Journal of Inequalities and Applications
Volume 5 (2000), Issue 4, Pages 351-365
doi:10.1155/S1025583400000187
A priori estimates for the existence of a solution for a multi-point boundary value problem
1Department of Mathematics, University of Nevada, Reno, Reno, NV 89557, USA
2Departamento de Matemática, Facultad de Ciencias, Universidade de Chile, Casilla, Santiago 653, Chile
Received 20 February 1999; Revised 10 July 1999
Copyright © 2000 Chaitan P. Gupta and Sergei Trofimchuk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let ai∈R, ζi∈(0,1), i=1,2,…,m−2, 0<ζ1<ζ2<⋯<ζm−2<1, with α=∑i=1m−2ai≠1 be given. Let x(t)∈W2,1(0,1) be such that x′(0)=0, x(1)=∑i=1m−2aix(ζi)(∗) be given. This paper is concerned with the problem of obtaining Poincaré type a priori estimates of the form ||x||∞≤C||x″||1. The study of such estimates is motivated by the problem of existence of a solution for the Caratheodory equation x″(t)=f(t,x(t),x′(t))+e(t), 0<t<1, satisfying boundary conditions (∗). This problem was studied earlier by Gupta et al. (Jour. Math. Anal. Appl. 189 (1995), 575–584) when the ai’s, all had the same sign.