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Letf:[0,1]×ℝ2→ℝbe a function satisfying Carathéodory's conditions ande(t)∈L1[0,1]. Letξi∈(0,1),ai∈ℝ,i=1,2,...,m−2,0<ξ1<ξ2<⋯<ξm−2<1be given. This paper is concerned with the problem of existence of a solution for them-point boundary value problemx″(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,doi:10.1155/s1085337599000093 fatcat:dbj57vovmvgalluuhp3yjmdgte