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

Abstract

We consider the sixth-order $m$-point boundary-value problem $$displaylines{ u^{(6)}(t)=fig(u(t), u''(t), u^{(4)}(t)ig),quad tin(0,1),cr u(0)=0, quad u(1)=sum_{i=1}^{m-2}a_iu(eta_i),cr u''(0)=0, quad u''(1)=sum_{i=1}^{m-2}a_iu''(eta_i),cr u^{(4)}(0)=0, quad u^{(4)}(1)=sum_{i=1}^{m-2}a_iu^{(4)}(eta_i), }$$ where $f: mathbb{R}imes mathbb{R}imes mathbb{R} o mathbb{R}$ is a sign-changing continuous function, $mgeq3$, $eta_iin(0,1)$, 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.

Keywords

• Nonlinear boundary value problems
• nodal solution
• eigenvalues
• bifurcation methods