\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations, {\em Electronic Journal of Differential Equations}, Conf. 17 (2009), pp. 71--80.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{71} \title[\hfilneg EJDE-2009/Conf/17\hfil Three solutions for BVPs] {Existence of three solutions for a higher-order boundary-value problem} \author[J. R. Graef, L. Kong, and Q. Kong\hfil EJDE/Conf/17 \hfilneg] {John R. Graef, Lingju Kong, Qingkai Kong} % in alphabetical order \address{John R. Graef \newline Department of Mathematics, the University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA} \email{John-Graef@utc.edu} \address{Lingju Kong \newline Department of Mathematics, the University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA} \email{Lingju-Kong@utc.edu} \address{Qingkai Kong \newline Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA} \email{kong@math.niu.edu} \thanks{Published April 15, 2009.} \subjclass[2000]{34B15, 34B18} \keywords{Solutions; boundary value problems; lower and upper solutions; \hfill\break\indent Nagumo condition; degree theory} \begin{abstract} We consider a higher-order multi-point boundary-value problem with a nonlinear boundary condition. Sufficient conditions are obtained for the existence of three solutions. In our problem, the differential equation has dependence on all lower order derivatives of the unknown function and the boundary condition covers many multi-point boundary conditions studied earlier by other authors. Our results extend some recent work in the literature. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this paper, we are concerned with the existence of solutions of the $n$th order boundary value problem (BVP) consisting of the equation $$\label{1.1} u^{(n)}+f\big(t,u,u',\dots,u^{(n-1)}\big)=0,\quad t\in (0,1),$$ and the general multi-point boundary conditions (BC) $$\label{1.2} \begin{gathered} u^{(i)}(0)=g_i\big(u^{(i)}(t_1),\dots,u^{(i)}(t_m)\big),\quad i=0,\dots,n-2,\\ u^{(n-2)}(1)=g_{n-1}\big(u^{(n-2)}(t_1),\dots,u^{(n-2)}(t_m)\big), \end{gathered}$$ where $n\geq 2$ and $m\geq 1$ are integers, $t_j\in [0,1]$ for $j=1,\dots,m$ with $0\leq t_1< t_2<\dots0$, $0<\eta<1$ with $0<\xi\eta<1$, and discussed the existence of three solutions. Our work is an improvement and extension of the result in \cite{dll}. In fact, our BC \eqref{1.2} is much more general than BC \eqref{1.3}; even for the special case of BC \eqref{1.3}, our result is new and better since the restriction $0<\xi\eta<1$ is removed; i.e., our result works not only for the nonresonance case covered in \cite{dll} but also for the resonance case. In the next section, we present our main theorem together with an illustrative example. The proof of the main theorem is given a separate section. \section{Main Result} In the sequel, for any $u\in C[0,1]$, we define $\|u\|_{\infty}=\max_{t\in [0,1]}|u(t)|$. Let $$\|u\|=\max\{\|u\|_{\infty},\|u'\|_{\infty},\dots,\|u^{(n-1)}\|_{\infty}\}$$ and $$\|u\|_p=\begin{cases} (\int_0^1|u(t)|^pdt)^{1/p},& 1\leq p<\infty,\\ \inf\{M : \text{meas}\{t : |u(t)|>M\}=0\},& p=\infty, \end{cases}$$ stand for the norms in $C^{n-1}[0,1]$ and $L^p(0,1)$, respectively, where $\text{mess}\{\cdot\}$ denotes the Lebesgue measure of a set. We first define strict lower and upper solutions of \eqref{1.1}, \eqref{1.2} and a Nagumo condition. \begin{definition}\label{d2.1} \rm A function $\alpha\in C^{n-1}[0,1]\cap C^{n}(0,1)$ is said to be a strict lower solution of \eqref{1.1}, \eqref{1.2} if $$\label{2.1} \alpha^{(n)}(t)+f\big(t,\alpha(t),\alpha'(t),\dots,\alpha^{(n-1)}(t)\big)> 0\quad \text{on } (0,1),$$ and $$\label{2.2} \begin{gathered} \alpha^{(i)}(0)< g_{i}\big(\alpha^{(i)}(t_1),\dots,\alpha^{(i)}(t_m)\big),\quad i=0,\dots,n-2,\\ \alpha^{(n-2)}(1) < g_{n-1}\big(\alpha^{(n-2)}(t_1),\dots,\alpha^{(n-2)}(t_m)\big). \end{gathered}$$ A function $\beta\in C^{n-1}[0,1]\cap C^{n}(0,1)$ is said to be a strict upper solution of \eqref{1.1}, \eqref{1.2} if $$\label{2.3} \beta^{(n)}(t)+f\big(t,\beta(t),\beta'(t),\dots,\beta^{(n-1)}(t)\big)< 0\quad \text{on } (0,1),$$ and $$\label{2.4} \begin{gathered} \beta^{(i)}(0)> g_{i}\big(\beta^{(i)}(t_1),\dots,\beta^{(i)}(t_m)\big),\quad i=0,\dots,n-2,\\ \beta^{(n-2)}(1)> g_{n-1}\big(\beta^{(n-2)}(t_1),\dots,\beta^{(n-2)}(t_m)\big). \end{gathered}$$ \end{definition} \begin{definition}\label{d2.2} \rm Let $\alpha$, $\beta\in C^{n-1}[0,1]$ satisfy $$\label{2.5} \alpha^{(i)}(t)\leq \beta^{(i)}(t)\quad \text{for }t\in [0,1]\text{ and } i=0,\dots,n-2.$$ We say that $f$ satisfies a Nagumo condition with respect to $\alpha$ and $\beta$ if for $$\label{2.6} \xi=\max\big\{\beta^{(n-2)}(1)-\alpha^{(n-2)}(0),\ \beta^{(n-2)}(0)-\alpha^{(n-2)}(1)\big\},$$ there exist a constant $C=C(\alpha,\beta)$ with $$\label{2.7} C> \max\big\{\xi,\ \|\alpha^{(n-1)}\|_{\infty},\ \|\beta^{(n-1)}\|_{\infty}\big\}$$ and functions $\phi\in C[0,\infty)$ and $w\in L^p(0,1)$, $1\leq p\leq\infty$, such that $\phi>0$ on $[0,\infty)$, $$\label{2.8} |f(t,x_0,\dots,x_{n-1})|\leq w(t)\phi(|x_{n-1}|)\quad \text{on } (0,1)\times\prod_{i=0}^{n-2}[\alpha^{(i)}(t),\beta^{(i)}(t)]\times\mathbb{R},$$ and $$\label{2.9} \int_{\xi}^C\frac{v^{(p-1)/p}}{\phi(v)}dv>\|w\|_p\eta^{(p-1)/p},$$ where $(p-1)/p\equiv 1$ for $p=\infty$ and $$\label{2.10} \eta=\max_{t\in [0,1]}\beta^{(n-2)}(t)-\min_{t\in [0,1]}\alpha^{(n-2)}(t).$$ \end{definition} \begin{remark}\label{r2.1} \rm Let $\alpha$, $\beta\in C^{n-1}[0,1]$ satisfy \eqref{2.5}. Assume that there exist $w\in L^p(0,1)$, $1\leq p\leq\infty$, and $0\leq\sigma\leq 1+(p-1)/p$ such that $$\label{2.11} |f(t,x_0,\dots,x_{n-1})|\leq w(t)(1+|x_{n-1}|^{\sigma})\quad \text{on } (0,1)\times\prod_{i=0}^{n-2}[\alpha^{(i)}(t),\beta^{(i)}(t)]\times\mathbb{R}.$$ Then $f$ satisfies a Nagumo condition with respect to $\alpha$ and $\beta$ with $\phi(v)=1+v^{\sigma}$. \end{remark} Now, we present the main result of this paper. \begin{theorem}\label{t2.1} Assume that the following conditions hold: \begin{itemize} \item[(H1)] BVP \eqref{1.1}, \eqref{1.2} has two strict lower solutions $\alpha_1$ and $\alpha_2$ and two strict upper solutions $\beta_1$ and $\beta_2$ satisfying \begin{equation*} \alpha_1^{(i)}(t)\leq\alpha_2^{(i)}(t)\leq\beta_2^{(i)}(t),\ \alpha_1^{(i)}(t)\leq\beta_1^{(i)}(t)\leq\beta_2^{(i)}(t),\ \text{and}\ \alpha_2^{(i)}(t)\not\le\beta_1^{(i)}(t) \end{equation*} for $t\in [0,1]$ and $i=0,\dots, n-2$; \item[(H2)] for $(t,x_0,\dots,x_{n-1})\in (0,1)\times\prod_{i=0}^{n-3}[\alpha^{(i)}(t),\beta^{(i)}(t)]\times\mathbb{R}^2$, $f(t,x_0,\dots,x_{n-1})$ is nondecreasing in each of the variables $x_0,\dots,x_{n-3}$; \item[(H3)] $f$ satisfies a Nagumo condition with respect to $\alpha_1$ and $\beta_2$ with $C=C(\alpha_1, \beta_2)$ being the constant given in Definition \ref{d2.2}; \item[(H4)] for $i=1,\dots,n-1$ and $(y_1,\dots,y_m)\in\mathbb{R}^m$, $g_i(y_1,\dots,y_m)$ is nondecreasing in each of its arguments. \end{itemize} Then \eqref{1.1}, \eqref{1.2} has at least three solutions $u_1(t)$, $u_2(t)$, and $u_3(t)$ satisfying $$\label{2.12} \alpha_j^{(i)}(t)\leq u_j^{(i)}(t)\leq\beta_j^{(i)}(t)\quad \text{for }t\in [0,1],\ i=0,\dots,n-2,\ \text{and}\ j=1,2,$$ and $$\label{2.13} \alpha_1^{(i)}(t)\leq u_3^{(i)}(t)\leq\beta_2^{(i)}(t),\ u_3^{(i)}(t)\not\le\beta_1^{(i)}(t),\ \text{and}\ u_3^{(i)}(t)\not\ge\alpha_2^{(i)}(t)$$ for $t\in [0,1]$ and $i=0,\dots,n-2$. \end{theorem} \begin{remark}\label{r2.2} \rm Notice that in (H2) we do not need the monotonicity of $f$ in the last two variables $x_{n-2}$ and $x_{n-1}$. In particular, for the case when $n=2$, no monotonicity is required on $f$. \end{remark} In the remainder of this section, we provide the following example to illustrate Theorem \ref{t2.1}. To the best of our knowledge, no existing criteria can be applied to this example. \medskip \noindent{\bf Example.} Consider the BVP consisting of the equation $$\label{2.38} u'''+t^{-1/2}h(u')+(u'')^2+1=0,\quad t\in (0,1),$$ and the BC $$\label{2.39} u(0)=u^{1/3}(1/2)+1,\quad u'(0)=u'(1)=u^{r}(1/2),$$ where $h\in C(\mathbb{R})$ satisfies $$\label{2.40} \begin{gathered} h(y)\geq 8\quad \text{for } y\in [-9,-8]\cup [2,3],\\ h(y)\leq -26\quad \text{for } y\in [-3,-2]\cup [8,9], \end{gathered}$$ and $r=a/b\in (\ln 2/\ln 3, \ln 9/\ln 8)$ with $a,b$ odd numbers. Clearly, the function $g(x)=x^r$ is nondecreasing and odd on $\mathbb{R}$. Let $$\label{2.42} \begin{gathered} \alpha_1(t)=-4t^3/3+2t^2-9t-2,\\ \alpha_2(t)=-4t^3/3+2t^2+2t+1,\\ \beta_1(t)=4t^3/3-2t^2-2t+2,\\ \beta_2(t)=4t^3/3-2t^2+9t+4. \end{gathered}$$ We claim that \eqref{2.38}, \eqref{2.39} has at least three solutions $u_1(t)$, $u_2(t)$, and $u_3(t)$ satisfying \eqref{2.12} and \eqref{2.13} with the above $\alpha_1(t)$, $\alpha_2(t)$, $\beta_1(t)$, and $\beta_2(t)$. In fact, with $n=3$, $m=1$, $t_1=1/2$, $f(t,x_0,x_1,x_2)=t^{-1/2}h(x_1)+x_2^2+1$, $g_0(x)=x^{1/3}+1$, and $g_1(x)=g_2(x)=x^r$, we see that BVP \eqref{2.38}, \eqref{2.39} is of the form of \eqref{1.1}, \eqref{1.2}. Clearly, (H2) and (H4) hold. From \eqref{2.42}, we have that for $t\in [0,1]$ $$\label{2.43} \begin{gathered} -9\leq \alpha_1'(t)=-4t^2+4t-9\leq -8,\\ 2\leq \alpha_2'(t)=-4t^2+4t+2\leq 3,\\ -3\leq \beta_1'(t)=4t^2-4t-2\leq -2,\\ 8\leq \beta_2'(t)=4t^2-4t+9\leq 9, \end{gathered}$$ and $$\label{2.44} \begin{gathered} -4\leq \alpha_1''(t)=-8t+4\leq 4,\\ -4\leq \alpha_2''(t)=-8t+4\leq 4,\\ -4\leq \beta_1''(t)=8t-4\leq 4,\\ -4\leq \beta_2''(t)=8t-4\leq 4. \end{gathered}$$ It follows from \eqref{2.43} and \eqref{2.44} that \begin{equation*} \alpha_1^{(i)}(t)\leq\alpha_2^{(i)}(t)\leq\beta_2^{(i)}(t),\ \alpha_1^{(i)}(t)\leq\beta_1^{(i)}(t)\leq\beta_2^{(i)}(t),\ \text{and}\ \alpha_2^{(i)}(t)\not\le\beta_1^{(i)}(t) \end{equation*} for $t\in [0,1]$ and $i=0, 1$. Moreover, from \eqref{2.40}--\eqref{2.44}, it is easy to verify that $\alpha_1(t)$ and $\alpha_2(t)$ are strict lower solutions of \eqref{2.38}, \eqref{2.39} and $\beta_1(t)$ and $\beta_2(t)$ are strict upper solutions of \eqref{2.38}, \eqref{2.39}. Hence, (H1) holds. In view of \eqref{2.43}, we see that \begin{equation*} |f(t,x_0,x_1,x_2)|\leq (1+t^{-1/2}\max_{y\in [-9,9]}|h(y)|)(1+x_2^2) \end{equation*} on $(0,1)\times [\alpha_1(t),\beta_2(t)]\times [\alpha_1'(t),\beta_2'(t)]$. Then, by Remark \ref{r2.1}, (H3) holds. The conclusion now follows from Theorem \ref{t2.1}. \begin{remark}\label{r2.3} \rm One example of a continuous function $h$ satisfying \eqref{2.40} is \begin{equation*} h(y)=\begin{cases} -34y/5-232/5, & y\in (-\infty,0),\\ 136y/5-232/5, & y\in [0,3],\\ -306y/25+1798/25, & y\in (3,\infty). \end{cases} \end{equation*} \end{remark} \section{Proof of the Main Result} In this section, we give a proof to Theorem \ref{t2.1}. Assume (H1)--(H4) hold. Let $\alpha$ and $\beta$ be strict lower and upper solutions of \eqref{1.1}, \eqref{1.2}, respectively, satisfying \eqref{2.5}. Let $C=C(\alpha,\beta)$ be given in Definition \ref{d2.2} and $f$ satisfy a Nagumo condition with respect to $\alpha$ and $\beta$. For $u\in C^{n-1}[0,1]$, define $$\label{2.14} \tilde{u}^{[i]}(\alpha,\beta)(t)=\max\big\{\alpha^{(i)}(t), \min\big\{u^{(i)}(t),\ \beta^{(i)}(t)\big\}\big\},\quad i=0,\dots,n-2$$ and $$\label{2.15} \tilde{u}^{[n-1]}(\alpha,\beta)(t)=\max\big\{-C(\alpha,\beta),\ \min\big\{u^{(n-1)}(t),\ C(\alpha,\beta)\big\}\big\}.$$ Then, for $i=0,\dots,n-1$, $\tilde{u}^{[i]}(\alpha,\beta)(t)$ is continuous on $[0,1]$, $$\label{2.16} \begin{gathered} \tilde{\alpha}^{[i]}(\alpha,\beta)(t)=\alpha^{(i)}(t),\ \tilde{\beta}^{[i]}(\alpha,\beta)(t)=\beta^{(i)}(t),\\ \alpha^{(i)}(t)\leq \tilde{u}^{[i]}(\alpha,\beta)(t)\leq \beta^{(i)}(t) \end{gathered}$$ for $t\in [0,1]$ and $i=0,\dots,n-2$, and $$\label{2.17} -C(\alpha,\beta)\leq \tilde{u}^{[n-1]}(\alpha,\beta)(t)\leq C(\alpha,\beta)\quad \text{on } [0,1].$$ Define a functional $F_{\alpha,\beta}: (0,1)\times C^{n-1}[0,1]\to \mathbb{R}$ by \label{2.18} \begin{aligned} F_{\alpha,\beta}(t,u(\cdot)) &=f\left(t,\tilde{u}^{[0]}(\alpha,\beta)(t),\tilde{u}^{[1]}(\alpha,\beta)(t), \dots,\tilde{u}^{[n-1]}(\alpha,\beta)(t)\right) \\ &\quad +\frac{\tilde{u}^{[n-2]}(\alpha,\beta)(t)-u^{(n-2)}(t)}{1+(u^{(n-2)}(t))^2}. \end{aligned} Then, for $u\in C^{n-1}[0,1]$ and $t\in (0,1)$, $F_{\alpha,\beta}(t,u(\cdot))$ is continuous in $u$, and from \eqref{2.8}, \eqref{2.16}, and \eqref{2.17}, we see that $$\label{2.19} |F_{\alpha,\beta}(t,u(\cdot))|\leq w(t)\max_{y\in [0,C(\alpha,\beta)]}\phi(y)+\|\alpha\|+\|\beta\|+1.$$ Consider the BVP consisting of the equation $$\label{2.20} u^{(n)}+F_{\alpha,\beta}(t,u(\cdot))=0,\quad t\in (0,1),$$ and the BC $$\label{2.21} \begin{gathered} u^{(i)}(0)=g_i\left(\tilde{u}^{[i]}(\alpha,\beta)(t_1),\dots,\tilde{u}^{[i]}(\alpha,\beta) (t_m)\right),\quad i=0,\dots,n-2,\\ u^{(n-2)}(1)=g_{n-1}\left(\tilde{u}^{[n-2]}(\alpha,\beta)(t_1),\dots, \tilde{u}^{[n-2]}(\alpha,\beta)(t_m)\right). \end{gathered}$$ It is well known that the Green's function for the BVP \begin{equation*} -u''(t)=0\quad \text{on } (0,1),\quad u(0)=u(1)=0, \end{equation*} is \begin{equation*} G(t,s)=\begin{cases} t(1-s), & 0\leq t\leq s\leq 1,\\ s(1-t), & 0\leq s\leq t\leq 1. \end{cases} \end{equation*} Let $G_1(t,s)=G(t,s)$ and for $j=2,\dots,n-1$, recursively define $$\label{2.22} G_j(t,s)=\int_0^tG_{j-1}(v,s)dv.$$ Lemma \ref{l2.1} below is taken from \cite[Lemma 3.2]{gkk} and Lemma \ref{l2.2} follows from \cite[Lemmas 3.4 and 3.5]{gkk}. \begin{lemma}\label{l2.1} The function $u(t)$ is a solution of \eqref{2.20}, \eqref{2.21} if and only if $u(t)$ is a solution of the integral equation \begin{equation*} u(t)=\sum_{i=0}^{n-1}g_i\big(\tilde{u}^{[i]}(\alpha,\beta)(t_1), \dots,\tilde{u}^{[i]}(\alpha,\beta)(t_m)\big)p_i(t) +\int_0^1G_{n-1}(t,s)F_{\alpha,\beta}(s,u(\cdot))ds, \end{equation*} where \begin{gather*} p_i(t)=\frac{t^i}{i!},\quad i=0,\dots,n-3, \\ p_{n-2}(t)=\frac{t^{n-2}}{(n-2)!}-\frac{t^{n-1}}{(n-1)!}, \\ p_{n-1}(t)=\frac{t^{n-1}}{(n-1)!}, \end{gather*} and $G_{n-1}(t,s)$ is given by \eqref{2.22} with $j=n-1$. \end{lemma} \begin{lemma}\label{l2.2} If $u(t)$ is a solution of \eqref{2.20}, \eqref{2.21}, then $u(t)$ satisfies $$\label{2.23} \alpha^{(i)}(t)\leq u^{(i)}(t)\leq\beta^{(i)}(t)\quad \text{for }t\in [0,1] \text{ and } i=0,\dots,n-2,$$ and $$\label{2.24} |u^{(n-1)}(t)|\leq C(\alpha,\beta)\quad \text{for }t\in [0,1].$$ Consequently, $u(t)$ is a solution of \eqref{1.1}, \eqref{1.2}. \end{lemma} \begin{proof}[Proof of Theorem \ref{t2.1}] Let $F_{\alpha_1, \beta_2}$ be defined by \eqref{2.18} with $(\alpha, \beta)$ replaced by $(\alpha_1, \beta_2)$. Define an operator $T_{\alpha_1,\beta_2}: C^{n-1}[0,1]\to C[0,1]$ by \label{2.25} \begin{aligned} T_{\alpha_1,\beta_2}u(t) &=\sum_{i=0}^{n-1}g_i\left(\tilde{u}^{[i]}(\alpha_1,\beta_2)(t_1),\dots,\tilde{u}^{[i]}(\alpha_1,\beta_2)(t_m)\right)p_i(t)\\ &\quad+\int_0^1G_{n-1}(t,s)F_{\alpha_1,\beta_2}(s,u(\cdot))ds. \end{aligned} Then, by Lemma \ref{l2.1}, $u(t)$ is a solution of \eqref{2.20}, \eqref{2.21} with $(\alpha, \beta)=(\alpha_1, \beta_2)$ if and only if $u$ is a fixed point of $T_{\alpha_1,\beta_2}$. In the following, we show that $T_{\alpha_1,\beta_2}$ is compact. Clearly, $T_{\alpha_1,\beta_2}$ is continuous. Let $S\subseteq C^{n-1}[0,1]$ be a bounded set; we will show that $T_{\alpha_1,\beta_2}(S)$ is relatively compact. For $u\in S$, in view of \eqref{2.16} where $(\alpha,\beta)=(\alpha_1,\beta_2)$, there exists $d>0$ such that $$\label{2.26} |g_i\left(\tilde{u}^{[i]}(\alpha_1,\beta_2)(t_1),\dots,\tilde{u}^{[i]} (\alpha_1,\beta_2)(t_m)\right)|\leq d\quad \text{for }i=0,\dots,n-1.$$ From \eqref{2.22}, we see that $$\label{2.27} 0\leq G_j(t,s)\leq 1\quad \text{for }(t,s)\in [0,1]\times [0,1]\ \text{and}\ j=1,\dots,n-1.$$ For $p_i(t)$ defined in Lemma \ref{l2.1}, $\|p_i\|\leq 1$, $i=0,\dots,n-1$. From \eqref{2.19} with $(\alpha,\beta)=(\alpha_1,\beta_2)$ and \eqref{2.25}--\eqref{2.27}, we obtain \label{2.28} \begin{aligned} |(T_{\alpha_1,\beta_2}u)^{(j)}(t)| &\leq \sum_{i=0}^{n-1}|g_i\left(\tilde{u}^{[i]}(\alpha_1,\beta_2)(t_1), \dots,\tilde{u}^{[i]}(\alpha_1,\beta_2)(t_m)\right)|\ |p_i^{(j)}(t)|\\ &\quad +\int_0^1G_{n-1-j}(t,s)|F_{\alpha_1,\beta_2}(s,u(\cdot))|ds\\ &\leq nd+\int_0^1|F_{\alpha_1,\beta_2}(s,u(\cdot))|ds\\ &\leq nd+\max_{y\in [0,C(\alpha_1,\beta_2)]}\phi(y)\int_0^1w(s)ds+\|\alpha\|+\|\beta\|+1<\infty \end{aligned} for $j=0,\dots,n-2$, and \begin{align}\label{2.29} |(T_{\alpha_1,\beta_2}u)^{(n-1)}(t)| &\leq \sum_{i=0}^{n-1}|g_i\left(\tilde{u}^{[i]}(\alpha_1,\beta_2)(t_1), \dots,\tilde{u}^{[i]}(\alpha_1,\beta_2)(t_m)\right)|\ |p_i^{(j)}(t)|\nonumber\\ &\quad +\int_0^ts|F_{\alpha_1,\beta_2}(s,u(\cdot))|ds+ \int_t^1(1-s)|F_{\alpha_1,\beta_2}(s,u(\cdot))|ds\nonumber\\ &\leq nd+\int_0^1|F_{\alpha_1,\beta_2}(s,u(\cdot))|ds\nonumber\\ &\leq nd + \max_{y\in [0,C(\alpha_1,\beta_2)]}\phi(y)\int_0^1w(s)ds+\|\alpha\|+\|\beta\|+1<\infty. \end{align} Then, $T_{\alpha_1,\beta_2}$ is uniformly bounded on $S$ and $(T_{\alpha_1,\beta_2}u)^{(j)}(t)$ is equicontinuous on $[0,1]$ for $j=0,\dots,n-2$. Moreover, since \begin{align*} (T_{\alpha_1,\beta_2}u)^{(n-1)}(t) &= -g_{n-2}\left(\tilde{u}^{[n-2]}(\alpha_1,\beta_2)(t_1),\dots,\tilde{u}^{[n-2]}(\alpha_1,\beta_2)(t_m)\right)\\ &\quad +g_{n-1}\left(\tilde{u}^{[n-2]}(\alpha_1,\beta_2)(t_1),\dots,\tilde{u}^{[n-2]}(\alpha_1,\beta_2)(t_m)\right)\\ &\quad +\int_t^1F_{\alpha_1,\beta_2}(s,u(\cdot))ds-\int_0^1sF_{\alpha_1,\beta_2}(s,u(\cdot))ds, \end{align*} the equicontinuity of $(T_{\alpha_1,\beta_2}u)^{(n-1)}(t)$ follows from the absolute continuity of the integrals. Thus, by the Arzel\`{a}-Ascola theorem, $T_{\alpha_1,\beta_2}$ is compact. Let $M$ be large enough so that $M>\max\big\{C(\alpha_1,\beta_2),\ nd+\max_{y\in [0,C(\alpha_1,\beta_2)]}\phi(y)\int_0^1w(s)ds+\|\alpha\|+\|\beta\|+1\big\}.$ Define \begin{equation*} \Omega=\{u\in C^{n-1}[0,1] : \|u\|\alpha_2^{(i)}(t) \text{ for }t\in [0,1] \text{ and } i=0,\dots,n-2\} \end{equation*} and $\Omega_{\beta_1}=\{u\in \Omega:u^{(i)}(t)<\beta_1^{(i)}(t) \text{ for }t\in [0,1] \text{ and } i=0,\dots,n-2\}.$ Since $\alpha_2^{(i)}(t)\not\le\beta_1^{(i)}(t)$, $\alpha_2^{(i)}(t)\geq \alpha_1^{(i)}(t)>-M$, and $\beta_1^{(i)}(t)\leq \beta_2^{(i)}(t)\alpha_2^{(i)}(t)\quad \text{for }t\in [0,1]\ \text{and}\ i=0,\dots,n-2; \item[(ii)] if$u(t)$is a solution of BVP \eqref{2.20}, \eqref{2.21} with$(\alpha,\beta)=(\alpha_1,\beta_2)$and satisfies \begin{equation*} u^{(i)}(t)\leq\beta_1^{(i)}(t)\quad \text{for }t\in [0,1]\ \text{and}\ i=0,\dots,n-2, \end{equation*} then we have the strict inequalities \begin{equation*} u^{(i)}(t)<\beta_1^{(i)}(t)\quad \text{for }t\in [0,1]\ \text{and}\ i=0,\dots,n-2. \end{equation*} \end{itemize} We first prove (i). By Lemma \ref{l2.2},$u(t)$is a solution of \eqref{1.1}, \eqref{1.2} satisfying \eqref{2.23} where$(\alpha,\beta)=(\alpha_1,\beta_2). Then, from \eqref{1.2}, \eqref{2.2}, (H4), and \eqref{2.31}, we have \begin{align*} \alpha^{(i)}_2(0) &< g_{i}\big(\alpha^{(i)}_2(t_1),\dots,\alpha^{(i)}_2(t_m)\big)\\ &\leq g_{i}\big(u^{(i)}(t_1),\dots,u^{(i)}(t_m)\big)=u^{(i)}(0),\quad i=0,\dots,n-2, \end{align*} and \begin{align*} \alpha^{(n-2)}_2(1) &< g_{n-1} \big(\alpha^{(n-2)}_2(t_1),\dots,\alpha^{(n-2)}_2(t_m)\big)\\ &\leq g_{n-1}\big(u^{(n-2)}(t_1),\dots,u^{(n-2)}(t_m)\big)=u^{(n-2)}(1); \end{align*} i.e., $$\label{2.33} u^{(i)}(0)>\alpha^{(i)}_2(0),\ i=0,\dots,n-2,\ u^{(n-2)}(1)>\alpha^{(n-2)}_2(1).$$ We now show that $$\label{2.34} u^{(n-2)}(t)>\alpha_2^{(n-2)}(t)\quad \text{for }t\in [0,1].$$ If \eqref{2.34} does not hold, then, in view of \eqref{2.31} and \eqref{2.33} withi=n-2$, there exists$t^*\in (0,1)$such that$u^{(n-2)}(t)-\alpha_2^{(n-2)}(t)$has the minimum value$0$at$t^*$. Thus,$u^{(n-2)}(t^*)=\alpha_2^{(n-2)}(t^*)$,$u^{(n-1)}(t^*)=\alpha_2^{(n-1)}(t^*)$, and$u^{(n)}(t^*)\geq \alpha_2^{(n)}(t^*). On the other hand, from \eqref{1.1}, \eqref{2.1}, (H2), and \eqref{2.31}, we obtain that \begin{align*} u^{(n)}(t^*) &= -f\big(t^*,u(t^*),u'(t^*),\dots,u^{(n-1)}(t^*)\big)\\ &\leq -f\big(t^*,\alpha_2(t^*),\alpha_2'(t^*),\dots,\alpha_2^{(n-1)}(t^*)\big)\\ &< \alpha_2^{(n)}(t), \end{align*} which is a contradiction. Thus, \eqref{2.34} holds. Integrating \eqref{2.34} and using \eqref{2.33}, we see thatu(t)$satisfies \eqref{2.32}. The proof for (ii) is similar and hence is omitted. Now, by the claim (see \eqref{2.31}--\eqref{2.32}), BVP \eqref{2.20}, \eqref{2.21} has no solution on$\partial\Omega_{\alpha_2}\cup\partial\Omega_{\beta_1}. Hence, \label{2.35} \begin{aligned} \deg(I-T_{\alpha_1,\beta_2}, \Omega, 0) &=\deg(I-T_{\alpha_1,\beta_2}, \Omega\setminus\{\overline{\Omega_{\alpha_2}\cup\Omega_{\beta_1}}\}, 0) \\ &\quad +\deg(I-T_{\alpha_1,\beta_2}, \Omega_{\alpha_2}, 0)+\deg(I-T_{\alpha_1,\beta_2}, \Omega_{\beta_1}, 0). \end{aligned} Next, we show that $$\label{2.36} \deg(I-T_{\alpha_1,\beta_2}, \Omega_{\alpha_2}, 0)=\deg(I-T_{\alpha_1,\beta_2}, \Omega_{\beta_1}, 0)=1.$$ LetF_{\alpha_2, \beta_2}$be defined by \eqref{2.18} with$(\alpha, \beta)$replaced by$(\alpha_2, \beta_2)$. Define an operator$T_{\alpha_2,\beta_2}: C^{n-1}[0,1]\to C[0,1]by \begin{align*} T_{\alpha_2,\beta_2}u(t) &=\sum_{i=0}^{n-1}g_i\left(\tilde{u}^{[i]}(\alpha_2,\beta_2)(t_1),\dots,\tilde{u}^{[i]}(\alpha_2,\beta_2)(t_m)\right)p_i(t)\nonumber\\ &\quad +\int_0^1G_{n-1}(t,s)F_{\alpha_2,\beta_2}(s,u(\cdot))ds. \end{align*} Then, by Lemma \ref{l2.1},u(t)$is a solution of \eqref{2.20}, \eqref{2.21} with$(\alpha, \beta)=(\alpha_2, \beta_2)$if and only if$u$is a fixed point of$T_{\alpha_2,\beta_2}$. It can also be shown that$T_{\alpha_2,\beta_2}$is compact. Arguing as before, it follows that$u(t)$is a solution of \eqref{2.20}, \eqref{2.21} with$(\alpha, \beta)=(\alpha_2, \beta_2)$only if$u\in\Omega_{\alpha_2}$. Then, \begin{equation*} \deg(I-T_{\alpha_2,\beta_2}, \Omega\setminus\overline{\Omega}_{\alpha_2}, 0)=0. \end{equation*} Moreover, as in \eqref{2.28} and \eqref{2.29}, it is easy to see that$T_{\alpha_2,\beta_2}(\overline{\Omega})\subseteq\Omega, which in turn implies that \begin{equation*} \deg(I-T_{\alpha_2,\beta_2}, \Omega, 0)=1. \end{equation*} Then, \begin{align*} \deg(I-T_{\alpha_1,\beta_2}, \Omega_{\alpha_2}, 0) &= \deg(I-T_{\alpha_2,\beta_2}, \Omega_{\alpha_2}, 0)\\ &= \deg(I-T_{\alpha_2,\beta_2}, \Omega\setminus\overline{\Omega}_{\alpha_2}, 0)+\deg(I-T_{\alpha_2,\beta_2}, \Omega_{\alpha_2}, 0)\\ &= \deg(I-T_{\alpha_2,\beta_2}, \Omega, 0)=1. \end{align*} Similarly, we can show that \begin{equation*} \deg(I-T_{\alpha_1,\beta_2}, \Omega_{\beta_1}, 0)=1. \end{equation*} Thus, \eqref{2.36} holds. From \eqref{2.30}, \eqref{2.35}, and \eqref{2.36}, we reach the conclusion that \begin{align}\label{2.37} \deg(I-T_{\alpha_1,\beta_2}, \Omega\setminus\{\overline{\Omega_{\alpha_2}\cup\Omega_{\beta_1}}\}, 0)=-1. \end{align} From \eqref{2.36}, \eqref{2.37}, and Lemma \ref{l2.2}, it follows that \eqref{1.1}, \eqref{1.2} has three solutions in\Omega_{\alpha_2}$,$\Omega_{\beta_1}$, and$\Omega\setminus\{\overline{\Omega_{\alpha_2}\cup\Omega_{\beta_1}}\}$, respectively, satisfying \eqref{2.12} and \eqref{2.13}. This completes the proof of the theorem. \end{proof} \begin{thebibliography}{00} \bibitem{att} R. P. Agarwal, H. B. Thompson, and C. C, Tisdell, On the existence of multiple solutions to boundary value problems for second order ordinary differential equations, {\it Dynam. Systems Appl.} {\bf 16} (2007), 595--609. \bibitem{dll} Z. Du, W. Liu, and X. Lin, Multiple solutions to a three-point boundary value problem for higher-order ordinary differential equations, {\it J. Math. Anal. 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