\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 17, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/17\hfil Nonlocal BVPs for n-th Order ODEs] {Nonlocal boundary-value problems for n-th order ordinary differential equations by matching solutions} \author[X. Liu\hfil EJDE-2011/17\hfilneg] {Xueyan Liu} \address{Xueyan (Sherry) Liu \newline Department of Mathematics \\ Baylor University \\ Waco, TX 76798-7328, USA} \email{Xueyan\_Liu@baylor.edu} \thanks{Submitted June 26, 2010. Published February 3, 2011.} \subjclass[2000]{34B15, 34B10} \keywords{Boundary value problem; nonlocal; matching solutions} \begin{abstract} We are concerned with the existence and uniqueness of solutions to nonlocal boundary-value problems on an interval $[a,c]$ for the differential equation $y^{(n)}=f(x,y,y',\dots,y^{(n-1)})$, where $n\geq 3$. We use the method of matching solutions, with some monotonicity conditions on $f$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this article, we are concerned with the existence and uniqueness of solutions of boundary-value problems (BVP's) for the differential equation \begin{gather}\label{e1} y^{(n)}(x)=f(x,y(x),y'(x),\dots,y^{(n-1)}(x)),\quad n\geq 3,\; x\in[a,c],\\ \label{e2} \begin{gathered} y(a)-\sum_{i=1}^{s}\alpha_iy(\xi_i)=y_1,\quad y^{(i)}(b)=y_{i+2},\quad 0\leq i\leq n-3,\\ \sum_{j=1}^{t}\beta_j y(\eta_j)-y(c)=y_n, \end{gathered} \end{gather} where $a<\xi_1<\xi_2<\dots<\xi_s 0$ for $1\leq i\leq s$, $\beta_j> 0$ for $1\leq j\leq t$, $\sum_{i=1}^{s}\alpha_i=1$, $\sum_{j=1}^{t}\beta_j=1$, and $y_1,y_2,\dots,y_n\in \mathbb{R}$. It is assumed throughout that $f:[a,c]\times \mathbb{R}^{n}\to \mathbb{R}$ is continuous and that solutions for the initial value problems (IVP's) for \eqref{e1} are unique and exist on $[a,c]$. Moreover $a<\xi_1<\xi_2<\dots<\xi_sg(x,v_0-u_0,v_1-u_1,\dots,v_{n-2}-u_{n-2},v-u)$ when $x\in (a,b]$, $(-1)^{n-i}v_i\geq (-1)^{n-i}u_i$, $0\leq i\leq n-3$, and $v_{n-2}>u_{n-2}$; or when $x\in [b,c)$, $v_i\geq u_i$, $0\leq i\leq n-3$, and $v_{n-2}>u_{n-2}$. \item[(B)] There exists $\delta_1>0$, such that for all $0<\delta<\delta_1$, the IVP \begin{gather}\label{e9} z^{(n)}=g(x,z,z',\dots, z^{(n-1)}),\\ \label{e10} z^{(i)}(b)=0,\quad 0\leq i\leq n-1,\quad i\neq n-2,\quad z^{(n-2)}(b)=\delta \end{gather} has a solution $z$ on $[a,c]$ such that $z^{(n-2)}(x)\geq 0$ on $[a,c]$. \item[(C)] There exists $\delta_2>0$, such that for all $0<\delta<\delta_2$, the IVP \begin{gather}\label{e11} z^{(n)}=g(x,z,z',\dots, z^{(n-1)}),\\ \label{e12} z^{(i)}(b)=0,\quad 0\leq i\leq n-2,\quad z^{(n-1)}(b)=\delta, (-\delta) \end{gather} has a solution $z$ on $[b,c]$ ($[a,b]$) such that $z^{(n-2)}(x)\geq 0$ on $[b,c]$, ($z^{(n-2)}(x)\geq 0$ on $[a,b]$). \item[(D)] For each $w\in \mathbb{R}$, $g(x,v_0,v_1,\dots,v_{n-2}, w)\geq g(x,u_0,u_1,\dots,u_{n-2}, w)$ when $x\in (a,b]$, $(-1)^{n-i}(v_i-u_i)\geq 0$, $i=0,1,\dots,n-3$, and $v_{n-2}>u_{n-2}\geq 0$, or when $x\in [b,c)$, $v_i\geq u_{i}$, $i=0,1,\dots, n-3$, and $v_{n-2}>u_{n-2}\geq 0$. \end{itemize} \section{Preliminaries} In this section, we give two lemmas which show the relationship between the value of the $n-2$nd order and the $n-1$st order of two solutions of \eqref{e1} at $b$ that satisfy the boundary conditions (2), respectively, on the interval $[a,b]$ and the interval $[b,c]$. All of the results in Section 3 are based on two lemmas. We basically prove the lemmas by using contradiction. \begin{lemma}\label{lemma1} Suppose $p$ and $q$ are solutions of \eqref{e1} on $[a,b]$ and $w=p-q$ satisfies the following boundary conditions: $$w(a)-\sum_{i=1}^{s}\alpha_iw(\xi_i)=0,\quad w^{(i)}(b)=0,\quad 0\leq i\leq n-3.$$ Then, $w^{(n-2)}(b)=0$ if and only if $w^{(n-1)}(b)=0$. Also, $w^{(n-2)}(b)>0$ if and only if $w^{(n-1)}(b)>0$. \end{lemma} \begin{proof} $(\Rightarrow)$ The necessity of the first part. Suppose $w^{(n-2)}(b)=0$ and $w^{(n-1)}(b)\neq 0$. Without loss of generosity, we assume $w^{(n-1)}(b)<0$. Since $0=w(a)-\sum_{i=1}^{s}\alpha_iw(\xi_i) =\sum_{i=1}^{s}\alpha_i(w(a)-w(\xi_i))$ and $\alpha_i> 0$, for some $i_1$, $w(a)\geq w(\xi_{i_1})$, and for some $i_2$, $w(a)\leq w(\xi_{i_2})$. Hence, there exists $r_1\in (a,b)$ such that $w'(r_1)=0$ and $(-1)^{n-1}w'(x)>0$ on $(r_1,b)$. By repeated applications of Rolle's Theorem, there exists $r_2 \in (r_1,b)$ such that $w^{(n-2)}(r_2)=0$ and $w^{(n-2)}(x)>0$, for $x\in (r_2,b)$. Hence, $(-1)^{n-j}w^{(j)}(x)> 0$, for $j=0,1,\dots,n-2$, on $(r_2,b)$. Let $\delta \in \mathbb{R}$ with $0<\delta<\min\{\delta_2, -w^{(n-1)}(b)\}$. Then, by Condition (C), we have a solution $z$ of \eqref{e11}-\eqref{e12} on $[a,b]$, such that $z^{(i)}(b)=0$, $0\leq i\leq n-2$, $z^{(n-1)}(b)=-\delta$, and $z^{(n-2)}(x)\geq 0$ on $[a,b]$. Let $h=w-z$. Then, we have \begin{gather*} h^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots, q^{(n-1)})-g(x,z,z',\dots,z^{(n-1)}),\\ h^{(i)}(b)=0, \; 0\leq i\leq n-2, \quad h^{(n-1)}(b)=w^{(n-1)}(b)-z^{(n-1)}(b)<0. \end{gather*} Notice $h^{(n-2)}(r_2)=w^{(n-2)}(r_2)-z^{(n-2)}(r_2)\leq 0$, $h^{(n-2)}(b)=0$ and $h^{(n-1)}(b)<0$. So there exists $r_3\in [r_2, b)$ such that $h^{(n-2)}(r_3)=0$ and $h^{(n-2)}(x)>0$ for $x\in (r_3,b)$. Then, it follows that $(-1)^{n-j}h^{(j)}(x)>0$ on $(r_3,b)$, for $j=0,1,\dots, n-2$. Therefore, by Rolle's Theorem, there is $r_4\in (r_3, b)$ such that $h^{(n-1)}(r_4)=0$. Since $h^{(n-1)}(b)<0$, there is $r_5\in [r_4, b)$ such that $h^{(n-1)}(r_5)=0$ and $h^{(n-1)}(x)<0$ for $x\in (r_5, b)$. Then, $h^{(n)}(r_5)=\lim_{x\to r_5^+} \frac{h^{(n-1)}(x)-h^{(n-1)}(r_5)}{x-r_5} \leq 0,$ whereas by Conditions (A) and (D), (note that $[r_5,b)\subset (r_3,b)\subset (r_2,b)$), \begin{align*} h^{(n)}(r_5) &= f(r_5,p,p',\dots, p^{(n-1)})-f(r_5,q,q',\dots, q^{(n-1)}) -g(r_5,z,z',\dots,z^{(n-1)})\\ &> g(r_5,w,w',\dots,w^{(n-1)})-g(r_5,z,z',\dots,z^{(n-1)}) \geq 0, \end{align*} which is a contradiction. Therefore, $w^{(n-1)}(b)= 0$. $(\Leftarrow)$ The sufficiency of the first part. Suppose $w^{(n-1)}(b)=0$ and $w^{(n-2)}(b)\neq 0$. Without loss of generality, we assume $w^{(n-2)}(b)>0$. Since $0=w(a)-\sum_{i=1}^{s}\alpha_iw(\xi_i) =\sum_{i=1}^{s}\alpha_i(w(a)-w(\xi_i))$ and $\alpha_i> 0$, there exists $r_1\in (a,b)$ such that $w'(r_1)=0$, and $(-1)^{n-1}w'(x)>0$ on $(r_1,b)$. By repeated applications of Rolle's Theorem, there exists $r_2 \in (r_1,b)$ such that $w^{(n-2)}(r_2)=0$ and $w^{(n-2)}(x)>0$ for $x\in (r_2,b)$. Hence, $(-1)^{n-j}w^{(j)}(x)> 0$, for $j=0,1,\dots,n-2$, on $(r_2,b)$. Now let $0<\delta<\min\{\delta_1,w^{(n-2)}(b)\}$, and let $z$ be a solution of \eqref{e9}-\eqref{e10} satisfying Condition (B) and $z^{(n-2)}(x)\geq 0$ on $[a,b]$. Let $h=w-z$. Then, \begin{gather*} h^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots, q^{(n-1)})-g(x,z,z',\dots,z^{(n-1)}),\\ h^{(i)}(b)=0, \; 0\leq i\leq n-1,\; i\neq n-2, \quad h^{(n-2)}(b)=w^{(n-2)}(b)-z^{(n-2)}(b)>0. \end{gather*} Note that $h^{(n-2)}(r_2)=w^{(n-2)}(r_2)-z^{(n-2)}(r_2)\leq 0$. Hence, there is $r_3\in [r_2,b)$ such that $h^{(n-2)}(r_3)=0$, $h^{(n-2)}(x)>0$ on $(r_3,b)$. By Rolle's Theorem, there is $r_4\in (r_3,b)$ such that $h^{(n-1)}(r_4)>0$ and $(-1)^{n-j}h^{(j)}(x)>0$ on $(r_4,b)$, for $j=0,1,\dots, n-2$. By Conditions (A) and (D), \begin{align*} h^{(n)}(b) &= f(b,p,p',\dots, p^{(n-1)})-f(b,q,q',\dots, q^{(n-1)}) -g(b,z,z',\dots,z^{(n-1)})\\ &> g(b,w,w',\dots,w^{(n-1)})-g(b,z,z',\dots,z^{(n-1)}) \geq 0. \end{align*} Together with $h^{(n-1)}(b)=0$, we have that $h^{(n-1)}(x)<0$ on a left neighborhood of $b$. Since $h^{(n-1)}(r_4)>0$, there is $r_5\in (r_4,b)$ such that $h^{(n-1)}(r_5)=0$ and $h^{(n-1)}(x)<0$ on $(r_5,b)$. Hence, $h^{(n)}(r_5)\leq 0$. However, (note that $[r_5,b)\subset(r_4,b)\subset(r_2,b)$), \begin{align*} h^{(n)}(r_5) &= f(r_5,p,p',\dots, p^{(n-1)})-f(r_5,q,q',\dots, q^{(n-1)}) -g(r_5,z,z',\dots,z^{(n-1)})\\ &> g(r_5,w,w',\dots,w^{(n-1)})-g(r_5,z,z',\dots,z^{(n-1)}) \geq 0, \end{align*} which is a contradiction. Hence, our assumption is false. $(\Rightarrow)$ The necessity of the second part. Assume $w^{(n-1)}(b)<0$ and $w^{(n-2)}(b)>0$. Similar to the proof of the first part, we have $r_1\in(a,b)$ such that $w^{(n-2)}(r_1)=0$ and $w^{(n-2)}(x)>0$, for $x\in (r_1,b)$ and $(-1)^{n-j}w^{(j)}(x)>0$ on $(r_1,b)$, for $j=0,1,\dots,n-2$. Now let $0<\delta<\min\{\delta_1,w^{(n-2)}(b)\}$, and let $z$ be a solution of \eqref{e9}-\eqref{e10} satisfying Condition (B) and $z^{(n-2)}(x)\geq 0$ on $[a,b]$. Let $h=w-z$. Then, \begin{gather*} h^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots, q^{(n-1)})-g(x,z,z',\dots,z^{(n-1)}),\\ h^{(i)}(b)=0,\; 0\leq i\leq n-3, \quad h^{(n-2)}(b)=w^{(n-2)}(b)-z^{(n-2)}(b)>0. \end{gather*} Note that $h^{(n-1)}(b)=w^{(n-1)}(b)-z^{(n-1)}(b)=w^{(n-1)}(b)<0$, $h^{(n-2)}(b)>0$ and $h^{(n-2)}(r_1)=w^{(n-2)}(r_1)-z^{(n-2)}(r_1)=-z^{(n-2)}(r_1)\leq 0$. So there exists $r_2\in [r_1,b)$ such that $h^{(n-2)}(r_2)=0$, $h^{(n-2)}(x)>0$, for $x\in (r_2,b)$. It follows that $(-1)^{n-j}h^{(j)}(x)>0$ on $(r_2,b)$, for $j=0,1,\dots, n-2$. By Rolle's Theorem and $h^{(n-1)}(b)<0$, there is $r_3\in(r_2,b)$ such that $h^{(n-1)}(r_3)=0$ and $h^{(n-1)}(x)< 0$ on $(r_3,b)$. Therefore, $h^{(n)}(r_3)\leq 0$, whereas by Conditions (A) and (D), (note that $[r_3,b)\subset (r_2,b)\subset (r_1,b)$), \begin{align*} h^{(n)}(r_3) &= f(r_3,p,p',\dots, p^{(n-1)})-f(r_3,q,q',\dots, q^{(n-1)}) -g(r_3,z,z',\dots,z^{(n-1)})\\ &> g(r_3,w,w',\dots,w^{(n-1)})-g(r_3,z,z',\dots,z^{(n-1)}) \geq 0, \end{align*} which is a contradiction. $(\Leftarrow)$ The sufficiency of the second part. We assume that $w^{(n-1)}(b)>0$ and $w^{(n-2)}(b)<0$. Then, we get the same situation as the proof of necessity with opposite signs of $w^{(n-1)}(b)$ and $w^{(n-2)}(b)$, which also implies a contradiction. Hence, the sufficiency is true. \end{proof} \begin{lemma}\label{lemma2} Suppose $p$ and $q$ are solutions of \eqref{e1} on $[b,c]$ and $w=p-q$ satisfies the following boundary conditions: $$w^{(i)}(b)=0,\; 0\leq i\leq n-3,\quad \sum_{j=1}^{t}\beta_jw(\eta_j)-w(c)=0.$$ Then, $w^{(n-2)}(b)=0$ if and only if $w^{(n-1)}(b)=0$. Also, $w^{(n-2)}(b)>0$ if and only if $w^{(n-1)}(b)<0$. \end{lemma} \begin{proof} $(\Rightarrow)$ The necessity of the first part. Assume $w^{(n-2)}(b)=0$ and for contradiction, without loss of generality, we assume $w^{(n-1)}(b)>0$. By $\sum_{j=1}^{t}\beta_jw(\eta_j)-w(c)=0$, there exists $r_1\in (b,c)$ such that $w'(r_1)=0$. By repeated applications of Rolle's Theorem, there exists $r_2\in (b,r_1)$ such that $w^{(n-2)}(r_2)=0$ and $w^{(n-2)}(x)>0$ on $(b,r_2)$. It follows that $w^{(j)}(x)>0$ on $(b,r_2)$, for $j=0,1,\dots,n-2$. Let $0<\delta<\min\{\delta_2, w^{(n-1)}(b)\}$. Then, by Condition (C), we have a solution $z$ of \eqref{e11}-\eqref{e12} on $[b,c]$ such that $z^{(i)}(b)=0$, $0\leq i\leq n-2$, $z^{(n-1)}(b)=\delta$, and $z^{(n-2)}(x)\geq 0$ on $[b,c]$. Then, $z^{(j)}(x)\geq 0$, for $j=0,1,\dots, n-2$, on $[b,c]$. Let $h=w-z$. Then, \begin{gather*} h^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots, q^{(n-1)})-g(x,z,z',\dots,z^{(n-1)}),\\ h^{(i)}(b)=0, ~0\leq i\leq n-2, ~~h^{(n-1)}(b)=w^{(n-1)}(b) -z^{(n-1)}(b)>0. \end{gather*} Note that $h^{(n-2)}(r_2)=w^{(n-2)}(r_2)-z^{(n-2)}(r_2)\leq 0$. Hence, there is $r_3\in (b,r_2]$ such that $h^{(n-2)}(r_3)=0$, $h^{(n-2)}(x)>0$ on $(b,r_3)$, and hence, $h^{(j)}(x)> 0$, for $j=0,1,\dots, n-2$ on $(b,r_3)$. By $h^{(n-2)}(b)=0$, Rolle's Theorem, and $h^{(n-1)}(b)>0$, there exists $r_4\in (b,r_3)$ such that $h^{(n-1)}(r_4)=0$ and $h^{(n-1)}(x)>0$ on $(b,r_4)$. Hence, $h^{(n)}(r_4)\leq0$, but by Conditions (A) and (D) and $(b,r_4]\subset(b,r_3)\subset(b,r_2)$, we have \begin{align*} h^{(n)}(r_4) &= f(r_4,p,p',\dots, p^{(n-1)})-f(r_4,q,q',\dots, q^{(n-1)}) -g(r_4,z,z',\dots,z^{(n-1)})\\ &> g(r_4,w,w',\dots,w^{(n-1)})-g(r_4,z,z',\dots,z^{(n-1)}) \geq 0, \end{align*} which is a contradiction. $(\Leftarrow)$ The sufficiency of the first part. Suppose $w^{(n-1)}(b)=0$ and $w^{(n-2)}(b)>0$. Similar to the above, we have $r_1\in (b,c)$ such that $w^{(n-2)}(r_1)=0$ and $w^{(j)}(x)>0$ on $(b,r_1)$ for $j=0,1,\dots,n-2$. Let $0<\delta<\min\{\delta_1, w^{(n-2)}(b)\}$. Then, by Condition (B), we have a solution $z$ of \eqref{e9}-\eqref{e10} on $[b,c]$ such that $z^{(i)}(b)=0$, $0\leq i\leq n-1$, $i\neq n-2$, $z^{(n-2)}(b)=\delta$, and $z^{(n-2)}(x)\geq 0$ on $[b,c]$. Then, $z^{(j)}(x)\geq 0$, for $j=0,1,\dots, n-2$, on $[b,c]$. Let $h=w-z$. Then, \begin{gather*} h^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots, q^{(n-1)})-g(x,z,z',\dots,z^{(n-1)})\\ h^{(i)}(b)=0, \; 0\leq i\leq n-1,\; i\neq n-1,\quad h^{(n-2)}(b)=w^{(n-2)}(b)-z^{(n-2)}(b)>0. \end{gather*} Note that $h^{(n-2)}(r_1)=w^{(n-2)}(r_1)-z^{(n-2)}(r_1)=-z^{(n-2)}(r_1)\leq 0$. So there is $r_2\in(b,r_1]$ such that $h^{(n-2)}(r_2)= 0$ and $h^{(n-2)}(x)> 0$, for $x\in (b,r_2)$, and $h^{(j)}(x)> 0$ on $(b,r_2)$, for $j=0,1,\dots,n-2$. By Rolle's Theorem, there is $r_3\in (b,r_2)$ such that $h^{(n-1)}(r_3)<0$. Note that \begin{align*} h^{(n)}(b) &= f(b,p,p',\dots, p^{(n-1)})-f(b,q,q',\dots, q^{(n-1)}) -g(b,z,z',\dots,z^{(n-1)})\\ &> g(b,w,w',\dots,w^{(n-1)})-g(b,z,z',\dots,z^{(n-1)}) \geq 0. \end{align*} Hence, there is $r_4\in(b,r_3)$ such that $h^{(n-1)}(r_4)=0$ and $h^{(n-1)}(x)> 0$ on $(b,r_4)$, which implies $h^{(n)}(r_4) \leq 0$. But by $(b,r_4]\subset(b,r_2)\subset(b,r_1)$ and Conditions (A) and (D), we have that \begin{align*} h^{(n)}(r_4) &= f(r_4,p,p',\dots, p^{(n-1)})-f(r_4,q,q',\dots, q^{(n-1)}) -g(r_4,z,z',\dots,z^{(n-1)})\\ &> g(r_4,w,w',\dots,w^{(n-1)})-g(r_4,z,z',\dots,z^{(n-1)}) \geq 0, \end{align*} which is a contradiction. $(\Rightarrow)$ The necessity of the second part. Suppose $w^{(n-2)}(b)>0$ and $w^{(n-1)}(b)>0$. Similar to the proof of the necessity of the first part, we also can get a contradiction. Hence, we omit the proof. Therefore, if $w^{(n-2)}(b)>0$, then $w^{(n-1)}(b)<0$. $(\Leftarrow)$ The sufficiency of the second part. Suppose $w^{(n-1)}(b)<0$. If $w^{(n-2)}(b)<0$, then similar to the proof of necessity, we can get $w^{(n-1)}(b)>0$, which is a contradiction. Hence, the sufficiency is also true. \end{proof} \section{Existence and uniqueness of solutions to \eqref{e1}-\eqref{e2}} Before discussing existence and uniqueness for \eqref{e1}-\eqref{e2}, we consider the uniqueness of solutions to each of the BVP's for \eqref{e1} satisfying any of \eqref{e3}, \eqref{e4}, \eqref{e5}, or \eqref{e6}. \begin{lemma} \label{lemma3} Let $y_1,y_2,\dots,y_n\in \mathbb{R}$ be given and assume Conditions {\rm (A)--(D)} are satisfied. Then, given $m\in \mathbb{R}$, each of the BVP's for \eqref{e1} satisfying any of conditions \eqref{e3}, \eqref{e4}, \eqref{e5}, or \eqref{e6} has at most one solution. \end{lemma} \begin{proof} The case of \eqref{e1}-\eqref{e3}: Suppose there are two distinct solutions $p(x)$ and $q(x)$ for some $m\in \mathbb{R}$. Let $w=p-q$. Then, $w$ satisfies \begin{gather*} w^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots, q^{(n-1)}),\\ w(a)-\sum_{i=1}^{s}\alpha_iw(\xi_i)=0, \quad w^{(i)}(b)=0, \quad 0\leq i\leq n-2. \end{gather*} By Lemma \ref{lemma1}, we can get that $w^{(n-1)}(b)=0$. Then, by the uniqueness of solutions of IVP's for \eqref{e1}, we can conclude that $p\equiv q$ on $[a,b]$. Hence, \eqref{e1}-\eqref{e3} has at most one solution on $[a,b]$. {The other cases:} By using similar ideas and Lemma \ref{lemma1} and Lemma \ref{lemma2}, we resolve the other cases. \end{proof} \begin{lemma}\label{lemma4} Let $y_1,y_2,\dots,y_n\in \mathbb{R}$ be given. Assume Conditions (A)-(D) are satisfied. Then, the BVP \eqref{e1}-\eqref{e2} has at most one solution. \end{lemma} \begin{proof} We argue by contradiction. Suppose for some values $y_1,y_2,\dots,y_n\in \mathbb{R}$, there exist distinct solutions $p$ and $q$ of \eqref{e1}-\eqref{e2}. Also, let $w=p-q$. Then, from Lemma \ref{lemma1} and Lemma \ref{lemma2}, we get $w^{(n-2)}(b)\neq 0$, $w^{(n-1)}(b)\neq 0$. Without loss of generality, we suppose $w^{(n-2)}(b)>0$. Then, by Lemma \ref{lemma1}, $w^{(n-1)}(b)> 0$. But by Lemma \ref{lemma2}, $w^{(n-1)}(b)<0$. This is a contradiction. Hence, $p\equiv q$ on $[a,c]$. \end{proof} Next, we show that solutions of \eqref{e1} satisfying each of \eqref{e3}, \eqref{e4}, \eqref{e5}, or \eqref{e6} respectively are monotone functions of $m$ at $b$. For notation purposes, given any $m\in \mathbb{R}$, let $\alpha(x,m)$, $u(x,m)$, $\beta(x,m)$, $v(x,m)$ denote the solutions, when they exist, of the boundary value problems of \eqref{e1} satisfying \eqref{e3}, \eqref{e4}, \eqref{e5}, or \eqref{e6}, respectively. \begin{lemma}\label{lemma5} Suppose that {\rm (A)--(D)} are satisfied and that for each $m\in \mathbb{R}$, there exist solutions of \eqref{e1} satisfying each of the conditions \eqref{e3}, \eqref{e4}, \eqref{e5}, \eqref{e6}, respectively. Then, $\alpha^{(n-1)}(b,m)$ and $\beta^{(n-1)}(b,m)$ are, respectively, strictly increasing and strictly decreasing functions of $m$ with ranges all of $\mathbb{R}$. \end{lemma} \begin{proof} The proof of $\{\alpha^{(n-1)}(b,m)|m\in \mathbb{R}\}=\mathbb{R}$ is the same as that in \cite[Theorem 2.4]{Henderson83}. We omit it here. \end{proof} Similarly, we obtain monotonicity conditions on $u^{(n-2)}(b,m)$ and $v^{(n-2)}(b,m)$. \begin{lemma}\label{lemma6} Under the assumption of Lemma \ref{lemma5}, the functions $u^{(n-2)}(b,m)$ and $v^{(n-2)}(b,m)$ are, respectively, strictly increasing and strictly decreasing functions of $m$, with ranges all $\mathbb{R}$. \end{lemma} Finally, we arrive at our existence result for \eqref{e1}-\eqref{e2}, which is obtained by solution matching. \begin{theorem} Assume the hypotheses of Lemma \ref{lemma5}. Then, \eqref{e1}-\eqref{e2} has a unique solution. \end{theorem} \begin{proof} We prove the existence from either Lemma \ref{lemma5} or Lemma \ref{lemma6}. Making use of Lemma \ref{lemma6}, there exists a unique $m_0\in \mathbb{R}$ such that $u^{(n-2)}(b,m_0)=v^{(n-2)}(b,m_0)$. Then, $$y(x)=\begin{cases} u(x,m_0), & a\leq x\leq b,\\ v(x,m_0), & b\leq x\leq c, \end{cases}$$ is a solution of \eqref{e1}-\eqref{e2} and by Lemma \ref{lemma4}, $y(x)$ is the unique solution. \end{proof} \section{Existence and uniqueness of solutions to \eqref{e1}-\eqref{e8}} We can obtain analogous results to those of Section 3 for \eqref{e1}-\eqref{e8} with $\tau, \sigma\in \{0,1,\dots,n-3\}$ fixed. We obtain solutions to \eqref{e1}-\eqref{e8} by matching solutions satisfying the following types of boundary conditions: \begin{gather}\label{e13} y^{(\tau)}(a)-\sum_{i=1}^{s}\alpha_iy^{(\tau)}(\xi_i)=y_1, \quad y^{(i)}(b)=y_{i+2},\; 0\leq i\leq n-3, \quad y^{(n-2)}(b)=m, \\ \label{e14} y^{(\tau)}(a)-\sum_{i=1}^{s}\alpha_iy^{(\tau)}(\xi_i)=y_1,\quad y^{(i)}(b)=y_{i+2},\; 0\leq i\leq n-3, \quad y^{(n-1)}(b)=m, \\ \label{e15} y^{(i)}(b)=y_{i+2},\; 0\leq i\leq n-3, \quad y^{(n-2)}(b)=m, \quad \sum_{j=1}^{t}\beta_j y^{(\sigma)}(\eta_j)-y^{(\sigma)}(c)=y_n, \\ \label{e16} y^{(i)}(b)=y_{i+2},\; 0\leq i\leq n-3, \quad y^{(n-1)}(b)=m, \quad \sum_{j=1}^{t}\beta_j y^{(\sigma)}(\eta_j)-y^{(\sigma)}(c)=y_n, \end{gather} where $m\in \mathbb{R}$, $a<\xi_1<\xi_2<\dots<\xi_s 0$ for $1\leq i\leq s$, $\beta_j> 0$ for $1\leq j\leq t$, $\sum_{i=1}^{s}\alpha_i=1$, $\sum_{j=1}^{t}\beta_j=1$ and $y_1,y_2,\dots,y_n\in \mathbb{R}$. We omit the proofs of the following results since they are essentially the same as those presented in Section 2 with only small modifications. \begin{lemma} \label{lemma7} Let $y_1,y_2,\dots,y_n\in \mathbb{R}$ be given and assume conditions {\rm (A)--(D)} are satisfied. Then, given $m\in \mathbb{R}$, each of the BVP's for \eqref{e1} satisfying any of conditions \eqref{e13}, \eqref{e14}, \eqref{e15}, or \eqref{e16} has at most one solution. \end{lemma} \begin{lemma} \label{lemma8} Let $y_1,y_2,\dots,y_n\in \mathbb{R}$ be given and assume conditions (A)-(D) are satisfied. Then \eqref{e1}-\eqref{e8} has at most one solution. \end{lemma} Now, given any $m\in \mathbb{R}$, let $\theta(x,m)$, $l(x,m)$, $\vartheta(x,m)$, $o(x,m)$ denote the solutions, when they exist, of the boundary value problems of \eqref{e1} satisfying \eqref{e13}, \eqref{e14}, \eqref{e15}, \eqref{e16}, respectively. \begin{lemma} \label{lemma9} Suppose that {\rm (A)--(D)} are satisfied and that for each $m\in \mathbb{R}$, there exist solutions of \eqref{e1} satisfying each of the conditions \eqref{e13}, \eqref{e14}, \eqref{e15}, \eqref{e16}. Then, $\theta^{(n-1)}(b,m)$ and $\vartheta^{(n-1)}(b,m)$ are respectively strictly increasing and strictly decreasing functions of $m$ with ranges all of $\mathbb{R}$. Also, $l^{(n-2)}(b,m)$ and $o^{(n-2)}(b,m)$ are respectively strictly increasing and strictly decreasing functions of $m$ with ranges all of $\mathbb{R}$. \end{lemma} \begin{theorem} Assume the hypotheses of Lemma \ref{lemma9}. 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