Fifth Mississippi State Conference on Differential Equations and Computational Simulations,
Electron. J. Diff. Eqns., Conf. 10, 2003, pp. 143-152.

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

Chaitan P. Gupta

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 $\xi_{i}\in (0,1)$, $a_{i}\in \mathbb{R}$, $i=1,2, \dots , m-2$, $\sum_{i=1}^{m-2}a_{i}\neq 1$, $0 lessthan \xi_{1} less than \xi_{2}less than \dots less than \xi_{m-2} less than 1$ be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary-value problem
$$\displaylines{ (\phi (x'(t)))'=f(t,x(t),x'(t))+e(t),\quad 0 less than t less than 1, \cr x(0)=0, \quad \phi (x'(1))=\sum_{i=1}^{m-2}a_{i}\phi(x'(\xi_{i})). }$$
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 $t\in \mathbb{R}$, this problem was studied earlier by Gupta, Trofimchuk in [2] and by Gupta, Ntouyas and Tsamatos in [1]. We give a priori estimates needed for this problem that are similar to a priori estimates obtained by Gupta, Trofimchuk in [2]. 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.
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.

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

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