Heriot-Watt Mathematics Report Series
HWM02-40, 26 Dec 2002
Solution curves of $2m$th order boundary value problems
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.
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