\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 20, pp. 1--43.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/20\hfil Non-homogeneous boundary-value problems] {Non-homogeneous boundary-value problems of higher order differential equations with $p$-Laplacian} \author[Y. Liu\hfil EJDE-2008/20\hfilneg] {Yuji Liu} \address{Yuji Liu \newline Department of Mathematics\\ Guangdong University of Business Studies \\ Guangzhou 510320, P. R. China} \email{liuyuji888@sohu.com} \thanks{Submitted September 27, 2007. Published February 21, 2008.} \thanks{Supported by grant 06JJ5008 from the Natural Science Foundation of Hunan Province \hfill\break\indent and by the Natural Science Foundation of Guangdong Province, P. R. China} \subjclass[2000]{34B10, 34B15, 35B10} \keywords{One-dimension $p$-Laplacian differential equation; positive solution; \hfill\break\indent multi-point boundary-value problem; non-homogeneous boundary conditions; \hfill\break\indent Mawhin's coincidence degree theory} \begin{abstract} We establish sufficient conditions for the existence of positive solutions to five multi-point boundary value problems. These problems have a common equation (in different function domains) and different boundary conditions. It is interesting note that the methods for solving all these problems and most of the reference are based on the Mawhin's coincidence degree theory. First, we present a survey of multi-point boundary-value problems and the motivation of this paper. Then we present the main results which generalize and improve results in the references. We conclude this article with examples of problems that can not solved by methods known so far. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} Multi-point boundary-value problems (BVPs) for differential equations were initialed by Il'in and Moiseev \cite{i1} and have received a wide attention because of their potential applications. There are many exciting results concerned with the existence of positive solutions of boundary-value problems of second or higher order differential equations with or without $p$-Laplacian subjected to the special homogeneous multi-point boundary conditions (BCs); we refer the readers to \cite{a1}--\cite{g2}, \cite{f1}--\cite{k3} \cite{k6}--\cite{m1}, \cite{m3}--\cite{m6}, \cite{n1}--\cite{z4}. The methods used for finding positive solutions of these problems at non-resonance cases, or solutions at resonance cases, are critical point theory, fixed point theorems in cones in Banach spaces, fixed point index theory, alternative of Leray-Schauder, upper and lower solution methods with iterative techniques, and so on. There are also several results concerned with the existence of positive solutions of multi-point boundary-value problems for differential equations with non-homogeneous BCs; see for example \cite{g3,g4,k4,k5,m2,m7} and the early paper \cite{z4}. For the reader's information and to compare our results with the known ones, we now give a simple survey. Multi-point boundary-value problems with homogeneous BCs consist of the second order differential equation and the multi-point homogeneous boundary conditions. The second order differential equation is either $$[\phi(x'(t))]'+f(t,x(t),x'(t))=0,\quad t\in (0,1),$$ or one of the following cases \begin{gather*} x''(t)+f(t,x(t),x'(t))=0,\quad t\in (0,1), \\ \big[\phi(x'(t))\big]'+f(t,x(t))=0,\quad t\in (0,1), \\ x''(t)+f(t,x(t))=0,\quad t\in (0,1). \end{gather*} The multi-point homogeneous boundary conditions are either \begin{gather*} x(0)-\sum_{i=1}^m\alpha_i x(\xi_i)=x(1)-\sum_{i=1}^n\beta_i x(\eta_i)=0, \\ x'(0)-\sum_{i=1}^m\alpha_i x'(\xi_i)=x(1)-\sum_{i=1}^n\beta_i x(\eta_i)=0, \\ x(0)-\sum_{i=1}^m\alpha_i x(\xi_i)=x'(1)-\sum_{i=1}^n\beta_i x'(\eta_i)=0,\\ x'(0)-\sum_{i=1}^m\alpha_i x'(\xi_i)=x'(1)-\sum_{i=1}^n\beta_i x'(\eta_i)=0, \\ x(0)-\sum_{i=1}^m\alpha_i x'(\xi_i)=x(1)-\sum_{i=1}^n\beta_i x'(\eta_i)=0,\\ x(0)-\sum_{i=1}^m\alpha_i x'(\xi_i)=x(1)-\sum_{i=1}^n\beta_i x(\eta_i)=0,\\ x(0)-\sum_{i=1}^m\alpha_i x'(\xi_i)=x'(1)-\sum_{i=1}^n\beta_i x'(\eta_i)=0, \end{gather*} or their special cases, where $0<\xi_1<\cdots<\xi_m<1$ and $0<\eta_i<\cdots<\eta_n<1$, $\alpha_i,\beta_j\in R$ are constants. These problems were studied extensively in papers \cite{a1}--\cite{y1} and the references therein. 1. For the second order differential equations, Gupta \cite{g7} studied the following multi-point boundary-value problem $$\label{e1} \begin{gathered} x''(t)=f(t,x(t),x'(t))+r(t),\quad t\in (0,1),\\ x(0)-\sum_{i=1}^m\alpha_i x(\xi_i)=x(1)-\sum_{i=1}^n\beta_i x(\eta_i)=0, \end{gathered}$$ and $$\label{e2} \begin{gathered} x''(t)=f(t,x(t),x'(t))+r(t),\quad t\in (0,1),\\ x(0)-\sum_{i=1}^m\alpha_i x(\xi_i)=x'(1)-\sum_{i=1}^n\beta_i x'(\eta_i)=0, \end{gathered}$$ where $0<\xi_i<\dots<\xi_m<1$, $0<\eta_1<\dots<\eta_n<1$, $\alpha_i,\beta_i\in \mathbb{R}$ with $(\sum_{i=1}^m\alpha_i\xi_i)(1-\sum_{i=1}^n\beta_i)\neq (1-\sum_{i=1}^m \alpha_i)(\sum_{i=1}^n\beta_i\eta_i-1)$ for \eqref{e1} and with $(1-\sum_{i=1}^m\alpha_i)(1-\sum_{i=1}^n\beta_i)\neq 0$ for \eqref{e2}. Some existence results for solutions of \eqref{e1} and \eqref{e2} were established in \cite{g5}. Liu \cite{l9} established the existence results of solutions of \eqref{e1} for the case $$\sum_{i=1}^m\alpha_i\xi_i=1-\sum_{i=1}^m\alpha_i=1-\sum_{i=1} \beta_i=1-\sum_{i=1}^m\beta_i\xi_i=0.$$ Liu and Yu \cite{l6,l7,l8,l10} studied the existence of solutions of \eqref{e1} and \eqref{e2} at some special cases. Zhang and Wang \cite{z3} studied the multi-point boundary-value problem $$\label{e3} \begin{gathered} x''(t)=f(t,x(t)),\quad t\in (0,1),\\ x(0)-\sum_{i=1}^m\alpha_i x(\xi_i)=x(1)-\sum_{i=1}^n\beta_i x(\eta_i)=0, \end{gathered}$$ where $0<\xi_i<\dots<\xi_m<1$, $\alpha_i,\beta_i\in [0,+\infty)$ with $0<\sum_{i=1}^m\alpha_i<1$ and $\sum_{i=1}^m\beta_i<1$. Under certain conditions on $f$, they established some existence results for positive solutions of \eqref{e3}. Liu in \cite{l5}, and Liu and Ge in \cite{l16} studied the four-point boundary-value problem $$\label{e4} \begin{gathered} x''(t)+f(t,x(t))=0,\quad t\in (0,1),\\ x(0)-\alpha x(\xi)=x(1)-\beta x(\eta)=0, \end{gathered}$$ where $0<\xi,\eta<1$, $\alpha,\beta\ge 0$, $f$ is a nonnegative continuous function. Using the Green's function of its corresponding linear problem, Liu established existence results for at least one or two positive solutions of \eqref{e4}. Ma in \cite{m3}, and Zhang and Sun in \cite{z2} studied the following multi-point boundary-value problem $$\label{e5} \begin{gathered} x''(t)+a(t)f(x(t))=0,\quad t\in (0,1),\\ x(0)=x(1)-\sum_{i=1}^m\alpha_i x(\xi_i)=0, \end{gathered}$$ where $0<\xi_i<1$, $\alpha_i\ge 0$ with $\sum_{i=1}^m\alpha_i\xi_i<1$, $a$ and $f$ are nonnegative continuous functions, there is $t_0\in [\xi_m,1]$ so that $a(t_0)>0$. Let $$\lim_{x\to 0}\frac{f(x)}{x}=l,\quad \lim_{x\to +\infty}\frac{f(x)}{x}=L.$$ It was proved that if $l=0,L=+\infty$ or $l=+\infty,L=0$, then \eqref{e5} has at least one positive solution. Ma and Castaneda \cite{m5} studied the problem $$\label{e6} \begin{gathered} x''(t)+a(t)f(x(t))=0,\quad t\in (0,1),\\ x'(0)-\sum_{i=1}^m\alpha_i x'(\xi_i)=x(1)-\sum_{i=1}^m\beta_i x(\xi_i)=0, \end{gathered}$$ where $0<\xi_i<\dots<\xi_m<1$, $\alpha_i,\beta_i\ge 0$ with $0<\sum_{i=1}^m\alpha_i<1$ and $0<\sum_{i=1}^m\beta_i<1$ and $a$ and $f$ are nonnegative continuous functions, there is $t_0\in [\xi_m,1]$ so that $a(t_0)>0$. Ma and Castaneda established existence results for positive solutions of \eqref{e6} under the assumptions $$\lim_{x\to 0}\frac{f(x)}{x}=0,\quad \lim_{x\to +\infty}\frac{f(x)}{x} =+\infty\hbox{ or }\lim_{x\to 0}\frac{f(x)}{x}=+\infty,\quad \lim_{x\to +\infty}\frac{f(x)}{x}=0.$$ 2. For second order differential equations with $p$-Laplacian, Drabek and Takc \cite{d1} studied the existence of solutions of the problem $$\label{e7} \begin{gathered} -(\phi(x'(t))'-\lambda\phi(x)=f(t),\quad t\in (0,T),\\ x(0)=x(T)=0, \end{gathered}$$ In a recent paper \cite{l1}, the author established multiplicity results for positive solutions of the problems \begin{gather*} \big[\phi_p(x'(t))\big]'+f(t,x(t))=0,\quad t\in (0,1) ,\\ x(0)=\int_0^1x(s)dh(s),\quad \phi_p(x'(1))=\int_0^1\phi_p(x'(s))dg(s), \end{gather*} and \begin{gather*} \big[\phi_p(x'(t))\big]'+f(t,x(t))=0,\quad t\in (0,1),\\ \phi_p(x'(0))=\int_0^1\phi_p(x'(s))dh(s),\quad x(1)=\int_0^1x(s)dg(s). \end{gather*} Gupta \cite{g8} studied the existence of solutions of the problem $$\label{e8} \begin{gathered} \big[\phi(x'(t))\big]'+f(t,x(t),x'(t))+e(t)=0,\quad t\in (0,1),\\ x(0)-\sum_{i=1}^m\alpha_i x(\xi_i)=x(1)-\sum_{i=1}^m\beta_i x(\xi_i)=0 \end{gathered}$$ by using topological degree and some a priori estimates. Bai and Fang \cite{b2} investigated the following multi-point boundary-value problem $$\label{e9} \begin{gathered} \big[\phi(x'(t))\big]'+a(t)f(t,x(t))=0,\quad t\in (0,1),\\ x(0)=x(1)-\sum_{i=1}^m\beta_i x(\xi_i)=0, \end{gathered}$$ where $0<\xi_i<\dots<\xi_m<1$, $\beta_i\ge 0$ with $\sum_{i=1}^m\beta_i\xi_i<1$, $a$ is continuous and nonnegative and there is $t_0\in [\xi_m,1]$ so that $a(t_0)>0$, $f$ is a continuous nonnegative function. The purpose of \cite{b2} is to generalize the results in \cite{m3}. Wang and Ge \cite{w1}, Ji, Feng and Ge \cite{j1}, Feng, Ge and Jiang \cite{f1}, Rynne \cite{r1} studied the existence of multiple positive solutions of the following more general problem \begin{gather*} \big[\phi(x'(t))\big]'+a(t)f(t,x(t))=0,\quad t\in (0,1),\\ x(0)-\sum_{i=1}^m\alpha_ix(\xi_i)=x(1)-\sum_{i=1}^m\beta_i x(\xi_i)=0 \end{gather*} by using fixed point theorems for operators in cones. Sun, Qu and Ge \cite{s4} using the monotone iterative technique established existence results of positive solutions of the problem \begin{gather*} \big[\phi(x'(t))\big]'+a(t)f(t,x(t),x'(t))=0,\quad t\in (0,1),\\ x(0)-\sum_{i=1}^m\alpha_ix(\xi_i)=x(1)-\sum_{i=1}^m\beta_i x(\xi_i)=0. \end{gather*} Bai and Fang \cite{b1} studied the problem $$\label{e10} \begin{gathered} \big[\phi(x'(t))\big]'+f(t,x(t))=0,\quad t\in (0,1),\\ x'(0)-\sum_{i=1}^m\alpha_i x'(\xi_i)=x(1)-\sum_{i=1}^m\beta_i x(\xi_i)=0, \end{gathered}$$ where $0<\xi_i<\dots<\xi_m<1$, $\alpha_i\ge 0,\beta_i\ge 0$ with $0<\sum_{i=1}^m\alpha_i<1$ and $0<\sum_{i=1}^m\beta_i<1$, $f$ is continuous and nonnegative. The purpose of \cite{b1} is to generalize and improve the results in \cite{m5}. In paper Ma, Du and Ge \cite{m8} studied \eqref{e6} by using the monotone iterative methods. The existence of monotone positive solutions of \eqref{e6} were obtained. Based upon the fixed point theorem due to Avery and Peterson \cite{a4}, Wang and Ge \cite{w2}, Sun, Ge and Zhao \cite{s3} established existence results of multiple positive solutions of the following problems \begin{gather*} \big[\phi(x'(t))\big]'+a(t)f(t,x(t),x'(t))=0,\quad t\in (0,1),\\ x'(0)-\sum_{i=1}^m\alpha_ix'(\xi_i)=x(1)-\sum_{i=1}^m\beta_i x(\xi_i)=0 \end{gather*} and \begin{gather*} \big[\phi(x'(t))\big]'+a(t)f(t,x(t),x'(t))=0,\quad t\in (0,1),\\ x(0)-\sum_{i=1}^m\alpha_ix(\xi_i)=x'(1)-\sum_{i=1}^m\beta_i x'(\xi_i)=0. \end{gather*} In \cite{l1,l10}, the authors studied the existence of solutions of the following BVPs at resonance cases $$\label{e11} \begin{gathered} x''(t)=f(t,x(t),x'(t))+e(t),\quad 0t\in (0,T),\\ x'(0)=\alpha x'(\xi),\;x'(1)=\sum_{i=1}^m\beta_i x'(\xi_i). \end{gathered}$$ In a recent paper \cite{g2}, the authors investigated the existence of solutions of the following problem for $p$-Laplacian differential equation $$\label{e12} \begin{gathered} (\phi(x'(t))'=f(t,x(t),x'(t)),\quad t\in (0,T),\\ x'(0)=0,\quad \theta(x'(1))=\sum_{i=1}^m\alpha_i\theta(x'(\xi_i)), \end{gathered}$$ where $\theta$ and $\phi$ are two odd increasing homeomorphisms from $R$ to $R$ with $\phi(0)=\theta(0)=0$. In the recent papers \cite{g10,k3,k4,l2,l9,p1,s2,s3,w1,w2,w3,w4,w9,z1}, the authors studied the existence of multiple positive solutions of \eqref{e8}, \eqref{e9}, \eqref{e10} or other more general multi-point boundary-value problems, respectively, by using of multiple fixed point theorems in cones in Banach spaces such as the five functionals fixed point theorem \cite{g10}, the fixed-point index theory \cite{s1}, the fixed point theorem due to Avery and Peterson, a two-fixed-point theorem \cite{g10,s3,w1,w2}, Krasnoselskii's fixed point theorem and the contraction mapping principle \cite{k1,l2,p1,s2,w9}, the Leggett-Williams fixed-point theorem \cite{k2,l9}, the generalization of polar coordinates \cite{w3}, using the solution of an implicit functional equation \cite{k1,k2}. 3. For higher order differential equations, there have been many papers discussed the existence of solutions of multi-point boundary-value problems for third order differential equations \cite{g6,m1,n1}. Ma \cite{m1} studied the solvability of the problem $$\label{e13} \begin{gathered} x'''(t)+k^2x'(t)+g(x(t),x'(t))=p(t),\quad t\in (0,\pi),\\ x(0)=0,\quad x'(0)=x'(\pi)=0, \end{gathered}$$ where $k\in N$, $g$ is continuous and bounded, $p$ is continuous. In \cite{g6,n1}, the authors investigated the solvability of the problem $$\label{e14} \begin{gathered} x'''(t)+k^2x'(t)+g(t,x(t),x'(t),x''(t))=p(t),\\ x'(0)=x'(1)=0,\quad x(0)=0, \end{gathered}$$ where $g$ and $p$ are continuous, $k\in \mathbb{R}$. It was supposed in \cite{n1} that $g$ is bounded and in \cite{g6} $g$ satisfies $g(t,u,v,w)v\ge 0$ for $t\in [0,1],\;(u,v,w)\in \mathbb{R}^3$, $\lim_{|v|\to \infty}\frac{g(t,u,v,w)}{v}<3\pi^2\quad \hbox{uniformly in }t,u,w .$ The upper and lower solution methods with monotone iterative technique are used to solve multi-point boundary-value problems for third or fourth order differential equations in papers \cite{z1} and \cite{w4}. In \cite{l13}, the authors studied the problem $$\label{e15} \begin{gathered}x^{(n)}(t))+\lambda f(x(t))=0,\quad t\in (0,1),\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3,\\ x^{(n-2)}(0)-\alpha x^{(n-2)}(\eta)=x^{(n-2)}(1)-\beta x^{(n-2)}(\eta)=0, \end{gathered}$$ the existence results for positive solutions of \eqref{e15} were established in \cite{l13} in the case that the nonlinearity $f$ changes sign. The existence of positive solutions of the following two problems: $$\label{e16} \begin{gathered} x^{(n)}(t))+\phi(t)f(t,x(t)),\dots,x^{(n-2)}(t))=0,\quad t\in (0,1),\\ x^{(i)}(0)=0,\quad i=0,\dots,n-2,\quad x^{(n-1)}(0)=0, \end{gathered}$$ and $$\label{e17} \begin{gathered} x^{(n)}(t))+\phi(t)f(t,x(t)),\dots,x^{(n-2)}(t))=0,\quad t\in (0,1),\\ x^{(i)}(0)=0,\quad i=0,\dots,n-2,\quad x^{(n-2)}(0)=0, \end{gathered}$$ were studied in \cite{a2,w11}. 4. For Sturm-Liouville type multi-point boundary conditions, Grossinho \cite{g3} studied the problem $$\label{e18} \begin{gathered} x'''(t)+f(t,x(t),x'(t),x''(t))=0,\quad t\in (0,1),\\ x(0)=0,\quad ax'(0)-bx''(0)=A,\quad cx'(1)+dx''(1)=B. \end{gathered}$$ By using theory of Leray-Schauder degree, it was proved that \eqref{e18} has solutions under the assumptions that there exist super and lower solutions of the corresponding problem. Agarwal and Wong \cite{a3}, Qi \cite{q1} investigated the solvability of the following problem with Sturm-Liouville type boundary conditions $$\label{e19} \begin{gathered} x^{(n)}(t)=f(t,x(t),\dots,x^{(n-2)}(t)),\quad t\in (0,1),\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3,\\ \alpha x^{(n-2)}(0)-\beta x^{(n-1)}(0)=\gamma x^{(n-2)}(1)+\tau x^{(n-1)}(1)=0, \end{gathered}$$ The authors in \cite{k3} studied the existence and nonexistence of solutions of a situation more general than \eqref{e18}. Lian and Wong \cite{l4} studied the existence of positive solutions of the following BVPs consisting of the $p$-Laplacian differential equation and Sturm-Liouville boundary conditions $$\label{e20} \begin{gathered} \big[\phi(x^{(n-1)}(t)\big]'+f(t,x(t),\dots,x^{(n-2)}(t))=0,\quad t\in (0,1),\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3,\\ \alpha x^{(n-2)}(0)-\beta x^{(n-1)}(0)=\gamma x^{(n-2)}(1)+\tau x^{(n-1)}(1)=0, \end{gathered}$$ In all above mentioned papers, all of the boundary conditions concerned are homogeneous cases. However, in many applications, BVPs are nonhomogeneous BVPc, for example, \begin{gather*} y''=\frac{1}{\lambda}(1+y^2)^{\frac{1}{2}},\quad t\in (a,b),\\ y(a)=a\alpha,\quad y(b)=\beta \end{gather*} and \begin{gather*} y''=-\frac{(1+y'(t))^2}{2(y(t)-\alpha)},\quad t\in (a,b),\\ y(a)=a\alpha,\quad y(b)=\beta \end{gather*} are very well known BVPs, which were proposed in 1690 and 1696, respectively. In 1964, The BVPs studied by Zhidkov and Shirikov in [USSR Computational Mathematics and Mathematical Physics, 4(1964)18-35] and by Lee in [Chemical Engineering Science, 21(1966)183-194] are nonhomogeneous BVPs too. There are also several papers concerning with the existence of positive solutions of BVPs for differential equations with non-homogeneous BCs. Ma \cite{m2} studied existence of positive solutions of the following BVP consisting of second order differential equations and three-point BC $$\label{e21} \begin{gathered}x''(t))+a(t)f(x(t))=0,\quad t\in (0,1),\\ x(0)=0,\quad x(1)-\alpha x(\eta)=b, \end{gathered}$$ In a recent paper \cite{k4,k5}, using lower and upper solutions methods, Kong and Kong established results for solutions and positive solutions of the following two problems $$\label{e22} \begin{gathered} x''(t))+f(t,x(t),x'(t))=0,\quad t\in (0,1),\\ x'(0)-\sum_{i=1}^m\alpha_i x'(\xi_i)=\lambda_1,\quad x(1)-\sum_{i=1}^m\beta_i x(\xi_i)=\lambda_2, \end{gathered}$$ and $$\label{e23} \begin{gathered} x''(t))+f(t,x(t),x'(t))=0,\quad t\in (0,1),\\ x(0)-\sum_{i=1}^m\alpha_i x(\xi_i)=\lambda_1,\quad x(1)-\sum_{i=1}^m\beta_i x(\xi_i)=\lambda_2, \end{gathered}$$ respectively. We note that the boundary conditions in \eqref{e17}, \eqref{e20}, \eqref{e21} and \eqref{e22} are two-parameter non-homogeneous BCs. The purpose of this paper is to investigate the more generalized BVPs for higher order differential equation with $p$-Laplacian subjected to non-homogeneous BCs, in which the nonlinearity $f$ contains $t,x,\dots,x^{(n-1)}$, i.e. the problems $$\label{e24} \begin{gathered} \big[\phi(x^{(n-1)}(t))\big]'+f(t,x(t),\dots,x^{(n-1)}(t))=0,\quad t\in (0,1),\\ x^{(n-2)}(0)-\sum_{i=1}^m\alpha_i x^{(n-2)}(\xi_i)=\lambda_1,\\ x^{(n-2)}(1)- \sum_{i=1}^m\beta_ix^{(n-2)}(\xi_i)=\lambda_2,\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3; \end{gathered}$$ $$\label{e25} \begin{gathered} \big[\phi(x^{(n-1)}(t))\big]'+f(t,x(t),\dots,x^{(n-1)}(t))=0,\quad t\in (0,1),\\ x^{(n-1)}(0)-\sum_{i=1}^m\alpha_i x^{(n-1)}(\xi_i)=\lambda_1,\\ x^{(n-2)}(1)- \sum_{i=1}^m\beta_ix^{(n-2)}(\xi_i)=\lambda_2,\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3; \end{gathered}$$ $$\label{e26} \begin{gathered} \big[\phi(x^{(n-1)}(t))\big]'+f(t,x(t),\dots,x^{(n-1)}(t))=0,\quad t\in (0,1),\\ x^{(n-2)}(0)-\sum_{i=1}^m\alpha_i x^{(n-2)}(\xi_i)=\lambda_1,\\ x^{(n-1)}(1)- \sum_{i=1}^m\beta_ix^{(n-1)}(\xi_i)=\lambda_2,\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3; \end{gathered}$$ $$\label{e27} \begin{gathered} \big[\phi(x^{(n-1)}(t))\big]'+f(t,x(t),\dots,x^{(n-1)}(t))=0,\quad t\in (0,1),\\ x^{(n-2)}(0)-\sum_{i=1}^m\alpha_i x^{(n-1)}(\xi_i)=\lambda_1,\\ x^{(n-2)}(1)+ \sum_{i=1}^m\beta_ix^{(n-1)}(\xi_i)=\lambda_2,\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3; \end{gathered}$$ and$$\label{e28} \begin{gathered} \big[\phi(x^{(n-1)}(t))\big]'+f(t,x(t),\dots,x^{(n-1)}(t))=0,\quad t\in (0,1),\\ \phi(x^{(n-1)}(0))-\sum_{i=1}^m\alpha_i\phi( x^{(n-1)}(\xi_i))=\lambda_1,\\ \theta(x^{(n-1)}(1))+ \sum_{i=1}^m\beta_i\theta(x^{(n-1)}(\xi_i))=\lambda_2,\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3; \end{gathered}$$ where $n\ge 2$, $0<\xi_i<\dots<\xi_m<1$, $\alpha_i,\beta_i\in \mathbb{R}$, $\lambda_1,\lambda_2\in \mathbb{R}$, $f$ is continuous, $\phi$ is called $p$-Laplacian, $\phi(x)=|x|^{p-2}x$ for $x\neq 0$ and $\phi(0)=0$ with $p>1$, its inverse function is denoted by $\phi^{-1}(x)$ with $\phi^{-1}(x)=|x|^{q-2}x$ for $x\neq 0$ and $\phi^{-1}(0)=0$, where $1/p+1/q=1$, $\theta$ is an odd increasing homeomorphisms from $R$ to $R$ with $\theta(0)=0$. We establish sufficient conditions for the existence of at least one positive solution of \eqref{e24}, \eqref{e25}, \eqref{e26}, \eqref{e27}, and at least one solution of \eqref{e28}, respectively. The first motivation of this paper is that it is of significance to investigate the existence of positive solutions of \eqref{e9} and \eqref{e10} since the operators defined in \cite{b1,b2,m2,m3} are can not be used; furthermore, it is more interesting to establish existence results for positive solutions of higher order BVPs with non-homogeneous BCs. The second motivation to study \eqref{e24}, \eqref{e25}, \eqref{e26}, \eqref{e27} and \eqref{e28} comes from the facts that \begin{itemize} \item[(i)] \eqref{e24} contains \eqref{e1}, \eqref{e3}, \eqref{e4}, \eqref{e5}, \eqref{e7} \eqref{e8}, \eqref{e9}, \eqref{e13}, \eqref{e14}, \eqref{e15}, \eqref{e17} and \eqref{e23} as special cases; \item[(ii)] \eqref{e25} contains \eqref{e6}, \eqref{e10} and \eqref{e22} as special cases; \item[(iii)] \eqref{e26} contains \eqref{e2} and \eqref{e16} as special cases; \item[(iv)] \eqref{e27} contains \eqref{e18} and \eqref{e19} as special cases; \item[(v)] \eqref{e28} contains \eqref{e11} and \eqref{e12} as special cases. \end{itemize} Furthermore, in most of the known papers, the nonlinearity $f$ only depends on a part of lower derivatives, the problem is that under what conditions problems have solutions when $f$ depends on all lower derivatives, such as in BVPs above, $f$ depends on $x,x',\dots,x^{(n-1)}$. The third motivation is that there exist several papers discussing the solvability of Sturm-Liouville type boundary-value problems for $p$-Laplacian differential equations, whereas there is few paper concerned with the solvability of Sturm-Liouville type multi-point boundary-value problems for $p$-Laplacian differential equations, such as \eqref{e27}. The fourth motivation comes from the challenge to find simple conditions on the function $f$, for the existence of a solution of \eqref{e28}, as the nonlinear homeomorphisms $\phi$ and $\theta$ generating, respectively, the differential operator and the boundary conditions are different. The techniques for studying the existence of positive solutions of multi-point boundary-value problems consisting of the higher-order differential equation with $p$-Laplacian and non-homogeneous BCs are few. Additional motivation is that the coincidence degree theory by Mawhin is reported to be an effective approach to the study the existence of periodic solutions of differential equations with or without delays, the existence of solutions of multi-point boundary-value problems at resonance case for differential equations; see for example \cite{l6,l8,l10,l12,l18} and the references therein, but there is few paper concerning the existence of positive solutions of non-homogeneous multi-point boundary-value problems for higher order differential equations with $p$-Laplacian by using the coincidence degree theory. The following of this paper is organized as follows: the main results and remarks are presented in Section 2, and some examples are given in Section 3. The methods used and the results obtained in this paper are different from those in known papers. Our theorems generalize and improve the known ones. \section{Main Results} In this section, we present the main results in this paper, whose proofs will be done by using the following fixed point theorem due to Mawhin. Let $X$ and $Y$ be real Banach spaces, $L:D(L)\subset X\to Y$ be a Fredholm operator of index zero, $P:X\to X$, $Q:Y\to Y$ be projectors such that $$\mathop{\rm Im}P=\mathop{\rm Ker}L,\quad \mathop{\rm Ker}Q =\mathop{\rm Im}L,\quad X=\mathop{\rm Ker}L\oplus \mathop{\rm Ker}P,\quad Y=\mathop{\rm Im}L \oplus \mathop{\rm Im}Q.$$ It follows that $$L|_{D(L)\cap \mathop{\rm Ker}P}:D(L)\cap \mathop{\rm Ker}P\to \mathop{\rm Im}L$$ is invertible, we denote the inverse of that map by $K_p$. If $\Omega$ is an open bounded subset of $X$, $D(L)\cap \overline{\Omega }\neq \emptyset$, 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{lemma}[\cite{g1}] \label{lem2.0.1} Let $L$ be a Fredholm operator of index zero and let $N$ be $L$-compact on $\Omega$. Assume that the following conditions are satisfied: \begin{itemize} \item[(i)] $Lx\neq \lambda Nx$ for every $(x,\lambda )\in [(D(L)\setminus\mathop{\rm Ker} L)\cap \partial \Omega ]\times (0,1)$; \item[(ii)] $Nx\notin \mathop{\rm Im} L$ for every $x\in \mathop{\rm Ker} L\cap \partial \Omega$; \item[(iii)] $\mathop{\rm deg}(\wedge QN\big|_{\mathop{\rm Ker} L},\;\Omega \cap \mathop{\rm Ker} L, 0)\neq 0$, where $\wedge:Y/\mathop{\rm Im} L\to \mathop{\rm Ker} L$ is an isomorphism. \end{itemize} Then the equation $Lx=Nx$ has at least one solution in $D(L)\cap \overline{\Omega}$. \end{lemma} In this paper, we choose $X=C^{n-2}[0,1]\times C^0[0,1]$ with the norm $$\|(x,y)\|=\max\{\|x\|_\infty,\dots,\|x^{(n-2)}\|_\infty,\|y\|_\infty\},$$ and $Y=C^0[0,1]\times C^0[0,1]\times R^2$ with the norm $$\|(x,y,a,b)\|=\max\{\|x\|_\infty,\|y\|_\infty,|a|,|b|\},$$ then $X$ and $Y$ are real Banach spaces. Let $$D(L)=\big\{(x_1,x_2)\in C^{n-1}[0,1]\times C^1[0,1]:x_1^{(i)}(0)=0,\; i=0,\dots,n-3\big\}.$$ Now we prove an important lemma. Then we will establish existence results for positive solutions of \eqref{e24}, \eqref{e25}, \eqref{e26}, \eqref{e27} and \eqref{e28} in sub-section 2.1, 2.2, 2.3, 2.4 and 2.5, respectively. \begin{lemma} \label{lem2.0.2} $\sum_{i=1}^ma_i^\sigma \le K_\sigma^{m-1}(\sum_{i=1}^ma_i)^\sigma$ for all $a_i\ge 0$ and $\sigma >0$, where $K_\sigma$ is defined by $K_\sigma=1$ for $\sigma\ge 1$ and $K_\sigma=2$ for $\sigma\in (0,1)$. \end{lemma} \begin{proof} {\bf Case 1.} $m=2$. Without loss of generality, suppose $a_1\ge a_2$. Let $g(x)=K_\sigma(1+x)^\sigma-(1+x^\sigma),x\in [1,+\infty)$, then $$g(1)=K_\sigma 2^\sigma-2 =\begin{cases} 2^\sigma-2\ge 0,&\sigma\ge 1,\\ 2^{\sigma+1}-2\ge 0,&\sigma\in (0,1) \end{cases}$$ and for $x\in [1,\infty)$, we get $$g'(x)=\sigma x^{\sigma-1}[K_\sigma(1+1/x)^{\sigma-1}-1] \ge \begin{cases} 0,&\sigma\ge1,\\ \sigma x^{\sigma-1}[2(1+1/1)^{0-1}-1]=0,&\sigma\in (0,1). \end{cases}$$ We get that $g(x)\ge g(1)$ for all $x\ge 1$ and so $1+x^\sigma \le K_\sigma(1+x)^\sigma$ for all $x\in [1,+\infty)$. Hence $a_1^\sigma+a_2^\sigma=a_2^\sigma [1+ (a_1/a_2)^\sigma]\le K_\sigma a_2^\sigma[1+a_1/a_2]^\sigma=K_\sigma(a_1+a_2)^\sigma$. \noindent {\bf Case 2.} $m>2$. It is easy to see that \begin{align*} \sum_{i=1}^ma_i^\sigma &= a_1^\sigma+a_2^\sigma+\sum_{i=3}^ma_i^\sigma \\ &\leq K_\sigma(a_1+a_2)^\sigma+\sum_{i=3}^ma_i^\sigma\\ &\leq K_\sigma\Big((a_1+a_2)^\sigma+\sum_{i=3}^ma_i^\sigma\Big)\\ &\leq K_\sigma\Big((a_1+a_2)^\sigma+a_3^\sigma+\sum_{i=4}^ma_i^\sigma\Big)\\ &\leq K_\sigma\Big(K_\sigma (a_1+a_2+a_3)^\sigma+\sum_{i=4}^ma_i^\sigma\Big)\\ &\leq K_\sigma^2\Big((a_1+a_2+a_3)^\sigma+\sum_{i=4}^ma_i^\sigma\Big)\\ &\leq \dots\\ &\leq K_\sigma^{m-1}\Big(\sum_{i=1}^ma_i\Big)^\sigma. \end{align*} The proof is complete. \end{proof} \begin{remark} \label{rmk1}\rm It is easy to see that $$\sum_{i=1}^m\phi(a_i)\le K_{p-1}^{m-1}\phi(\sum_{i=1}^ma_i),\quad \;\sum_{i=1}^m\phi^{-1}(a_i)\le K_{q-1}^{m-1}\phi^{-1}(\sum_{i=1}^ma_i).$$ \end{remark} \subsection{Positive solutions of Problem \eqref{e24}} Let $$f^*(t,x_0,\dots,x_{n-1})=f(t,\overline{x_0},\dots, \overline{x_{n-2}},x_{n-1}),(t,x_0,\dots,x_{n-1})\in [0,1]\times \mathbb{R}^n,$$ where $\overline{x}=\max\{0,x\}$. The following assumptions, which will be used in the proofs of all lemmas in this sub-section, are supposed. \begin{itemize} \item[(H1)] $f:[0,1]\times [0,+\infty)^{n-1}\times R\to [0,+\infty)$ is continuous with $f(t,0,\dots,0)\not\equiv 0$ on each sub-interval of [0,1]; \item[(H2)] $\lambda_1,\lambda_2\ge 0$, $\alpha_i\ge 0$, $\beta_i\ge 0$ satisfy $0<\sum_{i=1}^m\alpha_i<1,\quad 0<\sum_{i=1}^m\beta_i<1$ and $\lambda_1/(1-\sum_{i=1}^m\alpha_i)=\lambda_2/(1-\sum_{i=1}^m\beta_i)$; \item[(H3)] there exist continuous nonnegative functions $a$, $b_i$ and $c$ so that $$|f(t,x_0,\dots,x_{n-2},x_{n-1})|\le a(t)+\sum_{i=0}^{n-2}b_i(t)\phi(|x_i|)+c(t)\phi(|x_{n-1}|),$$ for $(t,x_0,\dots,x_{n-1})\in [0,1]\times R^n$; \item[(H4)] The following inequality holds \begin{align*} &K_{q-1}^{m-1}\phi\Big(1 +\frac{\sum_{i=1}^m\alpha_i\xi_i}{1-\sum_{i=1}^m\alpha_i}\Big)\Big[ \sum_{i=0}^{n-3}\phi\big(\frac{1}{(n-2-i)!}\big)\int_0^1b_i(s)ds\\ &+\int_0^1b_{n-2}(s)ds\Big]+\int_0^1c(s)ds<1. \end{align*} \end{itemize} We consider the problem $$\label{e29} \begin{gathered} \big[\phi(x^{(n-1)}(t))\big]'+f^*(t,x(t),\dots,x^{(n-1)}(t))=0,\quad t\in (0,1),\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3,\\ \;x^{(n-2)}(0)-\sum_{i=1}^m\alpha_i x^{(n-2)}(\xi_i)=\lambda_1,\\ x^{(n-2)}(1))- \sum_{i=1}^m\beta_ix^{(n-2)}(\xi_i)=\lambda_2. \end{gathered}$$ \begin{lemma} \label{lem2.1.1} If {\rm (H1)--(H2)} hold and $x$ is a solution of \eqref{e29}, then $x(t)>0$ for all $t\in (0,1)$, and $x$ is a positive solution of \eqref{e24}. \end{lemma} \begin{proof} (H1) implies that $[\phi(x^{(n-1)}(t)]'=-f^*(t,x(t),\dots,x^{(n-1)}(t))\le 0$ ($\not\equiv 0$), and then $x^{(n-1)}(t)$ is decreasing and so $x^{(n-2)}$ is concave on [0,1], thus $$\min_{t\in [0,1]}x^{(n-2)}(t)=\min\{x^{(n-2)}(0),x^{(n-2)}(1)\}.$$ Together with the boundary conditions in (29) and (H2), we get that $$\label{e30} x^{(n-2)}(0)=\sum_{i=1}^m\alpha_ix^{(n-2)}(\xi_i)+\lambda_1\ge \sum_{i=1}^m\alpha_i\min\{x^{(n-2)}(0),x^{(n-2)}(1)\},$$ and $$x^{(n-2)}(1)=\sum_{i=1}^m\beta_ix^{(n-2)}(\xi_i)+\lambda _2\ge \sum_{i=1}^m\beta_i\min\{x^{(n-2)}(0),x^{(n-2)}(1)\}.$$ Without loss of generality, assume that $\sum_{i=1}^m\alpha_i\ge \sum_{i=1}^m\beta_i$. If $\min\{x^{(n-2)}(0),x^{(n-2)}(1)\}<0$, then $$x^{(n-2)}(1)\ge \sum_{i=1}^m\beta_i\min\{x^{(n-2)}(0),x^{(n-2)}(1)\}\ge \sum_{i=1}^m\alpha_i\min\{x^{(n-2)}(0),x^{(n-2)}(1)\}.$$ Together with (30), we have $$\min\{x^{(n-2)}(0),x^{(n-2)}(1)\}\ge\sum_{i=1}^m\alpha_i \min\{x^{(n-2)}(0),x^{(n-2)}(1)\}.$$ Hence $\min\{x^{(n-2)}(0),x^{(n-2)}(1)\}\ge 0$. It follows that $\min\{x^{(n-2)}(0),x^{(n-2)}(1)\}\ge 0$. So (H1) implies that $x^{(n-2)}(t)>0$ for all $t\in (0,1)$. Then from the boundary conditions, we get $x^{(i)}(t)>0$ for all $t\in (0,1)$ and $i=0,\dots,n-3$. Then $f^*(t,x(t),\dots,x^{(n-1)}(t))=f(t,x(t),\dots,x^{(n-1)}(t))$. Thus $x$ is a positive solution of \eqref{e24}. The proof is complete. \end{proof} \begin{lemma} \label{lem2.1.2} If {\rm (H1)--(H2)} hold and $x$ is a solutions of \eqref{e29}, then there exists $\xi\in [0,1]$ such that $x^{(n-1)}(\xi)=0$. \end{lemma} \begin{proof} In fact, if $x^{(n-1)}(t)>0$ for all $t\in [0,1]$, then $$x^{(n-2)}(0)=\sum_{i=1}^m\alpha_ix^{(n-2)}(\xi_i)+\lambda_1> \sum_{i=1}^m\alpha_ix^{(n-2)}(0)+\lambda _1,$$ then $x^{(n-2)}(0)>\lambda_1/(1-\sum_{i=1}^m\alpha_i)$, it follows that $x^{(n-2)}(1)>\lambda_1/(1-\sum_{i=1}^m\alpha_i)$. On the other hand, $$x^{(n-2)}(1)=\sum_{i=1}^m\beta_ix^{(n-2)}(\xi_i)+\lambda_2< \sum_{i=1}^m\beta_ix^{(n-2)}(1)+\lambda_2,$$ thus $$x^{(n-2)}(1)<\lambda_2/(1-\sum_{i=1}^m\beta_i) =\lambda_1/(1-\sum_{i=1}^m\alpha_i) \sum_{i=1}^m\beta_ix^{(n-2)}(1)+\lambda_2,$$ then $x^{(n-2)}(1)>\lambda_2/(1-\sum_{i=1}^m\beta_i)$, it follows that $x^{(n-2)}(0)>\lambda_2/(1-\sum_{i=1}^m\beta_i)$. On the other hand, $$x^{(n-2)}(0)=\sum_{i=1}^m\alpha_ix^{(n-2)}(\xi_i)+\lambda_1< \sum_{i=1}^m\alpha_ix^{(n-2)}(0)+\lambda_1,$$ thus $$x^{(n-2)}(0)<\lambda_1/(1-\sum_{i=1}^m\alpha_i) =\lambda_2/(1-\sum_{i=1}^m\beta_i)0 so that \|x_2\|_\infty\le M. Since |x_1^{(i)}(t)|\le \frac{1}{(n-3-i)!}\|x_1^{(n-2)}\|_\infty and |x_1^{(n-2)}(t)|\le (1+\frac{\sum_{i=1}^m\alpha_i\xi_i}{1-\sum_{i=1}^m\alpha_i})\phi^{-1}(\|x_2\|_\infty)+ \frac{\lambda_1}{1-\sum_{i=1}^m\alpha_i}, there exist constants M_i>0 so that \|x_1^{(i)}\|_\infty\le M_i for all i=0,\dots,n-2. Then \Omega_0 is bounded. The proof is complete. \end{proof} \begin{lemma} \label{lem2.1.5} Suppose that (H2) holds. Then there exists a constant M_1'>0 such that for each x=(0,c)\in \mathop{\rm Ker}L, if N(0,c)\in \mathop{\rm Im}L, we get that |c|\le M_1'. \end{lemma} \begin{proof} For each x=(0,c)\in \mathop{\rm Ker}L, if N(0,c)\in \mathop{\rm Im}L, we get$$ \big(\phi^{-1}(c),-f^*(t,0,\dots,0,\phi^{-1}(c)),\lambda_1,\lambda_2\big) \in \mathop{\rm Im}L. Then \begin{align*} &\frac{1}{1-\sum_{i=1}^m\alpha_i}\Big(\sum_{i=1}^m\alpha_i\int_0^{\xi_i} \phi^{-1}(c)ds+\lambda_1\Big)\\ &+\frac{1}{1-\sum_{i=1}^m\beta_i}\Big(\int_0^1\phi^{-1}(c)ds -\sum_{i=1}^m\beta_i\int_0^{\xi_i}\phi^{-1}(c)ds-\lambda_2\Big) =0. \end{align*} It follows that \phi^{-1}(c)=\Big(\frac{\sum_{i=1}^m\alpha_i\xi_i}{1-\sum_{i=1}^m\alpha_i} +\frac{1-\sum_{i=1}^m\beta_i\xi_i} {1-\sum_{i=1}^m\beta_i}\Big)^{-1} \Big(\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i} +\frac{\lambda_1 }{1-\sum_{i=1}^m\alpha_i}\Big). So there exists a constant M_1'>0 such that |c|\le M_1'. The proof is complete. \end{proof} \begin{lemma} \label{lem2.1.6} Suppose that {\rm (H2)} holds. Then there exists a constant M_2'>0 such that for each x=(0,c)\in \mathop{\rm Ker}L, if \lambda\wedge^{-1}(0,c)+(1-\lambda)\mathop{\rm sgn}(\Delta)QN(0,c)=0, then |c|\le M_2'. \end{lemma} \begin{proof} For each x=(0,c)\in \mathop{\rm Ker}L, if \lambda\wedge^{-1}(0,c)+(1-\lambda)\mathop{\rm sgn}(\Delta)QN(0,c)=0, we get \begin{align*} \lambda c&=-(1-\lambda )\mathop{\rm sgn}(\Delta)\frac{1}{\Delta} \Big[\Big(\frac{\sum_{i=1}^m\alpha_i\xi_i}{1-\sum_{i=1}^m\alpha_i}+\frac{1-\sum_{i=1}^m\beta_i\xi_i} {1-\sum_{i=1}^m\beta_i}\Big)\phi^{-1}(c)\\ &\quad +\frac{\lambda_1}{1-\sum_{i=1}^m\alpha_i}+\frac{\lambda_2 }{1-\sum_{i=1}^m\beta_i}\Big]. \end{align*} Thus \begin{align*} \lambda c^2&= -(1-\lambda )\mathop{\rm sgn}(\Delta)\frac{1}{\Delta} \Big[\Big(\frac{\sum_{i=1}^m\alpha_i\xi_i}{1-\sum_{i=1}^m\alpha_i} +\frac{1-\sum_{i=1}^m\beta_i\xi_i} {1-\sum_{i=1}^m\beta_i}\Big)\phi^{-1}(c)c\\ &\quad +\Big(\frac{\lambda_1}{1-\sum_{i=1}^m\alpha_i}+\frac{\lambda_2}{1 -\sum_{i=1}^m\beta_i}\Big)c\Big]. \end{align*} If \lambda=1, then c=0. If \lambda\in [0,1), since q>1,\quad \frac{\sum_{i=1}^m\alpha_i\xi_i}{1-\sum_{i=1}^m\alpha_i} +\frac{1-\sum_{i=1}^m\beta_i\xi_i} {1-\sum_{i=1}^m\beta_i}>0, one sees, for sufficiently large |c|, that \begin{align*} \lambda c^2&= -(1-\lambda)\mathop{\rm sgn}(\Delta)\frac{1}{\Delta} \Big[\Big(\frac{\sum_{i=1}^m\alpha_i\xi_i}{1-\sum_{i=1}^m\alpha_i} +\frac{1-\sum_{i=1}^m\beta_i\xi_i} {1-\sum_{i=1}^m\beta_i}\Big)|c|^q \\ &\quad +\Big(\frac{\lambda_1}{1-\sum_{i=1}^m\alpha_i} +\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}\Big)c\Big]<0 \end{align*} a contradiction. So there exists a constant M_2'>0 such that |c|\le M_2'. The proof is complete. \end{proof} \begin{theorem} \label{thm1} Suppose {\rm (H1)--(H4)} hold. Then \eqref{e24} has at least one positive solution. \end{theorem} \begin{proof} Let \Omega\supseteq \Omega_0 be a bounded open subset of X centered at zero with its diameter greater than \max\{M_1',M_2'\}. It follows from Lemmas \ref{lem2.1.4}, \ref{lem2.1.5}, \ref{lem2.1.6} that Lx\neq \lambda Nx for all (x,\lambda )\in [(D(L)\setminus \mathop{\rm Ker} L)\cap \partial \Omega ] \times (0,1); Nx\notin \mathop{\rm Im} L for every x\in \mathop{\rm Ker} L\cap \partial \Omega ; \mathop{\rm deg}(\wedge QN\big|_{\mathop{\rm Ker} L},\;\Omega \cap \mathop{\rm Ker} L, 0)\neq 0. Since (H1) holds, L be a Fredholm operator of index zero and N be L-compact on \Omega. It follows from Lemma \ref{lem2.0.1} that Lx=Nx has at least one solution x=(x_1,x_2). Then x_1 is a solution of \eqref{e29}. We note that x_1^{(i)}(t)\ge 0 for t\in [0,1] and i=0,\dots,n-2, so $f^*\big(t,x_1(t),\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2(t))\big) =f\big(t,x_1(t),\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2(t))\big).$ Hence x_1 is a positive solution of \eqref{e24}. The proof is complete. \end{proof} \begin{remark} \label{rmk2.1} \rm The operator defined in \cite{b2} can not be used, so we follow a different method. Theorem \ref{thm1} also generalizes and improves the results in \cite{d1,g5,l5,l13,m3}. \end{remark} \subsection{Positive solutions of Problem \eqref{e25}} Let f^*(t,x_0,\dots,x_{n-1})=f(t,\overline{x_0},\dots,\overline{x_{n-2}}, \underline{x_{n-1}}),\quad (t,x_0,\dots,x_{n-1})\in [0,1]\times \mathbb{R}^n, and \overline{x}=\max\{0,x\} and \underline{y}=\min\{0,y\}. We consider the problem $$\label{e32} \begin{gathered} \big[\phi(x^{(n-1)}(t))\big]'+f^*(t,x(t),\dots,x^{(n-1)}(t))=0,\quad t\in (0,1),\\ x^{(n-1)}(0)-\sum_{i=1}^m\alpha_i x^{(n-1)}(\xi_i)=\lambda_1,\\ x^{(n-2)}(1)- \sum_{i=1}^m\beta_ix^{(n-2)}(\xi_i)=\lambda_2,\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3, \end{gathered}$$ Suppose (H3) and the following assumptions, which will be used in the proof of all lemmas in this sub-section. \begin{itemize} \item[(H5)] f:[0,1]\times [0,+\infty)^{n-1}\times (-\infty,0]\to [0,+\infty) is continuous and f(t,0,\dots,0)\not\equiv 0 on each sub-interval of [0,1]; \item[(H6)] \lambda_1\le 0, \lambda_2\ge 0, \alpha_i,\beta_i\ge 0 with \sum_{i=1}^m\phi(\alpha_i)<1/\phi(K^m_{q-1}), \sum_{i=1}^m\alpha_i<1 and \sum_{i=1}^m\beta_i<1. \item[(H7)] The following inequality holds \begin{align*} &\Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\Big)\\ &\times \Big[\|b_{n-2}\|_\infty\phi(K_{q-1}) \phi\Big(1+\frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\Big)\\ &+\sum_{i=0}^{n-3}\|b_i\|_\infty\phi(K_{q-1})\phi\big(\frac{1}{(n-2-i)!}\big) \phi\Big(1+\frac{\sum_{i=1}^m\beta_i(1-\xi_i)}{1-\sum_{i=1}^m\beta_i}\big) +\|c\|_\infty\Big]<1. \end{align*} \end{itemize} \begin{lemma} \label{lem2.2.1} If (x_1,x_2) is a solution of the problem $$\label{e33} \begin{gathered} x_1^{(n-1)}(t)=\phi^{-1}(x_2(t)),\quad t\in [0,1],\\ x_2'(t)=-f^*(t,x_1(t),\dots,x^{(n-2)}_1(t),\phi^{-1}(x_2(t))),\quad t\in [0,1],\\ \phi^{-1}(x_2(0))-\sum_{i=1}^{m}\alpha_i\phi^{-1}(x_2(\xi_i))=\lambda_1,\\ x_1^{(n-2)}(1)- \sum_{i=1}^{n}\beta_ix_1^{(n-2)}(\xi_i)=\lambda_2,\\ x_1^{(i)}(0)=0,\quad i=0,\dots,n-3, \end{gathered}$$ then x_1 is a solution of \eqref{e32}. \end{lemma} The proof of the above lemma is simple and is omitted. \begin{lemma} \label{lem2.2.2} If {(H5)--(H6)} hold and x is a solution of \eqref{e32}, then x(t)>0 for all t\in (0,1), and x is a positive solution of \eqref{e25}. \end{lemma} \begin{proof} Firstly, since [\phi(x^{(n-1)}(t)]'=-f^*(t,x(t),\dots,x^{(n-1)}(t))\le 0 and \alpha_i\ge 0,\sum_{i=1}^m\alpha_i<1 and \lambda_1\le 0, we have, using \eqref{e6}, that x^{(n-1)}(0)=\sum_{i=1}^m\alpha_ix^{(n-1)}(\xi_i)+\lambda_1\le \sum_{i=1}^m\alpha _ix^{(n-1)}(0). Hence x^{(n-1)}(0)\le 0 and (H6). We get x^{(n-1)}(t)\le 0 for all t\in [0,1]. Since x^{(n-1)}(t)\le 0 for all t\in [0,1], we get x^{(n-2)}(1)=\sum_{i=1}^m\beta_ix^{(n-2)}(\xi_i)+\lambda_2\ge \sum_{i=1}^m\beta_ix^{(n-2)}(1). So one gets x^{(n-2)}(1)\ge 0. Thus we get x^{(n-2)}(t)>0 for all t\in (0,1) since x^{(n-1)}(t)\le 0 for all t\in [0,1]. It follows from the boundary conditions that x^{(i)}(t)>0 for all t\in (0,1), i=0,\dots, n-3. Then f^*(t,x(t),\dots,x^{(n-1)}(t))=f(t,x(t),\dots,x^{(n-1)}(t)). Thus x is a positive solution of \eqref{e25}. The proof is complete. \end{proof} Let \lambda\in (0,1), consider the problem $$\label{e34} \begin{gathered} x_1^{(n-1)}(t)=\lambda \phi^{-1}(x_2(t)),\quad t\in [0,1],\\ x_2'(t)=-\lambda f^*(t,x_1(t),\dots,x_1^{(n-2)}(t), \phi^{-1}(x_2(t))),\quad t\in [0,1],\\ 0=\lambda (\phi^{-1}(x_2(0))-\sum_{i=1}^{m} \alpha_i\phi^{-1}(x_2(\xi_i))-\lambda_1),\\ x_1^{(n-2)}(1)-\sum_{i=1}^{n}\beta_ix_1^{(n-2)}(\xi_i) =\lambda \lambda_2,\\ x_1^{(i)}(0)=0,\quad i=0,\dots,n-3. \end{gathered}$$ \begin{lemma} \label{lem2.2.3} Suppose {\rm (H5)--(H6)} hold. If (x_1,x_2) is a solution of \eqref{e34}, then \begin{align*} &\|x_2\|_\infty \\ &\le \|x_2'\|_\infty\Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\Big) +\frac{\phi(K^m_{q-1})|\lambda_1|}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}. \end{align*} \end{lemma} \begin{proof} Since (H5)--(H6) and \eqref{e34} imply that x_2'(t)\le 0 for all t\in [0,1]. Similar to the discussion of Lemma \ref{lem2.2.2}, \lambda_1\le 0 and using \eqref{e34}, we have \phi^{-1}(x_2(0))=\sum_{i=1}^m\alpha_i\phi^{-1}(x_2(\xi_i)) +\lambda_1\le\sum_{i=1}^m\alpha_i\phi^{-1}(x_2(\xi_i))\le \sum_{i=1}^m\alpha_i\phi^{-1}(x_2(0)). $$This together with (H6), one sees that x_2(0)\le 0. Then x_2(t)\le 0 for all t\in [0,1]. It follows from$$ \phi^{-1}(x_2(0))-\sum_{i=1}^{m}\alpha_i\phi^{-1}(x_2(\xi_i))-\lambda_1=0 $$and Lemma \ref{lem2.0.2} that$$ -\phi^{-1}(x_2(0))\le K^m_{q-1}\phi^{-1}(-\sum_{i=1}^m\phi(\alpha_i)x_2(\xi_i)-\phi(\lambda_1)). Hence we get -x_2(0)\le-\phi(K^m_{q-1})\Big(\sum_{i=1}^m\phi(\alpha_i)x_2(\xi_i) -\phi(\lambda_1)\Big). Thus, from (H6), we see, there is \eta_i\in [0,\xi_i], that \begin{align*} -x_2(0)&= \frac{1}{1-\phi(K^m_{q-1})\sum_{i=1}^m\phi(\alpha_i)} \Big(-x_2(0)+\phi(K^m_{q-1})\sum_{i=1}^m\phi(\alpha_i)x_2(0)\Big)\\ &\leq \frac{1}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\Big(-\phi(K^m_{q-1})\sum_{i=1}^m \phi(\alpha_i)x_2(\xi_i) \\ &\quad +\phi(K^m_{q-1})\sum_{i=1}^m\phi(\alpha_i)x_2(0) +\phi(K^m_{q-1}|\lambda_1|)\Big)\\ &= \frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i[-x_2'(\eta_i)] +\frac{\phi(K^m_{q-1}|\lambda_1|)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\\ &\leq \|x_2'\|_\infty\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i +\frac{\phi(K^m_{q-1}|\lambda_1|)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}. \end{align*} Hence we get \begin{align*} |x_2(t)| &\leq |x_2(t)-x_2(0)|+|x_2(0)| \\ &\leq \|x_2'\|_\infty+\|x_2'\|_\infty\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\\ &\quad +\frac{\phi(K^m_{q-1}|\lambda_1|)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}. \end{align*} Thus \begin{align*} &\|x_2\|_\infty\\ &\le \|x_2'\|_\infty\Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\Big) +\frac{\phi(K^m_{q-1}|\lambda_1|)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}. \end{align*} \end{proof} \begin{lemma} \label{lem2.2.4} Suppose {\rm (H5)--(H6)} hold. If (x_1,x_2) is a solution of \eqref{e34}, then \|x_1^{(n-2)}\|_\infty\le\phi^{-1}(\|x_2\|_\infty) \Big(1+\frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\Big)+\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}. \end{lemma} \begin{proof} In fact, \begin{align*} |x_1^{(n-2)}(1)| &= \frac{1}{1-\sum_{i=1}^m\beta_i}\left|x_1^{(n-2)}(1) -\sum_{i=1}^m\beta_ix_1^{(n-2)}(1)\right|\\ &\leq \frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\|x_1^{(n-1)}\|_\infty +\frac{\lambda_2}{1 -\sum_{i=1}^m\beta_i}\\ &\leq \frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\phi^{-1}(\|x_2\|_\infty) +\frac{\lambda_2}{1 -\sum_{i=1}^m\beta_i}. \end{align*} Then we get \begin{align*} |x_1^{(n-2)}(t)| &\leq |x_1^{(n-2)}(t)-x_1^{(n-2)}(1)|+|x_1^{(n-2)}(1)|\\ &\leq \frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\phi^{-1}(\|x_2\|_\infty) +\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}+\|x_1^{(n-1)}\|_\infty\\ &\leq \frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\phi^{-1}(\|x_2\|_\infty) +\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}+\phi(\|x_2\|_\infty). \end{align*} Then \|x_1^{(n-2)}\|_\infty\le\phi^{-1}(\|x_2\|_\infty) \Big(1+\frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\Big)+\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}. and for i=0,\dots,n-3, \begin{align*} \|x_1^{(i)}\|_\infty&\leq \frac{1}{(n-i-2)!}\|x_1^{(n-2)}\|_\infty\\ &\leq \frac{1}{(n-i-2)!}\Big[\phi^{-1}(\|x_2\|_\infty) \Big(1+\frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\Big)\\ &\quad +\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}\Big]. \end{align*} Define the operators \begin{gather*} L(x_1,x_2)=((x_1^{(n-1)},x_2',0,x_1^{(n-2)}(1) -\sum_{i=1}^{n}\beta_ix_1^{(n-2)}(\xi_i)),\quad (x_1,x_2)\in X\cap D(L), \\ \begin{aligned} N(x_1,x_2)&=(\phi^{-1}(x_2),-f^*(t,x_1,\phi^{-1}(x_2), \phi^{-1}(x_2(0))\\ &\quad -\sum_{i=1}^{m}\alpha_i\phi^{-1}(x_2(\xi_i)) -\lambda_1,\lambda_2),\quad(x_1,x_2)\in X. \end{aligned} \end{gather*} Suppose {\rm (H5)--(H6)} hold. It is easy to show the following results: \begin{itemize} \item[(i)] \mathop{\rm Ker}L=\{(0,c):c\in \mathbb{R}\} and \mathop{\rm Im}L=\{(y_1,y_2,a,b):a=0\}; \item[(ii)] L is a Fredholm operator of index zero; \item[(iii)] There are projectors P:X\to X and Q:Y\to Y such that \mathop{\rm Ker} L=\mathop{\rm Im} P and \mathop{\rm Ker} Q=\mathop{\rm Im} L. Furthermore, let \Omega \subset X be an open bounded subset with \overline{\Omega }\cap D(L)\neq \emptyset, then N is L-compact on \overline{\Omega}; \item[(iv)] x=(x_1,x_2) is a solution of \eqref{e33} if and only if x is a solution of the operator equation Lx=Nx in D( L). \end{itemize} We present the projectors P and Q as follows: P(x_1,x_2)=(0,x_2(0)) for all x=(x_1,x_2)\in X and Q(y_1,y_2,a,b)=(0,0,a,0). The generalized inverse of L:D(L)\cap \mathop{\rm Ker}P\to \mathop{\rm Im}L is defined by \begin{align*} K_P(y_1,y_2,a,b) &=\Big(\int_0^t\frac{(t-s)^{n-2}}{(n-2)!}y_1(s)ds-\frac{t^{n-2}}{(n-2)!} \frac{1}{1-\sum_{i=1}^m\beta_i}\Big(\int_0^1y_1(s)ds\\ &\quad -\sum_{i=1}^m\beta_i\int_0^{\xi_i}y_1(s)ds+b\Big), \int_0^ty_2(s)ds\Big), \end{align*} the isomorphism \wedge:Y/\mathop{\rm Im}L\to \mathop{\rm Ker}L is defined by \wedge(0,0,c,0)=(0,c). \end{proof} \begin{lemma} \label{lem2.2.5} Suppose {\rm (H3), (H5)--(H7)} hold. Then the set \Omega_0=\big\{(x_1,x_2)\in D(L)\setminus \mathop{\rm Ker}L:L(x_1,x_2)=\lambda N(x_1,x_2) \text{ for some }\lambda \in (0,1)\big\} is bounded. \end{lemma} \begin{proof} It follows from \eqref{e34}, (H3), Lemmas \ref{lem2.0.2}, \ref{lem2.2.3} and \ref{lem2.2.4} that \begin{align*} &\|x_2\|_\infty\\ &\leq \|x_2'\|_\infty\Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\Big) +\frac{\phi(K^m_{q-1}|\lambda_1|)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\\ &\leq \max_{t\in [0,1]}|f^*(t, x_1(t),\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2(t))|\\ &\quad\times \Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\Big) +\frac{\phi(K^m_{q-1}|\lambda_1|)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\\ &\leq \Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\Big)\\ &\quad \times \Big(\|a\|_\infty+ \sum_{i=0}^{n-2}\|b_i\|_\infty\phi(\|x_1^{(i)}\|_\infty)+ \|c\|_\infty\phi(\phi^{-1}(\|x_2\|_\infty))\Big)\\ &\quad +\frac{\phi(K^m_{q-1}|\lambda_1|)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\\ &\leq \Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\Big)\\ &\quad\times \Big\{\|a\|_\infty+\|b_{n-2}\|_\infty \phi\Big(\phi^{-1}(\|x_2\|_\infty) \big(1+\frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\big) \\ &\quad +\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}\Big) \\ &\quad +\sum_{i=0}^{n-3}\|b_i\|_\infty \phi\Big(\frac{1}{(n-2-i)!}\Big[\phi^{-1}(\|x_2\|_\infty) \big(1+\frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\big)\\ &\quad +\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}\Big]\Big) +\|c\|_\infty\|x_2\|_\infty\Big\} +\frac{\phi(K^m_{q-1}|\lambda_1|)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\\ &\leq \Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\Big)\\ &\quad\times \Big[\|a\|_\infty+ \Big(\|b_{n-2}\|_\infty\phi(K_{q-1})\phi\big(1+\frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\big)+\|c\|_\infty\Big)\|x_2\|_\infty \\ &\quad +\|b_{n-2}\|_\infty\phi(K_{q-1}) \phi\big(\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}\big) +\sum_{i=0}^{n-3}\|b_i\|_\infty\phi(K_{q-1}) \phi\big(\frac{1}{(n-2-i)!}\big)\|x_2\|_\infty\\ &\quad\times \phi\Big(1+ \frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\Big) \\ &\quad +\sum_{i=0}^{n-3}\|b_i\|_\infty\phi\big(\frac{1}{(n-2-i)!}\big) \phi(K_{q-1})\phi(\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i})\Big]\\ &\quad +\frac{\phi(K^m_{q-1}|\lambda_1|)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}. \end{align*} We get \begin{align*} &\Big\{1-\Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\Big)\\ &\times\Big[\|b_{n-2}\|_\infty\phi(K_{q-1}) \phi\Big(1+\frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\Big)\\ & +\sum_{i=0}^{n-3}\|b_i\|_\infty\phi(K_{q-1}) \phi\big(\frac{1}{(n-2-i)!}\big) \phi\Big(1+ \frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\Big) \\ &+\|c\|_\infty\Big]\Big\}\|x_2\|_\infty \\ &\leq \Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}\sum_{i=1}^m\phi(\alpha_i)\xi_i\Big) \Big(\|a\|_\infty+\phi(K_{q-1}) \phi\big(\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}\big)\\ &\quad\times\|b_{n-2}\|_\infty +\sum_{i=0}^{n-3}\|b_i\|_\infty\phi\big(\frac{1}{(n-2-i)!}\big) \phi(K_{q-1})\phi\big(\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}\big)\Big)\\ &\quad +\frac{\phi(K^m_{q-1}|\lambda_1|)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\alpha_i)}. \end{align*} It follows from (H7) that there is a constant M>0 so that \|x_2\|_\infty\le M. Thus, from Lemma \ref{lem2.2.4}, \|x_1^{(n-2)}\|_\infty\le\phi^{-1}(M) \Big(1+\frac{1}{1-\sum_{i=1}^m\beta_i}\sum_{i=1}^m \beta_i(1-\xi_i)\Big)+\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i}=:M_{n-2}, $$and there exist constants M_i>0 such that \|x_1^{(i)}\|_\infty\le M_i for i=0,\dots,n-3. So \Omega_0 is bounded. The proof is complete. \end{proof} \begin{lemma} \label{lem2.2.6} Suppose {\rm (H5)--(H7)} hold. If (0,c)\in \mathop{\rm Ker}L and N(0,c)\in \mathop{\rm Im}L, then there exists a constant M_1'>0 such that |c|\le M_1'. \end{lemma} \begin{proof} If (0,c)\in \mathop{\rm Ker}L and N(0,c)\in \mathop{\rm Im}L, we get \phi^{-1}(c)-\sum_{i=1}^m\alpha_i\phi^{-1}(c)=\lambda_1. Then (H6) implies that there is M_1'>0 such that |c|\le M_1'. \end{proof} \begin{lemma} \label{lem2.2.7} Suppose {\rm (H5)--(H7)} hold. If (0,c)\in \mathop{\rm Ker}L with \lambda \wedge^{-1}(0,c)+(1-\lambda)QN(0,c)=0, then there exists a constant M_2'>0 such that |c|\le M_2'. \end{lemma} \begin{proof} If (0,c)\in \mathop{\rm Ker}L with \lambda \wedge^{-1}(0,c)+(1-\lambda)QN(0,c)=0, then$$ \lambda c=-(1-\lambda )\Big(\phi^{-1}(c)-\sum_{i=1}^m\alpha_i\phi^{-1}(c)-\lambda_1\Big). $$It follows that$$ \lambda c^2=-c\phi^{-1}(c)(1-\lambda )\Big(1-\sum_{i=1}^m\alpha_i-\frac{\lambda_1}{\phi^{-1}(c)}\Big). $$It is easy to see that there is M_2'>0 so that |c|\le M_2'. \end{proof} \begin{theorem} \label{thm2} Suppose {\rm (H3), (H5)--(H7)} hold. Then \eqref{e25} has at least one positive solution. \end{theorem} \begin{proof} Let \Omega\supseteq \Omega_0 be a bounded open subset of X centered at zero with its diameter greater than \max\{M_1',M_2'\}. Then Lemmas \ref{lem2.2.5}, \ref{lem2.2.6} and \ref{lem2.2.7} imply that Lx\neq \lambda Nx for all (x,\lambda )\in [(D(L)\setminus \mathop{\rm Ker} L)\cap \partial \Omega ] \times (0,1); Nx\notin \mathop{\rm Im} L for every x\in \mathop{\rm Ker}L\cap\partial \Omega ; \mathop{\rm deg}(\wedge QN\big|_{\mathop{\rm Ker} L}, \Omega \cap \mathop{\rm Ker} L, 0)\neq 0. Since (H5) holds, we let L be a Fredholm operator of index zero and N be L-compact on \Omega. It follows from Lemma \ref{lem2.0.1} that Lx=Nx has at least one solution x=(x_1,x_2). Then x_1 is a solution of \eqref{e32}. We note that x_1^{(i)}(t)\ge 0 for t\in [0,1] and i=0,\dots,n-2, and x_2(t)\le 0 for all t\in [0,1], so $f^*\big(t,x_1(t),\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2(t))\big) =f\big(t,x_1(t),\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2(t))\big).$ Hence x_1 is a positive solution of \eqref{e25}. \end{proof} \begin{remark} \label{rmk2.2}\rm The operator defined in \cite{b1} can not be used, so follow a different method. Theorem \ref{thm2} generalizes and improves the theorems in \cite{b1,g5,k5,m5}. \end{remark} \subsection{Positive solutions of Problem \eqref{e26}} Let$$ f^*(t,x_0,\dots,x_{n-1})=f(t,\overline{x_0},\dots,\overline{x_{n-2}}, \overline{x_{n-1}}),\quad (t,x_0,\dots,x_{n-1})\in [0,1]\times \mathbb{R}^n, and \overline{x}=\max\{0,x\}. We consider the problem $$\label{e35} \begin{gathered} \big[\phi(x^{(n-1)}(t))\big]'+f^*(t,x(t),\dots,x^{(n-1)}(t))=0,\quad t\in (0,1),\\ x^{(n-2)}(0)-\sum_{i=1}^m\alpha_i x^{(n-2)}(\xi_i)=\lambda_1,\\ x^{(n-1)}(1)-\sum_{i=1}^m\beta_ix^{(n-1)}(\xi_i)=\lambda_2,\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3, \end{gathered}$$ Problem \eqref{e31} can be transformed into $$\label{e36} \begin{gathered} x_1^{(n-1)}(t)=\phi^{-1}(x_2(t)),\quad t\in [0,1],\\ x_2'(t)=-f^*(t,x_1(t),\dots,x^{(n-2)}_1(t),\phi^{-1}(x_2(t)),\quad t\in [0,1],\\ x_1^{(n-2)}(0)-\sum_{i=1}^{m}\alpha_ix_1^{(n-2)}(\xi_i)=\lambda_1,\\ 0=\phi^{-1}(x_2(1))- \sum_{i=1}^{n}\beta_i\phi^{-1}(x_2(\xi_i))-\lambda_2,\\ x_1^{(i)}(0)=0,\quad i=0,\dots,n-3. \end{gathered}$$ Suppose that (H3) and the following assumptions hold. \begin{itemize} \item[(H8)] f:[0,1]\times [0,+\infty)^{n}\to [0,+\infty) is continuous with f(t,0,\dots,0)\not\equiv 0 on each sub-interval of [0,1]; \item[(H9)] \lambda_1\ge 0, \lambda_2\ge 0, \alpha_i,\beta_i\ge 0 with \sum_{i=1}^m\phi(\beta_i)<1/\phi(K^m_{q-1}), \sum_{i=1}^m\alpha_i<1 and \sum_{i=1}^m\beta_i<1. \item[(H10)] The following inequality holds \begin{align*} &\Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\beta_i)}\sum_{i=1}^m\phi(\beta_i)(1-\xi_i)\Big)\\ &\Big[\|b_{n-2}\|_\infty\phi(K_{q-1}) \phi\Big(1+\frac{1}{1-\sum_{i=1}^m\alpha_i}\sum_{i=1}^m \alpha_i\xi_i\Big)\\ &+\sum_{i=0}^{n-3}\|b_i\|_\infty\phi(K_{q-1}) \phi\big(\frac{1}{(n-2-i)!}\big) \phi\big(\frac{\sum_{i=1}^m\alpha_i\xi_i}{1-\sum_{i=1}^m\alpha_i}\big) +\|c\|_\infty\Big]<1. \end{align*} \end{itemize} Define the operators \begin{gather*} L(x_1,x_2)=\Big(x_1^{(n-1)},x_2',x_1^{(n-2)}(0) -\sum_{i=1}^{n}\alpha_ix_1^{(n-2)} (\xi_i),0\Big),\quad (x_1,x_2)\in X\cap D(L),\\ \begin{aligned} N(x_1,x_2)&=\Big(\phi^{-1}(x_2),-f^*(t,x_1(t),\dots,x_1^{(n-2)}(t)), \lambda_1, \phi^{-1}(x_2(1))\\ &\quad -\sum_{i=1}^{m}\beta_i\phi^{-1}(x_2(\xi_i) -\lambda_2)\Big) \end{aligned} \end{gather*} for (x_1,x_2)\in X. It is easy to show the following results: \begin{itemize} \item[(i)] \mathop{\rm Ker}L=\{(0,c):c\in \mathbb{R}\} and \mathop{\rm Im}L=\{(y_1,y_2,a,b):b=0\}; \item[(ii)] L is a Fredholm operator of index zero; \item[(iii)] There are projectors P:X\to X and Q:Y\to Y such that \mathop{\rm Ker} L=\mathop{\rm Im} P and \mathop{\rm Ker} Q=\mathop{\rm Im} L. Furthermore, let \Omega \subset X be an open bounded subset with \overline{\Omega }\cap D(L)\neq \emptyset, then N is L-compact on \overline{\Omega}; \item[(iv)] x=(x_1,x_2) is a solution of \eqref{e36} if and only if x is a solution of the operator equation Lx=Nx in D( L). \end{itemize} We define the projectors P and Q as follows: P(x_1,x_2)=(0,x_2(0)) for all x=(x_1,x_2)\in X and  Q(y_1,y_2,a,b)=(0,0,0,b). The generalized inverse of L is defined by \begin{align*} K_P(y_1,y_2,a,b) &=\Big(\int_0^t\frac{(t-s)^{n-2}}{(n-2)!}y_1(s)ds\\ &\quad +\frac{t^{n-2}}{(n-2)!} \frac{1}{1-\sum_{i=1}^m\alpha_i} \Big(\sum_{i=1}^m\alpha_i\int_0^{\xi_i} y_1(s)ds+a\Big), \int_0^ty_2(s)ds\Big), \end{align*} the isomorphism \wedge:Y/\mathop{\rm Im}L\to \mathop{\rm Ker}L is defined by \wedge(0,0,0,b)=(0,b). Similar to the lemmas in sub-section 2.2, it is easy to prove the following Lemmas. \begin{lemma} \label{lem2.3.1} If (x_1,x_2) is a solution of problem \eqref{e36}, then x_1 is a solution of \eqref{e35}. \end{lemma} The proof is easy; it is omitted. \begin{lemma} \label{lem2.3.2} Suppose that {\rm (H8)--(H9)} hold. If x is a solution of the problem \eqref{e35}, then x(t)>0 for all t\in (0,1). \end{lemma} The proof is similar to that of Lemma \ref{lem2.2.2}; it is omitted. Let \lambda\in (0,1), consider the problem $$\label{e37} \begin{gathered} x_1^{(n-1)}(t)=\lambda \phi^{-1}(x_2(t)),\quad t\in [0,1],\\ x_2'(t)=-\lambda f^*(t,x_1(t),\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2(t))), \quad t\in [0,1],\\ 0=\lambda (\phi^{-1}(x_2(1))-\sum_{i=1}^{m}\beta_i\phi^{-1}(x_2(\xi_i)) -\lambda_2),\\ x_1^{(n-2)}(0)-\sum_{i=1}^{n}\alpha_ix_1^{(n-2)}(\xi_i)=\lambda \lambda_1,\\ x_1^{(i)}(0)=0,\quad i=0,\dots,n-3. \end{gathered}$$ \begin{lemma} \label{lem2.3.3} Suppose that {\rm (H8)--(H9)} hold. If (x_1,x_2) is a solution of \eqref{e37}, then \begin{align*} \|x_2\|_\infty &\leq \|x_2'\|_\infty\Big(1+\frac{\phi(K^m_{q-1})}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\beta_i)}\sum_{i=1}^m\phi(\beta_i)(1-\xi_i)\Big)\\ &\quad +\frac{\phi(K^m_{q-1}\lambda_2)}{1-\phi(K^m_{q-1}) \sum_{i=1}^m\phi(\beta_i)}. \end{align*} \end{lemma} \begin{lemma} \label{lem2.3.4} Suppose that {\rm (H8)--(H9)} hold. If (x_1,x_2) is a solution of \eqref{e33}, then \|x_1^{(n-2)}\|_\infty\le\phi^{-1}(\|x_2\|_\infty) \Big(1+\frac{1}{1-\sum_{i=1}^m\alpha_i}\sum_{i=1}^m \alpha_i\xi_i\Big)+\frac{\lambda_1}{1-\sum_{i=1}^m\alpha_i}, and for i=0,\dots,n-3, \begin{align*} \|x_1^{(i)}\|_\infty&\leq \frac{1}{(n-i-3)!}\|x_1^{(n-2)}\|_\infty\\ &\leq \frac{1}{(n-i-3)!}(\phi^{-1}(\|x_2\|_\infty) \big(1+\frac{1}{1-\sum_{i=1}^m\alpha_i}\sum_{i=1}^m \alpha_i(1-\xi_i)\big)\\ &\quad +\frac{\lambda_1}{1-\sum_{i=1}^m\alpha_i}). \end{align*} \end{lemma} Similar to Theorem \ref{thm2}, we obtain the following theorem. \begin{theorem} \label{thm3} Suppose {\rm (H3), (H8)--(H10)} hold. Then \eqref{e26} has at least one positive solution. \end{theorem} We remark that Theorem \ref{thm3} generalizes the theorems in \cite{a2,g5,w11}. \subsection{Solutions of Problem \eqref{e27}} We consider \eqref{e27}, (H1), (H3) and the following assumptions are supposed in this sub-section. \begin{itemize} \item[(H11)] \alpha_i,\beta_i\ge 0 for all i=1,\dots,m and \lambda_1,\lambda_2\in \mathbb{R}; \item[(H12)] The following inequality holds \begin{align*} &\phi(K_{q-1})\Big[\sum_{i=0}^{n-3}\phi\big(\frac{1 +\sum_{i=1}^m\alpha_i}{(n-2-i)!}\big)\int_0^1b_i(s)ds\\ &+\phi\big(1+\sum_{i=1}^m\alpha_i\big)\int_0^1b_{n-2}(s)ds\Big] +\int_0^1c(s)ds<1\,. \end{align*} \end{itemize} Let x_1=x and x_2=\phi(x_1), then \eqref{e24} is transformed into $$\label{e38} \begin{gathered} x_1^{(n-1)}(t)=\phi^{-1}(x_2(t)),\quad t\in [0,1],\\ x_2'(t)=-f(t,x_1(t),\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2(t))),\quad t\in [0,1],\\ x_1^{(n-2)}(0)=\sum_{i=1}^{m}\alpha_i\phi^{-1}(x_2(\xi_i))+\lambda _1,\\ x_1^{(n-2)}(1)=-\sum_{i=1}^{n}\beta_i\phi^{-1}(x_2(\xi_i))+\lambda_2,\\ x_1^{(i)}(0)=0,\quad i=0\dots,n-3, \end{gathered}$$ Suppose \lambda\in (0,1), we consider the problem $$\label{e39} \begin{gathered} x_1^{(n-1)}(t)=\lambda \phi^{-1}(x_2(t)),\quad t\in [0,1],\\ x_2'(t)=-\lambda f(t,x_1(t),\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2(t))), \quad t\in [0,1],\\ x_1^{(n-2)}(0)=\lambda(\sum_{i=1}^{m}\alpha_i\phi^{-1}(x_2(\xi_i))+\lambda _1),\\ x_1^{(n-2)}(1)=\lambda(-\sum_{i=1}^{n}\beta_i\phi^{-1}(x_2(\xi_i))+\lambda_2),\\ x_1^{(i)}(0)=0,\quad i-0\dots,n-3, \end{gathered}$$ Define the operators \begin{gather*} L(x_1,x_2)= \big(x_1^{(n-1)},x_2',x_1^{(n-2)}(0),x_1^{(n-2)}(1)\big),\quad (x_1,x_2)\in X\cap D(L),\\ \begin{aligned} N(x_1,x_2)=& \Big(\phi^{-1}(x_2),-f^*(t,x_1(t),\dots,x_1^{(n-2)}(t), \phi^{-1}(x_2(t))),\\ &\sum_{i=1}^{m}\alpha_i\phi^{-1}(x_2(\xi_i))+\lambda_1, -\sum_{i=1}^{n}\beta_i\phi^{-1}(x_2(\xi_i))+\lambda_2\Big), \end{aligned} \end{gather*} for (x_1,x_2)\in X. Suppose (H11)--(H12) hold. It is easy to show the following results: \begin{itemize} \item[(i)] \mathop{\rm Ker}L=\{(0,c):c\in \mathbb{R}\} and \mathop{\rm Im}L=\{(y_1,y_2,a,b):\int_0^1y_1(s)ds=b-a\}; \item[(ii)] L is a Fredholm operator of index zero; \item[(iii)] There are projectors P:X\to X and Q:Y\to Y such that \mathop{\rm Ker} L=\mathop{\rm Im} P and \mathop{\rm Ker} Q=\mathop{\rm Im} L. Furthermore, let \Omega \subset X be an open bounded subset with \overline{\Omega }\cap D(L)\neq \emptyset, then N is L-compact on \overline{\Omega}; \item[(iv)] x=(x_1,x_2) is a solution of \eqref{e39} if and only if x is a solution of the operator equation Lx=\lambda Nx in D(L). \end{itemize} We define the projectors P and Q as follows:  P(x_1,x_2)=(0,x_2(0)) for all x=(x_1,x_2)\in X and  Q(y_1,y_2,a,b)=(\int_0^1y_1(s)ds-b+a,0,0,0). The generalized inverse of L is K_P(y_1,y_2,a,b)=\Big(\frac{a}{(n-2)!}t^{n-2} +\int_0^t\frac{(t-s)^{n-2}}{(n-2)!}y_1(s)ds,\int_0^ty_2(s)ds\Big), $$the isomorphism \wedge:Y/\mathop{\rm Im}L\to \mathop{\rm Ker}L is defined by \wedge(c,0,0,0)=(0,c). Similar to Lemmas in sub-section 2.2, it is easy to prove the following Lemmas. \begin{lemma} \label{lem2.4.1} Suppose {\rm (H11), (H12)} hold. If x=(x_1,x_2) is a solution of \eqref{e39}, then there exists \xi\in [0,1] such that$$ |\phi^{-1}(x^{(n-1)}(\xi))|\le M=:\begin{cases} \frac{|\lambda_1-\lambda_2|}{\sum_{i=1}^m\alpha_i+\sum_{i=1}^m\beta_i}, &\sum_{i=1}^m\alpha_i+\sum_{i=1}^m\beta_i\neq 0,\\[4pt] |\lambda_1-\lambda_2|,&\sum_{i=1}^m\alpha_i+\sum_{i=1}^m\beta_i=0. \end{cases} $$\end{lemma} \begin{proof} {\bf Case 1.} \sum_{i=1}^m\alpha_i+\sum_{i=1}^m\beta_i=0. Then \alpha_i=\beta_i=0. In this case, x_1^{(n-2)}(0)=\lambda\lambda_1 and x_1^{(n-2)}(1)=\lambda\lambda_2, it is easy to see that there is \xi\in [0,1] so that |x^{(n-1)}(\xi)|=\lambda|\lambda_1-\lambda_2|. Then$$ \lambda |\phi^{-1}(x_2(\xi))|=\lambda |\lambda_1-\lambda_2|. So |\phi^{-1}(x_2(\xi))|=|\lambda_1-\lambda_2|. \noindent {\bf Case 2.} \sum_{i=1}^m\alpha_i+\sum_{i=1}^m\beta_i\neq 0. In this case, if |\phi^{-1}(x_2(t))|>M for all t\in (0,1), then x_1^{(n-2)}(0) \lambda\sum_{i=1}^m\alpha_iM+\lambda\lambda_1\\ &\ge\lambda\sum_{i=1}^m\alpha_i \frac{-\lambda_1+\lambda_2}{\sum_{i=1}^m\alpha_i +\sum_{i=1}^m\beta_i}+\lambda\lambda_1 \\ &=\lambda\frac{\lambda_2\sum_{i=1}^m\alpha_i +\lambda_1\sum_{i=1}^m\beta_i}{\sum_{i=1}^m\alpha_i+\sum_{i=1}^m\beta_i}. \end{align*} On the other hand, x_1^{(n-2)}(1)<-\lambda\sum_{i=1}^m\beta_iM+\lambda\lambda _2\le \lambda\sum_{i=1}^m\beta_i\frac{\lambda_1-\lambda_2}{\sum_{i=1}^m\alpha_i+\sum_{i=1}^m\beta_i}+\lambda\lambda _20$so that$\|x_2\|_\infty\le A$. Hence $$\|x_1^{(n-2)}\|_\infty\le\big(1+\sum_{i=1}^m\alpha_i\big) \phi^{-1}(A)+|\lambda_1|.$$ And for$i=0,\dots,n-3$, we get $$\|x_1^{(i)}\|_\infty\le \frac{1}{(n-2-i)!}\big[\big(1+\sum_{i=1}^m\alpha_i\big)\phi^{-1}(A) +|\lambda_1|\big].$$ The above inequalities imply that$\Omega_0$is bounded. \end{proof} \begin{lemma} \label{lem2.4.4} Let$\Omega_1=\{x\in \mathop{\rm Ker}L:Nx\in \mathop{\rm Im}L\}$. Then$\Omega_1$is bounded. \end{lemma} \begin{proof} In fact,$(0,c)\in \Omega_1$, then$(0,c)\in \mathop{\rm Ker}L$and$N(0,c)\in \mathop{\rm Im}L$, then we get $$\phi^{-1}(c)=-\sum_{i=1}^m\alpha_i\phi^{-1}(c)-\sum_{i=1}^m\beta_i\phi^{-1}(c)+\lambda_2-\lambda_1.$$ Hence there is$M_1>0$so that$|c|\le M_1$. \end{proof} \begin{lemma} \label{lem2.4.5} Let$\Omega_2=\{x\in \mathop{\rm Ker}L:\lambda \wedge ^{-1}x+(1-\lambda )QNx=0\}$. Then$\Omega_2$is bounded. \end{lemma} \begin{proof} In fact, if$(0,c)\in \Omega_2$, then $$\lambda c=-(1-\lambda)\Big(\sum_{i=1}^m\alpha_i\phi^{-1}(c) +\sum_{i=1}^m\beta_i\phi^{-1}(c)-\lambda_2+\lambda_1\Big).$$ Thus $$\lambda c^2=-(1-\lambda)c\phi^{-1}(c)\Big(\sum_{i=1}^m\alpha_i +\sum_{i=1}^m\beta_i-\frac{\lambda_2-\lambda_1}{\phi^{-1}(c)}\Big).$$ Hence there is$M_1>0$so that$|c|\le M_1$. \end{proof} The following theorem has proof similar to that of Theorem \ref{thm1}; its proof is omitted. \begin{theorem} \label{thm4} Suppose {\rm (H1), (H3), (H11), (H12)} hold. Then \eqref{e27} has at least one solution. \end{theorem} \begin{remark} \label{rmk2.4.1} \rm Consider the problems $$\label{e40} \begin{gathered} \big[\phi(x^{(n-1)}(t))\big]'+f(t,x(t),\dots,x^{(n-1)}(t))=0,\quad t\in (0,1),\\ x^{(n-2)}(0)-\alpha x^{(n-1)}(0)=\lambda_1,\\ x^{(n-2)}(1)+ \beta x^{(n-1)}(1)=\lambda_2,\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3, \end{gathered}$$ where$\alpha\ge 0,\beta\ge 0$,$\lambda_1\ge 0,\lambda_2\ge 0$, and$f$is nonnegative and continuous. If$x(t)$is a solution of \eqref{e40}, then$x^{(n-1)}$is decreasing on$[0,1]$. \noindent{\bf Case 1.}$x^{(n-1)}(0)\ge 0$and$x^{(n-1)}(1)\ge 0$; At this case, we see that$x^{(n-2)}(t)$is increasing on$[0,1]$. It follows from $$x^{(n-2)}(0)=\alpha x^{(n-1)}(0)+\lambda_1\ge0$$ that$x^{(n-2)}(t)>0$for all$t\in (0,1)$. Then$x(t)$is a positive solution of \eqref{e40}. \noindent{\bf Case 2.}$x^{(n-1)}(0)\ge 0$and$x^{(n-1)}(1)\le 0$; At this case, one sees that $$x^{(n-2)}(1)=-\beta x^{(n-1)}(1)+\lambda_2\ge0$$ and $$x^{(n-2)}(0)=-\alpha x^{(n-1)}(0)+\lambda_2\ge0.$$ It follows from$x^{(n)}(t)\le 0$that$x^{(n-2)}(t)\ge 0$for all$t\in [0,1]$. Then$x$is a positive solution of \eqref{e40}. \noindent{\bf Case 2.}$x^{(n-1)}(0)\le 0$and$x^{(n-1)}(1)\le 0$; At this case, one sees that$x^{(n-2)}(t)$is decreasing on$[0,1]$. It follows from $$x^{(n-2)}(1)=-\beta x^{(n-1)}(1)+\lambda_2\ge0$$ that$x^{(n-2)}(t)>0$for all$t\in (0,1)$. Then$x(t)$is a positive solution of \eqref{e40}. We can establish similar results for the existence of positive solutions of \eqref{e40} and the details are omitted. \end{remark} \begin{remark} \label{rmk2.4.2} \rm Consider the problems $$\label{e41} \begin{gathered} \big[\phi(x^{(n-1)}(t))\big]'+f(t,x(t),\dots,x^{(n-1)}(t))=0,\quad t\in (0,1),\\ x^{(n-2)}(0)-\alpha x^{(n-1)}(0)=\lambda_1,\\ x^{(n-2)}(1)+ \sum_{i=1}^m\beta_i x^{(n-1)}(\xi_i)=\lambda_2,\\ x^{(i)}(0)=0,\quad i=0,\dots,n-3, \end{gathered}$$ where$\alpha\ge 0,\beta_i\ge 0$,$\lambda_1\ge 0,\lambda_2\ge 0$, and$f$is nonnegative and continuous. \eqref{e41} need not has positive solution. It is easy to show that the problem \begin{gather*} x''+8+6t=0,\quad t\in (0,1),\\ x(0)-x'(0)=1,\\ x(1)+ \theta x'(\frac{1}{8})=0 \end{gather*} has no positive solution since the solution of the above problem is $$x(t)=-4t^2-t^3+\frac{4+\frac{67}{64}\theta}{2+\theta}t +\frac{4+\frac{67}{64}\theta}{2+\theta}+1.$$ Then $$x(1)=-5+2\frac{4+\frac{67}{64}\theta}{2+\theta}+1 =-4+2\frac{4+\frac{67}{64}\theta}{2+\theta}<0\quad \text{if }\theta>0.$$ \end{remark} \subsection{Solutions of \eqref{e28}} We consider \eqref{e28}, assuming (H3) and the following conditions: \begin{itemize} \item[(H13)] there are nonnegative numbers$\alpha,\theta_i$and$L$so that$|f(t,x_0,\dots,x_{n-1})| \ge \alpha\phi(|x_{n-2}|)-\sum_{i=0}^{n-3}\theta_i\phi(|x_i|) -\theta_{n-1}\phi(|x_{n-1}|)-L; \item[(H14)] $$\lim_{|a|\to +\infty}\frac{|f(t,\frac{t^{n-2}}{(n-2)!}a, \dots,a,\phi(\frac{\lambda_1}{1-\sum_{i=1}^m\alpha_i}))|}{\phi(|a|)}=\mu$$ and the following three inequalities hold: \begin{gather*} \phi^{-1}(K^{n-1}_{p-1})\sum_{i=0}^{n-3} \phi(\frac{\theta_i}{\alpha})\phi\big(\frac{1}{(n-2-i)!}\big)<1, \\ \mu+\sum_{i=0}^{n-3}\theta_i\phi\big(\frac{1}{(n-2-i)!}\big)<\alpha, \end{gather*} and \begin{align*} &(1+\frac{\sum_{i=1}^m\alpha_i\xi_i}{1-\sum_{i=1}^m\alpha_i}) (\sum_{i=0}^{n-3}\|b_i\|_\infty\phi\big(\frac{1}{(n-2-i)!}\big) +\|b_{n-2}\|_\infty) \\ &\times \phi\big(\frac{1}{ (1-\phi^{-1}(K^{n-1}_{p-1}))\sum_{i=1}^m\phi^{-1}(\frac{\beta_i}{\alpha})}\frac{1}{(n-2-i)!}\big) \phi(K_{p-1}) \\ &\quad\times\phi\big(1+\phi^{-1}(K_{p-1})\big) \phi^{-1}(\frac{\beta_{n-1}}{\alpha}) +\big(1+\frac{\sum_{i=1}^m\alpha_i\xi_i}{1-\sum_{i=1}^m\alpha_i}\big) \|c\|_\infty<1. \end{align*} \item[(H15)]\phi\big(\theta^{-1}(\frac{\lambda_2}{1-\sum_{i=1}^m\beta_i})\big) =\frac{\lambda_1}{1-\sum_{i=1}^m\alpha_i}$; \item[(H16)]$\lambda_1,\lambda_2\in \mathbb{R},\alpha_i\ge 0,\beta_i\ge0$for$i=1,\dots,m$with$\sum_{i=1}^m\alpha_i<1$and$\sum_{i=1}^m\beta_i<1$. \end{itemize} Let$x_1=x$and$x_2=\phi(x_1)$, then \eqref{e28} is transformed into $$\label{e42} \begin{gathered} x_1^{(n-1)}(t)=\phi^{-1}(x_2(t)),\quad t\in [0,1],\\ x_2'(t)=-f(t,x_1(t),\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2(t))),\quad t\in [0,1],\\ 0=x_2(0)-\sum_{i=1}^{m}\alpha_ix_2(\xi_i)-\lambda _1,\\ 0=\theta(\phi^{-1}(x_2(1)))-\sum_{i=1}^{m}\beta_i\theta(\phi^{-1}(x_2(\xi_i)))-\lambda_2,\\ x_1^{(i)}(0)=0,\quad i=0\dots,n-3, \end{gathered}$$ Suppose$\lambda\in (0,1)$, we consider the following problem $$\label{e43} \begin{gathered} x_1^{(n-1)}(t)=\lambda \phi^{-1}(x_2(t)),\quad t\in [0,1],\\ x_2'(t)=-\lambda f(t,x_1(t),\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2(t))),\quad t\in [0,1],\\ 0=\lambda (x_2(0)-\sum_{i=1}^{m}\alpha_ix_2(\xi_i)-\lambda_1),\\ 0=\lambda (\theta(\phi^{-1}(x_2(1)))-\sum_{i=1}^{m}\beta_i\theta(\phi^{-1}(x_2(\xi_i)))-\lambda_2),\\ x_1^{(i)}(0)=0,\quad i=0\dots,n-3, \end{gathered}$$ Define the operators \begin{gather*} L(x_1,x_2)=(x_1^{(n-1)},x_2',0,0),\quad (x_1,x_2)\in X\cap D(L),\\ N(x_1,x_2)=\begin{pmatrix}\phi^{-1}(x_2)\\ -f^*(t,x_1,\dots,x_1^{(n-2)}(t),\phi^{-1}(x_2))\\ x_2(0)-\sum_{i=1}^{m}\alpha_ix_2(\xi_i)-\lambda_1\\ \theta(\phi^{-1}(x_2(1)))-\sum_{i=1}^{n}\beta_i\theta(\phi^{-1}(x_2(\xi_i))) -\lambda_2\end{pmatrix}^T,\quad (x_1,x_2)\in X. \end{gather*} Suppose that (H14), (H15), (H16) hold. It is easy to show the following results: \begin{itemize} \item[(i)]$\mathop{\rm Ker}L=\{(\frac{t^{n-2}}{(n-2)!}a,b): a,b\in \mathbb{R}\}$and$\mathop{\rm Im}L=\{(y_1,y_2,a,b):a=b=0\}$; \item[(ii)]$L$is a Fredholm operator of index zero; \item[(iii)] There are projectors$P:X\to X$and$Q:Y\to Y$such that$\mathop{\rm Ker} L=\mathop{\rm Im} P$and$\mathop{\rm Ker} Q=\mathop{\rm Im} L$. Furthermore, let$\Omega \subset X$be an open bounded subset with$\overline{\Omega }\cap D(L)\neq \emptyset$, then$N$is$L$-compact on$\overline{\Omega}$; \item[(iv)]$x=(x_1,x_2)$is a solution of \eqref{e42} if and only if$x$is a solution of the operator equation$Lx=Nx$in$D( L)$. \end{itemize} We present the projectors$P$and$Q$as follows:$P(x_1,x_2)=(\frac{t^{n-2}}{(n-2)!}x_1^{(n-2)}(0),x_2(0))$for all$x=(x_1,x_2)\in X$and$ Q(y_1,y_2,a,b)=(0,0,a,b)$. The generalized inverse of$L$is $$K_P(y_1,y_2,a,b)=(\int_0^t\frac{(t-s)^{n-2}}{(n-2)!}y_1(s)ds, \int_0^ty_2(s)ds),$$ the isomorphism$\wedge:Y/\mathop{\rm Im}L\to \mathop{\rm Ker}L$is defined by$\wedge(0,0,a,b)=(\frac{t^{n-2}}{(n-2)!}a,b)$. \begin{lemma} \label{lem2.5.1} If$(x_1,x_2)$is a solution of problem (42), then$x_1$is a solution of \eqref{e28}. \end{lemma} \begin{lemma} \label{lem2.5.2} Suppose that$(H_{14}),(H_{15}),(H_{16})$hold. If$(x_1,x_2)$is a solution of problem \eqref{e43}, then there is a$\xi\in (0,1)$so that$x'_2(\xi)=0$. \end{lemma} \begin{proof} In fact, if$x_2'(t)>0$for all$t\in (0,1)$, we get $$x_2(0)=\sum_{i=1}^m\alpha_ix_2(\xi_i)+\lambda_1> \sum_{i=1}^m\alpha_ix_2(0)+\lambda_1.$$ So$x_2(0)>\lambda_1/(1-\sum_{i=1}^m\alpha_i)$. It follows from$x_2'(t)>0$that$x_2(1)>\lambda_1/(1-\sum_{i=1}^m\alpha_i)$. On the other hand, $$\theta(\phi^{-1}(x_2(1))) =\sum_{i=1}^m\beta_i\theta(\phi^{-1}(x_2(\xi_i)))+\lambda_2 <\sum_{i=1}^m\beta_i\theta(\phi^{-1}(x_2(1)))+\lambda_2.$$ Then$\theta(\phi^{-1}(x_2(1)))<\lambda_2/(1-\sum_{i=1}^m\beta_i)$. So we get$x_2(1)<\phi(\theta^{-1}(\lambda_2/(1-\sum_{i=1}^m\beta_i))) =\lambda_1/(1-\sum_{i=1}^m\alpha_i)0$so that$\|x_2\|_\infty\le M. Hence \begin{align*} \|x_1^{(n-2)}\|_\infty &\leq \Big(1-\phi^{-1}(K_{p-1})\sum_{i=0}^{n-3}\phi^{-1}(\theta_i/\alpha) \frac{1}{(n-2-i)!}\Big)^{-1}\\ &\quad \times \big[(1+\phi^{-1}(K_{p-1})\phi^{-1}(\theta_{n-1}/\alpha)) \phi^{-1}(M)+\phi^{-1}(K_{p-1})\phi^{-1}(L/\alpha)\big]\\ &=:M_{n-2}, \end{align*} and fori=0,\dots,n-3$, we get $$\|x_1^{(i)}\|_\infty\le\frac{1}{(n-2-i)!}\|x_1^{(n-2)}\|_\infty\le \frac{1}{(n-2-i)!}M_{n-2}=:M_i.$$ Hence$\Omega_0$is bounded. \end{proof} \begin{lemma} \label{lem2.5.5} Suppose that {\rm (H3), (H14), (H15), (H16)} hold. Let$\Omega_1=\{x\in \mathop{\rm Ker} L,\;Nx\in \mathop{\rm Im} L\}$. Then$\Omega_1$is bounded. \end{lemma} \begin{proof} In fact, if$x=(\frac{t^{n-2}}{(n-2)!}a,b)\in \mathop{\rm Ker}L$and$Nx\in \mathop{\rm Im}L$, then we get $$b-\sum_{i=1}^m\alpha_ib-\lambda_1=0,\quad \theta(\phi^{-1}(b))-\sum_{i=1}^m\beta_i\theta(\phi^{-1}(b))-\lambda_2=0.$$ So we have$b=\lambda_1/(1-\sum_{i=1}^m\alpha_i)$. From (H14), choose$\epsilon>0$so that $$\sum_{i=0}^{n-3}\theta_i\phi\big(\frac{1}{(n-2-i)!}\big) +(\mu+\epsilon)<\alpha,$$ and then there is a$\delta>0so that $$\big|f\big(t,\frac{t^{n-2}}{(n-2)!}x,\dots,x, \phi^{-1}(\lambda_1/(1-\sum_{i=1}^m\alpha_i))\big)\big| <(\mu+\epsilon)\phi(|x|),\quad |x|>\delta.$$ Let $$A=\max_{t\in [0,1],|x|\le \delta}\big|f\big(t,\frac{t^{n-2}}{(n-2)!}x,\dots,x, \phi^{-1}(\lambda_1/(1-\sum_{i=1}^m\alpha_i))\big)\big|.$$ Then one sees that \begin{align*} (\mu+\epsilon)\phi(|a|)+A &\ge \big|f\big(t,\frac{t^{n-2}}{(n-2)!}a,\dots, a,\phi^{-1}(\lambda_1/(1-\sum_{i=1}^m\alpha_i))\big)\big|\\ &\ge \alpha\phi(|a|)-\sum_{i=0}^{n-3}\theta_i \phi\big(\frac{t^{n-2-i}}{(n-2-i)!}|a|\big) -\theta_{n-1}|\lambda_1|/\big(1-\!\sum_{i=1}^m\alpha_i\big)-L. \end{align*} So \begin{align*} \theta_{n-1}|\lambda_1|/(1-\sum_{i=1}^m\alpha_i)+L &\ge \phi(|a|)\Big(\alpha-\sum_{i=0}^{n-3}\theta_i \phi(\frac{t^{n-2-i}}{(n-2-i)!})-(\mu+\epsilon)\Big)\\ &\ge \phi(|a|)\Big(\alpha-\sum_{i=0}^{n-3}\theta_i \phi\big(\frac{1}{(n-2-i)!}\big)-(\mu+\epsilon)\Big) \end{align*} Then there isM_1'>0$so that$|a|\le M_1'$. Hence$|a|,|b|$are bounded. Then$\Omega_1$is bounded. \end{proof} \begin{lemma} \label{lem2.5.6} Suppose that {\rm (H3), (H14), (H15), (H16)} hold. Then the set$\Omega_2=\{x\in \mathop{\rm Ker} L,\;\lambda \wedge^{-1} x+(1-\lambda )QNx=0,\;\lambda \in [0,1]\}$is bounded. \end{lemma} \begin{proof} In fact, if$\Omega_2$is unbounded, then there are sequences$\{\lambda_n\in [0,1]\}$and$\{x_n=(\frac{t^{n-2}}{(n-2)!}a_n,b_n)\}$such that $$\lambda _n (0,0,a_n,b_n)+(1-\lambda_n) \Big(0,0,b_n-\sum_{i=1}^m\alpha_ib_n-\lambda_1, \theta(\phi^{-1}(b_n))-\sum_{i=1}^m\beta_i \theta(\phi^{-1}(b_n))-\lambda_2\Big)$$ and either$|b_n|\to +\infty$as$n$tends to infinity or$\{b_n\}$is bounded and$|a_n|\to +\infty$as$n$tends to infinity. It follows that \begin{gather} \label{e44} \lambda_n a_n=-(1-\lambda_n )(b_n-\sum_{i=1}^m\alpha_ib_n-\lambda_1), \\ \label{e45} \lambda_n b_n=-(1-\lambda_n )(\theta(\phi^{-1}(b_n))-\sum_{i=1}^m\beta_i\theta(\phi^{-1}(b_n)) -\lambda_2). \end{gather} Then $$\lambda_n b_n^2=-(1-\lambda_n )\theta(\phi^{-1}(b_n))b_n\big[\big(1-\sum_{i=1}^m\beta_i\big) -\frac{\lambda_2}{\theta(\phi^{-1}(b_n))}\big]$$ implies that there is a constant$B>0$so that$|b_n|\le B$since$\phi^{-1}(b_n))b_n>0$. Thus we get that$|a_n|\to +\infty$as$n$tends to infinity. It follows from \eqref{e44} that$\lambda_n\to 0$as$ntends to infinity. Thus \eqref{e45} implies that $$b_n\to b_0=\phi(\theta^{-1}(\lambda_2/(1-\sum_{i=1}^m\beta_i)))=\lambda_1/(1-\sum_{i=1}^m\alpha_i).$$ Then \begin{align*} (\mu+\epsilon)\phi(|a_n|)+A &\ge \big|f\big(t,\frac{t^{n-2}}{(n-2)!}a_n,\dots,a_n, \phi^{-1}(\lambda_1/(1-\sum_{i=1}^m\alpha_i))\big)\big|\\ &\ge \alpha\phi(|a_n|)-\sum_{i=0}^{n-3}\theta_i \phi\big(\frac{t^{n-2-i}}{(n-2-i)!}\phi(|a_n|)\big)\\ &\quad -\theta_{n-1}|\lambda_1|/(1-\sum_{i=1}^m\alpha_i)-L. \end{align*} So \begin{align*} \theta_{n-1}|\lambda_1|/(1-\sum_{i=1}^m\alpha_i)+L &\ge \phi(|a_n|)(\alpha-\sum_{i=0}^{n-3}\theta_i \phi(\frac{t^{n-2-i}}{(n-2-i)!})-(\mu+\epsilon))\\ &\ge \phi(|a_n|)(\alpha-\sum_{i=0}^{n-3}\theta_i \phi\big(\frac{1}{(n-2-i)!}\big)-(\mu+\epsilon)). \end{align*} It follows from $$\sum_{i=0}^{n-3}\theta_i\phi\big(\frac{1}{(n-2-i)!}\big) +(\mu+\epsilon)<\alpha$$ that there is a constantC>0$so that$|a_n|\le C$, a contradiction. Hence$\Omega_2$is bounded. \end{proof} \begin{theorem} \label{thm5} Suppose that {\rm (H3), (H13)--(H16)} hold. Then \eqref{e28} has at least one solution. \end{theorem} \begin{proof} Let$\Omega\supseteq \Omega_0\cup\Omega_1\cup\Omega_2$be a bounded open subset of$X$centered at zero Then$Lx\neq \lambda Nx$for all$(x,\lambda )\in [(D(L)\setminus \mathop{\rm Ker} L)\cap \partial \Omega ] \times (0,1)$;$Nx\notin \mathop{\rm Im} L$for every$x\in \mathop{\rm Ker} L\cap \partial \Omega $;$\mathop{\rm deg}(\wedge QN\big|_{\mathop{\rm Ker} L},\;\Omega \cap \mathop{\rm Ker} L, 0)\neq 0$. It follows from Lemma \ref{lem2.0.1} that$Lx=Nx$has at least one solution$x=(x_1,x_2)$. Then$x_1$is a solution of \eqref{e42}. Hence$x_1$is a solution of \eqref{e28}. \end{proof} We remark that Theorem \ref{thm5} generalizes the results in \cite{g2,l9}. \section{Examples} Now, we present some examples to illustrate the main results. These BVPs can not be solved by known results. \begin{example} \label{exa1}\rm Consider the problem $$\label{e46} \begin{gathered} x''(t)+b(t)|x'(t)|+c(t)x(t)+r(t)=0,\quad \;t\in (0,1),\\ x(0)=\frac{1}{2}x(1/4)+2,\quad x(1)=\frac{1}{2}x(\frac{1}{2})+2, \end{gathered}$$ where$b$,$c$and$r$are nonnegative continuous functions. Corresponding to \eqref{e24}, it is easy to find that (H1), (H2), (H3) hold. We find from Theorem \ref{thm1} that if$\frac{5}{4}\int_0^1c(s)ds+\int_0^1b(s)ds<1$, then \eqref{e46} has at least one positive solution for each$r\in C[0,1]$with$r(t)\ge0$and$ \not\equiv 0$on each subinterval of [0,1]. \end{example} \begin{example} \label{exa2}\rm Consider the problem $$\label{e47} \begin{gathered} (\phi_{3}(x'))'+a(t)\phi_{3}(x)+b(t)\phi_{3}(|x'|)+r(t)=0,\quad t\in (0,1),\\ x(0)=\frac{1}{2}x(1/2)+6,\quad x(1)=\frac{1}{4}x(1/4)+\frac{1}{3}x(1/2)+7, \end{gathered}$$ where$a$,$b$and$r$are nonnegative continuous functions. We find$p=3$and$q=3/2$. Then by application of Theorem \ref{thm1}, \eqref{e47} has at least one positive solution if$\phi_3(2)\phi_3(\frac{3}{2})\int_0^1a(s)ds+\int_0^1b(s)ds<1$for each$r\in C[0,1]$with$r(t)\ge 0$and$\not\equiv 0$on each subinterval of [0,1]. \end{example} \begin{example} \label{exa3}\rm Consider the problem $$\label{e48} \begin{gathered} (\phi_{3}(x'))'+a(t)\phi_{3}(x)+b(t)\phi_{3}(|x'|)+r(t)=0,\quad t\in (0,1),\\ x'(0)=\frac{1}{2}x'(1/2)-3,\quad x(1)=\frac{1}{4}x(1/4)+\frac{1}{3}x(1/2)+4, \end{gathered}$$ where$a$,$b$and$r$are nonnegative continuous functions. We find$p=3,q=3/2,m=2$. Then by application of Theorem \ref{thm2}, \eqref{e48} has at least one positive solution if $$\Big(1+\frac{\phi_3(4)}{1-\phi_3(4)\phi_3(1/2)}\frac{\phi_3(1/2)}{2}\Big) \Big[\|b\|_\infty+\phi_3(2)\phi_3(1+\frac{12}{5}\frac{7}{48})\|a\|_\infty \Big] <1$$ for each$r\in C[0,1]$with$r(t)\ge 0$and$\not\equiv 0$on each subinterval of [0,1]. \end{example} \begin{example} \label{exa4} \rm Consider the problem $$\label{e49} \begin{gathered} (\phi_{3}(x'))'+a(t)\phi_{3}(x)+b(t)\phi_{3}(|x'|)+r(t)=0,\quad t\in (0,1),\\ x(0)=\frac{1}{2}x(1/2)+\frac{1}{3}x(3/4)+6,\quad x'(1)=\frac{1}{4}x(1/2)+\frac{1}{3}x'(3/4)+7, \end{gathered}$$ where$a$,$b$and$rare nonnegative continuous functions. Then by application of Theorem \ref{thm3}, \eqref{e49} has at least one positive solution if \begin{align*} &\Big(1+\frac{\phi(4)}{1-\phi(4)(\phi(1/2)+\phi(1/3))} (\phi(1/4)\frac{1}{2}+\phi(1/3)\frac{1}{4})\Big)\\ &\times \Big[\|b\|_\infty+\phi(2)\phi_3(41/13)\|a\|_\infty\Big]<1 \end{align*} for eachr\in C[0,1]$with$r(t)\ge 0$and$\not\equiv 0$on each subinterval of [0,1]. \end{example} \begin{example} \label{exa5} \rm Consider the problem $$\label{e50} \begin{gathered}x''(t)+a(t)|x'(t)|+b(t)x(t)+r(t)=0,\quad t\in (0,1),\\ x(0)=\frac{1}{2}x'(0)+6,\quad x(1)=4x'(1)+7, \end{gathered}$$ where$a$,$b$and$r$are nonnegative continuous functions. Then by Theorem \ref{thm3}, \eqref{e50} has at least one positive solution if $$\frac{3}{2}\int_0^1b(t)dt+\int_0^1a(t)dt<1$$ for each$r\in C[0,1]$with$r(t)\ge 0$and$\not\equiv 0$on each subinterval of [0,1]. \end{example} \subsection*{Acknowledgements} The author would like to thank the anonymous referee and the editors for the very careful reading of the original manuscript and the helpful suggestions. \begin{thebibliography}{00} \bibitem{a1} R. P. Agarwal, D. O'Regan, P. J. Y. Wong, Positive solutions of differential, difference and integral equations. Kluwer Academic, Dordrecht. 1999. \bibitem{a2} R. P. Agarwal, D. O'Regan, V. Lakshmikanthams, Singular (n-p,p) focal and (n,p) higher order BVPs. Nonl. Anal. 42(2000), 215-228. \bibitem{a3} R. P. Agarwal, F. Wong, Existence of positive solutions for non-positive higher-order BVPs. J. Comput. Appl. Math. 88(1998), 3-14. \bibitem{a4} R. I. Avery, A. C. Peterson, Three positive fixed points of nonlinear operators on ordered banach spaces. Comput. Math. Appl. 42(2004), 313-322 \bibitem{b1} C. Bai, J. Fang, Existence of multiple positive solutions for nonlinear$m$-point boundary-value problems. Appl. Math. Comput. 140(2003), 297-305. \bibitem{b2} C. Bai, J. Fang, Existence of multiple positive solutions for nonlinear multi-point boundary-value problems. J. Math. Anal. Appl., 281(2003), 76-85. \bibitem{b3} G. Bognar. J. Cepika, P. Drabek, P. Necesal, E. Rozgonyi; Necessary and sufficient conditions for the existence of solution to three-point BVP. Nonl. Anal. doi:10.1016/j.na.2007.08.072. \bibitem{d1} P. Drabek and P. Takac, A counterexample to the Fredholm alternative for the$p$-Laplacian. Proc. Amer. Math. Soc. 127(1999), 1079-1087. \bibitem{f1} H. Feng, W. Ge, M. Jiang, Multiple positive solutions for m-point boundary-value problems with a one-dimensional p-Laplacian. Nonl. Anal. 68(2008), 2269-2279. \bibitem{g1} R. E. Gaines, J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations. Lecture Notes in Math. 568, Springer, Berlin, 1977. \bibitem{g2} M. Garcia-Huidobro, C.P.Gupta, R. Manasevich, An multi-point boundary-value problem of Neumann type for$p$-Laplacian like operator. Nonl. Anal. 56(2004), 1071-1089. \bibitem{g3} M. R. Grossinho, F. M. Minhos, A. I. Santos, Solvability of some third-order boundary value problems with asymmetric unbounded nonlinearities. Nonl. Anal., 62(2005), 1235-1250. \bibitem{g4} M. D. R. Grossinho, F. M. Minhos, A. I. Santos, A third-order boundary value problem with one-sided Nagumo condition. Nonl. Anal. 63(2005), 247-256. \bibitem{g5} Y. Guo, W. Ge, Three positive solutions for the one dimension p-Laplacian. J. Math. Anal. Appl., 286(2003), 491-508. \bibitem{g6} Y. Guo, Y. Ji, X. Liu, Multiple positive solutions for some multi-point boundary value problems with p-Laplacian. J. Comput. Appl. Math. doi:10.1016/j.cam.2007.04.023. \bibitem{g7} C. P. Gupta, A non-resonant generalized multi-point boundary-value problem of Dirichelet type involving a p-laplacian type operator, Sixth Mississippi State Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns. Conference 15 (2007), 127-139. \bibitem{g8} C. P. Gupta, A generalized multi-point boundary value problem for second order ordinary differential equations. Appl. Math. Comput. 89(1998), 133-146. \bibitem{g9} C. P. Gupta, On a third order boundary value problem at resonance. Differential Integral Equations. 2(1989), 1-12. \bibitem{g10} C. P. Gupta, A non-resonance multi-point boundary value problem for a$p$-Laplacian type operator. Electronic Journal Of Differential Equations, Conference. 10(2003), 143-152. \bibitem{i1} V. A. Il'in, E. I. Moiseev, Nonlocal boundary value problem of the second kind for Sturm-Liouville operator. Diff. Eqs. 23(1987), 979-987. \bibitem{j1} D. Ji, H. Feng, W. Ge, The existence of symmetric positive solutions for some nonlinear equation systems. Appl. Math. Comput. 197(2008), 51-59. \bibitem{k1} G. L. Karakostas, Triple positive solutions for the$\phi$-Laplacian when$\phi$is a sup-multiplicative-like function. Elec. J. Differential Equations. 69(2004), 1-13. \bibitem{k2} G. L. Karakostas, Triple positive solutions for the$\phi$-Laplacian when$\phi$is a sup-multiplicative-like function. Elec. J. Differential Equations. 68(2004), 1-12. \bibitem{k3} G. Karakostas, P. Tsamatos, Nonlocal boundary vector value problems for ordinary differential systems of higher order. Nonl. Anal. 5(1)(2002), 1421-1427. \bibitem{k4} L. Kong, Q. Kong, Second-order boundary value problems with nonhomogeneous boundary conditions(II). J. Math. Anal. Appl. 330(2007), 1393-1411. \bibitem{k5} L. Kong, Q. Kong, Multi-point boundary value problems of second order differential equations(I). Nonl. Anal., 58(2004), 909-931. \bibitem{k6} L. Kong, J. Wang, Multiple positive solutions for one-dimension$p$-Laplacian. Nonl. Anal. 42(2000), 1327-1333. \bibitem{l1} K. Lan, Multiple positive solutions of semi-linear differential equations with singularities. J. London Math. Soc. 63(2001), 690-704. \bibitem{l2} H. Lian, W. Ge, Positive solutions for a four-point boundary value problem with the p-Laplacian. Nonl. Anal. doi:10.1016/j.na.2007.03.042. \bibitem{l3} W. Lian, F. Wong, Existence of positive solutions for higher-order generalized p-Laplacian BVPs. Appl. Math. Letters. 13(2000), 35-43. \bibitem{l4} W. Lian, F. Wong, Existence of positive solutions for higher-order generalized p-Laplacian BVPs. Appl. Math. letters. 13(2000), 35-43. \bibitem{l5} B. Liu, Positive solutions of a nonlinear four-point boundary value problems. Appl. Math. Comput. 155(2004), 179-203. \bibitem{l6} B. Liu, Solvability of multi-point boundary value problems at resonance (III). Appl. Math. Comput. 129(2002), 119-143. \bibitem{l7} B. Liu, Solvability of multi-point boundary value problems at resonance (IV). Appl. Math. Comput. 143(2003), 275-299. \bibitem{l8} B. Liu, Solvability of multi-point boundary value problems at resonance (II). Appl. Math. Comput. 136(2003), 353-377. \bibitem{l9} B. Liu, J. Yu, Solvability of multi-point boundary value problems at resonance (I). India J. Pure Appl. Math. 33(4)(2002), 475-494. \bibitem{l10} B. Liu, Z. Zhao, A note on multi-point boundary value problems. Nonl. Anal. 67(2007), 2680-2689. \bibitem{l11} Y. Liu, Existence of multiple positive solutions of$p$-Laplacian boundary value problems. Math. Slovak. 57(2007), 225-242. \bibitem{l12} Y. Liu, W. Ge, Solvability of nonlocal boundary value problems for ordinary differential equations of higher order. Nonl. Anal. 57(2004), 435-458. \bibitem{l13} Y. Liu and Weigao Ge, Positive solutions for (n-1,1)three-point BVPs with coefficient that changes sign. J. Math. Anal. Appl. 282(2003), 816-825. \bibitem{l14} Y. Liu, W. Ge, Multiple positive solutions to a three-point boundary value problem with p-Laplacian. J. Math. Anal. Appl. 277(2003), 293-302. \bibitem{l15} Y. Liu and W. Ge, Multiple positive solutions of a three-point BVP with p-Laplacian. J. Math. Anal. Appl. 277(2003), 293-302. \bibitem{l16} Y. Liu, W. Ge, Existence and non-existence of positive solutions for nonlinear three-point boundary value problems. Fasciculi Mathematici. 34(2004), 157-167. \bibitem{l17} Y. Liu and W. Ge, Solvability of nonlocal boundary value problems for ordinary differential equations of higher order. Nonl. Anal. 57(2004), 435-458. \bibitem{l18} Y. Liu, P. Yang and W. Ge, Periodic solutions of higher order delay differential equations. Nonl. Anal. 63(2005), 136-152. \bibitem{l19} H. Lu, D. O'Regan, C. Zhong, Multiple positive solutions for one dimensional singular p-Laplacian. Appl. Math. Comput. 133(2002), 407-422. \bibitem{m1} R. Ma, Multiplicity results for third order boundary value problem at resonance. Nonl. Anal. 32(1998), 493-499. \bibitem{m2} R. Ma, Positive solutions for second order three-point boundary value problems. Appl. Math. letters. 24(2001), 1-5. \bibitem{m3} R. Ma, Positive solutions of non-linear m-point boundary value problems, Comput. Math. with Appl., 42(2001)755-765. \bibitem{m4} R. Ma, Multiple positive solutions for nonlinear m-point boundary value problems. Appl. Math. Comput. 148(2004), 249-262. \bibitem{m5} R. Ma, N. Castaneda, Existence of solutions of nonlinear$m$-point boundary value problems. J. Math. Anal. Appl. 256(2001), 556-567. \bibitem{m6} R. Ma, B. Thompson, Positive solutions for nonlinear m-point eigenvalue problems. J. Math. Anal. Appl. 297(2004), 24-37. \bibitem{m7} J. Marcos, S. Lorca, P. Ubilla, Multiplicity of solutions for a class of non-homogeneous fourth-order boundary value problems. Appl. Math. Letters. doi:10.1016/j.aml.2007.02.025. \bibitem{m8} D. Ma, Z. Du, W. Ge, Existence and iteration of monotone positive solutions for multipoint boundary value problem with p-Laplacian operator. Comput. Math. Appl. 50(2005), 729-739. \bibitem{n1} R. K. Nagle, K.L. Pothoven, On a third order nonlinear boundary value problem at resonance. J. Math. Anal. Appl. 195(1995), 149-159. \bibitem{p1} H. Pang, H. Lian, W. Ge, Multiple positive solutions for second-order four-point boundary value problem Comput. Math. Appl. doi:10.1016/j.camwa.2007.03.014. \bibitem{q1} S. Qi, Multiple positive solutions to boundary value problems for higher-order nonlinear differential equations in Banach spaces. Acta Math. Appl. Sinica 17(2)(2001), 271-278. \bibitem{r1} B. P. Rynne, Spectral properties and nodal solutions for second-order, m-point, boundary value problems. Nonl. Anal. 67(2007), 3318-3327. \bibitem{s1} H. Su, Z. Wei, X. Zhang and J. Liu, Positive solutions of n-order and m-order multi-point singular boundary value system. Appl. Math. Comput. 188(2007), 1234-1243. \bibitem{s2} W. Sun, S. Chen, Q. Zhang, C. Wang, Existence of positive solutions to n-point nonhomogeneous boundary value problem. J. Math. Anal. Appl. 330(2007), 612-621. \bibitem{s3} B. Sun, W. Ge, D. Zhao, Three positive solutions for multipoint one-dimensional p-Laplacian boundary value problems with dependence on the first order derivative. Math. Comput. Modelling. 45(2007), 1170-1178. \bibitem{s4} B. Sun, Y. Qu, W. Ge, Existence and iteration of positive solutions for a multipoint one-dimensional p-Laplacian boundary value problem. Appl. Math. Comput. 197(2008), 389-398. \bibitem{w1} J. Wang, W. Gao, A singular boundary value problem for one-dimension$p$-Laplacian. J. Math. Anal. Appl. 201(1996), 851-866. \bibitem{w2} Y. Wang, W. Ge, Positive solutions for multi-point boundary value problems with a one-dimensional$p$-Laplacian. Nonl. Anal. 66(2007), 1246-1256. \bibitem{w3} Y. Wang, W. Ge, Existence of periodic solutions for nonlinear differential equations with a p-Laplacian-like operator. Appl. Math. Letters. 19(2006), 251-259. \bibitem{w4} Y. Wang, W. Ge, Existence of solutions for a third order differential equation with integral boundary conditions. Comput. Math. Appl. 53(2007), 144-154. \bibitem{w5} Y. Wang, W. Ge, Existence of triple positive solutions for multi-point boundary value problems with a one dimensional p-Laplacian. Comput. Math. Appl. 54(2007), 793-807. \bibitem{w6} Y. Wang, W. Ge, Existence of multiple positive solutions for multipoint boundary value problems with a one-dimensional p-Laplacian. Nonl. Anal. 67(2007), 476-485. \bibitem{w7} Y. Wang, W. Ge, Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian. J. Math. Anal. Appl. 327(2007), 1381-1395. \bibitem{w8} Y. Wang, C. Hou, Existence of multiple positive solutions for one-dimensional$p$-Laplacian. J. Math. Anal. Appl. 315(2006), 144-153. \bibitem{w9} Y. Wang, W. Zhao, W. Ge, Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian. J. Math. Anal. Appl. 326(2007), 641-654. \bibitem{w10} F.H. Wong, An application of Schauder's fixed point theorem with respect to higher order BVPs. Proc. Amer. Math. Soc. 126(1998), 2389-2397. \bibitem{w11} P. J. Wong, R. P. Agarwal, Singular differential equations with (n,p) boundary conditions. Mathl. Comput. Modelling. 28(1)(1998), 37-44. \bibitem{w12} F. Xu, Positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian operator. Appl. Math. Comput. doi:10.1016/j.amc.2007.04.118. \bibitem{y1} X. Yang, Positive solutions for the one-dimensional$p\$-Lpalacian. Math. Comput. Modelling. 35(2002), 129-135. \bibitem{z1} Q. Zhang, S. Chen, J. Lu, Upper and lower solution method for fourth-order four-point boundary value problems. J. Comput. Appl. Math. 196(2006), 387-393. \bibitem{z2} G. Zhang, J. Sun, Positive solutions of m-point boundary value problems. J. Math. Anal. Appl. 291(2004), 406-418. \bibitem{z3} Z. Zhang, J. Wang, On the existence and multiplicity of positive solutions to a singular multi-point boundary value problem. J. Math. Anal. Appl. 295(2004), 502-512. \bibitem{z4} E. P. Zhidkov, V. P. Shirikov, Boundary value problem for ordinary second order differential equations, USSR Computational Mathematics and Mathematical Physics, 4(5)(1964), 18-35. \end{thebibliography} \end{document}