Electronic Journal of Differential Equations,
Conference 07 (2001), pp. 47-59.

Title: A new a priori estimate for multi-point boundary-value problems.

Authors: Chaitan P. Gupta (Univ. of Nevada, Reno, NV, USA)
 
Abstract: 
  Let $f:[0,1]\times \mathbb{R}^2\to \mathbb{R}$ be a function satisfying
  Caratheodory's conditions and $e(t)\in L^{1}[0,1]$. Let $0<\xi
  _1<\xi_2<\dots <\xi_{m-2}<1$ and $a_i\in \mathbb{R}$ for
  $i=1,2,\dots ,m-2$ be given. A priori estimates of the
  form  
  $$
  \|x\|_{\infty }\leq C\| x''\|_1, \quad \|x'\|_{\infty }\leq
  C\|x''\|_1, $$
  are needed to obtain the existence
  of a solution for the multi-point bound\-ary-value problem
  \begin{gather*}
  x''(t)=f(t,x(t),x'(t))+e(t),\quad 0<t<1, \\
  x(0)=0,\quad x(1)=\sum_{i=1}^{m-2}a_ix(\xi_i),
  \end{gather*}
  using Leray Schauder continuation theorem. The purpose of this
  paper is to obtain a new a priori estimate of the form $\| x\|
  _{\infty }\leq C\| x''\|_1$. This new estimate then enables us to
  obtain a new existence theorem. Further, we obtain a new a priori estimate
  of the form $\| x\|_{\infty }\leq C\| x''\|_1$
  for the three-point boundary-value problem
  \begin{gather*}
  x''(t)=f(t,x(t),x'(t))+e(t),\quad 0<t<1, \\
  x'(0)=0,\quad x(1)=\alpha x(\eta ),
  \end{gather*}
  where $\eta \in (0,1)$ and $\alpha \in \mathbb{R}$ are given.
  The estimate obtained for the three-point boundary-value problem
  turns out to be sharper than the one obtained by particularizing
  the $m$-point boundary value estimate to the three-point case.

Published July 20, 2001.
Math Subject Classifications: 34B10, 34B15, 34G20.
Key Words: Three-point boundary-value problem; m-point boundary-value
 problem; a-priori estimates; Leray-Schauder Continuation theorem;
 Caratheodory's conditions.