Electronic Journal of Differential Equations (Feb 2003)
A non-resonant multi-point boundary-value problem for a p-Laplacian type operator
Abstract
Let $phi $ be an odd increasing homeomorphism from $mathbb{R}$ onto $mathbb{R}$ with $phi (0)=0$, $f:[0$,$1]imes mathbb{R}^{2}o mathbb{R}$ be a function satisfying Caratheodory's conditions and $e(t)in L^{1}[0,1]$. Let $xi_{i}in (0,1)$, $a_{i}in mathbb{R}$, $i=1,2, dots , m-2$, $sum_{i=1}^{m-2}a_{i}eq 1$, $0<xi_{1}<xi_{2}<dots<xi_{m-2}<1$ be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary-value problem {gather*} (phi (x'(t)))'=f(t,x(t),x'(t))+e(t),quad 0<t<1, x(0)=0, quad phi (x'(1))=sum_{i=1}^{m-2}a_{i}phi(x'(xi_{i})). end{gather*} This paper gives conditions for the existence of a solution for the above boundary-value problem using some new Poincare type a priori estimates. In the case $phi (t)equiv t$ for $tin mathbb{R}$, this problem was studied earlier by Gupta, Trofimchuk in cite{gt3} and by Gupta, Ntouyas and Tsamatos in cite{gnt1}. We give a priori estimates needed for this problem that are similar to a priori estimates obtained by Gupta, Trofimchuk in cite{gt3}. We then obtain existence theorems for the above multi-point boundary-value problem using the a priori estimates and Leray-Schauder continuation theorem.
