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

Abstract

Let f:[0,1]×ℝ2→ℝ be a function satisfying Carathéodory's conditions and e(t)∈L1[0,1]. Let ξi∈(0,1),ai∈ℝ,i=1,2,…,m−2,0<ξ1<ξ2<⋯<ξm−2<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=1m−2ai x′(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∈ℝ,i=1,2,…,m−2, had the same sign. The results of this paper give considerably better existence conditions even in the case when all of the ai∈ℝ,i=1,2,…,m−2, have the same sign. Some examples are given to illustrate this point.