\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 114, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/114\hfil Conjugate boundary-value problems] {Solvability of (k,n-k) conjugate boundary-value problems at resonance} \author[W. Jiang, J. Qiu \hfil EJDE-2012/114\hfilneg] {Weihua Jiang, Jiqing Qiu} % in alphabetical order \address{Weihua Jiang \newline College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, China} \email{weihuajiang@hebust.edu.cn} \address{Jiqing Qiu \newline College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, China} \email{qiujiqing@263.net} \thanks{Submitted May 1, 2012. Published July 5, 2012.} \subjclass[2000]{35B34, 34B10, 34B15} \keywords{Resonance; Fredholm operator; boundary-value problem} \begin{abstract} Using the coincidence degree theory due to Mawhin and constructing suitable operators, we prove the existence of solutions for $(k,n-k)$ conjugate boundary-value problems at resonance. \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 non-resonance has been studied in many papers (see \cite{1,2,3,4,7,8,11,12,13,14,17,22,26,27,28,31,32,33}). For example, using fixed point theorem in a cone, Jiang \cite{13} obtained the existence of positive solutions for $(k,n-k)$ conjugate boundary-value problem \begin{gather*} (-1)^{n-k}y^{(n)}(t)=f(t,y(t)),\quad 00$, 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} First, we introduce some notation and state a theorem to be used later. For more details see \cite{24}. 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 the inverse by $K_{P}$. Assume that $\Omega$ is 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{24}] \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)] $\deg(QN|_{\ker L},\Omega\cap \ker L,0)\neq 0$, 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 the operator $Ly(t)=(-1)^{n-k}y^{(n)}(t)$ with \begin{align*} \operatorname{dom}L=\{&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-2,\; y^{(n-1)}(1)=\sum_{i=1}^{m}\alpha_iy^{(n-1)}(\xi_i)\}. \end{align*} Let $N:X\to Y$ be defined as $$ Ny(t)=f(t,y(t),y'(t),\dots,y^{(n-1)}(t))+\varepsilon(t),\quad t\in [0,1]. $$ Then problem \eqref{e1.1}, \eqref{e1.2} becomes $Ly=Ny$. \section{Main results} By Cramer's rule, we can get the following lemmas. \begin{lemma} \label{lem3.1} For given $u\in Y$, the system of linear equations \begin{equation} \begin{gathered} \frac{x_k}{k!}+\frac{x_{k+1}}{(k+1)!}+\dots+\frac{x_{n-2}}{(n-2)!} +\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-2}}{(n-3)!} +\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-2)]!}+\frac{x_{k+1}}{[k+1-(n-k-2)]!} +\dots+ \frac{x_{n-2}}{[n-2-(n-k-2)]!}\\ &+ \frac{(-1)^{n-k}}{[n-1-(n-k-2)]!}\int_0^1(1-s)^{k+1}u(s)ds=0 \end{aligned} \end{gathered}\label{e3.1} \end{equation} has an only one solution, $(x_k,x_{k+1},\dots,x_{n-2})$ with \begin{align*} x_m&=\int_0^1\frac{(-1)^{n-k-1}m!}{(m-k)!(k-1)!(n-m-2)!} \sum_{i=0}^{m-k}(-1)^{m-k-i}\frac{C_{m-k}^i}{m-i}\\ &\quad \times \Big[\sum_{j=0}^{n-m-2}(-1)^{j}C_{n-m-2}^j \frac{(1-s)^{n-1-i-j}}{n-1-i-j}\Big]u(s)ds,\quad m=k,k+1,\dots,n-2. \end{align*} \end{lemma} \begin{lemma} \label{lem3.2} The system of linear equations \begin{equation} \begin{gathered} \frac{x_k}{k!}+\frac{x_{k+1}}{(k+1)!}+\dots+\frac{x_{n-2}}{(n-2)!}+\frac{1}{(n-1)!}=0 \\ \frac{x_k}{(k-1)!}+\frac{x_{k+1}}{k!}+\dots+\frac{x_{n-2}}{(n-3)!}+\frac{1}{(n-2)!}=0\\ \dots\\ \begin{aligned} &\frac{x_k} {[k-(n-k-2)]!}+\frac{x_{k+1}}{[k+1-(n-k-2)]!} +\dots \\ &+\frac{x_{n-2}}{[n-2-(n-k-2)]!}+ \frac{1}{[n-1-(n-k-2)]!}=0 \end{aligned} \end{gathered} \label{e3.2} \end{equation} has an only one solution, $(x_k,x_{k+1},\dots,x_{n-2})$ with \begin{align*} x_m&=-\frac{m!}{(m-k)!(k-1)!(n-m-2)!}\sum_{i=0}^{m-k}(-1)^{m-k-i} \frac{C_{m-k}^i}{m-i}\\ &\quad \times \Big(\sum_{j=0}^{n-m-2}(-1)^{j}C_{n-m-2}^j\frac{1}{n-1-i-j}\Big), \quad m=k,k+1,\dots,n-2. \end{align*} \end{lemma} Let $(B_k(u),B_{k+1}(u),\dots,B_{n-2}(u))$ denote the only solution of \eqref{e3.1}, and let $(A_k,A_{k+1},\dots,A_{n-2})$ denote the only solution of \eqref{e3.2}, and let $A_{n-1}=1$. In order to obtain our main results, we firstly present and prove the following lemmas. \begin{lemma} \label{lem3.3} Suppose {\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 $$ Qu=\frac{1}{1-\sum_{i=1}^{m}\alpha_i\xi_i} \sum_{i=1}^{m}\alpha_i\int_{\xi_i}^1u(s)ds, $$ and the linear operator $K_P:\operatorname{Im}L\to \operatorname{dom}L\cap \ker P$ can be written as $$ K_Pu=\sum_{i=k}^{n-2}\frac{B_i(u)}{i!}t^{i} +\frac{(-1)^{n-k}}{(n-1)!}\int_0^t(t-s)^{n-1}u(s)ds. $$ \end{lemma} \begin{proof} By simple calculations, we obtain that $$ \ker L=\big\{y:y=c\Big(\sum_{i=k}^{n-1}\frac{A_i}{i!}t^{i}\Big),\; c\in \mathbb{R}\big\}. $$ Define linear operator $P:X\to X$ as follows $$ Py(t)=\Big(\sum_{i=k}^{n-1}\frac{A_i}{i!}t^{i}\Big)y^{(n-1)}(0). $$ 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{e3.3} \end{equation} We will show that $$ \operatorname{Im}L=\big\{u\in Y: \sum_{i=1}^{m}\alpha_i\int _{\xi_i}^1u(s)ds=0 \big\}. $$ In fact, if $u\in \operatorname{Im}L$, there exists $y\in \operatorname{dom}L$ such that $u=Ly\in Y$. So, we have $$ y=\sum_{i=k}^{n-1}\frac{c_i}{i!}t^{i}+\frac{(-1)^{n-k}}{(n-1)!} \int_0^t(t-s)^{n-1}u(s)ds. $$ Since $\sum_{i=1}^{m}\alpha_i=1$ and $y^{(n-1)}(1)=\sum_{i=1}^{m}\alpha_iy^{(n-1)}(\xi_i)$, we have $$ \int_0^1u(s)ds=\sum_{i=1}^{m}\alpha_i\int_0^{\xi_i}u(s)ds; $$ i.e., $\sum_{i=1}^{m}\alpha_i\int _{\xi_i}^1u(s)ds=0$. On the other hand, if $u\in Y$ satisfies $\sum_{i=1}^{m}\alpha_i\int _{\xi_i}^1u(s)ds=0$, we take $$ y=\sum_{i=k}^{n-2}\frac{B_i(u)}{i!}t^{i} +\frac{(-1)^{n-k}}{(n-1)!}\int_0^t(t-s)^{n-1}u(s)ds. $$ Obviously, $Ly=u$ and $y^{(n-1)}(1)=\sum_{i=1}^{m}\alpha_iy^{(n-1)}(\xi_i)$. By Lemma \ref{lem3.1}, we obtain that $y\in \operatorname{dom}L$; i.e., $u\in \operatorname{Im}L$. Define operator $Q:Y\to Y$ as follows $$ Qu=\frac{1}{1-\sum_{i=1}^m\alpha_i\xi_i} \Big(\sum_{i=1}^{m}\alpha_i\int _{\xi_i}^1u(s)ds\Big). $$ Obviously, $Q^2y=Qy$ and $\operatorname{Im}L=\ker Q$. For $y\in Y$, set $y=(y-Qy)+Qy$. Then $y-Qy\in \ker Q=\operatorname{Im}L$, $Qy\in \operatorname{Im}Q$. It follows from $\ker Q= \operatorname{Im}L$ and $Q^2y=Qy$ that $\operatorname{Im}Q\cap \operatorname{Im}L=\{0\}$. So we have $$ Y=\operatorname{Im}L\oplus \operatorname{Im}Q. $$ This, together with \eqref{e3.3}, means that $L$ is a Fredholm operator of index zero. Define operator $K_P:Y\to X $ as follows $$ K_Pu=\sum_{i=k}^{n-2}\frac{B_i(u)}{i!}t^{i} +\frac{(-1)^{n-k}}{(n-1)!}\int_0^t(t-s)^{n-1}u(s)ds. $$ Now we show that $K_P(\operatorname{Im}L)\subset \operatorname{dom}L\cap \ker P$. Take $u\in \operatorname{Im}L$. Obviously, $(K_P(u))^{(n-1)}(0)=0$. This implies that $K_P(u)\in \ker P$. It is easy to see that $(K_P(u))^{(i)}(0)=0,~0\leq i\leq k-1$. It follows from Lemma \ref{lem3.1} that $(K_P(u))^{(j)}(1)=0$, $0\leq j\leq n-k-2$. From $u\in \operatorname{Im}L$, we obtain $$ (K_P(u))^{(n-1)}(1)=\sum_{i=1}^m\alpha_i(K_P(u))^{(n-1)}(\xi_i). $$ So, $K_P(u)\in \operatorname{dom}L$. Now we 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-2}\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-2}\big(\frac{B_i(Ly)-y^{(i)}(0)}{i!}\big)t^i+y(t). \end{align*} Since $y$ and $K_PLy\in \operatorname{dom}L$, we have $(K_PLy)^{(j)}(1)=y^{(j)}(1)=0,~0\leq j\leq n-k-2$. This means that $(B_k(Ly)-y^{(k)}(0),~B_{k+1}(Ly)-y^{(k+1)}(0),\dots,B_{n-2}(Ly)-y^{(n-2)}(0))$ 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-2}}{(n-2)!}=0\\ \frac{x_k}{(k-1)!}+\frac{x_{k+1}}{k!}+\dots+\frac{x_{n-2}}{(n-3)!}=0\\ \dots\\ \frac{x_k} {[k-(n-k-2)]!}+\frac{x_{k+1}}{[k+1-(n-k-2)]!} +\dots+ \frac{x_{n-2}}{[n-2-(n-k-2)]!}=0. \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} \begin{lemma} \label{lem3.4} 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} Obviously, $QN(\overline{\Omega})$ is bounded. Now we will show that $K_P(I-Q)N:\overline{\Omega}\to X$ is compact. It follows from (H2) 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 (H2) and Lebesgue Dominated Convergence theorem, we get that $K_P(I-Q)N:\overline{\Omega}\to X$ is continuous. Since $\big\{\int_0^t(t-s)^j(I-Q)Ny(s)ds,~y\in \overline{\Omega}\big\}$, $j=0,1\dots,n-1$ 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 conditions. \begin{itemize} \item[(H3)] There exists a constant $M>0$ such that if $|y^{(n-1)}(t)|>M$, $t\in [\xi_m,1]$ then $$ \sum_{i=1}^{m}\alpha_i\int _{\xi_i}^1\big[f(s,y(s),y'(s), \dots,y^{(n-1)}(s))+\varepsilon(s)\big]ds\neq 0. $$ \item[(H4)] There exist functions $g,h,\psi_i\in L^1[0,1]$, $i=1,2,\dots,n$, with $\sum_{i=1}^{n}\|\psi_i\|_1<1/2$, $\theta\in[0,1)$, some $1\leq j\leq n$ 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[(H5)] There exists a constant $c_0>0$ such that, if $|c|>c_0$, one of the following two conditions holds \begin{gather} c\sum_{i=1}^m\alpha_i\int_{\xi_i}^1 \Big[f\Big(s,c\Big(\sum_{i=k}^{n-1}\frac{A_i}{i!}s^i\Big), c \Big(\sum_{i=k}^{n-1}\frac{A_i}{(i-1)!}s^{i-1}\Big),\dots, c\Big)+\varepsilon(s)\Big]ds>0,\label{e3.4} \\ c\sum_{i=1}^m\alpha_i\int_{\xi_i}^1 \Big[f\Big(s,c\Big(\sum_{i=k}^{n-1}\frac{A_i}{i!}s^i\Big), c\Big(\sum_{i=k}^{n-1}\frac{A_i}{(i-1)!}s^{i-1}\Big),\dots, c\Big)+\varepsilon(s)\Big]ds<0.\label{e3.5} \end{gather} \end{itemize} \begin{lemma} \label{lem3.5} Assume {\rm (H1)--(H4)}. Then the set $$\Omega_1=\big\{y\in \operatorname{dom}L\setminus \ker L: Ly=\lambda Ny,\;\lambda\in (0,1)\big\} $$ is bounded. \end{lemma} \begin{proof} Take $y\in \Omega_1$. Since $Ny\in \operatorname{Im}L$, we have \begin{equation} \sum_{i=1}^{m}\alpha_i\int _{\xi_i}^1\left[f(s,y(s),y'(s), \dots,y^{(n-1)}(s))+\varepsilon(s)\right]ds=0.\label{e3.6} \end{equation} Since $Ly=\lambda Ny$ and $y\in \operatorname{dom}L$, it follows that \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}\left[f(s,y(s),y'(s),\dots,y^{(n-1)}(s)) +\varepsilon(s)\right]ds,\label{e3.7} \end{equation} where $c_i$, $i=k,k+1,\dots,n-1$ 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} \big[f(s,y(s),y'(s),\dots,y^{(n-1)}(s))+\varepsilon(s)\big]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} \big[f(s,y(s),y'(s),\dots,y^{(n-1)}(s))+\varepsilon(s)\big]ds\\ \dots\\ \begin{aligned} \sum_{i=k}^{n-1}\frac{c_i}{[i-(n-k-2)]!} &=-\frac{(-1)^{n-k}}{[n-1-(n-k-2)]!}\lambda\int_0^1(1-s)^{k+1}\\ &\quad \times \big[f(s,y(s),y'(s),\dots,y^{(n-1)}(s))+\varepsilon(s)\big]ds. \end{aligned} \end{gather*} It follows from $y^{(i)}(0)=y^{(j)}(1)=0$, $0\leq i\leq k-1$, $0\leq j\leq n-k-2$ that there exists at least one point $\delta_i\in [0,1]$ such that $y^{(i)}(\delta_i)=0$, $i=0,1,\dots,n-2$. So, we have $$ y^{(i)}(t)=\int_{\delta_i}^ty^{(i+1)}(s)ds,\quad i=0,1,\dots,n-2. $$ Therefore, \begin{equation} \|y^{(i)}\|_\infty\leq \|y^{(i+1)}\|_1\leq \|y^{(i+1)}\|_\infty,\quad i=0,1,\dots,n-2.\label{e3.8} \end{equation} By \eqref{e3.6} and (H3), there exists $t_0\in [\xi_m,1]$ such that $|y^{(n-1)}(t_0)|\leq M$. This, together with \eqref{e3.7}, implies \begin{equation} |c_{n-1}|\leq M+\int_0^1\left|f(s,y(s),y'(s),\dots,y^{(n-1)}(s))\right|ds +\|\varepsilon\|_1.\label{e3.9} \end{equation} It follows from \eqref{e3.7}-\eqref{e3.9} and (H4) that \begin{align*} \|y^{(n-1)}\|_\infty &\leq M+2\int_0^1\big|f(s,y(s),y'(s),\dots,y^{(n-1)}(s))\big|ds +2\|\varepsilon\|_1\\ &\leq M+2[\|g\|_1+\sum_{i=1}^{n}\|\psi_i\|_1\|y^{(i-1)}\|_\infty +\|h\|_1\|y^{(j-1)}\|_\infty^\theta]+2\|\varepsilon\|_1\\ &\leq M+2\|g\|_1+2\sum_{i=1}^{n}\|\psi_i\|_1\|y^{(n-1)}\|_\infty +2\|h\|_1\|y^{(n-1)}\|_\infty^\theta+2\|\varepsilon\|_1. \end{align*} So, we obtain $$ \|y^{(n-1)}\|_\infty \leq\frac{M+2\|g\|_1+2\|\varepsilon\|_1}{1-2\sum_{i=1}^{n}\|\psi_i\|_1}+ \frac{2\|h\|_1}{1-2\sum_{i=1}^{n}\|\psi_i\|_1}\|y^{(n-1)}\|_\infty^\theta. $$ Then $\theta\in [0,1)$ implies that $\{\|y^{(n-1)}\|_\infty|: y\in \Omega_1\}$ is bounded. Considering of \eqref{e3.8}, we obtain that $\Omega_1$ is bounded. \end{proof} \begin{lemma} \label{lem3.6} Assume {\rm (H1), (H2), (H5)}. Then the set $$ \Omega_2=\{y:y\in \ker L,\;Ny\in \operatorname{Im}L\} $$ is bounded. \end{lemma} \begin{proof} Take $y\in \Omega_2$, then $y(t)= c\big(\sum_{i=k}^{n-1}\frac{A_i}{i!}t^i\big)$, $c\in \mathbb{R}$ and $Ny\in \operatorname{Im}L$. So, we have $$ c \sum_{i=1}^m\alpha_i\int_{\xi_i}^1\Big[f\Big(s,c\Big(\sum_{i=k}^{n-1} \frac{A_i}{i!}s^i\Big),c\Big(\sum_{i=k}^{n-1}\frac{A_i}{(i-1)!}s^{i-1}\Big), \dots,c\Big) +\varepsilon(s)\Big]ds=0. $$ By (H5), we obtain that $|c|\leq c_0$. So, $\Omega_2$ is bounded. \end{proof} \begin{lemma} \label{lem3.7} Assume {\rm (H1), (H2), (H5)}. Then the set $$ \Omega_3=\{y\in \ker L:\lambda Jy+(1-\lambda)\theta QNy=0,\,\lambda\in [0,1]\} $$ is bounded, where $J:\ker L\to \operatorname{Im}Q$ is a linear isomorphism given by $$ J\Big(c\sum_{i=k}^{n-1}\frac{A_i}{i!}t^i\Big) = \frac{c}{1-\sum_{i=1}^{m}\alpha_i\xi_i} ,\quad c \in \mathbb{R} $$ and $\theta=\begin{cases} 1 & \text{if \eqref{e3.4} holds},\\ -1,&\text{if \eqref{e3.5} holds}. \end{cases}$. \end{lemma} \begin{proof} For $y\in \Omega_3$, we get $y=c\big(\sum_{i=k}^{n-1}\frac{A_i}{i!}t^i\big)$ with $$ \lambda c+(1-\lambda)\theta \sum_{i=1}^m\alpha_i\int_{\xi_i}^1 \Big[f\Big(s,c\Big(\sum_{i=k}^{n-1}\frac{A_i}{i!}s^i\Big), c\Big(\sum_{i=k}^{n-1}\frac{A_i}{(i-1)!}s^{i-1}\Big),\dots,c\Big) +\varepsilon(s)\Big]ds=0. $$ If $\lambda=0$, by (H5), we get $|c|\leq c_0$. If $\lambda=1$, $c=0$. For $\lambda\in (0,1)$, if $|c|\geq c_0$, then \begin{align*} \lambda c^2&=-(1-\lambda)\theta c\sum_{i=1}^m\alpha_i\int_{\xi_i}^1 \Big[f\Big(s,c\Big(\sum_{i=k}^{n-1}\frac{A_i}{i!}s^i\Big), c\Big(\sum_{i=k}^{n-1}\frac{A_i}{(i-1)!}s^{i-1}\Big),\dots, c\Big) \\ &\quad +\varepsilon(s)\Big]ds<0. \end{align*} This is a contradiction. So, $\Omega_3$ is bounded. \end{proof} \begin{theorem} \label{thm3.1} Assume {\rm (H1)--(H5)} Then problem \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{lem3.4} that $N$ is $L-$compact on $\overline{\Omega}$. By Lemmas \ref{lem3.5} and \ref{lem3.6}, we obtain: (1) $Ly\neq \lambda Ny$ for every $(y,\lambda)\in [(\operatorname{dom}L\setminus \ker L)\cap\partial\Omega]\times(0,~1)$; and (2) $Ny\notin \operatorname{Im}L$ for every $y\in \ker L\cap\partial\Omega$. We need to prove only (3) $\deg(QN|_{\ker L},~\Omega\cap \ker L,~0)\neq0$. To do this, we take $$ H(y,\lambda)=\lambda Jy+\theta(1-\lambda)QNy. $$ According to Lemma \ref{lem3.7}, we know $H(y,\lambda)\neq 0$ for $y\in \partial\Omega\cap \ker L$. By the homotopy of degree, we obtain \begin{align*} \deg(QN|_{\ker L}, \Omega \cap \ker L,0) &=\deg(\theta H(\cdot,0),\Omega \cap \ker L,0)\\ &=\deg(\theta H(\cdot,1),\Omega \cap \ker L,0)\\ &=\deg(\theta J, \Omega \cap \ker L,0)\neq 0. \end{align*} By Theorem \ref{thm2.1}, we 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} This research was supported by grants 11171088 from the Natural Science Foundation of China, QD201020 from the Doctoral Program Foundation of Hebei University of Science and Technology, and XL201136 from the the Foundation of Hebei University of Science and Technology. \begin{thebibliography}{10} \bibitem{1} 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{2} 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{3} R. P. Agarwal, F. H. Wong; \emph{Existence of solutions to (k,n-k-2) boundary-value problems}, Appl. Math. Comput. 104 (1999), 33-50. \bibitem{4} R. P. Agarwal, S. R. Grace, D. O'Regan; \emph{Semipositone higher-order differential equations}, Appl. Math. Lett. 17 (2004), 201-207. \bibitem{5} 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{6} 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{7} P. W. Eloe, J. Henderson; \emph{Singular nonlinear (k,n-k) conjugate boundary-value problems}, J. Differential Equations 133 (1997), 136-151. \bibitem{8} P. W. Eloe, J. Henderson; \emph{Positive solutions for (n-1,1) conjugate boundary-value problems}, Nonlinear Anal. 30 (1997), 3227-3238. \bibitem{9} 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{10} C. P. Gupta; \emph{Solvability of multi-point boundary-value problem at resonance}, Results Math. 28 (1995), 270-276. \bibitem{11} X. He, W. Ge; \emph{Positive solutions for semipositone (p, n-p) right focal boundary-value problems}, Appl. Anal. 81 (2002), 227-240. \bibitem{12} W. Jiang, J. Zhang; \emph{Positive solutions for (k, n-k) conjugate eigenvalue problems in Banach spaces}, Nonlinear Anal. 71 (2009), 723-729. \bibitem{13} D. Jiang; \emph{Positive solutions to singular (k,n-k) conjugate boundary-value problems} (in Chinese), Acta Mathematica Sinica, 44 (2001), 541-548. \bibitem{14} 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{15} G. L. Karakostas, P. Ch. Tsamatos; \emph{On a Nonlocal Boundary Value Problem at Resonance}, J. Math. Anal. Appl. 259 (2001), 209-218. \bibitem{16} N. Kosmatov; \emph{Multi-point boundary-value problems on an unbounded domain at resonance}, Nonlinear Anal., 68 (2008), 2158-2171. \bibitem{17} K. Q. Lan; \emph{Multiple positive solutions of conjugate boundary-value problems with singularities}, Appl. Math. Comput. 147 (2004), 461-474. \bibitem{18} 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{19} 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{20} B. Liu; \emph{Solvability of multi-point boundary-value problem at resonance (II)}, Appl. Math. Comput. 136 (2003), 353-377. \bibitem{21} 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{22} R. Ma; \emph{Positive solutions for semipositone (k, n-k) conjugate boundary-value problems}, J. Math. Anal. Appl. 252 (2000), 220-229. \bibitem{23} R. Ma; \emph{Existence results of a $m$-point boundary-value problem at resonance}, J. Math. Anal. Appl. 294 (2004), 147-157. \bibitem{24} 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{25} B. Prezeradzki, R. Stanczy; \emph{Solvability of a multi-point boundary-value problem at resonance}, J. Math. Anal. Appl. 264 (2001), 253-261. \bibitem{26} H. Su, Z. Wei; \emph{Positive solutions to semipositone (k, n-k) conjugate eigenvalue problems}, Nonlinear Anal. 69 (2008), 3190-3201. \bibitem{27} P. J. Y. Wong, R. P. Agarwal; \emph{Singular differential equation with (n,p) boundary conditions}, Math. Comput. Modelling, 28 (1998), 37-44. \bibitem{28} J. R. L. Webb; \emph{Nonlocal conjugate type boundary-value problems of higher order}, Nonlinear Anal. 71 (2009), 1933-1940. \bibitem{29} 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{30} 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{31} X. Yang; \emph{Green's function and positive solutions for higher-order ODE}, Appl. Math. Comput. 136 (2003), 379-393. \bibitem{32} 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{33} G. Zhang, J. Sun; \emph{Positive solutions to singular (k, n-k) multi-point boundary-value problems} (in Chnise), Acta Mathematica Sinica, 49 (2006), 391-398. \bibitem{34} 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}