### Abstract:

For the third order ordinary differential equation,
$y'''=f(x,y,y',y'')$, it is assumed that, for some $m\geq 4$,
solutions of nonlocal boundary value problems satisfying
\[y(x_1)=y_1,\ y(x_2)=y_2,\] \[y(x_m)-\sum_{i=3}^{m-1}
y(x_{i})=y_3,\] $a<x_1<x_2<\cdots<x_m<b$, and
$y_1,y_2,y_3\in\mathbb{R}$, are unique when they exist. It is
proved that, for all $3\leq k \leq m$, solutions of nonlocal
boundary value problems satisfying \[y(x_1)=y_1,\ y(x_2)=y_2,\]
\[y(x_k)-\sum_{i=3}^{k-1} y(x_{i})=y_3,\]
$a<x_1<x_2<\cdots<x_k<b$, and $y_1,y_2,y_3\in\mathbb{R}$, are
unique when they exist. It is then shown that solutions do indeed
exist.