We shall provide conditions on the function
f
(
t
,
u
1
,
⋯
,
u
n
−
1
)
f(t,u_{1},\cdots , u_{n-1})
. The higher order boundary value problem
({BVP})
{
(
E
)
u
(
n
)
(
t
)
+
f
(
t
,
u
(
t
)
,
u
(
1
)
(
t
)
,
⋯
,
u
(
n
−
2
)
(
t
)
)
=
0
f
o
r
t
∈
(
0
,
1
)
a
n
d
n
≥
2
,
(
B
C
)
{
u
(
i
)
(
0
)
=
0
,
0
≤
i
≤
n
−
3
,
α
u
(
n
−
2
)
(
0
)
−
β
u
(
n
−
1
)
(
0
)
=
0
,
γ
u
(
n
−
2
)
(
1
)
+
δ
u
(
n
−
1
)
(
1
)
=
0
\begin{equation*}\begin {cases}(E)~~ u^{(n)}(t)+ f(t, u(t),u^{(1)}(t),\cdots ,u^{(n-2)}(t))=0~~~~~\mathrm {~for~}~~~~~t\in (0,1)~~~~\mathrm {and}~~~~~~n\ge 2,\ (BC)~~ \begin {cases}u^{(i)}(0)=0,~~~~~0\le i \le n-3,\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\ \gamma u^{(n-2)}(1)+\delta u^{(n-1)}(1)=0\end{cases} \end{cases} \tag {{BVP}}\end{equation*}
has at least one solution.