For the
$n$
-th order nonlinear differential equation,
${{y}^{(n)}}\,=\,f(x,\,y,\,y\prime ,\ldots ,\,{{y}^{(n-1)}})$
, we consider uniqueness implies uniqueness and existence results for solutions satisfying certain
$(k\,+\,j)$
-point boundary conditions for
$1\,\le \,j\,\le \,n\,-\,1$
and
$1\,\le \,k\,\le \,n\,-\,j$
. We define
$(k;\,j)$
-point unique solvability in analogy to
$k$
-point disconjugacy and we show that
$(n\,-\,{{j}_{0}};\,{{j}_{0}})$
-point unique solvability implies
$(k;\,j)$
-point unique solvability for
$1\,\le \,j\,\le \,{{j}_{0}}$
, and
$1\,\le \,k\,\le \,n\,-\,j$
. This result is analogous to
$n$
-point disconjugacy implies
$k$
-point disconjugacy for
$2\,\le \,k\,\le \,n\,-\,1$
.