Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 32, pp. 1-16.
Title: Solution curves of 2m-th order boundary-value problems
Author: Bryan P. Rynne (Heriot-Watt Univ., Edinburgh, Scotland)
Abstract:
We consider a boundary-value problem of the form $L u = \lambda f(u)$,
where $L$ is a $2m$-th order disconjugate ordinary differential operator
($m \ge 2$ is an integer), $\lambda \in [0,\infty)$,
and the function $f : \mathbb{R} \to \mathbb{R}$ is $C^2$ and
satisfies $f(\xi) > 0$, $\xi \in \mathbb{R}$.
Under various convexity or concavity type assumptions on $f$
we show that this problem has a smooth curve, $\mathcal{S}_0$, of solutions
$(\lambda,u)$, emanating from $(\lambda,u) = (0,0)$,
and we describe the shape and asymptotes of $\mathcal{S}_0$.
All the solutions on $\mathcal{S}_0$ are positive
and all solutions for which $u$ is stable lie on $\mathcal{S}_0$.
Submitted December 15, 2003. Published March 3, 2004.
Math Subject Classifications: 34B15.
Key Words: Ordinary differential equations; nonlinear boundary value problems.