Abstract
We establish the existence, uniqueness, and global behavior of a positive solution for the following superlinear fractional boundary value problem:
D
α
u
(
x
)
=
u
(
x
)
φ
(
x
,
u
(
x
)
)
,
x
∈
(
0
,
1
)
,
lim
x
→
0
+
D
α
−
1
u
(
x
)
=
−
a
,
u
(
1
)
=
b
, where
1
<
α
≤
2
,
D
α
is the standard Riemann-Liouville fractional derivative,
a
,
b
are nonnegative constants such that
a
+
b
>
0
and
φ
(
x
,
t
)
is a nonnegative continuous function in
(
0
,
1
)
×
[
0
,
∞
)
that is required to satisfy some appropriate conditions related to a certain class of functions
K
α
. Our approach is based on estimates of the Green’s function and on perturbation arguments.
MSC:
34A08, 34B18, 34B27.