_0 \mbox{ is a zero operator, and }  QN_{\lambda}x=0\Leftrightarrow QNx=0,
 \lambda\in(0,1), \eqno(1.5)
$$
$$
KM=I-P,\  K(I-Q)N_{\lambda}:\overline{\Omega}\to X_2\subset X \ {\hbox{is compact}},
\eqno(1.6)
$$
$$
M[P+K(I-Q)N_{\lambda}]=(I-Q)N_{\lambda}. \eqno(1.7)
$$
\end{itemize}

\begin{theorem}[\cite{g2}] \label{thm1.1}
 Let $X$ and $Z$ be two Banach spaces with the norms
 $\|\cdot\|_X$ and $\|\cdot\|_Z$, respectively, and
$\Omega\subset X$ an open and bounded set. Suppose $M: X\cap \mathop{\rm dom}M\to Z $
is a quasi-linear operator  and $N_{\lambda}: \overline{\Omega}\to Z$ is
admissibly compact with respect to $M$. In addition, if
  \begin{itemize}
\item[(C1)] $Mx\neq N_{\lambda}x, \lambda\in(0,1)$,  $x\in\partial\Omega$
\item[(C2)] $\deg\{JQN,\Omega\cap \mathop{\rm Ker}M, 0\}\neq0$, where
$ N=N_1$ and  $J:Z_1\to X_1$ is a homeomorphism  with $J(0)=0$;
\end{itemize}
then the abstract equation $Mx=Nx$ has at least one solution in
$\overline{\Omega}$.
\end{theorem}

\section{General Results}

\subsection{Results for boundary-value problem (1.1), (1.2)}
Let $ X=\{x\in C[0,1]:\ x'(0)=0$,
$\phi_p(x'(1))=\sum_{i=1}^{m-2}\beta_i\phi_p(x'(\eta_i))\}$  with
norm $\|x\|_1=\max\{\|x\|,\|x'\|\}$  and $Z=C[0,1]$ with norm
$\|w\|=\max_{t\in[0,1]}|w(t)|$. Let $M$ be the operator from
$\mathop{\rm dom}M\subset X $ to $Z$ with
 $$
\mathop{\rm dom}M=\{x\in C^1[0,1]: \phi_p(x')\in C^1[0,1]\}
$$
 and $Mx=(\phi_p(x'))'$. For any open and bounded $\Omega\subset X$, we define
$ N_{\lambda}:\overline{\Omega}\to Z$ by
$$
(N_{\lambda}x)=\lambda f(t,x(t),x'(t)),\quad t\in[0,1]
$$
Then (1.1), (1.2) can be written as
$$ Mx=Nx,\quad \mbox{where } N=N_1.
$$

\begin{lemma} \label{lm2.1}
 If $0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$,  then there exists integer
$k\in\{1,2,\dots,m-1\}$ such that
 $\sum_{i=1}^{m-2}\beta_i\eta^k_i\neq1$.
\end{lemma}

\begin{proof}  If this were not the case, we have
 $$
 \begin{pmatrix}
 \eta_1&\eta_2&\dots&\eta_{m-2}\\
\eta^2_1&\eta^2_2&\dots&\eta^2_{m-2}\\
\vdots&\vdots&\ddots&\vdots\\
\eta^{m-1}_1&\eta^{m-1}_2&\dots&\eta^{m-1}_{m-2}
\end{pmatrix}
\begin{pmatrix}
 \beta_1\\
 \beta_2\\
 \vdots\\
 \beta_{m-2}
 \end{pmatrix}=
\begin{pmatrix}
 1\\
 1\\
 \vdots\\
 1
 \end{pmatrix}.$$
This implies
 $$
\begin{pmatrix}
 \eta_1&\eta_2&\dots&\eta_{m-2}&1\\
\eta^2_1&\eta^2_2&\dots&\eta^2_{m-2}&1\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
\eta^{m-1}_1&\eta^{m-1}_2&\dots&\eta^{m-1}_{m-2}&1
\end{pmatrix}
\begin{pmatrix}
 \beta_1\\
 \beta_2\\
 \vdots\\
 \beta_{m-2}\\
 -1
 \end{pmatrix}=
 \begin{pmatrix}
 0\\
 0\\
 \vdots\\
 0\\
 0
 \end{pmatrix}.$$
Putting
$$
A=\begin{pmatrix}
 \eta_1&\eta_2&\dots&\eta_{m-2}&1\\
\eta^2_1&\eta^2_2&\dots&\eta^2_{m-2}&1\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
\eta^{m-1}_1&\eta^{m-1}_2&\dots&\eta^{m-1}_{m-2}&1
\end{pmatrix}
$$
we have
$$ \det A=\eta_1\eta_2\eta_{m-2}\left|\begin{matrix}
1&1&\dots&1&1\\
\eta_1&\eta_2&\dots&\eta_{m-2}&1\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
\eta^{m-2}_1&\eta^{m-2}_2&\dots&\eta^{m-2}_{m-2}&1
\end{matrix}\right|\neq 0.
$$
Hence,
$(\beta_1,\beta_2,\ldots,\beta_{m-2},-1)^T=(0,0,\ldots,0,0)^T$,
which leads to a contradiction, $-1=0$.
\end{proof}

\begin{lemma} \label{lm2.2}
If  $\sum_{i=1}^{m-2}\beta_i=1$, then the operator $M: \mathop{\rm dom}M\cap X\to Z$
is a quasi-linear and $N_{\lambda}:\overline{\Omega}\to Z$
is admissibly compact in $\overline{\Omega} $ with respect to $M$.
\end{lemma}

\begin{proof} Obviously,  $X_1=\mathop{\rm Ker}M=\{\ x=A,\; A\in \mathbb{R}\}$
 and
$$
Z_2=\mathop{\rm Im}M=\big\{y\in Z,\ \sum_{i=1}^{m-2}\int_{\eta_i}^1
\beta_i y(s)ds=0\big\}.
$$
 We have $\dim \mathop{\rm Ker}M=1<\infty$ and $M(X\cap \mathop{\rm dom}M)\subset Z$
closed. Hence, $M$ is a quasi-linear operator. Define  projectors $P: X\to X_1 $ as
 $Px=x(0)$ for all $x\in X$
 and  $Q: Z\to Z_1$ as
 $$
Qy=\Big[\frac{k}{1-\sum_{i=1}^{m-2}\beta_i\eta^k_i}\sum_{i=1}^{m-2}\beta_i
\int_{\eta_i}^1 y(s)ds\Big]t^{k-1},\quad \forall y\in Z.
$$
 where $Z_1$ is the complement space of $Z_2$ in $Z$. (Indeed, $Z_1\cong \mathbb{R})$.
Thus  $\dim X_1=\dim Z_1=1$.

Taking $K: \mathop{\rm Im} M\to X_2$ as follows:
 $$
K(I-Q)h(t)=\int_0^t\phi_p^{-1}\Big(\int_0^s((I-Q)h(\tau)d\tau\Big)ds,\quad
 \forall h(t)\in Z.
$$
 here $X_2$ is the complement space of $X_1$ in $X$. It is clear that
 $K\theta=\theta$ and (1.4), (1.5) hold. For for all $x\in X\cap \mathop{\rm dom}M$,
we have
 $$
M[P+K(I-Q)N_\lambda]x=(I-Q)N_\lambda x
$$
 and
 $$
KMx=K(I-Q)Mx=\int_0^t\phi_p^{-1}(\int_0^s(\phi_p(x'(\tau)))'d\tau)ds=x(t)-x(0)
=(I-P)x.
$$
For any open and  bounded set
$\Omega\subset X, \lambda\in[0,1]$, by using the Ascoli-Arzela
Theorem, we can prove that
$[K(I-Q)N_{\lambda}](\overline{\Omega})$ is relatively compact
and continuous. This implies that (1.6), (1.7) are valid. Hence,
$N_{\lambda}$ is admissibly compact in $\overline{\Omega}$ with
respect to $M$.
\end{proof}

For the next theorem we have the assumptions:
\begin{itemize}
\item[(H1)]  There exist functions $a,b,r\in C([0,1],[0,+\infty))$ such that
for all $(x,y)\in \mathbb{R}^2$, $t\in[0,1]$,
 $$|f(t,x,y)|\leq\phi_p[a(t)|x|+b(t)|y|+r(t)].
$$
\item[(H2)] There exists a constant $C>0$ such that for any  $x\in X$,
 if $|x(t)|>c$, for all $t\in[0,1]$, one has
$$
\mathop{\rm sgn}(x(t))\sum_{i=2}^{m-2}\beta_i\int_{\eta_i}^1f(t,x(t),x'(t))dt<0
\eqno(2.1)
$$
or
$$
\mathop{\rm sgn}(x(t))\sum_{i=2}^{m-2}\beta_i\int_{\eta_i}^1f(t,x(t),x'(t))dt>0
\eqno(2.2)
$$
\end{itemize}

\begin{theorem} \label{thm2.1}
 Suppose  $f:[0,1]\times \mathbb{R}^2\to \mathbb{R}$ is a continuous function
and  $0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$,
$\beta_i\in \mathbb{R}$ ($1\leq i\leq m-2$) are given.
Under Assumptions  (H1) and (H2),
the boundary-value problem (1.1), (1.2) with $\sum_{i=1}^{m-2}\beta_i=1$ has at
least one solution provided that $\|a\|+\|b\|<1$.
\end{theorem}

\begin{proof}
Set $\Omega_1=\{x\in X\cap \mathop{\rm dom}M,\;
Mx=N_{\lambda}x,\ \lambda\in(0,1)\}$,
then for $x\in \Omega_1$, $Mx=N_{\lambda}x$.
Thus, we have  $Nx\in \mathop{\rm Im}M=\mathop{\rm Ker}Q$  and
$$
\sum_{i=2}^{m-2}\beta_i\int_{\eta_i}^1f(t,x(t),x'(t))dt=0.
$$
 By (H2), there exists $t_0\in[0,1]$ such that $|x(t_0)|\leq C$.
Hence, we have $\|x\|\leq C+\|x'\|$,  in view of the relation
$x(t)=x(t_0)+\int_{t_0}^tx'(s)ds$.  At the same time,
$$
|x'(t)|\leq\phi_p^{-1}(\|Nx\|)\leq\|a\|\|x\|+\|b\|\|x'\|+\|r\|.
$$
Noticing $\ \|a\|+\|b\|<1,\ $ we know the set $\Omega_1$\ is
bounded. Let $\|x\|_1\leq C^*$ for all $x\in \Omega_1$.

If (2.1) holds, taking $H(x,\lambda)=-\lambda Ix+(1-\lambda)JQNx$,
$\lambda\in[0,1]$ and
$$
\Omega_2=\{x\in \mathop{\rm Ker}M,\; H(x,\lambda)=0\},
$$
where $J: \mathop{\rm Im}Q\to \mathop{\rm Ker}M$ is a homomorphism such that
$J(At^{k-1})=A$, for all $A\in \mathbb{R}$. We claim that for any
$x\in \Omega_2$, $\|x\|_1\leq C$.
 In fact, if this were not the case, there exist $\lambda_0\in[0,1]$ and
$|x_0|>C$ such that $H(x_0,\lambda_0)=0$. Without loss of generality, suppose
$x_0>C$ and  $\sum_{i=1}^{m-2}\beta_i\eta^k_i<1$. If  $\lambda_0=1$, then
$x_0=0$ otherwise, it implies that
$$
0\leq\lambda_0x_0=(1-\lambda_0)\frac{k}{1-\sum_{i=1}^{m-2}\beta_i\eta^k_i}
\sum_{i=2}^{m-2}\beta_i\int_{\eta_i}^1f(t,x_0,0)dt<0,
$$
which leads to a contradiction,  $0<0$.

When (2.2) holds, putting $H(x,\lambda)=\lambda Ix+(1-\lambda)JQNx$.
By a similar argument, we can obtain the same conclusion.

Next, we shall prove that all  conditions of Theorem \ref{thm1.1}
are satisfied. Let $\Omega$ be an  open bounded subset
of $X$ such that $\bigcup\limits_{i=1}^2\overline{\Omega}_i\subset\Omega$.
 Clearly, we have
$$
Mx\neq N_{\lambda}x,\quad \lambda\in(0,1),\; x\in \partial\Omega
$$
and
$$
H(x,\lambda)\neq0, \quad \lambda\in[0,1],\;
x\in \partial\Omega\cap \mathop{\rm Ker}M.
$$
In view of the homotopy property of degree,
$$
\deg\{JQN,\Omega\cap\mathop{\rm Ker}M,0\}
=\deg\{-I,\Omega\cap \mathop{\rm Ker}M,0\}=-1.
$$
Theorem \ref{thm1.1} can be generalized to obtain the existence of at least one
solution for (1.1), (1.2) with $\sum_{i=2}^{m-2}\beta_i=1$.
\end{proof}

\subsection{Results for Problem (1.1), (1.3)}

Let $X=\{x\in C[0,1]:\phi_p(x'(0))=\sum_{j=1}^{n-2}\alpha_j\phi_p(x'(\xi_j)),\
x'(1)=0\}$. $Z, M, \mathop{\rm dom}M\ $and $\ N_{\lambda}\ $ be defined as
above.

\begin{lemma} \label{lm2.3}
If $\sum_{j=1}^{n-2}\alpha_j=1$ and $\sum_{j=1}^{n-2}\alpha_j\xi_j\neq 0$
then operator $M: \mathop{\rm dom}M\cap X\to Z$ is a quasi-linear and
$N_{\lambda}:\overline{\Omega}\to Z$  is admissibly compact in $\overline{\Omega}$
with respect to $M$.
\end{lemma}

Note that $\mathop{\rm Ker}M=\{ x=A, A\in \mathbb{R}\}$ and
$\mathop{\rm Im}M=\{y\in Z,\;\sum_{j=1}^{n-2}\alpha_j\int_0^{\xi_j}y(s)ds=0\}$.
We define $P:X\to X_1$, and $Q: Z\to Z_1$ as follows:
$$
(Px)(t)=x(0),\quad
(Qy)(t)=\frac{1}{\sum_{j=1}^{n-2}\alpha_j\xi_j}\sum_{j=1}^{n-2}\alpha_j
\int_0^{\xi_j}y(s)ds.
$$
Let $K(I-Q)h(t)=-\int_0^t\phi_p^{-1}\big[\int_s^1((I-Q)h(\tau)d\tau\big]ds$,
for $h(t)\in Z$. Using the method in the proof of Lemma \ref{lm2.2},
we can show the theorem bellow. However, we need the assumption:
\begin{itemize}
\item[(H3)] There exists a constant $D>0$ such that for all $x\in X$,
if $\ |x(t)|>D$ for some $t\in[0,1]$, one has
$$
\mathop{\rm sgn}(x(t))\sum_{j=1}^{n-2}\alpha_j\int_0^{\xi_j}f(t,x(t),x'(t))dt<0
\eqno(2.3)$$
or
$$
\mathop{\rm sgn}(x(t))\sum_{j=1}^{n-2}\alpha_j\int_0^{\xi_j}f(t,x(t),x'(t))dt>0
\eqno(2.4)$$
\end{itemize}

\begin{theorem} \label{thm2.2}
 Suppose  that $f:[0,1]\times \mathbb{R}^2\to \mathbb{R}$ is
 continuous,  $0<\xi_1<\xi_2<\dots<\xi_{n-2}<1$, and
$\alpha_j\in \mathbb{R}$ ($1\leq j\leq n-2$) are given.
Under Assumptions (H1), (H3), the boundary-value problem (1.1), (1.3)
with $\sum_{j=1}^{n-2}\alpha_j=1$,
$\sum_{j=1}^{n-2}\alpha_j\xi_j\neq 0$ has at least one solution
provided that $\|a\|+\|b\|<1$.
\end{theorem}

The proof of this theorem is similar to that of Theorem \ref{thm2.1}; therefore,
we omit it here.

\section{Examples}

\subsection*{Example 3.1} Consider the boundary-value problem,
at resonance,
\[
\begin{gathered}
(\phi_p(x'))'=\frac13x+\frac{(\cos(x'))^2}{2(t+1)}-3t^2,\quad t\in[0,1]\\
x'(0)=0,\quad  \phi_p(x'(1))=\sum_{i=1}^{m-2}\beta_i\phi_p(x'(\eta_i))
\end{gathered}\eqno(3.1)
\]
where  $p>2$, $0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$,
$\sum_{i=1}^{m-2}\beta_i=1$, $\sum_{i=1}^{m-2}\beta_i\eta_i\neq1$ and
$\beta_i(1\leq i\leq m-2)\geq 0$.

Let $f(t,x,y)=\frac13x+\frac{(\cos(x'))^2}{2(t+1)}-3t^2$, then
$|f(t,x,y)|\leq\frac13|x|+\frac12|y|+3t^2\leq\phi_p(\frac13|x|+\frac12|y|+3)$.
Clearly, there exists a constant $C>0$ such that if $|x(t)|>C$,
for any $t\in [0,1]$, one has
$$
\mathop{\rm sgn(x(t))}\sum_{i=2}^{m-2}\beta_i\int_{\eta_i}^1f(t,x(t),x'(t))dt>0.
$$
Applying Theorem \ref{thm2.1}, we obtain that (3.1) has at least one
solution.

\subsection*{Example 3.2} Consider the boundary-value problem
\[
\begin{gathered}
(\phi_p(x'))'=\frac14x+\frac1{12}x\sin(x')+t\\
\phi_p(x'(0))=3\phi_p(x'(\frac14))-2\phi_p(x'(\frac12)),\ x'(1)=0,
\end{gathered}\eqno(3.2)
\]
where $p>2$, $\xi_1=\frac14$, $\xi_2=\frac12$, $\alpha_1=3$,
$\alpha_2=-2$. Taking $f(t,x,y)=\frac14x+\frac1{12}x\sin(x')+t$,
 then $|f(t,x,y)|\leq\frac1{12}|x|+\frac14|x|+1\leq\phi_p(\frac13|x|+1)$ and
 for any $A\in \mathbb{R}$,
 $$
 3\int_0^{\frac14}f(t,A,0)dt-2\int_0^{\frac12}f(t,A,0)dt=-\frac{A^3}4-\frac5{32}.
 $$
Hence, there exists a constant $D>0$ such that for  $t\in[0,1]$
$$
\sum_{j=1}^{n-2}\alpha_j\int_0^{\xi_j}f(t,D,0)dt<0<\sum_{j=1}^{n-2}\alpha_j
\int_0^{\xi_j}f(t,-D,0)dt.
$$
From Theorem \ref{thm2.2}, Problem (3.2) has at least one solution.


\begin{thebibliography}{00}

\bibitem{b2} J. Bouchala, \emph{Resonance problems for p-laplacian},
Math. Compu. Simulation, 61(2003), 599-604.

\bibitem{g1}  R. Gaines, J. Mawhin; \emph{Coincide degree and nonlinear
differential equation}, Springer-Verlag, Berlin, 1977.

\bibitem{g2}  W. G. Ge, J. L. Ren; \emph{An extension of Mawhin's continuation
theorem and its application to boundary value problems with a
p-laplacian}, (preprint).

\bibitem{l1} B. Liu, J. S. Yu; \emph{Solvability of multi-point boundary
value problems at resonance (I)}, India. J. Pure. Math, 33(2002), 475-494.

\bibitem{l2} B. Liu, \emph{Solvability of multi-point boundary value problem at
resonance (II)}, Appl. Math. Compu, 136(2003), 353-377.

\bibitem{l3} B. Liu, J. S. Yu; \emph{Solvability of multi-point boundary value
problem at resonance (III)}, Appl. Math. Compu, 129(2002), 119-143.

\bibitem{m1} R. Y. Ma, \emph{Multilicity results for a third order boundary value
problem at resonance}, Nonlinear Anal. Theo. Meth. Appl, 32(1998), 493-499.

\bibitem{p1} B. Przeradzki, \emph{Nonlinear equations at resonance with two solutions},
Nonlinear Anal, 38(1999),  1023-1031.


\end{thebibliography}
\end{document}