Electronic Journal of Differential Equations (Apr 2009)
A third-order m-point boundary-value problem of Dirichlet type involving a p-Laplacian type operator
Abstract
Let $phi $, be an odd increasing homeomorphisms from $mathbb{R}$ onto mathbb{R} satisfying $phi (0)=0$, and let $f:[0,1]imes mathbb{R}imes mathbb{R}imes mathbb{R}mapsto mathbb{R}$ be a function satisfying Caratheodory's conditions. Let $alpha _{i}in {mathbb{R}}$, $xi _{i}in (0,1)$, $i=1,dots ,m-2$, $0<xi _{1}<xi _{2}<dots <xi _{m-2}<1$ be given. We are interested in the existence of solutions for the $m$-point boundary-value problem: $$displaylines{ (phi (u''))'=f(t,u,u',u''), quad tin (0,1), cr u(0)=0,quad u(1)=sum_{i=1}^{m-2}alpha _{i}u(xi _{i}), quad u''(0)=0, }$$ in the resonance and non-resonance cases. We say that this problem is at emph{resonance} if the associated problem $$ (phi (u''))'=0, quad tin (0,1), $$ with the above boundary conditions has a non-trivial solution. This is the case if and only if $sum_{i=1}^{m-2}alpha _{i}xi _{i}=1$. Our results use topological degree methods. In the non-resonance case; i.e., when $sum_{i=1}^{m-2}alpha_{i}xi _{i} eq 1$ we note that the sign of degree for the relevant operator depends on the sign of $sum_{i=1}^{m-2}alpha _{i}xi _{i}-1$.
