| Summary: | Let $f:[0,1]imes mathbb{R}^2o mathbb{R}$ be a function satisfying Caratheodory's conditions and $e(t)in L^{1}[0,1]$. Let $0<xi _1<xi_2<dots <xi_{m-2}<1$ and $a_iin mathbb{R}$ for $i=1,2,dots ,m-2$ be given. A priori estimates of the form $$ |x|_{infty }leq C| x''|_1, quad |x'|_{infty }leq C|x''|_1, $$ are needed to obtain the existence of a solution for the multi-point bound-ary-value problem {gather*} x''(t)=f(t,x(t),x'(t))+e(t),quad 0<t<1, x(0)=0,quad x(1)=sum_{i=1}^{m-2}a_ix(xi_i), end{gather*} using Leray Schauder continuation theorem. The purpose of this paper is to obtain a new a priori estimate of the form $| x| _{infty }leq C| x''|_1$. This new estimate then enables us to obtain a new existence theorem. Further, we obtain a new a priori estimate of the form $| x|_{infty }leq C| x''|_1$ for the three-point boundary-value problem {gather*} x''(t)=f(t,x(t),x'(t))+e(t),quad 0<t<1, x'(0)=0,quad x(1)=alpha x(eta ), end{gather*} where $eta in (0,1)$ and $alpha in mathbb{R}$ are given. The estimate obtained for the three-point boundary-value problem turns out to be sharper than the one obtained by particularizing the $m$-point boundary value estimate to the three-point case.
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