Positive Solutions of Boundary Value Problem for System of Nonlinear nTh-Order Ordinary Differential Equations
$n$阶非线性常微分方程组边值问题的正解

In this paper, the existence and multiplicity of positive solutions of boundary value problems are considered for the system of $n$th-order nonlinear ordinarydifferential equations\\left\{\begin{array}{l}-u^{(n)}=f_1(x,u,v), \q -v^{(n)}=f_2(x,u,v),\\2mm] u^{(i)}(0)=u^{(p)}(1)=v^{(i)}(0)=v^{(p)}(1)=0,\end{array}\right.\]where $n\geq 2$, $i = 0,1,\cdots,n-2$, $p \in \{1,2,\cdots,n-1\}$, and $f_i\in C(0,1]\times\mathbb R^+\times \mathbb R^+,\mathbb R^+)(i=1,2)$, by using the fixed point index theory. Concave functions are utilized to characterize coupling behaviors of $f_1$ and $f_2$, so $f_1$ and $f_2$ can be one of the following cases: (1) $f_1$ and $f_2$ are both superlinear; (2) $f_1$ and $f_2$ are both sublinear; (3) one is superlinear, and another is sublinear.