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.