### B P Rynne

#### Abstract

For any integer $m \ge 2$, we consider the $2m$th order boundary value problem \begin{gather*} (-1)^m u^{(2m)}(x) + \sum_{i=0}^{m-1} (-1)^i p_{i} u^{(2i)}(x) = \lambda f(u(x)) , \quad x \in (-1,1), \\ u^{(i)}(-1) = u^{(i)}(1) = 0, \quad i=0,\dots, m-1, \end{gather*} where $p_i \ge 0$, $i=0,\dots, m-1$, are constants and $u^{(i)}$ is the $i$th derivative of $u \in C^{2m}[-1,1]$, the number $\lambda \in R_+ := [0,\infty)$, and the function $f : R \to R$ is $C^2$ and satisfies %$$%f(0) > 0, \quad f'(\xi) \ge 0, \quad f''(\xi) \ge 0, \quad \xi \ge 0.%$$ $f(\xi) > 0$, $\xi \in R$. Under various additional assumptions on $f$ we show that this problem has a curve of solutions $(\lambda,u)$ in $R_+ \X C^{2m}[-1,1]$, emanating from $(\lambda,u) = (0,0)$, and we describe the shape of this curve. All the solutions on this curve are positive (that is, $u$ is positive on $(-1,1)$), and any solutions for which $u$ is stable lie on this curve.