Seventh Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 17 (2009), pp. 81-94.
Title: A third-order m-point boundary-value problem of Dirichlet type
involving a p-Laplacian type operator
Author: Chaitan P. Gupta (Univ. of Nevada, Reno, NV, USA)
Abstract:
Let $\phi $, be an odd increasing homeomorphisms from $\mathbb{R}$ onto
\mathbb{R}
satisfying $\phi (0)=0$, and let
$f:[0,1]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ 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:
$$\displaylines{
(\phi (u''))'=f(t,u,u',u''), \quad t\in (0,1), \cr
u(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha _{i}u(\xi _{i}), \quad
u''(0)=0,
}$$
in the resonance and non-resonance cases. We say that this problem is at
\emph{resonance} if the associated problem
$$
(\phi (u''))'=0, \quad t\in (0,1),
$$
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
$\sum_{i=1}^{m-2}\alpha_{i}\xi _{i}\neq 1$ we note that the sign of
degree for the relevant operator depends on the sign of
$\sum_{i=1}^{m-2}\alpha _{i}\xi _{i}-1$.
Published April 15, 2009.
Math Subject Classifications: 34B10, 34B15, 34L30.
Key Words: m-point boundary value problems; p-Laplace type operator;
non-resonance; resonance; topological degree.