Electron. J. Diff. Eqns.,Vol. 2004(2004), No. 32, pp. 1-16.

Solution curves of 2m-th order boundary-value problems

Bryan P. Rynne

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 $f(\xi) greater than 0$, $\xi \in \mathbb{R}$. 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.

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Bryan P. Rynne
Mathematics Department
Heriot-Watt University
Edinburgh EH14 4AS, Scotland
email: bryan@ma.hw.ac.uk

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