Abstract and Applied Analysis
Volume 4 (1999), Issue 2, Pages 71-81

Solvability of a multi-point boundary value problem of Neumann type

Chaitan P. Gupta1 and Sergei Trofimchuk2

1Department of Mathematics, University of Nevada, Reno, NV 89557, USA
2Departamento de Matemáticas, Facultad de Ciencias, Universidade de Chile, Casilla, Santiago 653, Chile

Received 5 April 1999

Copyright © 1999 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.


Let f:[0,1]×2 be a function satisfying Carathéodory's conditions and e(t)L1[0,1]. Let ξi(0,1),ai,i=1,2,,m2,0<ξ1<ξ2<<ξm2<1 be given. This paper is concerned with the problem of existence of a solution for the m-point boundary value problem x(t)=f(t,x(t),x(t))+e(t),0<t<1;x(0)=0,x(1)=i=1m2aix(ξi). This paper gives conditions for the existence of a solution for this boundary value problem using some new Poincaré type a priori estimates. This problem was studied earlier by Gupta, Ntouyas, and Tsamatos (1994) when all of the ai,i=1,2,,m2, had the same sign. The results of this paper give considerably better existence conditions even in the case when all of the ai,i=1,2,,m2, have the same sign. Some examples are given to illustrate this point.