We consider, for
q
>
1
q>1
, the one-dimensional Kirchhoff-type problem
−
A
(
∫
0
1
(
u
′
(
s
)
)
q
d
s
)
u
(
t
)
a
m
p
;
=
λ
f
(
u
(
t
)
)
,
t
∈
(
0
,
1
)
u
(
0
)
a
m
p
;
=
0
u
′
(
1
)
a
m
p
;
=
0
\begin{equation} \begin {split} -A\left (\int _0^1\big (u’(s)\big )^q\ ds\right )u(t)&=\lambda f\big (u(t)\big )\text {, }t\in (0,1)\\ u(0)&=0\\ u’(1)&=0\notag \end{split} \end{equation}
and demonstrate the existence of at least one positive solution to this problem. The main contribution is to show that by using a nonstandard order cone together with topological fixed point theory much weaker conditions than usual can be imposed on the coefficient function
A
A
.