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

Chaitan P. Gupta1 and Sergei Trofimchuk2

 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

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.