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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 226, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/226\hfil Existence of solutions]
{Existence of solutions for $(k,n-k-2)$ conjugate boundary-value
 problems at Resonance with $\dim\ker  L=2$}

\author[W. Jiang \hfil EJDE-2013/226\hfilneg]
{Weihua Jiang}

\address{Weihua Jiang \newline
College of Sciences, Hebei University of Science and Technology,
Shijiazhuang, 050018, Hebei, China}
\email{weihuajiang@hebust.edu.cn}

\thanks{Submitted July 24, 2012. Published October 11, 2013.}
\subjclass[2000]{35B34, 34B10}
\keywords{Resonance; Fredholm operator; boundary value problem}

\begin{abstract}
 By constructing suitable project operators and using the coincidence
 degree theory due to Mawhin, the existence of solutions for
 $(k,n-k-2)$ conjugate boundary-value problems at
 resonance with dim$kerL=2$ is obtained.
\end{abstract}

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

\section{Introduction}

The existence of solutions for $(k,n-k)$ conjugate
boundary-value problems at nonresonance has been studied in many
papers (see \cite{a1,a2,a3,e1,f1,h1,j1,j2,l1,k2,m1,s1,w1,w3,y1,y2,z1}). 
The solvability of boundary-value problems at resonance has been investigated
by many authors (see \cite{d1,d2,g1,j3,k1,k3,l2,l3,l4,l5,x1,m2,p1,w2,z2}).
In \cite{j3}, the existence of solutions for $(k,n-k)$ conjugate
boundary-value problems at resonance  with $\dim\ker L=1$ has been
studied. To the best of our knowledge,  no paper discusses the existence 
of solutions for $(k,n-k-2)$ conjugate
boundary-value problems at resonance with $\dim\ker L=2$. We will
fill this gap in the literature.

In this article, we investigate the existence of solutions for the
$(k,n-k-2)$ conjugate boundary-value problem at resonance
\begin{gather}
(-1)^{n-k}y^{(n)}(t)=f\left(t,y(t),y'(t),\dots,y^{(n-1)}(t)\right)
+\varepsilon(t), \quad \text{a.e. }t\in [0,1], \label{e1.1}\\
\begin{gathered}
y^{(i)}(0)=y^{(j)}(1)=0,\quad  0\leq i\leq k-1,\; 0\leq j\leq n-k-3,\\
y^{(n-2)}(1)=\sum_{j=1}^{l}\beta_jy^{(n-2)}(\eta_j),\quad
y^{(n-1)}(1)=\sum_{i=1}^{m}\alpha_iy^{(n-1)}(\xi_i),
\end{gathered}\label{e1.2}
\end{gather}
where $1\leq k\leq n-3$, $0<\eta_1<\eta_2<\dots<\eta_{l}<1$,
$0<\xi_1<\xi_2<\dots<\xi_{m}<1$.

 In this article, we assume that the following conditions hold.
\begin{itemize}
\item[(H1)]
 $\sum_{i=1}^{m}\alpha_i=1$, $\sum_{j=1}^{l}\beta_j=1$,
$\sum_{j=1}^{l}\beta_j\eta_j=1$.

\item[(H2)] $e=\begin{vmatrix} e_1& e_2\\ e_3& e_4 \end{vmatrix} \neq 0$, 
where
\begin{gather*}
e_1=1-\sum_{i=1}^ma_i\xi_i, \quad
e_2=\frac{1}{2}\Big(1-\sum_{j=1}^l\beta_j\eta_j^2\Big), \\
e_3=\frac{1}{2}\Big(1-\sum_{i=1}^ma_i\xi_i^2\Big),\quad
e_4=\frac{1}{6}\Big(1-\sum_{j=1}^l\beta_j\eta_j^3\Big).
\end{gather*}

\item[(H3)] $\varepsilon(t)\in L^\infty[0,1]$, 
$f:[0,1]\times \mathbb{R}^n \to \mathbb{R}$ satisfies
Carath\'eodory conditions; i.e., $f(\cdot,x)$ is
measurable for each fixed $x\in \mathbb{R}^n$, $f(t,\cdot)$ is
continuous for a.e. $t\in[0,1]$, and for each $r>0$, there exists
$\Phi_r\in L^\infty[0,1]$ such that
$|f(t,x_1,x_2,\dots,x_n)|\leq\Phi_r(t)$ for all 
$|x_i|\leq r$, $i=1,2,\dots,n$, a.e. $t\in[0,1]$.
\end{itemize}

\section{Preliminaries}
\label{s2}

For convenience, we introduce some notation and a
theorem. For more details see \cite{m3}.
 Let $X$ and $Y$ be real Banach spaces and
$L:\operatorname{dom} L\subset X\to Y$ be a Fredholm operator with index
zero, $P:X\to X$, $Q:Y\to Y$ be projectors such that
$$
\operatorname{Im}P=\ker L,\quad \ker Q=\operatorname{Im}L,\quad
X=\ker L\oplus \ker P,\quad Y=\operatorname{Im}L\oplus \operatorname{Im}Q.
$$
It follows that
$$L\big|_{\operatorname{dom}L\cap \ker P}:\operatorname{dom}L\cap \ker 
P\to \operatorname{Im}L
$$
is invertible. We denote its inverse by $K_P$.

 Let $\Omega$ be an open bounded subset of
$ X, \operatorname{dom}L\cap\overline{\Omega}\neq\emptyset$, the map
$N:X\to Y$ will be called $L$-compact on
$ \overline{\Omega}$ if $QN(\overline{\Omega})$ is bounded and
$K_P(I-Q)N:\overline{\Omega}\to X$ is compact.

\begin{theorem}[\cite{m3}] \label{thm2.1} 
 Let $L: \operatorname{dom}L\subset X\to Y$ be a
Fredholm operator of index zero and $N: X\to Y$
$L$-compact on $\overline{\Omega}$. Assume that the following
conditions are satisfied:
\begin{itemize}
\item[(1)] $Lx\neq \lambda Nx$ for every 
$(x,\lambda)\in [(\operatorname{dom}L\setminus
\ker L)\cap\partial\Omega]\times(0, 1)$;

\item[(2)] $Nx\notin \operatorname{Im}L$ for every
$x\in \ker L\cap\partial\Omega$;

\item[(3)] $\deg(QN|_{\ker L}, \Omega\cap \ker L, 0)\neq0$, where
$Q: Y\to Y$ is a projection such that $\operatorname{Im}L=\ker Q$.

\end{itemize}
Then the equation $Lx=Nx$ has at least one solution in
$\operatorname{dom}L\cap\overline\Omega$.
\end{theorem}

Take $X=C^{n-1}[0,1]$ with norm
$\|u\|=\max\{\|u\|_{\infty}, \|u'\|_{\infty},\dots,\|u^{(n-1)}\|_{\infty}\}$,
where $\|u\|_{\infty}=\max_{t\in[0,1]}|u(t)|$, $Y=L^1[0,1]$
with norm $\|x\|_1=\int_0^1|x(t)|dt$. Define operator
$Ly(t)=(-1)^{n-k}y^{(n)}(t)$ with
\begin{align*}
\operatorname{dom}L=\Big\{&y\in X: y^{(n)}\in Y,\; y^{(i)}(0)=y^{(j)}(1)=0,\; 
0\leq i\leq k-1,\;\\
& 0\leq j\leq n-k-3,\; y^{(n-2)}(1)=\sum_{j=1}^{l}\beta_jy^{(n-2)}(\eta_j),\\\
& y^{(n-1)}(1)=\sum_{i=1}^{m}\alpha_iy^{(n-1)}(\xi_i)
\Big\}.
\end{align*}
 Let $N:X\to Y$ be defined as
$$
Ny(t)=f\Big(t,y(t),y'(t),\dots,y^{(n-1)}(t)\Big)+\varepsilon(t), \quad
t\in [0,1].
$$
Then problem \eqref{e1.1}, \eqref{e1.2} becomes $Ly=Ny$.

We use convention that $1/k!=0$, for $k=-1,-2,\dots$.
By simple calculation, we can get the following results.
\begin{align*}
&\begin{vmatrix}
 \frac{1}{k!} & \frac{1}{(k+1)!}&\dots & \frac{1}{(n-3)!}\\
\frac{1}{(k-1)!} &\frac{1}{k!}  &\dots &  \frac{1}{(n-4)!}\\
 & \dots & \dots & \\
\frac{1}{[k-(n-k-3)]!}& \frac{1}{[k+1-(n-k-3)]!}& \dots
&\frac{1}{[n-3-(n-k-3)]!}
\end{vmatrix}\\[4pt]
&=\frac{(n-k-3)!}{k!}\cdot\frac{(n-k-4)!}{(k+1)!}\dots\frac{1}{(n-3)!}
\neq 0.
\end{align*}
So, the following lemmas hold.

\begin{lemma} \label{lem2.1} 
The system of linear equations
\begin{gather*}
\frac{x_k}{k!}+\frac{x_{k+1}}{(k+1)!}+\dots+\frac{x_{n-3}}{(n-3)!}
 +\frac{1}{(n-2)!}=0,\\
\frac{x_k}{(k-1)!}+\frac{x_{k+1}}{k!}+\dots+\frac{x_{n-3}}{(n-4)!}
 +\frac{1}{(n-3)!}=0,\\
\dots\\
\begin{aligned}
&\frac{x_k}{[k-(n-k-3)]!}+\frac{x_{k+1}}{[k+1-(n-k-3)]!} +\dots\\
&+ \frac{x_{n-3}}{[n-3-(n-k-3)]!}
+ \frac{1}{[n-2-(n-k-3)]!}=0
\end{aligned}
\end{gather*}
has only one solution, its denoted by $(a_k,a_{k+1},\dots,a_{n-3})$.
\end{lemma}

\begin{lemma} \label{lem2.2} 
The system of linear equations
\begin{gather*}
\frac{x_k}{k!}+\frac{x_{k+1}}{(k+1)!}+\dots+\frac{x_{n-3}}{(n-3)!}
  +\frac{1}{(n-1)!}=0,\\
\frac{x_k}{(k-1)!}+\frac{x_{k+1}}{k!}+\dots+\frac{x_{n-3}}{(n-4)!}
 +\frac{1}{(n-2)!}=0,\\
\dots \\
\begin{aligned}
&\frac{x_k}{[k-(n-k-3)]!}+\frac{x_{k+1}}{[k+1-(n-k-3)]!} +\dots\\
&+ \frac{x_{n-3}}{[n-3-(n-k-3)]!}
+\frac{1}{[n-1-(n-k-3)]!}=0
\end{aligned}
\end{gather*}
has only one solution, it is denoted by $(b_k,b_{k+1},\dots,b_{n-3})$.
\end{lemma}

 \begin{lemma} \label{lem2.3} 
For given $u\in Y$, the system of linear equations
\begin{gather*}
\frac{x_k}{k!}+\frac{x_{k+1}}{(k+1)!}+\dots+\frac{x_{n-3}}{(n-3)!}
 +\frac{(-1)^{n-k}}{(n-1)!}\int_0^1(1-s)^{n-1}u(s)ds=0,
\\
\frac{x_k}{(k-1)!}+\frac{x_{k+1}}{k!}+\dots+\frac{x_{n-3}}{(n-4)!}
+\frac{(-1)^{n-k}}{(n-2)!}\int_0^1(1-s)^{n-2}u(s)ds=0,
\\
\dots\\
\begin{aligned}
&\frac{x_k} {[k-(n-k-3)]!}+\frac{x_{k+1}}{[k+1-(n-k-3)]!} +\dots
+ \frac{x_{n-3}}{[n-3-(n-k-3)]!}\\
&+\frac{(-1)^{n-k}}{[n-1-(n-k-3)]!}\int_0^1(1-s)^{n-1-(n-k-3)}u(s)ds=0
\end{aligned}
\end{gather*}
has only one solution, its  denoted by
$(B_k(u),B_{k+1}(u),\dots,B_{n-3}(u))$.
\end{lemma}

Define the operators $T_1, T_2, Q_1, Q_2:Y\to R$ as follows:
\begin{gather*}
T_1u(t)=\sum_{i=1}^{m}\alpha_i\int_{\xi_i}^{1}u(s)ds,\\
T_2u(t)=\sum_{j=1}^{l}\beta_j\Big[\int_{\eta_j}^{1}(1-s)u(s)ds+(1-\eta_j)
\int^{\eta_j}_{0}u(s)ds\big],\\
Q_1u=\frac{1}{e}(e_4T_1u-e_3T_2u),\quad 
Q_2u=\frac{1}{e}(-e_2T_1u+e_1T_2u).
\end{gather*}
Obviously, $e_1=T_1(1)$, $e_2=T_2(1)$, $e_3=T_1(t)$, $e_4=T_2(t)$.

\begin{lemma} \label{lem2.4} Assume that {\rm (H1)} holds, then
 $L:\operatorname{dom}L\subset X\to Y$ is a Fredholm operator of index 
zero and the linear continuous projector $Q:Y\to Y$ can be defined as
$$
Qu=Q_1u+t\cdot Q_2u, 
$$
and the linear operator 
$K_P:\operatorname{Im}L\to \operatorname{dom}L\cap \ker P$ can be
written as
$$
K_Pu=\sum_{i=k}^{n-3}\frac{B_i(u)}{i!}t^{i}
+\frac{(-1)^{n-k}}{(n-1)!}\int_0^t(t-s)^{n-1}u(s)ds.
$$
\end{lemma}

\begin{proof}
  Take $y\in \ker L$. We obtain
 $y=\sum_{i=k}^{n-1}\frac{x_i}{i!}t^i$
 satisfying
\begin{gather*}
\frac{x_k}{k!}+\frac{x_{k+1}}{(k+1)!}+\dots+\frac{x_{n-2}}{(n-2)!}
  +\frac{x_{n-1}}{(n-1)!}=0,\\
\frac{x_k}{(k-1)!}+\frac{x_{k+1}}{k!}+\dots+\frac{x_{n-2}}{(n-3)!}
+\frac{x_{n-1}}{(n-2)!}=0,\\
\dots\\
\begin{aligned}
&\frac{x_k} {[k-(n-k-3)]!}+\frac{x_{k+1}}{[k+1-(n-k-3)]!} +\dots\\
&+\frac{x_{n-2}}{[n-2-(n-k-3)]!}
+\frac{x_{n-1}}{[n-1-(n-k-3)]!}=0.
\end{aligned}
\end{gather*}

Setting $x_{n-2}=1$, $x_{n-1}=0$, and $x_{n-2}=0$, $x_{n-1}=1$,
respectively, by Lemmas \ref{lem2.1}, \ref{lem2.2}, we have
$$
y=\sum_{i=k}^{n-3}\frac{ca_i+db_i}{i!}t^{i}+\frac{c}{(n-2)!}t^{n-2}
+\frac{d}{(n-1)!}t^{n-1}, c, d\in \mathbb{R}.
$$
Therefore,
$$
\ker L=\big\{ y:y=\sum_{i=k}^{n-3}\frac{ca_i+db_i}{i!}t^{i}+\frac{c}{(n-2)!}t^{n-2}
+\frac{d}{(n-1)!}t^{n-1}, c, d\in R\big\}.
$$
Define the linear operator $P:X\to X$ as follows
$$
Py(t)=\sum_{i=k}^{n-3}\frac{y^{(n-2)}(0)a_i+y^{(n-1)}(0)b_i}{i!}t^{i}
+\frac{y^{(n-2)}(0)}{(n-2)!}t^{n-2}+\frac{y^{(n-1)}(0)}{(n-1)!}t^{n-1}.
$$
Obviously, $\operatorname{Im}P=\ker L$ and $P^2y=Py$. For any $y\in X$, 
it follows from $y=(y-Py)+Py$ that $X=\ker P+\ker L$. By simple calculation, we
can get that $\ker L\cap \ker P=\{0\}$. So, we have
\begin{equation}
X=\ker L\oplus \ker P.\label{e2.1}
\end{equation}
We will show that
\begin{align*}
\operatorname{Im}L=\big\{&u\in Y:
 \sum_{i=1}^{m}\alpha_i\int_{\xi_i}^{1}u(s)ds=0,\\
& \sum_{j=1}^{l}\beta_j
\Big[\int_{\eta_j}^{1}(1-s)u(s)ds+(1-\eta_j)\int^{\eta_j}_{0}u(s)ds\Big]=0
\big\}.
\end{align*}
In fact, if $u\in \operatorname{Im}L$, there exists $y\in \operatorname{dom}L$
such that $u=Ly\in Y$. This, together with $y^{i}(0)=0$, $0\leq i\leq k-1$,
implies that
$$
y(t)=\sum_{i=k}^{n-1}\frac{c_i}{i!}t^{i}+\frac{(-1)^{n-k}}{(n-1)!}
\int_0^t(t-s)^{n-1}u(s)ds.
$$
Since $\sum_{i=1}^{m}\alpha_i=1$ and
$y^{(n-1)}(1)=\sum_{i=1}^{m}\alpha_iy^{(n-1)}(\xi_i)$, we obtain
\begin{equation}
\sum_{i=1}^{m}\alpha_i\int_{\xi_i}^1u(s)ds=0.\label{e2.2}
\end{equation}
Since $\sum_{j=1}^{l}\beta_j=1$, $\sum_{j=1}^{l}\beta_j\eta_j=1$ and
$y^{(n-2)}(1)=\sum_{j=1}^{l}\beta_jy^{(n-2)}(\eta_j)$, we obtain
\begin{equation}
\sum_{j=1}^{l}\beta_j
\Big[\int_{\eta_j}^{1}(1-s)u(s)ds+(1-\eta_j)\int^{\eta_j}_{0}u(s)ds\Big]=0.
\label{e2.3}
\end{equation}
On the other hand, if $u\in Y$ satisfies \eqref{e2.2} and \eqref{e2.3},
take
$$
y=\sum_{i=k}^{n-3}\frac{B_i(u)}{i!}t^{i}
+\frac{(-1)^{n-k}}{(n-1)!}\int_0^t(t-s)^{n-1}u(s)ds.
$$
It follows from \eqref{e2.2}, \eqref{e2.3} and Lemma \ref{lem2.3}
 that $y\in \operatorname{dom} L$.
Obviously, $Ly=u$. So, we get $u\in \operatorname{Im}L$.

Now we will prove that $Q:Y\to Y$ is a projector such that
$\ker Q=\operatorname{Im}L$, $Y=\operatorname{Im}L\oplus \operatorname{Im}Q$.
For $u\in Y$, since
\begin{gather*}
Q_1(1)=\frac{1}{e}[e_4T_1(1)-e_3T_2(1)]=1, \quad
Q_1(t)=\frac{1}{e}[e_4T_1(t)-e_3T_2(t)]=0,\\
Q_2(1)=\frac{1}{e}[-e_2T_1(1)+e_1T_2(1)]=0, \quad
Q_2(t)=\frac{1}{e}[-e_2T_1(t)+e_1T_2(t)]=1,
\end{gather*}
we have
\begin{gather*}
Q_1(Qu)=Q_1(Q_1u+t\cdot Q_2u)=Q_1u\cdot Q_1(1)+Q_2u\cdot
Q_1(t)=Q_1u, \\
Q_2(Qu)=Q_2(Q_1u+t\cdot Q_2u)=Q_1u\cdot Q_2(1)+Q_2u\cdot
Q_2(t)=Q_2u.
\end{gather*}
Thus,
$$
Q^2u=Q_1(Qu)+t\cdot Q_2(Qu)=Q_1u+t\cdot Q_2u=Qu.
$$
Since $u\in \ker Q$, we have
\begin{gather*}
 e_4T_1u-e_3T_2u=0,\\
 -e_2T_1u+e_1T_2u=0.
\end{gather*}
It follows from (H2) that $T_1u=T_2u=0$. So, $u\in \operatorname{Im}L$;
 i.e., $\ker Q\subset \operatorname{Im}L$. Clearly, 
$\operatorname{Im}L\subset \ker Q$. So, $\operatorname{Im}L= \ker Q$. 
This, together with $Q^2y=Qy$, means that 
$\operatorname{Im}L\cap \operatorname{Im} Q=\{0\}$.
 Thus, we have $Y=\operatorname{Im}L\oplus \operatorname{Im}Q$. 
Considering \eqref{e2.1}, we
know that $L$ is a Frdholm operator of index zero.

Define the operator $K_P:Y\to X $ as follows
$$
K_Pu=\sum_{i=k}^{n-3}\frac{B_i(u)}{i!}t^{i}
+\frac{(-1)^{n-k}}{(n-1)!}\int_0^t(t-s)^{n-1}u(s)ds.
$$
For $u\in \operatorname{Im}L$, by Lemma \ref{lem2.3}, we have
$K_Pu\in \operatorname{dom}L$. 
Clearly, $K_Pu\in \ker P$. So, we get that 
$K_P(\operatorname{Im}L)\subset \operatorname{dom}L\cap \ker P$.
Now we will prove that $K_P$ is the inverse of $L|_{\operatorname{dom}L\cap
\ker P}$.

Obviously, $LK_Pu=u$, for $u\in \operatorname{Im}L$. On the other hand, for
$y\in \operatorname{dom}L\cap \ker P$, we have
\begin{align*}
K_PLy(t)
&=\sum_{i=k}^{n-3}\frac{B_i(Ly)}{i!}t^{i}+\frac{(-1)^{n-k}}{(n-1)!}
\int_0^t(t-s)^{n-1}(-1)^{n-k}y^{(n)}(s)ds\\
&=\sum_{i=k}^{n-3}\big(\frac{B_i(Ly)-y^{(i)}(0)}{i!}\big)t^i+y(t).
\end{align*}
Since $K_P(Ly)\in \operatorname{dom}L$ and $y\in \operatorname{dom}L$, we obtain
$(K_PLy)^{(j)}(1)=y^{(j)}(1)=0$, $0\leq j\leq n-k-3$. Thus
$(B_k(Ly)-y^{(k)}(0), B_{k+1}(Ly)-y^{(k+1)}(0),\dots,B_{n-3}(Ly)-y^{(n-3)}(0))$
is the only zero solution of the system of linear equations
\begin{gather*}
\frac{x_k}{k!}+\frac{x_{k+1}}{(k+1)!}+\dots+\frac{x_{n-3}}{(n-3)!}=0,\\
\frac{x_k}{(k-1)!}+\frac{x_{k+1}}{k!}+\dots+\frac{x_{n-3}}{(n-4)!}=0,\\
\dots\\
\begin{aligned}
&\frac{x_k} {[k-(n-k-2)]!}+\frac{x_{k+1}}{[k+1-(n-k-2)]!} +\dots\\
&+ \frac{x_{n-3}}{[n-3-(n-k-3)]!}=0.
\end{aligned}
\end{gather*}
 So, we have $K_PLy=y$, for $y\in \operatorname{dom}L\cap \ker P$.
Thus, $K_P=(L|_{\operatorname{dom}L\cap \ker P})^{-1}$. The proof is
complete.
\end{proof}

\section{Main results}

\begin{lemma} \label{lem3.1} 
Assume $\Omega\subset X$ is an open bounded
subset and $\operatorname{dom}L\cap \overline{\Omega}\neq \emptyset$, 
then $N$ is $L$-compact on $ \overline{\Omega}$.
\end{lemma}

\begin{proof} By (H3), we have that $QN(\overline{\Omega})$ is
bounded. Now we will show that
$K_P(I-Q)N:\overline{\Omega}\to X$ is compact.

It follows from (H3) that there exists constant $M_0>0$ such
that $|(I-Q)Ny|\leq M_0$, a.e. $t\in [0,1], y\in \overline{\Omega}$. 
Thus, $K_P(I-Q)N(\overline{\Omega})$ is
bounded. By (H3) and Lebesgue Dominated Convergence theorem, we
get that $K_P(I-Q)N:\overline{\Omega}\to X$ is continuous.
Since 
$\{\int_0^t(t-s)^j(I-Q)Ny(s)ds, y\in \overline{\Omega}\}$, $j=0,1\dots,n-1$ 
are equi-continuous,
and $t^j$, $j=0,1\dots,n-1$ are uniformly continuous on [0,1],
using Ascoli-Arzela theorem, we obtain that
$K_P(I-Q)N:\overline{\Omega}\to X$ is compact. The proof
is complete.
\end{proof}

To obtain our main results, we need the following assumptions.
\begin{itemize}
\item[(H4)] There exist constants $M_1>0, M_2>0$ such that if
$|y^{(n-1)}(t)|>M_1$, $t\in [\xi_m,1]$ then
$$
\sum_{i=1}^{m}\alpha_i\int _{\xi_i}^1Ny(s)ds\neq 0,
$$
and if $|y^{(n-2)}(t)|>M_2$, $t\in [0,\eta_1]$ then
$$
\sum_{j=1}^{l}\beta_j
\Big[\int_{\eta_j}^{1}(1-s)Ny(s)ds+(1-\eta_j)\int^{\eta_j}_{0}Ny(s)ds\Big]
\neq 0.
$$

\item[(H5)] There exist functions $g, h, \psi_i\in L^1[0,1]$, 
$i=1,2,\dots,n$, with
$\|\psi_n\|_1:=r_1<1/2$, $\sum_{i=1}^{n-1}\|\psi_i\|_1:=r_2<\frac{1-2r_1}{4}$,
$ \theta\in[0,1)$, and some $1\leq j\leq n-1$ such that
$$
|f(t,x_1,x_2,\dots,x_n)|\leq g(t)+\sum_{i=1}^{n}\psi_i(t)|x_i|+h(t)|x_j|^\theta.
$$

\item[(H6)] There exist constants $c_0>0, d_0>0$ such that, for
\[
y=\sum_{i=k}^{n-3}\frac{ca_i+db_i}{i!}t^{i}+\frac{c}{(n-2)!}t^{n-2}
+\frac{d}{(n-1)!}t^{n-1}\in \ker L,
\]
 one of the following two conditions holds
\begin{itemize}
\item[(1)] $c\cdot T_1Ny<0$, if $|c|>c_0$, $d\cdot T_2Ny<0$, if $|d|>d_0$, 
\item[(2)] $c\cdot T_1Ny>0$, if $|c|>c_0$, $d\cdot T_2Ny>0$, if $|d|>d_0$,
\end{itemize}
\end{itemize}

\begin{lemma} \label{lem3.2} 
Suppose {\rm (H1)--(H5)} hold, then the set
$$
\Omega_1=\{y\in \operatorname{dom}L\setminus \ker L :
 Ly=\lambda Ny, \lambda\in (0,1)\}
$$
is bounded.
\end{lemma}

\begin{proof} Take $y\in \Omega_1$. By $Ny\in \operatorname{Im}L$,
we have
\begin{gather}
\sum_{i=1}^{m}\alpha_i\int _{\xi_i}^1Ny(s)ds=0,\label{e3.1}\\ 
\sum_{j=1}^{l}\beta_j \Big[\int_{\eta_j}^{1}(1-s)Ny(s)ds+(1-\eta_j)
 \int^{\eta_j}_{0}Ny(s)ds\Big]=0.\label{e3.2}
\end{gather}
Since $Ly=\lambda Ny$ and $y\in \operatorname{dom}L$, we obtain
\begin{equation}
y(t)=\sum_{i=k}^{n-1}\frac{c_i}{i!}t^{i}+\frac{(-1)^{n-k}}{(n-1)!}
\lambda\int_0^t(t-s)^{n-1}Ny(s)ds,\label{e3.3}
\end{equation}
where $c_{k}, c_{k+1},\dots,c_{n-1}$  satisfy
\begin{gather*}
 \sum_{i=k}^{n-1}\frac{c_i}{i!}=-\frac{(-1)^{n-k}}{(n-1)!}
\lambda\int_0^1(1-s)^{n-1}Ny(s)ds,\\
  \sum_{i=k}^{n-1}\frac{c_i}{(i-1)!}=-\frac{(-1)^{n-k}}{(n-2)!}
\lambda\int_0^1(1-s)^{n-2}Ny(s)ds,\\
 \dots   \\
 \sum_{i=k}^{n-1}\frac{c_i}{[i-(n-k-3)]!}=-\frac{(-1)^{n-k}}{[i-(n-k-3)]!}
\lambda\int_0^1(1-s)^{i-(n-k-3)}Ny(s)ds.
\end{gather*}
 It follows from $y^{(i)}(0)=y^{(j)}(1)=0$, $0\leq i\leq k-1$,
$0\leq j\leq n-k-3$ that there exist points
$\delta_i\in [0,1]$ such that $y^{(i)}(\delta_i)=0$, $i=0,1,\dots,n-3$.
So, we have
$$
y^{(i)}(t)=\int_{\delta_i}^ty^{(i+1)}(s)ds,\quad i=0,1,\dots,n-3.
$$
Therefore,
\begin{equation}
\|y^{(i)}\|_\infty\leq \|y^{(i+1)}\|_1\leq
\|y^{(i+1)}\|_\infty, i=0,1,\dots,n-3.\label{e3.4}
\end{equation}
 By \eqref{e3.1} and (H4), there exists $t_0\in [\xi_m,1]$ such that
$|y^{(n-1)}(t_0)|\leq M_1$. This, together with \eqref{e3.3}, implies that
$$
|c_{n-1}|\leq M_1+\int_0^1\big|f(s,y(s),y'(s),\dots,y^{(n-1)}(s))\big|ds
+\|\varepsilon\|_1.
$$
By \eqref{e3.2} and (H4), we get that there exists $t_1\in [0,\eta_1]$
such that $|y^{(n-2)}(t_1)|\leq M_2$. It follows from \eqref{e3.3} that
\begin{align*}
|c_{n-2}|
&\leq M_2+|c_{n-1}|+\int_0^1\big|f(s,y(s),y'(s),\dots,y^{(n-1)}(s))\big|ds
+\|\varepsilon\|_1\\
&\leq M_1+M_2+2\int_0^1\big|f(s,y(s),y'(s),\dots,y^{(n-1)}(s))\big|ds
+2\|\varepsilon\|_1.
\end{align*}
Thus,
\begin{gather*}
\|y^{(n-1)}\|_\infty
\leq M_1+2\int_0^1\big|f(s,y(s),y'(s),\dots,y^{(n-1)}(s))\big|ds
 +2\|\varepsilon\|_1,
\\
\|y^{(n-2)}\|_\infty
\leq 2M_1+M_2+4\int_0^1\big|f(s,y(s),y'(s),\dots,y^{(n-1)}(s))\big|ds
+4\|\varepsilon\|_1.
\end{gather*}
By (H5) and \eqref{e3.4}£¬ we have
$$
\|y^{(n-1)}\|_\infty\leq r_3+2r_2\|y^{(n-2)}\|_\infty
+2r_1\|y^{(n-1)}\|_\infty+2\|h\|_1\|y^{(n-2)}\|_\infty^\theta
$$
and
\begin{equation}
\|y^{(n-2)}\|_\infty\leq 2r_3+M_2+4r_2\|y^{(n-2)}\|_\infty
+4r_1\|y^{(n-1)}\|_\infty+4\|h\|_1\|y^{(n-2)}\|_\infty^\theta,\label{e3.5}
\end{equation}
where $r_3=M_1+2\|g\|_1+2\|\varepsilon\|_1$.
So, we obtain
\begin{equation}
\|y^{(n-1)}\|_\infty\leq \frac{1}{1-2r_1}
[r_3+2r_2\|y^{(n-2)}\|_\infty+2\|h\|_1\|y^{(n-2)}\|_\infty^\theta].\label{e3.6}
\end{equation}
By \eqref{e3.5} and \eqref{e3.6}, we have
$$
\|y^{(n-2)}\|_\infty\leq
\frac{2r_3}{1-2r_1}+M_2+\frac{4r_2}{1-2r_1}\|y^{(n-2)}\|_\infty
+\frac{4\|h\|_1}{1-2r_1}\|y^{(n-2)}\|_\infty^\theta.
$$
Therefore,
$$
\|y^{(n-2)}\|_\infty\leq \frac{1}{1-2r_1-4r_2}[2r_3+(1-2r_1)M_2+4\|h\|_1
\|y^{(n-2)}\|_\infty^\theta].
$$
It follows from $\theta\in[0,1)$ that
$\{\|y^{(n-2)}\|_\infty: y\in \Omega_1\}$ is bounded. By \eqref{e3.4} and
\eqref{e3.6}, we get that $\Omega_1$ is bounded.
\end{proof}

\begin{lemma} \label{lem3.3} 
Suppose {\rm (H1)--(H3), (H6)} hold. Then the set
$$
\Omega_2=\{y\in \ker L: Ny\in \operatorname{Im}L\}
$$
is bounded.
\end{lemma}

\begin{proof} Take $y\in \Omega_2$, then 
\[
y(t)= \sum_{i=k}^{n-3}\frac{ca_i+db_i}{i!}t^{i}+\frac{c}{(n-2)!}t^{n-2}
+\frac{d}{(n-1)!}t^{n-1}.
\]
By $Ny\in \operatorname{Im}L$, we have $T_1Ny=0, T_2Ny=0$. By (H6), we get
that $|c|\leq c_0,  |d|\leq d_0$. This means that $\Omega_2$ is
bounded.
\end{proof}

\begin{lemma} \label{lem3.4} 
Suppose {\rm (H1)--(H3), (H6)} hold. Then the
set
$$
\Omega_3=\{y\in \ker L : \lambda Jy+(1-\lambda)\omega QNy=0, 
\lambda\in [0,1]\}
$$
is bounded, where $J:\ker L\to \operatorname{Im}Q$ is a linear isomorphism
given by
 $$ 
J\Big(
\sum_{i=k}^{n-3}\frac{ca_i+db_i}{i!}t^{i}+\frac{c}{(n-2)!}t^{n-2}
+\frac{d}{(n-1)!}t^{n-1}\Big)
= \frac{1}{e}(e_4c-e_3d)+\frac{1}{e}(-e_2c+e_1d)t ,
$$ 
where $ c, d\in \mathbb{R}$ and
$$
\omega=\begin{cases}
-1, &\text{if (H6)(1) holds},\\
1,  &\text{if (H6)(2) holds}.
\end{cases}
$$
\end{lemma}

\begin{proof} Take $y\in \Omega_3$. $y\in \ker L$ implies that
\[
y= \sum_{i=k}^{n-3}\frac{ca_i+db_i}{i!}t^{i}+\frac{c}{(n-2)!}t^{n-2}
 +\frac{d}{(n-1)!}t^{n-1}, c, d\in \mathbb{R}.
\]
Since $\lambda Jy+(1-\lambda)\omega QNy=0$, we obtain
$$
\lambda c=-(1-\lambda)\omega T_1Ny,\quad  
\lambda d=-(1-\lambda)\omega T_2Ny. 
$$ 
If $\lambda=0$, by (H6), we get 
$|c|\leq c_0, |d|\leq d_0$. If $\lambda=1$, then $c=d=0$. 
For $\lambda\in (0,1)$, if $|c|> c_0$ or $|d|> d_0$, then
$$
\lambda c^2=-(1-\lambda)\omega c\cdot T_1Ny<0
$$
or
$$
\lambda d^2=-(1-\lambda)\omega d\cdot T_2Ny<0.
$$
A contradiction. So, $\Omega_3$ is bounded.
\end{proof}

\begin{theorem} \label{thm3.1} 
Suppose {\rm (H1)--(H6)} hold. Then  
\eqref{e1.1}--\eqref{e1.2} has at least one solution in $X$.
\end{theorem}

\begin{proof} 
Let $\Omega\supset\cup_{i=1}^{3}\overline{\Omega_i}\cup\{0\}$
be a bounded open subset of $X$. It follows from Lemma \ref{lem3.1} 
that $N$ is $L$-compact on $\overline{\Omega}$.
 By Lemmas \ref{lem3.2} and \ref{lem3.3}, we obtain
\begin{itemize}
\item[(1)] $Ly\neq \lambda Ny$ for every 
$(y,\lambda)\in [(\operatorname{dom}L\setminus \ker L)
\cap\partial\Omega]\times(0, 1)$;

\item[(2)] $Ny\notin \operatorname{Im}L$ for every
$y\in \ker L\cap\partial\Omega$.
\end{itemize}
We need  to prove only that:
\[
\deg(QN|_{\ker L}, \Omega\cap \ker L, 0)\neq0.
\]
Take
$$
H(y,\lambda)=\lambda Jy+\omega(1-\lambda)QNy.
$$
According to Lemma \ref{lem3.4}, we know that 
$H(y,\lambda)\neq 0$ for $y\in \partial\Omega\cap \ker L$, $\lambda\in [0,1]$. 
By the homotopy of degree, we obtain
\begin{align*}
\deg(QN|_{\ker L}, \Omega \cap \ker L,0)
&=\deg(\omega H(\cdot,0),\Omega \cap \ker L,0)\\
&=\deg(\omega H(\cdot,1),\Omega \cap \ker L,0)\\
&=\deg(\omega J, \Omega \cap \ker L,0)\neq 0.
\end{align*}
By Theorem \ref{thm2.1}, we can obtain that $Ly=Ny$ has at least one
solution in $\operatorname{dom}L\cap\overline{\Omega}$; i.e.,
\eqref{e1.1}--\eqref{e1.2} has
at least one solution in $X$. The prove is complete.
\end{proof}

\subsection*{Acknowledgments} The author is grateful to anonymous
referees for their constructive comments and suggestions which led
to improvement of the original manuscript.

This work is supported by the Natural Science Foundation
of China (11171088), the Natural Science Foundation of Hebei
Province (A2013208108) and the Doctoral Program Foundation of
Hebei University of Science and Technology (QD201020).

\begin{thebibliography}{00}

\bibitem{a1} R.P. Agarwal, D. O'Regan;
\emph{Positive solutions for $(p, n-p)$  conjugate boundary value problems}, 
J. Differential Equations, 150 (1998), 462-473.

\bibitem{a2} R. P. Agarwal, D. O'Regan;
\emph{Multiplicity results for singular conjugate, focal, and $(N, P)$ problems},
 J. Differential  Equations,  170 (2001), 142-156.

\bibitem{a3} R. P. Agarwal, S. R. Grace, D. O'Regan;
\emph{Semipositone higher-order differential equations}, 
Appl. Math. Lett. 17 (2004) 201-207.

\bibitem{d1} B. Du, X. Hu;
\emph{A new continuation theorem for the existence
      of solutions to $P$-Lpalacian BVP at resonance}, Appl. Math. Comput.
      208 (2009), 172-176.

\bibitem{d2} Z. Du, X. Lin, W. Ge;
\emph{Some higher-order multi-point boundary value problem at resonance},
J. Comput. Appl. Math.  177 (2005), 55-65.

\bibitem{e1} P. W. Eloe, J. Henderson;
\emph{Singular nonlinear $(k,n-k)$ conjugate boundary value problems},
J. Differential Equations 133 (1997), 136-151.

 \bibitem{f1} W. Feng, J. R. L. Webb;
\emph{Solvability of $m$-point  boundary value problems with nonlinear growth},
 J. Math. Anal. Appl.  212 (1997), 467-480.

\bibitem{g1} C. P. Gupta;
\emph{Solvability of multi-point boundary value
      problem at resonance}, Results Math. 28 (1995), 270-276.

\bibitem{h1} X. He, W. Ge;
\emph{Positive solutions for semipositone $(p, n-p)$
      right focal boundary value problems}, Appl. Anal. 81 (2002),
      227-240.

\bibitem{j1} D. Jiang;
\emph{Positive solutions to singular $(k,n-k)$ conjugate boundary
      value problems}, Acta Mathematica Sinica, 44 (2001), 541-548(in Chinese)

\bibitem{j2} W. Jiang, J. Zhang;
\emph{Positive solutions for $(k, n-k)$
      conjugate eigenvalue problems in Banach spaces},
Nonlinear Anal. 71 (2009),  723-729.

\bibitem{j3} W. Jiang, J. Qiu;
\emph{Solvability of $(k,n-k)$ Conjugate Boundary Value Problems at
     Resonance}, ElectroN. J. Differential Equations, 2012 (2012),
No. 114, pp. 1-10.

\bibitem{k1} G. L. Karakostas, P. Ch. Tsamatos;
\emph{On a Nonlocal Boundary Value Problem at Resonance},
J. Math. Anal. Appl. 259(2001), 209-218.

\bibitem{k2} L. Kong, J. Wang;
\emph{The Green's function for $(k, n-k)$ conjugate boundary value problems
and its applications}, J. Math.   Anal. Appl. 255 (2001), 404-422.

\bibitem{k3} N. Kosmatov;
\emph{Multi-point boundary value problems on an unbounded domain at resonance},
       Nonlinear  Anal., 68 (2008), 2158-2171.

\bibitem{l1} K. Q. Lan;
\emph{Multiple positive solutions of conjugate boundary
      value problems with singularities}, Appl. Math. Comput. 147 (2004),
      461-474.

\bibitem{l2} H. Lian, H. Pang, W. Ge;
\emph{Solvability for second-order three-point boundary value problems
at resonance on a half-line}, J. Math. Anal. Appl.
      337 (2008), 1171-1181.

\bibitem{l3} B. Liu;
\emph{Solvability of multi-point boundary value problem
at resonance (II)}, Appl. Math. Comput. 136(2003) 353-377.

\bibitem{l4} Y. Liu, W. Ge;
\emph{Solvability of nonlocal boundary value  problems  for ordinary
differential equations of higher order}, Nonlinear
      Anal., 57 (2004), 435-458.

\bibitem{l5} S. Lu, W. Ge;
\emph{On the existence of $m$-point boundary value
      problem at resonance for higher order differential equation},
J. Math. Anal. Appl.  287 (2003), 522-539.

\bibitem{m1} R. Ma;
\emph{Positive solutions for semipositone $(k, n-k)$
      conjugate boundary value problems}, J. Math.
      Anal. Appl. 252 (2000), 220-229.

\bibitem{m2} R. Ma;
\emph{Existence results of a $m$-point boundary value problem at resonance},
J. Math. Anal. Appl. 294 (2004), 147-157.

\bibitem{m3} J. Mawhin;
\emph{Topological degree methods in nonlinear boundary
      value problems}, in NSFCBMS Regional Conference Series in
      Mathematics, American Mathematical Society, Providence, RI, 1979.

\bibitem{p1} B. Prezeradzki, R. Stanczy;
\emph{Solvability of a multi-point boundary value problem at
      resonance}, J. Math. Anal. Appl. 264 (2001), 253-261.

\bibitem{s1} H. Su, Z. Wei;
\emph{Positive solutions to semipositone $(k, n-k)$
      conjugate eigenvalue problems}, Nonlinear Anal. 69 (2008),
      3190-3201.

\bibitem{x1} C. Xue, W. Ge;
\emph{The existence of solutions for multi-point
      boundary value problem at resonance}, ACTA Mathematica Sinica,
      48 (2005), 281-290.

\bibitem{w1} J. R. L. Webb;
\emph{Nonlocal conjugate type boundary value problems of higher
       order}, Nonlinear Anal. 71 (2009), 1933-1940.

\bibitem{w2} J. R. L. Webb, M. Zima;
\emph{Multiple positive solutions of resonant and non-resonant nonlocal 
boundary value problems}, Nonlinear Anal. 71 (2009),  1369-1378.

\bibitem{w3} P. J. Y. Wong, R. P. Agarwal;
\emph{Singular differential equation with (n,p) boundary conditions},
Math. Comput. Modelling, 28 (1998), 37-44.

\bibitem{y1} X. Yang;
\emph{Green's function and positive solutions for
      higher-order ODE}, Appl. Math. Comput. 136 (2003), 379-393.

\bibitem{y2} H. Yang, F. Wang;
\emph{The existence of solutions of $(k, n-k)$
      conjugate eigenvalue problems in Banach spaces}, Chin. Quart. J. of
      Math. 23 (2008), 470-474.

\bibitem{z1} G. Zhang, J. Sun;
\emph{Positive solutions to singular $(k, n-k)$
      multi-point boundary value problems},
 Acta Mathematica Sinica, 49 (2006), 391-398 (in Chinese).

\bibitem{z2} X. Zhang, M. Feng, W. Ge;
\emph{Existence result of second-order  differential equations with
integral boundary conditions at resonance}, J. Math. Anal. Appl.
      353 (2009), 311-319.

\end{thebibliography}

\end{document}