Electronic Journal of Differential Equations,
Conf. 10 (2002), pp. 143--152.

Title: A non-resonant multi-point boundary-value problem for a $p$-Laplacian
 type operator

Author: Chaitan P. Gupta (Univ. of Nevada, Reno, NV, USA)
 
Abstract: 
 Let $\phi $ be an odd increasing homeomorphism from $\mathbb{R}$ onto
 $\mathbb{R}$ with $\phi (0)=0$,
 $f:[0$,$1]\times \mathbb{R}^{2}\to \mathbb{R}$ be a
 function satisfying Caratheodory's conditions and $e(t)\in L^{1}[0,1]$.
 Let $\xi_{i}\in (0,1)$, $a_{i}\in \mathbb{R}$, $i=1,2, \dots , m-2$,
 $\sum_{i=1}^{m-2}a_{i}\neq 1$,
 $0<\xi_{1}<\xi_{2}<\dots<\xi_{m-2}<1$ be given.
 This paper is concerned with the problem of existence of a solution for
 the multi-point boundary-value problem
 \begin{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 Poincar\'{e} type
 a priori estimates. In the case $\phi (t)\equiv t$ for $t\in \mathbb{R}$,
 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.


Published February 28, 2003.
Math Subject Classifications: 34B10, 34B15, 34G20.
Key Words: Multi-point boundary-value problem; three-point boundary-value problem;
 $p$-Laplacian; Leray Schauder Continuation theorem;  Caratheodory's conditions.