Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 217, pp. 1-18.
Title: Nodal solutions for sixth-order m-point boundary-value
problems using bifurcation methods
Authors: Yude Ji (Hebei Univ. of Science and Tech., China)
Yanping Guo (Hebei Univ. of Science and Tech., China)
Yukun Yao (Hebei Medical Univ., China)
Yingjie Feng (Hebei Vocational Technical College, China)
Abstract:
We consider the sixth-order $m$-point boundary-value problem
$$\displaylines{
u^{(6)}(t)=f\big(u(t), u''(t), u^{(4)}(t)\big),\quad t\in(0,1),\cr
u(0)=0, \quad u(1)=\sum_{i=1}^{m-2}a_iu(\eta_i),\cr
u''(0)=0, \quad u''(1)=\sum_{i=1}^{m-2}a_iu''(\eta_i),\cr
u^{(4)}(0)=0, \quad u^{(4)}(1)=\sum_{i=1}^{m-2}a_iu^{(4)}(\eta_i),
}$$
where $f: \mathbb{R}\times \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ is
a sign-changing continuous function, $m\geq3$,
$\eta_i\in(0,1)$, and $a_i>0$ for $i=1,2,\dots,m-2$ with
$\sum_{i=1}^{m-2}a_i<1$. We first show that the spectral
properties of the linearisation of this problem are similar to the
well-known properties of the standard Sturm-Liouville problem with
separated boundary conditions. These spectral properties are then
used to prove a Rabinowitz-type global bifurcation theorem for a
bifurcation problem related to the above problem. Finally, we obtain
the existence of nodal solutions for the problem, under various
conditions on the asymptotic behaviour of nonlinearity $f$ by using
the global bifurcation theorem.
Submitted June 20, 2012. Published November 29, 2012.
Math Subject Classifications: 34B15.
Key Words: Nonlinear boundary value problems; nodal solution;
eigenvalues; bifurcation methods.