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

Abstract

Let $f:\left[0,1\right]\times {ℝ}^2\to ℝ$ be a function satisfying Carathéodory′s conditions and $e\left(t\right)\in L^1\left[0,1\right]$. Let be given. This paper is concerned with the problem of existence of a solution for the $m$-point boundary value problem . 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 $a_i\in ℝ,i=1,2,\dots ,m-2$, had the same sign. The results of this paper give considerably better existence conditions even in the case when all of the $a_i\in ℝ,i=1,2,\dots ,m-2$, have the same sign. Some examples are given to illustrate this point.