Let $f:[0,1]\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a function satisfying Carathéodory′s conditions and $e(t)\in L^{1}[0,1]$ . Let $\xi_{i}\in(0, 1),a_{i}\in\mathbb{R},i=1,2,\dotsc,m-2, 0 < \xi _{1} < \xi_{2} < \cdots < \xi _{m-2} < 1$ be given. This paper is concerned with the problem of existence of a solution for the $m$ -point boundary value problem $x^{\prime\prime}(t)=f(t,x(t),x^{\prime}(t))+ e(t),0 < t < 1;x(0)=0,x^{\prime}(1)=\sum_{i=1}^{m-2}a_{i}x^{\prime}(\xi_{i})$ . 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 $a_{i}\in\mathbb{R},i = 1,2,\dotsc,m - 2$ , had the same sign. The results of this paper give considerably better existence conditions even in the case when all of the $a_{i}\in\mathbb{R},i = 1,2,\dotsc,m - 2$ , have the same sign. Some examples are given to illustrate this point.