The differential equation considered is
y
(
n
)
+
Σ
p
i
(
x
)
y
(
i
)
=
0
{y^{(n)}} + \Sigma {{p_i}(x){y^{(i)}}} = 0
, where
σ
i
p
i
(
x
)
⩾
0
,
i
=
0
,
…
,
n
−
1
,
σ
i
=
±
1
{\sigma _i}{p_i}(x) \geqslant 0,i = 0,\ldots ,n - 1,{\sigma _i} = \pm 1
. The focal point
ζ
(
a
)
\zeta (a)
is defined as the least value of s,
s
>
a
s > a
, such that there exists a nontrivial solution y which satisfies
y
(
i
)
(
a
)
=
0
,
σ
i
σ
i
+
1
>
0
{y^{(i)}}(a) = 0,{\sigma _i}{\sigma _{i + 1}} > 0
and
y
(
i
)
(
s
)
=
0
{y^{(i)}}(s) = 0
,
σ
i
σ
i
+
1
>
0
{\sigma _i}{\sigma _{i + 1}} > 0
. Our method is based on a characterization of
ζ
(
a
)
\zeta (a)
by solutions which satisfy
σ
i
y
(
i
)
>
0
,
i
=
0
,
…
,
n
−
1
{\sigma _i}{y^{(i)}} > 0,i = 0,\ldots ,n - 1
, on
[
a
,
b
]
[a,b]
,
b
>
ζ
(
a
)
b > \zeta (a)
. We study the behavior of the function
ζ
\zeta
and the dependence of
ζ
(
a
)
\zeta (a)
on
p
0
,
…
,
p
n
−
1
{p_0},\ldots ,{p_{n - 1}}
when at least a certain
p
i
(
x
)
{p_i}(x)
does not vanish identically near a or near
ζ
(
a
)
\zeta (a)
. As an application we prove the existence of an eigenvalue of a related boundary value problem.