Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 81-94.

A third-order m-point boundary-value problem of Dirichlet type involving a p-Laplacian type operator

Chaitan P. Gupta

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.

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Chaitan P. Gupta
Department of Mathematics, 084
University of Nevada
Reno, NV 89557, USA
email: gupta@unr.edu

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