Electron. J. Diff. Equ., Vol. 2012 (2012), No. 217, pp. 1-18.

Nodal solutions for sixth-order m-point boundary-value problems using bifurcation methods

Yude Ji, Yanping Guo, Yukun Yao, Yingjie Feng

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.

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Yude Ji
Hebei University of Science and Technology
Shijiazhuang, 050018, Hebei, China
email: jiyude-1980@163.com
Yanping Guo
Hebei University of Science and Technology
Shijiazhuang, 050018, Hebei, China
email: guoyanping65@sohu.com
Yukun Yao
Hebei Medical University
Shijiazhuang, 050200, Hebei, China
email: yaoyukun126@126.com
Yingjie Feng
Hebei Vocational Technical College of Chemical and Medicine
Shijiazhuang, 050026, Hebei, China
email: fengyingjie126@126.com

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